Models of Neurovascular Coupling in the Brain

byAlexandra E. Witthoft

B.A., Physics, Mount Holyoke College, USA, 2008

A dissertation submitted in partial fulfillment of therequirements for the degree of Doctor of Philosophyin the School of Engineering at Brown University

May 2015

c© Copyright 2015 by Alexandra E. Witthoft

This dissertation by Alexandra E. Witthoft is accepted in its present formby the School of Engineering as satisfying the

dissertation requirement for the degree of Doctor of Philosophy.

Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Professor George Em Karniadakis, Advisor

Recommended to the Graduate Council

Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Professor Petia M. Vlahovska, Reader

Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Professor Christopher I. Moore, Reader

Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Professor Stephanie R. Jones, Reader

Approved by the Graduate Council

Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Peter M. Weber, Dean of the Graduate School

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Vitae

Education

B.A. Physics, Mount Holyoke College, USA, 2008

Awards and Fellowships

Jan Tauc Named Fellowship, Brown University 2008-2009

Publications

1. Witthoft, A., J. A. Filosa, and G. E. Karniadakis. 2013. “Potassium buffering in theneurovascular unit: Models and sensitivity analysis.” Biophysical journal. 105:20462054

2. Witthoft, A., and G. E. Karniadakis. 2012.“A bidirectional model for communicationin the neurovascular unit. Journal of Theoretical Biology.” 311:8093

Conference Presentations and Invited Talks

1. Witthoft, A., J. A. Filosa, G. E. Karniadakis. “Modeing Astrocyte Potassium Bufferingand Bidirectional Neurovascular Signaling.” IMAG Multiscale Modeling (MSM) Consor-tium Meeting, Bethesda, MD, September 2014

2. Witthoft, A., G. E. Karniadakis. “Bidirectional neurovascular communication: Mod-eling the vascular influence on astrocytic and neural function.” Workshop on CerebralBlood Flow (CBF) and Models of Neurovascular Coupling, Fields Institute, University ofToronto, July 2014

3. Witthoft, A., J. A. Filosa, G. E. Karniadakis. “A Computational Model of AstrocytePotassium Buffering and Bidirectional Signaling in the Neurovascular Unit.” Biophysical

iv

Society 58th Annual Meeting, San Francisco, California, February 2014

4. Witthoft, A., J. A. Filosa, G. E. Karniadakis. “Modeling Bidirectional Communicationin the Neurovascular Unit.” ICERM, Brown University, January 2014

5. Witthoft, A., J. A. Filosa, G. E. Karniadakis. “Potassium Transport in the Neurovas-cular Unit.” University of Utah, Invited Lecture, May 2013

6. Witthoft, A., J. A. Filosa, G. E. Karniadakis. “Bidirectional Modeling of the Neu-rovascular Unit.” SIAM Conference on Applications of Dynamical Systems (DS13), May2013

v

Acknowledgements

I would like to thank my advisor, George Em Karniadakis. I’m lucky to have worked with

such a visionary advisor. When I first joined his research group, it was his idea for me to

combine their cutting edge cerebral blood flow modeling work with neural models. Before

this, I hadn’t even considered studying the brain, but it immediately became a fascinating

and exciting topic to me. George is someone who opened the door and really encouraged

me to collaborate with some great experimental researchers like Professor Chris Moore

and Professor Jessica Filosa. He also has a remarkable talent for being able to communi-

cate ideas with people across fields, and his uniquely vast interdisciplinary interests and

motivations have really inspired me in how I have pursued my own research.

I would like to thank Petia Vlahovska, one of the earliest members of my committee.

She gave really wonderful insight from the beginning. Her ideas really helped shape the

foundations of my thesis, and I am grateful to her.

I would also like to Chris Moore for being a part of my committee. Chris Moore has

been incredibly supportive through my time at Brown, inviting me to participate in journal

club meetings, giving me oportunities to give practice talks in front of his lab, and has

been generous with sharing feedback and intuition that have been extremely valuable. He

is also a wonderful person to be able to interact with, and talking with him makes science

and research truly exciting.

I am very grateful to Stephanie Jones for being in my thesis committee and for her

guidance. She has been a wonderful mentor and kind enough to include me in her lab

meetings and introduce me to students and post docs in her research group, from whom

I learned a great deal. During many chaotic meetings about designing research projects,

Stephanie Jones has been the voice of reason and has been invaluable in refining and

evaluating the details, which would otherwise be hastily ignored and left to cause problems

later.

vi

I cannot adequately thank our collaborator Jessica Filosa, who was so generous to host

me for six weeks at her laboratory at Georgia Regents University in Augusta, GA. Her

mentorship was extremely valuable, and lead to the writing of our second paper, which

she coauthored. Nothing can replace the experience I was so lucky to receive visiting her

laboratory and interacting with her research group. I especially want to thank her graduate

students Jennifer Iddings and Wenting Du, as well as her post docs Helena Morrison and Ki

Jung Kim. Helena, in addition to her amazing intuitions and feedback during lab meetings,

selflessly gave me a place to stay for one week in her house while I mentally recovered from

a traumatic incident with a cockroach in the apartment where I was staying. Aside from

that one roach (which crawled up to my face one night while I was lying in bed) I also

thank my roommate in Augusta, Wagner Reis, for opening up the spare bedroom in his

home to me during my stay. Wagner was a post doc in Jessica Filosa’s department at the

university, and he helped orient me to the town and the university.

I also need to thank our other collaborators, Chiara Bellini and Jay Humphrey, along

with the entire Humphrey research group at Yale. Jay Humphrey and Chiara have been

extremely generous in sharing their experimental data with us. Chiara has been kind

(and very patient) in helping us understand her methods and the continuum modeling of

arteries. I am also grateful to Jay Humphrey for his valuable guidance and thoughtful

feedback, as well as his kind encouragement.

I owe a deep gratitude to the entire Chris Moore lab, especially Tyler Jones. I also want

to thank the entire Stephanie Jones lab, especially Shane Lee. Shane Lee, while juggling

the demands of his post doc work, has shown bottomless generosity: sharing with me

his parallel-neuron/python code; installing it (through many struggles) on my computer,

teaching me how to use it and creating a special branch of the code for me on bitbucket;

for that matter, teaching me how to use git, and even giving me a one-on-one crash course

on python, plus supreme-expert guidance on LaTeX!

I express my gratitude to all the members of the CRUNCH group that I have had

the pleasure to interact with. I thank them all for creating a supportive and friendly

atmosphere. I have enjoyed many helpful discussions with my coworkers that have sped

up my research in ways I couldn’t have done alone. I owe particular thanks to Alireza

vii

Yazdani and Zhangli Peng for all of their help with solid mechanics and DPD. Without

their help, I would never have been able to complete the DPD arteriole model. I also

want to thank Daniele Venturi, for many helpful conversations, and for his immeasurable

patience in helping me work through many different problems.

This work was supported by the following grants:

• NSF grant OCI-0904288

• NIH grant C14A11773 (A09383)

Most importantly, I would like to thank my family. My parents Julie and Carl have

offered me tremendous and unending support throughout my entire academic career. I

also want to thank my little brother, Luke, who I know is currently doing wonderful things

in his own PhD research.

viii

To my family:

my parents Julie and Carl,

and my brother Luke.

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Abstract of “Models of Neurovascular Coupling in the Brain” by Alexandra E. Witthoft,Ph.D., Brown University, May 2015

We develop three new models of neurovascular coupling at each interface of the neurovas-

cular unit. The neurovascular unit comprises neurons, microvessels, and astrocytes, a type

of glial cell that mediates neurovascular communication. We first develop a bidirectional

dynamical model of an astrocyte that both controls and responds to dilations of an arte-

riole. The astrocyte induces dilation by releasing potassium near the vessel in response

to increased neural activity, a phenomenon known as functional hyperemia. In the re-

verse direction, the astrocyte responds to the arteriole movement via mechanosensitive ion

channels on its membrane which contacts the arteriole wall. We perform several sensitivity

studies of the model, employing both global parameter sensitivity analysis using stochas-

tic collocation, and various model sensitivity studies. In the second model, we consider

the neuron-vessel interface, where we simulate a small network of cortical interneurons

in contact with a dilating vessel. These perivascular interneurons express mechanosen-

sitive pannexin channels that respond to vessel dilations and constrictions. We use our

model to explore how changes in the neural network structure affect the function of the

neurovascular connectivity. Our third model is a discrete particle model of a multi-layer

fiber-reinforced anisotropic arterial wall, which we develop using the Dissipative Particle

Dynamics (DPD) method. The model is constructed based on the true microstructure of

the wall and provides an accurate description of the biaxial mechanical behavior of arter-

ies, which we validate with experimental results provided by collaborators. In addition,

we add an active mechanism to the discrete particle wall in order to model the arteriolar

smooth muscle cell contraction in response to changes in internal pressure (causing the

arteriole to constrict with rising pressure) as well as extracellular potassium. We combine

the DPD model with the dynamical astrocyte model as a bidirectional system: the vessel

dilates with astrocytic potassium release, and the adjacent astrocyte reacts to changes in

vessel dilation. The DPD arteriole model provides a bridge between neurovascular model

and complex blood flow simulations in DPD, in which existing DPD red blood cell models

can be leveraged.

Contents

Vitae iv

Acknowledgments vi

1 Background on astrocyte and neurovascular modeling 11.1 Background on astrocyte modeling . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Heterogeneity of astrocytes . . . . . . . . . . . . . . . . . . . . . . . 41.2 Previous astrocyte model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Synaptic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Astrocytic Intracellular Space . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Perivascular Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Previous arteriole model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Arteriole Smooth Muscle Cell Intracellular Space . . . . . . . . . . . 11

1.4 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Bidirectional astrocyte model 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Potassium buffering . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Mechanosensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Results — Bidirectional signalling . . . . . . . . . . . . . . . . . . . . . . . 302.4 Results — Potassium buffering . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Effect of Astrocyte K+ Buffering on Neurovascular Coupling . . . . 352.4.2 Kir channel blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6 Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.1 Astrocytic TRPV4 channels . . . . . . . . . . . . . . . . . . . . . . . 442.6.2 Role of astrocyte BK channels on neurovascular coupling . . . . . . 452.6.3 Adult brain astrocyte model . . . . . . . . . . . . . . . . . . . . . . 46

2.7 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Neuronal response to hemodynamics 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

x

3.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 Model 1 — single perivascular FS cell . . . . . . . . . . . . . . . . . 583.3.2 Model 2 — perivascular FS interneuron network . . . . . . . . . . . 593.3.3 Model 3 — networks of perivascular and peripheral FS cells . . . . . 61

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Discrete particle model of arteriole 654.1 Background on arteriole structure . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Dissipative Particle Dynamics model of flexible arteriole — single layer . . . 68

4.2.1 Dissipative Particle Dynamics (DPD) method . . . . . . . . . . . . . 694.2.2 Single layer DPD arteriole model with triangular mesh . . . . . . . . 704.2.3 Single layer DPD arteriole with square mesh . . . . . . . . . . . . . 76

4.3 Multilayer arteriole model in DPD . . . . . . . . . . . . . . . . . . . . . . . 834.3.1 Results and verification – uniaxial stretch . . . . . . . . . . . . . . . 904.3.2 Results and verification – biaxial stretch . . . . . . . . . . . . . . . . 934.3.3 Results and verification – biaxial stretch of four-fiber model . . . . . 103

5 Multiphysics Neurovascular Coupling and Future Directions 1135.1 Example of Neurovascular Coupling with Multiphysics DPD Vessel . . . . . 114

5.1.1 Modeling Framework and Constitutive Equations . . . . . . . . . . . 1165.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Future Directions for Multiscale and Multiphysics Neurovascular Models . . 127

A Simulation Parameters 139

B Manual for LAMMPS code 144

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List of Tables

3.1 Px1 physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Arteriole mechanical properties . . . . . . . . . . . . . . . . . . . . . . 684.2 Parameters for axial stretch of triangular mesh cylinder . . . . . . . . 744.3 Mesh values used for square sheet with square mesh . . . . . . . . . . 834.4 Parameters for uniaxial stretch of fiber-reinforced sheet . . . . . . . . 904.5 Parameters for pressurization of thin-walled tube . . . . . . . . . . . . 934.6 Mesh values for square sheet in Figure 4.13 . . . . . . . . . . . . . . . 954.7 Mesh values for square sheets with fiber angle 30 in Figure 4.14 . . . 974.8 Mesh values for square sheets in Figure 4.15 . . . . . . . . . . . . . . 984.9 Mesh values for square sheets in Figure 4.16 . . . . . . . . . . . . . . 1004.10 Mesh values for square sheets with fiber angle 30 in Figure 4.18 . . . 1024.11 Parameters and mesh values for square sheets in Figure 4.20 . . . . . 1054.12 Parameters and mesh values for thick walled tube in Figure 4.21 . . . 1084.13 Original and adjusted matrix layer parameters for Figure 4.23 . . . . . 111

5.1 Parameters for myogenic arteriole . . . . . . . . . . . . . . . . . . . . 1225.2 Parameters for DPD arteriole used in neurovascular coupling . . . . . 1245.3 Pulsatile flow measurements in various arterioles . . . . . . . . . . . . 1295.4 Summary of neurovascular coupling mechanisms . . . . . . . . . . . . 136

A.1 Ωs Synaptic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.2 Ωastr Astrocytic Intracellular Space . . . . . . . . . . . . . . . . . . . 140A.3 ΩP Perivascular Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.4 ΩSMC KirSMC Channels in Vascular Smooth Muscle Cell . . . . . . . . 142A.5 ΩSMC Vascular Smooth Muscle Cell Space . . . . . . . . . . . . . . . 143

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List of Figures

1.1 Previous NVU model overview. . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Model overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Astrocyte response to mechanical stretching of vessel. . . . . . . . . . . 312.3 Astrocyte response to drug induced vasodilation. . . . . . . . . . . . . 312.4 Astrocytic and vascular bidirectional response during neural stimulation. 332.5 Astrocyte Kir effect on neurovascular coupling. . . . . . . . . . . . . . 362.6 K+ undershoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Astrocyte response to K+ channel blocker with short stimulus spike . . 392.8 Sensitivity of K+ undershoot and effects of Kir blockade . . . . . . . . 402.9 Sensitivity of baseline and maximum extracellular potassium . . . . . . 422.10 Effects of TRPV4 K+ and Na+ effluxes . . . . . . . . . . . . . . . . . 442.11 Astrocyte BK channel effect on neurovascular coupling. . . . . . . . . 452.12 Comparison of young and “adult brain” astrocyte models . . . . . . . . 47

3.1 Illustration of perivascular FS spiking behavior . . . . . . . . . . . . . 543.2 Single FS cell response to gradual dilation . . . . . . . . . . . . . . . 583.3 Effect of dilation on single FS cell response to sensory input . . . . . 583.4 Schematic of perivascular FS cell network . . . . . . . . . . . . . . . 593.5 Perivascular FS cells response to dilation in network . . . . . . . . . 613.6 2x3 FS network response to dilation . . . . . . . . . . . . . . . . . . . 623.7 2x5 FS network response to dilation . . . . . . . . . . . . . . . . . . . 63

4.1 Parenchymal arteriole structure . . . . . . . . . . . . . . . . . . . . . 674.2 Triangulated arteriole wall . . . . . . . . . . . . . . . . . . . . . . . . 714.3 Passive arteriole with axial stretch . . . . . . . . . . . . . . . . . . . . 734.4 Pressurization of passive arteriole . . . . . . . . . . . . . . . . . . . . 754.5 Representative area element of square DPD grid . . . . . . . . . . . . 764.6 Uniaxial stretch of anisotropic square mesh . . . . . . . . . . . . . . . 824.7 Structural schematic of two-layer fiber-reinforced arterial wall . . . . . 844.8 Adhesion between fiber and matrix particles . . . . . . . . . . . . . . . 894.9 Thickness map for uniaxial stretch of axial and circumferential sheets 914.10 Uniaxial stretch of axial and circumferential sheets . . . . . . . . . . . 924.11 Pressurization of cylinder with two fibers . . . . . . . . . . . . . . . . 924.12 Alignment of fibers and matrix triangles . . . . . . . . . . . . . . . . . 944.13 Biaxial stretch of square sheet with α = 40.02 . . . . . . . . . . . . . 944.14 Biaxial stretch of square sheet with α = 30 . . . . . . . . . . . . . . . 96

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4.15 Biaxial stretch of square sheet with fine grained fiber layer . . . . . . . 994.16 Biaxial stretch of square sheet with coarse grained fiber layer . . . . . . 1004.17 Structural schematic of two-layer fiber mesh . . . . . . . . . . . . . . . 1014.18 Biaxial stretch of square sheet with α = 30 for two-layer fiber mesh . 1034.19 Structural schematic of two-layer fiber-reinforced arterial wall with four

fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.20 Biaxial stretch of four-fiber family square sheet with α = 30 . . . . . . 1064.21 Pressure vs. diameter at three axial stretch levels . . . . . . . . . . . . 1094.22 Force vs. length tests at four levels of internal pressure . . . . . . . . . 1104.23 Biaxial stretch tests of four-fiber vessel . . . . . . . . . . . . . . . . . . 112

5.1 SMC equilibrium length vs pressure in DPD . . . . . . . . . . . . . . 1175.2 Myogenic response in DPD . . . . . . . . . . . . . . . . . . . . . . . . 1195.3 Dynamical astrocyte simulation with DPD vessel . . . . . . . . . . . . 1235.4 Sensitivity analysis of neurovascular coupling in DPD . . . . . . . . . 1265.5 Immunolabeling image of cortical perivascular astrocytes from [173] . 1305.6 Modular astrocyte model in arteriole tree . . . . . . . . . . . . . . . . 1375.7 Models of astrocyte intercellular communication . . . . . . . . . . . . 138

xiv

Chapter One

Background on astrocyte and

neurovascular modeling

2

1.1 Background on astrocyte modeling

Neural modeling has advanced to the point where entire brain sections are simulated using

networks of hundreds of millions of neurons. Scientists are now beginning to appreciate

two other highly complex, adaptive networks in the brain that interact bidirectionally

with neurons and influence their behavior and function: the cerebrovasculature and spe-

cialized glial cells called astrocytes. Together, these comprise a complex interactive system

known as the neurovascular unit (NVU). The NVU is essential to the health and function

of the central nervous system and serves many critical regulatory functions in the brain.

Among these are maintenance and modulation of the blood brain barrier [74, 197, 100],

autoregulation of cerebral blood flow [103, 57, 178, 72], and both long-term and short-

term modulation of neuronal function [188, 1, 152, 128]. The definition of the NVU has

evolved slightly over time, as outlined by the authors of [100]. Originally, the neurovas-

cular unit was defined as the cerebral blood vessels, neurons and astrocytes, but this has

been updated to specify specific cellular components of the cerebral vasculature, namely

arterial smooth muscle and endothelial cells that form the outer layers of the vessel wall,

and pericytes, small contractile cells located sparsely along capillaries. One of the most

important functions of the NVU is a phenomenon known as functional hyperemia, in which

increased synaptic activity induces dilations of local arterioles, which then brings increased

blood flow to the active region in the brain. This function is believed to be mediated by

astrocytes and arteriole smooth muscle cells, as the astrocytes respond to local synaptic

activity and signal to local arteriole smooth muscle cells, which release their constrictions

and allow the vessel to dilate [56, 178, 103, 42, 72]. In this and the next chapter, we focus

on astrocytes and arterial smooth muscle cells and introduce a coupled model of their

biochemical interactions.

Astrocytes are star-shaped glial cells with long arms called processes that wrap around

neural synapses. Astrocytes typically have at least one process that terminates with end-

foot that encloses an arteriole. While astrocytes have been observed to signal to one another

[71, 123], their processes do not overlap with those of other astrocytes, so the brain is spa-

tially mapped into a 3D grid of distinct “astrocyte domains,” where each astrocyte may

3

have sole influence over all the synapses within its domain [135, 71].

Computational modeling of astrocytes is still a relatively new field. A common feature

of all astrocyte models is the signaling cascade in which active neural synapses produce

a rise in astrocyte intracellular calcium (Ca2+): synaptic glutamate binds to receptors

on the synapse-adjacent process, initiating a G protein cascade in which inositol 1,4,5-

trisphosphate (IP3) is produced in the cell wall, causing release of intracellular stores of

Ca2+ via IP3- and Ca2+-sensitive channels on the endoplasmic reticulum (ER).

Roth et al. [169] developed one of the earliest astrocyte models. They modeled the

irregular calcium propagation in an astrocyte using the assumption that there are discrete

loci of activation along the cell, separated by regions of passive diffusion.

A model that connected astrocyte activation to vasodilation was developed by Bennet,

Farnell, and Gibson [17]. This group developed the model for over a decade, investigating

chemical diffusion and electrical propagation along a row of astrocytes with side-by-side

endfeet wrapping a capillary or arteriole. These models assumed that, in response to high

neural activity, EET was released by the astrocyte endfoot onto the arteriole, causing

dilation [114, 14, 64, 17]. Building off this work, Farr and David [46] added potassium

uptake and release mechanisms to the astrocyte, including an inward K+ pump at the

perisynaptic processes and a high-conductance voltage- and Ca2+-sensitive potassium ion

channel (BK channel) at the perivascular endfoot. Unlike its predecessor, the model by

[46] assumed that potassium release from the astrocyte was responsible for the arteriole

dilation and that EET was not released from the cell but activated the BK channels.

Another approach to astrocyte modeling focused on the interaction between the synapse

and the astrocyte, rather than the astrocyte and the vessel. A detailed model of the

tripartite synapse was first introduced by Nadkarni and Jung [131, 132]. The tripartite

synapse comprises the pre- and post- synaptic neuron and the astrocyte process encircling

the synapse. This astrocyte model did not include EET production, K+ transfer, or any

mechanism involved in signaling to the vasculature; instead, it modeled the release of

glutamate from the astrocyte into the synaptic space, which can both inhibit and enhance

the synapse. Postnov et al. [161] extend a similar model into a spatially expansive network

of astrocytes and neurons.

4

There has also been some work in modeling calcium dynamics across networks of as-

trocytes wherein a calcium rise in one astrocyte leads to similar rises in its neighboring

astrocytes. Bennett, Farnell, and Gibson, the same group that pionereed the model of

astrocyte-induced vasodilation, also modeled astrocyte network calcium waves through

purinergic transmission: astrocytes release the chemical transmitter adenosine triphos-

phate (ATP) at intracellular junctions, which activates purinergic P2Y receptors on neigh-

boring astrocytes, that then experience both a release of ATP in an autocrine manner and

a rise in intracellular Ca2+ via a similar signaling pathway to the glutamate receptors [15].

A similar model is also presented by Hofer et al. in [82], which uses the same basic path-

ways but also includes the role of Ca2+ influx from extracellular space across the astrocyte

membrane as well as the influence of intracellular Ca2+ on IP3 production in the astrocyte.

1.1.1 Heterogeneity of astrocytes

The study of astrocytes is still a new field, and astrocytes have not extensively been

considered as a diverse group of cells. However, there have been studies emerging [23,

166, 9, 165, 144, 86, 59, 196] indicating that astrocytes across and within brain regions

have significant differences in their morphologies, functions, and even network structures

and communication mechanisms. From a modeling perspective, it is critical to consider

these differences, because a simulation of an astrocyte may not be physiologically accurate

or meaningful if the model is informed by data from two distinct astrocyte subtypes.

Moreover, network and communication mechanisms in an astrocyte model need to be

appropriate for the specific astrocyte type in the model.

A review of several astrocyte studies identified that various astrocyte types differ in

gene expression, physiology, electrical properties (resting membrane potential ranging from

-85 to -25 mV), response to disease, and even glutamate uptake [196]. In fact, astrocytes

in the supraoptic nucleus of the rat hypothalamus were found to lack glutamate uptake

currents and glutamate receptor responses completely [93].

Two main astrocyte subtypes are protoplasmic and fibrous. Fibrous astrocytes are

typically in white matter with their somata arranged in rows between axon bundles, and

are common in the optic nerve and the nerve fiber layer of mammalian vascularized retinae

5

[151, 166, 61, 125]. Fibrous astrocytes in the optic nerve have been found to possess sodium

ion channels as well as Kir channels [8].

Protoplasmic astrocytes occur in grey matter, where they are the predominant type

and are found in high density in the cortex [23, 121, 166, 165, 125]. They each have at

least one process terminating in a perivascular endfoot. Each has processes that spread

radially from the somata with many fine, complex side branches, which establish its own

primarily exclusive territory where it interacts with the local synapses [23, 166, 165].

This thesis is primarily concerned with protoplasmic astrocytes, as these grey matter

cells are likely to be a primary link between cerebral blood flow and neural processing.

1.2 Previous astrocyte model

KirSMC

Ca2+

endoplasmic reticulum

process endfootsoma

εIP3glutamate

Na/K, Kir

Astrocyte (II)Synaptic

Space (I)

Perivascular

Space (III)

Vessel

SMC (IV)

BK

CaEET

K+

mGluR

K+K+

Figure 1.1: Previous NVU model overview. (I) Synaptic Space — Active synapses release glutamate and

K+. K+ enters the astrocyte through Na-K pump and Kir channels. Glutamate binds to metabotropic

receptors on the astrocyte endfoot. (II) Astrocyte Intracellular Space — Bound glutamate receptors effect

IP3 production inside the astrocyte wall, leading to release of Ca2+ from internal stores, causing EET pro-

duction. Ca2+ and EET open BK channels at the perivascular endfoot, releasing K+ into the perivascular

space (III). (IV) Arteriole Smooth Muscle Cell Intracellular Space — Kir channels in the arteriolar SMC

are activated by the increase in extracellular K+. The resulting drop in membrane potential closes Ca2+

channels, reducing Ca2+ influx, leading to SMC relaxation, and arteriole dilation (strain, ǫ). Note that the

diagram here is not to scale. The perivascular endfoot is actually in contact with the arteriole and wraps

around its circumference, but we show them separated here in order to detail the ion flow at the endfoot-

vessel interface. Dashed arrows indicate ion movement; solid arrows indicate causal relationships; dotted

arrows indicate inhibition. Thin dashed arrows in the Kir channels indicate ion flux direction at baseline,

or the change in flux direction when extracellular K+ exceeds 10 mM (see text). Coloring is meant to help

distinguish biochemical pathways: potassium transfer is indicated in orange; calcium and EET response to

glutamate is indicated in blue.

The work in this thesis builds off of the astrocyte models by Bennet, Farnell, and

Gibson [17] and Farr and David [46], which are detailed here, and the arteriole smooth

6

muscle model of Gonzalez-Fernandez and Ermentrout [67], which we present below in

Section 1.3.1. These model equations concern a single astrocyte domain without astroctye-

to-astrocyte signalling, and consider only the net neural synaptic activity across the entire

astrocyte domain, and assume this activity to be uniform throughout the domain.

A conceptual diagram of the model is shown in Figure 1.1. The synaptic space and

astrocyte perisynaptic process in the diagram represent the sum of the activities of all the

synapses and perisynaptic processes associated with the astrocyte. During high synaptic

activity in the region, the neurons release K+ ions and glutamate at the synapses (I). K+

flows into the adjacent astrocytic endfoot depolarizing the astrocyte membrane. It should

be noted that [46] refer to this current as the sodium-potassium (Na-K) pump current,

but this is not entirely accurate. The Na-K pump is a mechanism present in the astrocyte

synapse-adjacent processes that exchanges sodium (Na+) ions for K+ ions: three Na+ ions

are released for every two K+ ions that enter the astrocyte. Thus, the electrical flux from

the Na-K pump would actually hyperpolarize the membrane. The depolarization happens

because there is an additional K+ influx through Kir channels present in the astrocyte

processes ([8, 79, 81]). When [46] refer to the Na-K flux, JNaK , they are probably referring

to the combined potassium flux from the Na-K pump and the Kir channels. Here, for

clarity, we will call this combined flux JΣK .

The synaptic glutamate binds to metabotropic receptors (mGluR) on the astrocyte

endfoot, initiating a G-protein cascade in which IP3 is produced inside the astrocyte wall

(II). Inside the astrocyte, IP3 binds to receptors (IP3R) on the endoplasmic reticulum

(ER), releasing internal stores of calcium ions (Ca2+). The rise in intracellular Ca2+

enables production of EET, and EET and Ca2+ activate BK channels in the vessel-adjacent

astrocyte endfoot, releasing K+ into the perivascular space (III). The K+ buildup in the

perivascular space activates Kir channels in the arteriolar smooth muscle cell (SMC) (IV).

Unlike the astrocyte Kir channels on the synapse-adjacent processes, SMC Kir channels

have a reversal potential much lower than the SMC membrane potential, so the K+ flows

outward. The resulting membrane voltage drop closes inward Ca2+ channels, and the

intracellular Ca2+ concentration in the SMC drops. Because Ca2+ is required for myosin-

actin crossbridge attachment, the crossbridges then detach, allowing the SMC to relax and

7

the arteriole to expand.

1.2.1 Synaptic Space

When neurons are activated, they release K+ and glutamate into the synaptic space. The

governing equation for the potassium in the synaptic space is ([46])

d[K+]Sdt

= JKS − JΣK , (1.1)

where JKS is a smooth pulse approximation of potassium release from active neurons. The

combined Na-K pump and Kir flux out of the synaptic space and into the astrocyte, JΣK ,

is

JΣK = JΣK,maxkNa[K+]s

[K+]s +KKoa. (1.2)

This equation actually models the Na-K pump flux ([46]), but their chosen value for

JΣK,max, the maximum flux, gives a large enough value for JΣK to represent the com-

bined activity of the Na-K pump and Kir channels. Thus, they are using the Na-K pump

model as a lumped model for both fluxes. The potassium concentration in the synaptic

space is [K+]s, and KKoa is the threshold value for [K+]s. For simplicity, the intercellular

sodium concentration, [Na+]s is assumed to be constant, so the parameter kNa comes from

kNa = [Na+]1.5s /([Na+]1.5s +KNa1.5s ), where KNas is the threshold value for [Na+]s ([46]).

The synaptic glutamate release is assumed to be a smooth pulse, and the ratio of active

to total G-protein due to mGluR binding on the astrocyte endfoot is given by

G∗ =ρ+ δ

KG + ρ+ δ, (1.3)

where ρ = [Glu]/(KGlu + [Glu]) is the ratio of bound to unbound receptors, and δ is

the ratio of the activities of bound and unbound receptors, which allows for background

activity in the absence of a stimulus ([17]).

8

1.2.2 Astrocytic Intracellular Space

The influx of synaptic potassium into the astrocyte causes a membrane depolarization.

Meanwhile, astrocytic IP3 production occurs inside the cell wall in response to synaptic

glutamate binding to metabotropic receptors. IP3 causes release of intracellular Ca2+,

which triggers EET production. Both Ca2+ and EET activate the astrocytic BK channels,

which release K+ into the perivascular space.

The IP3 production in the astrocyte is based on the model by [17] as modified by [46]:

d[IP3]

dt= r∗hG

∗ − kdeg[IP3], (1.4)

where r∗h is the IP3 production rate, and kdeg is the degradation rate.

The astrocytic intracellular Ca2+ comes from both external influx and release of internal

stores in the endoplasmic reticulum (ER):

d[Ca2+]

dt= β(JIP3

− Jpump + Jleak), (1.5)

where [Ca2+] is the cytosolic calcium concentration; β is the factor describing Ca2+ buffer-

ing, The calcium stores in the ER have three mechanisms for calcium transport: (1) IP3R

receptors on the ER bind to intracellular IP3, initiating Ca2+ outflux from the ER, JIP3

into the intracellular space; (2) a pump uptakes Ca2+ from the cytosol into the ER, Jpump,

and (3) a leak flux Jleak from the ER into the intracellular space ([17]). The IP3-dependent

current is

JIP3= Jmax

[(

[IP3]

[IP3] +KI

)(

[Ca2+]

[Ca2+] +Kact

)

h

]3(

1− [Ca2+]

[Ca2+]ER

)

, (1.6)

where Jmax is the maximum rate; KI is the dissociation constant for IP3R binding; Kact is

the dissociation constant for Ca2+ binding to an activation site on the IP3R, and [Ca2+]ER

is the Ca2+ concentration in the ER. The gating variable h is governed by

dh

dt= kon[Kinh − ([Ca2+] +Kinh)h], (1.7)

9

where kon and Kinh are the Ca2+ binding rate and dissociation constant, respectively, at

the inhibitory site on the IP3R. The pump flux is

Jpump = Vmax[Ca2+]

2

[Ca2+]2 +K2p

, (1.8)

where Vmax is the maximum pump rate, and Kp is the pump constant. The leak channel

flux is

Jleak = PL

(

1− [Ca2+]

[Ca2+]ER

)

, (1.9)

where PL is determined by the steady-state flux balance.

The rise in intracellular Ca2+ in the astrocyte leads to EET production inside the cell.

The EET production is governed by

d[EET]

dt= VEET ([Ca

2+]− [Ca2+]min)− kEET [EET], (1.10)

where VEET is the EET production rate; [Ca2+]min is the minimum [Ca2+] required for

EET production, and kEET is the EET decay rate. Following [46], we assume that EET

acts only on the astrocyte BK channels in the perivascular endfoot, rather than acting

directly on the arteriole SMC as in [17].

Astrocytic BK channels, which occur on the perivascular endfeet, are affected by both

EET and Ca2+, as described by [46]

IBK = gBKnBK(VA − vBK), (1.11)

where gBK is the channel conductance; vBK is the reversal potential, and nBK is governed

by

dnBKdt

= φBK(nBK∞ − nBK), (1.12)

with

φBK = ψBK cosh

(

VA − v3,BK2v4,BK

)

, (1.13)

n∞,BK = 0.5

(

1 + tanh

(

VA + EETshift[EET]− v3,BKv4,BK

))

. (1.14)

10

Also v3,BK is the potential associated with 1/2 open probability, which depends on [Ca2+]:

v3,BK = −v5,BK2

tanh

(

[Ca2+]− Ca3,BKCa4,BK

)

+ v6,BK , (1.15)

where v4,BK , v5,BK , v6,BK , Ca3,BK , Ca4,BK , and ψBK are constants, and EETshift deter-

mines the EET-dependent shift of the channel reversal potential.

The astrocyte membrane potential is described by

dVAdt

=1

Castr(−IBK − Ileak − IΣK), (1.16)

where IΣK is the electrical current carried by the K+ influx at the perisynaptic process:

IΣK = −JΣKCastrγ (see Eq. (1.2)). The leak current, Ileak is

Ileak = gleak(VA − vleak), (1.17)

where gleak and vleak are the leak conductance and reversal potential, respectively.

1.2.3 Perivascular Space

The perivascular space experiences a buildup of K+ due to outflow from the astrocyte and

smooth muscle cell intracellular spaces. The perivascular K+ activates SMC Kir channels.

The potassium accumulates in the perivascular space due to outflow from astrocytic

BK channels and arteriolar smooth muscle cell Kir channels. The equation governing

perivascular K+ comes from [46]:

d[K+]Pdt

= − JBKV Rpa

− JKir,SMC

V Rps−Rdecay([K

+]P − [K+]P,min), (1.18)

where V Rpa and V Rps are the volume ratios of perivascular space to astrocyte and SMC,

respectively, and [K+]P,min is the resting state equilibrium K+ concentration in the perivas-

cular space. Rdecay is the rate at which perivascular K+ concentration decays to its base-

line state due to a combination of mechanisms including uptake in background cellular

activity and diffusion through the extracellular space. The potassium flow from the as-

11

trocyte and SMC are JBK and JKir,SMC , respectively, given as JBK = −IBK/(Castrγ),

and JKir,SMC = −IKir,SMC/(CSMCγ) (Eqs. (1.11) and (1.20)), where CSMC is the SMC

capacitance.

The perivascular Ca2+ concentration obeys

d[Ca2+]Pdt

= −JCa − Cadecay([Ca2+]P − [Ca2+]P,0), (1.19)

where JCa is the calcium current from the the arteriole SMC (see Eq. (1.32) below), and

Cadecay is the decay rate of perivascular Ca2+ concentration (similar to Rdecay).

1.3 Previous arteriole model

1.3.1 Arteriole Smooth Muscle Cell Intracellular Space

The arteriole tone depends on the level of intracellular Ca2+ in the smooth muscle cell

(SMC) layer of the vessel wall. SMC is responsible for a mechanism called the myogenic

response, in which increased pressure in the vessel actually results in constriction rather

than dilation ([37, 122, 171, 99, 94, 140]). This is primarily due to stretch-sensitive Ca2+ ion

channels in the SMC that are activated by pressure, allowing increased Ca2+ influx. SMC

contain parallel myosin and actin filaments that slide together – causing muscle contrac-

tion – when myosin crossbridges attach to the actin. Phosphorylation of the mysosin-actin

crossbridges, which is required for attachment, depends on intracellular Ca2+. The myo-

genic response occurs because when the pressure increases and opens SMC Ca2+ channels,

the active SMC constriction due to Ca2+ influx exceeds the pressure driven dilation of the

viscoelastic arteriole wall. It is worth mentioning now that when we refer to an “active”

vessel, we are referring to a vessel in which there is a non-zero Ca2+ concentration in the

SMC; a “passive” vessel is purely mechanical, as it has no SMC Ca2+ (for example, an

isolated arteriole in a solution without Ca2+). Likewise “active” dilation and constriction

refers to vessel wall movement due to detatchment and attatchment, respectively, of SMC

myosin-actin crossbridges, whereas “pasive” perturbations are purely mechanical in nature.

When the SMC Kir channels are activated due to perivascular K+, the SMC mem-

12

brane potential experiences a hyperpolarization, which closes Ca2+ channels. The reduced

Ca2+ influx results in a higher concentration of Ca2+ in the perivascular space. Also, the

decreased Ca2+ level in the SMC intracellular space results in a dilation (ǫ).

We combine the astrocyte model above with a biochemical model for arterioles devel-

oped by Gonzalez-Fernandez and Ermentrout [67] which includes an explicit biochemical

description of the contraction and relaxation of smooth muscle cells. This model describes

vasomotion as a result of pressure-sensitive Ca2+ ion channel activity in the SMC (see

Eqs. (1.33) and (1.34), below). Other models assume that vasomotion is also affected by

endothelial cell activity ([104, 46]). Both the SMC and endothelial cells are likely to have

a contribution to vasomotion. However, there have been observations of vasomotion in

arterioles in which the endothelium was removed ([70]), indicating that the endothelium is

not required for vasomotion, even if it can have an effect. Thus, for simplicity, the role of

endothelial cells is not addressed here, but can be added to the model at a later time.

Ion currents

The potassium buildup in the perivascular space activates the SMC Kir channels according

to ([46])

IKir,SMC = gKir,SMCk(Vm − vKir,SMC), (1.20)

where the channel conductance, gKir,SMC , reversal potential, vKir,SMC , and open proba-

bility, k, all depend on the perivascular K+ concentration:

gKir,SMC = gKir,0

√

[K+]P , (1.21)

where [K+]P is in units of mM, and gKir,0 is the conductance when the perivascular K+

concentration is 1 mM;

vKir,SMC = vKir,1 log [K+]P − vKir,2, (1.22)

13

where [K+]P is again in units of mM, and vKir,1 and vKir,2 are constants, and

dk

dt=

1

τk(k∞ − k), (1.23)

where τk = 1/(αk + βk), and k∞ = αk/(αk + βk), in which

αk =αKir

1 + exp(Vm−vKir+av1

av2)

(1.24)

βk = βKir exp(bv2(Vm − vKir + bv1), (1.25)

where αKir, βKir, av1 , av2 , bv1 , and bv2 are constants ([46]). The SMC membrane potential

is Vm (see Eq. (1.26), below).

The equations for the SMC dynamics are taken from [67] except that the membrane

potential is modified to include the Kir current (Eq. (1.20)) from [46]:

dVmdt

=1

CSMC(−IL − IK − ICa − IKir,SMC), (1.26)

where CSMC is the cell capacitance, and IL, IK , and ICa are the leak, K+ channel potas-

sium, and calcium currents, respectively. The leak current is simply IL = gL(Vm − vL),

where gL is the leak conductance, and vL is the leak reversal potential. The K+ channel

current is

IK = −gKn(Vm − vK), (1.27)

where gK and vK are the channel conductance and reversal potential, respectively. The

fraction of K+ channel open states, n is described by

dn

dt= λn(n∞ − n), (1.28)

with

n∞ = 0.5

(

1 + tanhVm − v3v4

)

, (1.29)

14

and

λn = φn coshVm − v3

2v4, (1.30)

where v4 is the spread of the open state distribution with respect to voltage, and v3 is the

voltage associated with the opening of half the population, and is dependent on the Ca2+

concentration in the SMC:

v3 = −v52tanh

[Ca2+]SMC − Ca3Ca4

+ v6. (1.31)

The parameters Ca3 and Ca4 affect the shift and spread of the distribution, respectively,

of v3 with respect to Ca2+, and v5, v6 are constants. The Ca2+ channel current is

ICa = gCam∞(Vm − vCa), (1.32)

where gCa and vCa are the channel conductance and reversal potential, respectively. Since

fast kinetics are assumed for the Ca2+ channel, the distribution of open channel states is

equal to the equilibrium distribution

m∞ = 0.5

(

1 + tanhVm − v1v2

)

, (1.33)

with v1 and v2 having the same effect as Ca3 and Ca4 in Eq. (1.31) and v3 and v4 in Eq.

(1.29). Note that in this case, v1 is a variable that depends on the transmural pressure.

We represent the relationship using this linear approximation of the data from [67]:

v1 = −17.4− 12∆P/200, (1.34)

where ∆P is in units of mmHg, and v1 is in mV. For our model, we chose a value of 60

mmHg for ∆P , which we found to be consistent with experimental observations (e.g. Fig.

5 in [88]) for arterioles around 50 µm in diameter, the size used in our simulations.

The SMC myogenic contractile behavior, which constricts the vessel, depends on the

15

Ca2+ concentration in the SMC. The vessel circumference, x, is described by

dx

dt=

1

τ(f∆P − fx − fu), (1.35)

where τ is the time constant, and f∆P fx fu are the forces due to transmural pressure,

viscoelasticity of the material, and myogenic response, respectively (see Eq. (1.46), below).

The myogenic force, fu depends on the SMC Ca2+ concentration, which changes based on

the Ca2+ ion channel current.

Vessel SMC calcium concentration

The Ca2+ concentration in the SMC is governed by

d[Ca2+]SMC

dt= −ρ( 1

2αICa + kCa[Ca

2+]SMC), (1.36)

where ICa is the Ca2+ ion current (Eq. (1.32)); α is the Faraday constant times cytosol

volume (see Table A.5); kCa is the constant ratio of Ca2+ outflux to influx, and ρ is

ρ =(Kd + [Ca2+]SMC)

2

(Kd + [Ca2+]SMC)2 +KdBT

, (1.37)

with Kd being the rate constant in the calcium buffer reaction, and BT is the total buffer

concentration.

Vessel mechanics

We consider a section of the vessel of length 1cm (several orders of magnitude larger

than the vessel diameter, which is µm scaled). The force on the vessel due to transmural

pressure, ∆p, is

f∆p =1

2∆p(

x

π− A

x) (1.38)

where the cross-sectional area is A = π(r2o − r2i ), and the mean radius is r = (ro + ri)/2,

giving the mean circumference x = 2πr.

For the muscle mechanics, consider a segment of the vessel as a cylindrical element

16

with thickness ro − ri, 0 ≤ θ ≤ 2π, and unit length along the axis. The longitudinal cross

sectional surface area, S, is then S = 1(ro − ri). The stresses on S are described using a

Maxwell model along x (the mean circumference) that consists of a contractile component

of length y, a series elastic component of length u, a parallel elastic component of length

x = u+ y, and a parallel viscous component (details in [67]). The hoop stresses associated

with x, y, and u are σx, σy, and σu, respectively. The normalized hoop stresses in terms

of the normalized lengths are

σ′x = x′3

(

1 + tanhx′ − x′1x′2

)

+ x′4(x′ − x′5)− x′8

(

x′6x′ − x′7

)2

− x′9, (1.39)

σ′u = u′2 exp(u′1u

′)− u′3, (1.40)

and

σ′y =σy0

σ#0

exp

[(

−(y′−y′0)2

2[y′1/(y′+y′2)]

2y′4

)

− y′3

]

1− y′3. (1.41)

Here we consider nondimensional variables x′ = x/x0, y′ = y/x0 u

′ = u/x0, y′0 = y0/x0,

and similarly σ′x = σx/σ#0 , σ′y = σy/σ

#0 , σ′u = σu/σ

#0 . The muscle-activation level σy0

comes from the attachment of myosin actin crossbridges (Eq. (1.44)).

Myogenic stress

The myogenic contraction occurs after the attachment of myosin and actin crossbridges,

which involves the Ca2+-dependent phosphorylation of the myosin chains. The ratio, ψ, of

phosphorylated to total myosin chains is

ψ =[Ca2+]

qSMC

Caqm + [Ca2+]qSMC

, (1.42)

where Cam and q are constants. The fraction of attached crossbridges, ω is governed by

dω

dt= kψ

(

ψ

ψm + ψ− ω

)

, (1.43)

17

where kpsi is the rate constant, and ψm is a constant. If the value of experimental

[Ca2+]SMC associated with reference activation is Caref , then

σy0 =σ#y0ωref

ω, (1.44)

where ωref = ψ(Caref )/(ψm + ψ([Ca2+]SMC,ref )).

dy′

dt=

−ν ′refψ

ψrefa′

1−σ′uσ′y

a′+σ′uσ′y

, 0 ≤ σ′

uσ′

y≤ 1

c′[

exp(

b′(

σ′

uσ′

y− d′

))

− exp(b′(1− d′))]

, 1 ≤ σ′

uσ′

y,

(1.45)

where ν is the velocity of contraction of the contractile component at zero load, and ν ′ =

ν/x0, with ν′ref is ν ′ at the reference muscle activation level. Similarly, ψref = ψ(Caref ).

The hoop forces on S due to the viscoelastic and myogenic stress are

fx = weSσ′xσ

#0 , fu = wmSσ

′uσ

#0 , (1.46)

respectively. The weights, we and wm represent the contributions of the viscoelastic and

myogenic hoop forces, respectively.

The circumferential contraction or dilation resulting from the forces on S is then

dx

dt=

1

τ(f∆p − fx − fu), (1.47)

where the time constant, τ , is associated with the wall internal friction.

1.4 Outline of Dissertation

This section provides an outline of the thesis and a brief description of each chapter.

Chapter 2: Dynamical astrocyte model with bidirectional neurovascular interaction. It

includes

• conceptual model for bidirectional neurovascular interactions mediated by astrocyte

signaling

18

• dynamical equations for potassium buffering between the synaptic space, astrocyte

intracellular space, and perivascular space

• mathematical description of astrocytic mechanosensation of vascular dilation

• simulations of astrocyte depolarization and calcium response to vessel dilation in

comparison to in vivo and in vitro experiments [26]

• simulation of bidirectional signaling between astrocytes and vessels during functional

hyperemia (vessel dilation in response to neural activity via astrocyte mediated signaling)

• simulation of astrocytic modulation of extracellular potassium at the synaptic space

in comparison with in vivo experimental results [32]

• simulation of effects of Kir potassium channel blockade in astrocytes in comparison

with experimental results [7]

• global parameter sensitivity analysis of astrocyte model using ANOVA functional

decomposition and stochastic collocation

• analysis of model uncertainty

Chapter 3: Model of direct neuronal response to vascular movement via mechanosensitive

pannexin ion channels. It includes

• basic model for cortical fast-spiking interneurons with mechanosensitive pannexin

channels

• simulation of a single perivascular interneuron response to vasodilation

• simulations of two synaptically connected fast-spiking interneurons in contact with a

dilating microvessel

• simulations of small networks of fast-spiking interneurons, in which only part of the

network is in direct contact with a dilating vessel.

Chapter 4: Discrete particle models of flexible anisotropic arterioles using Dissipative

Particle Dynamics (DPD) method. It includes

• two single-layer models of an arteriole using different mesh geometries

• two-layer orthotropic arteriole model using fiber-reinforced elastin matrix with sym-

metric diagonal fibers

19

• derivation of DPD spring forces for fibers from continuum strain energies in [62, 83]

• uniaxial stretch simulations in comparison with continuum results [62]

• biaxial stretch simulations for square sheets and sensitivity to mesh orientation

• two-layer orthotropic arteriole model using fiber-reinforced elastin matrix with sym-

metric diagonal fibers and two additional fiber families oriented in the circumferential and

axial directions

• biaxial stretch test of square sheet using four fiber model

• biaxial stretch simulations of four-fiber thick walled cylinder in comparison with

experimental results for left common carotid artery (lCCA) provided to us by our collab-

orators Chiara Bellini and Jay Humphrey

Chapter 5: Multiphysics neurovascular coupling in DPD and future directions. It includes

• lumped model for pressure-induced myogenic constriction and potassium-induced

dilation in DPD

• simulation of myogenic response for DPD vessel at baseline and with increased ex-

tracellular potassium with qualitative comparison to experiment [171, 102]

• simulation of bidirectional neurovascular coupling in DPD using astrocyte dynamical

equations combined with DPD myogenic arteriole model

• sensitivity analysis of neurovascular coupling model in DPD for two key parameters

• review of additional mechanisms of neurovascular communication to be included in

future generations of the model including astrocyte release of neurotransmitter, astrocyte

intercellular communication, randomness and distributed modeling. Relevant length and

time scales are provided along with mechanistic diagrams and examples of multicellular

modular network models

The thesis includes two appendices. Appendix A provides the complete set of parame-

ters for the model defined in Chapter 2, while Appendix B contains documentation of the

LAMMPS code for DPD simulations of the two- and four-fiber arteriole.

Chapter Two

Bidirectional astrocyte model

21

2.1 Introduction

The conventional view of the brain has long been a large network of neurons. Other

cerebral cell types and vasculature were originally considered as having supporting roles.

It is now accepted that astrocytes (a specific type of glial cell) and cerebral vasculature

may play a critical role in neural behavior, giving rise to the idea of a neurovascular unit

(NVU). Astrocytes are believed to mediate “neurovascular coupling,” also called “func-

tional hyperemia,” the phenomenon in which synaptic activity induces dilation in nearby

microvasculature, allowing increased blood flow.

A central function of cerebral astrocytes is spatial potassium (K+) buffering: transport

of K+ from extracellular regions of high to low concentration via active uptake and release.

Uptake usually occurs at the astrocyte-neural interfaces, where active neurons release K+,

which at high extracellular levels can be excitatory to neurons; release typically occurs

at the perivascular space, the extracellular region between the astrocyte endfoot and the

abluminal surface of an arteriole, which dilates in response to K+. Thus, the buffering is

a regulatory mechanism that both protects neurons from excessive excitation and dilates

arterioles to increase blood supply to areas of increased neural activity. There may also be

a functional role, as subtle changes to the neuronal extracellular K+ can have behavioral

consequences in terms of synaptic activity. To study the neurovascular unit as an inter-

connected, interactional system, a quantitative mechanistic understanding of K+ spatial

buffering is critical.

Astrocytes express potassium inward rectifier (Kir) channels on their perisynaptic pro-

cesses and perivascular endfeet [92, 105, 137, 32, 81, 24, 79], and these channels have been

reported to play a major role in potassium uptake and release involved in spatial buffering.

Calcium sensitive BK channels in the perivascular endfeet are also a critical means of potas-

sium release [57, 162, 65]. There are also active K+ uptake mechanisms in the perisynaptic

processes including a sodium-potassium (Na-K) pump and a sodium-potassium-chloride

cotransport (NKCC) [148, 181, 110, 147].

In this chapter, we present a model of the neurovascular unit in the cortex with a

detailed mechanistic description of astrocytic potassium buffering. The present model

22

describes the potassium dynamics in the astrocyte intracellular space and in the extracel-

lular spaces at the synaptic and perivascular interfaces. Astrocyte potassium uptake at the

synaptic space is carried by potassium inward rectifier (Kir) channels, potassium-sodium

(Na-K) exchange and a potassium-sodium-chloride cotransporter (NKCC), on the astro-

cyte perisynaptic process. From here on, KirAS refers to the Kir channel on the Astrocyte

at the Synapse-adjacent process. The perivascular endfoot expresses Kir, here referred to

as KirAV, for Astrocytic at the Vessel-adjacent endfoot, and calcium-sensitive BK chan-

nels. While astrocytes express other ion channels, these are not included explicitly, but are

accounted for collectively by a nonspecific leak channel. This model is specific to cortical

astrocytes in the developing brain, as we discuss further, below.

Our model builds off the previous models detailed in Chapter 1.2. In addition to adding

potassium buffering, we make the model bidirectional by including a signaling mechanism

in the reverse direction: from the vessel to the astrocyte. While there are likely several

such mechanisms, the one we focus on is astrocyte mechanosensation of arteriole movement

via stretch-gate TRPV4 channels on the astrocyte perivascular endfoot.

2.2 Mathematical model

A conceptual diagram of the model is shown in Figure 2.1. The model equations concern

the small spatial region of the developing brain cortex occupied by a single astrocyte and

the synapses and arteriole segment it contacts. Astrocyte-to-astrocyte signaling is left

out, and the synaptic space represents the net neural synaptic activity across the entire

astrocyte domain, which is assumed to be spatially uniform within the region.

During high synaptic activity, neurons release K+ and glutamate at the synapses (I).

K+ flows into the adjacent astrocytic process through KirAS channels, Na-K, and NKCC on

the perisynaptic process. The Na-K pump exchanges three sodium (Na+) ions for two K+

ions. The NKCC is an electrically neutral import of one Na+ ion, one K+ ion, and two Cl−

ions; however, the Na+ intake affects the Na-K pump activity, which is hyperpolarizing.

The KirAS current is larger in magnitude than the outward Na+ current from the Na-K

pump, resulting in an overall depolarization of the astrocyte membrane. The NKCC and

23

KirSMC

Ca2+

endoplasmic reticulum

process endfootsoma

εIP3glutamate

KirAS

K+

NKCC

K+

Na/K

K+

Na+

K+

Astrocyte (II)Synaptic

Space (I)

Perivascular

Space (III)

Vessel

SMC (IV)

BK

KirAV

TRP

Ca

Ca2+

EETEET

K+

K+

mGluR

Figure 2.1: Model overview. (I) Synaptic Space — Active synapses release glutamate and K+. (II),

Astrocyte Intracellular Space — K+ enters the astrocyte through Na-K pump, NKCC, and KirAS channels.

Na+ enters via NKCC and exits via Na-K pump. Glutamate binds to metabotropic receptors on the

astrocyte endfoot effecting IP3 production inside the astrocyte wall, leading to release of Ca2+ from internal

stores, causing EET production. Ca2+ and EET open BK channels at the perivascular endfoot, releasing

K+ into the perivascular space (III). Meanwhile, the buildup of intracellular K+ in the astrocytes results in

K+ efflux through the perivascular endfoot KirAV. (IV), Arteriole Smooth Muscle Cell Intracellular Space

— KirSMC channels are activated by the increase in extracellular K+. The resulting drop in membrane

potential closes Ca2+ channels, reducing Ca2+ influx, leading to SMC relaxation, and arteriole dilation

(strain, ǫ). The arteriole dilation (strain, ǫ) stretches the membrane of the enclosing astrocyte perivascular

endfoot , which activates Ca2+ influx through TRPV4 channels. The prohibition sign on the channel

is meant to indicate the inhibition mechanism of the channel, as the TRPV4 channel is inhibited by

intracellular and extracellular Ca2+. Note that the diagram here is not to scale. The perivascular endfoot

is actually wrapped around the arteriole, but we show them separated here in order to detail the ion

flow at the endfoot-vessel interface. Dashed arrows indicate ion movement; solid arrows indicate causal

relationships; dotted arrows indicate inhibition. Thin dashed arrows in the Kir channels indicate ion flux

direction at baseline, or in the vessel Kir, the change in flux direction when extracellular K+ exceeds 10

mM. New mechanisms we have developed for this model are indicated in color, while mechanisms borrowed

from previous models are shown in grey. Coloring is meant to distinguish biochemical pathways: potassium

transfer is shown in orange; sodium movement is shown in green; mechanosensation is indicated in red.

Na-K pumps have slow dynamics, making them potentially less efficient for K+ buffering.

Still, they are likely critical to the astrocyte’s role in regulating K+ in the synaptic space

(see Section 2.4, below).

Cortical astrocytes in young brains express glutamate receptors (mGluR5) on their

perisynaptic processes (II) that initiate intracellular IP3 production in response to synaptic

glutamate release. IP3 binds to receptors (IP3R) on the endoplasmic reticulum (ER),

releasing calcium (Ca2+) from internal stores. This is most likely specific to astrocytes

in the young brain, as [177] recently found that mGluR5 is expressed in cortical and

hippocampal astrocytes from young (<2 week old) mice brains, but not in adult mouse

or human brains, and further, that glutamate-dependent astrocytic Ca2+ rises may be

24

unlikely in the adult brain.

The mGluR5-dependent rise in intracellular Ca2+ causes epoxyeicosatrienoic acid (EET)

production. EET and Ca2+ activate BK channels in the astrocyte endfoot, releasing K+

into the perivascular space (III). It is unclear whether EET acts directly on the BK chan-

nels; it may act indirectly by activating TRPV4 channels [52, 143]. This would result in

a Ca2+ influx and membrane depolarization, both of which activate BK channels. For the

moment, we follow the model of [46] which is an empirical description of the relationship

between EET and BK activity, but a more mechanistic description can be added later as

more data become available. K+ is also released through the endfeet KirAV channels.

The K+ buildup in the perivascular space activates arteriolar smooth muscle cell (SMC)

Kir channels, here on referred to as KirSMC (IV). The resting SMC membrane potential

is higher than the KirSMC reversal potential, so the K+ flows outward. The resulting

SMC membrane voltage drop closes inward Ca2+ channels, and the intracellular Ca2+

concentration in the SMC drops. Because Ca2+ is required for myosin-actin crossbridge

attachment, the crossbridges then detach, allowing the SMC to relax and the arteriole to

expand.

As the vessel dilates, it stretches the perivascular astrocyte endfoot encircling it (II),

opening stretch-gated Ca2+-permeable TRPV4 channels in the endfoot. TRPV4 channels

are also sensitive to intra- and extracellular Ca2+ concentration [190, 12, 142]. There is

experimental evidence that TRPV4 channels are activated by a diverse range of chemical

and physical factors including heat [12, 190, 143, 109], EET and IP3 [52, 142], and they

are modulated by phosphorylation [142, 143]; for simplicity we leave these mechanisms

out for the moment. The astrocyte then experiences a depolarizing Ca2+ influx through

active TRPV4, thus maintaining BK activation, which prolongs the K+ signal (III) to the

arteriole (IV).

Below, we summarize the new ODEs we have developed and added to this model. All

other equations are given in detail in Chapter 1. All parameters (for the complete set of

equations) are given in Appendix A.

25

2.2.1 Potassium buffering

We describe the neurovascular K+ movement between three regions in the NVU: the synap-

tic space, astrocytic intracellular space, and perivascular space. In this section and through-

out the chapter, we use the letters J and I to distinguish between ionic flux (the change in

ionic concentration over time) and electrical current carried by that ionic flux. For example

JKir refers to the molar concentration of potassium ions that flow through the Kir channel,

whereas IKir is merely the electrical current of the channel. Potassium concentrations in

these three regions obey

d[K+]Sdt

= JKs − (JNaK,K + JNKCC + JKir,AS)1

V Rsa−RdcK+,S([K

+]S − [K+]S,0), (2.1)

in the synaptic space,

d[K+]Adt

= JNaK,K+JNKCC+JKir,AS+JBK+JKir,AV −RdcK+,A([K+]A− [K+]A,0), (2.2)

in the astrocyte intracellular space, and

d[K+]PVdt

= −JBK + JKir,AVV Rpa

− JKir,SMC

V Rps−Rdc([K

+]PV − [K+]P,0), (2.3)

in the perivascular space.

We assume in this model that the astrocyte membrane potential is uniform across the

entire cell; it is likely that this is not the case, as similar membrane structures (e.g. neuronal

dendritic trees) are highly lossy. However, for simplicity, we consider the astrocyte as a

single electrical compartment in which the membrane potential obeys

dVAdt

=1

Cast(−INaK − IKir,AS − IBK − ITRP − IKir,AV − Ileak), (2.4)

where Cast is the astrocyte cell capacitance. The individual flux terms and parameters in

Eqs. (2.1) – (2.4) that come from the previous models are all discussed in detail in Chapter

1, but we discuss here the new astrocytic flux terms that we have introduced to this model.

The electrical current through the Na-K pump is carried by both Na+ and K+ ions,

26

thus we treat it as the sum of these components:

INaK = INaK,K + INaK,Na, (2.5)

where the potassium current is carried by an influx of K+ ions which is described by the

Na-K potassium flux from [46]:

JNaK,K = JNaK,max[K+]S

[K+]S +KKoa

[Na+]1.5

[Na+]1.5 +KNa1.5i, (2.6)

where JNaK,max is the maximum K+ flux through the channel; the potassium concentration

in the synaptic space is [K+]S , and KKoa is the threshold value for [K+]S . [Na+] is

the intracellular sodium concentration, and KNai is the threshold value. V Rsa is the

volume ratio of the astrocyte intracellular space to the synaptic space. All astrocyte

cation currents, Ii+ , are related to the corresponding cation concentration flux, Ji+, as

Ii+ = −Ji+Castγ, where Cast is the astrocyte cell capacitance, and γ is a scaling factor for

relating the net movement of ion fluxes to the membrane potential [104]. (For anion flux,

the factor of −1 is removed). Thus, the potassium current in the astrocyte due to Na/K is

INaK,K = −JNaK,KCastγ. The Na-K pump exchanges 3 sodium ions for every 2 potassium

ions, so the Na+ current is

INaK,Na = −3

2INaK,K . (2.7)

The flux from the NKCC is adapted from [147]:

JNKCC = JNKCC,max log

[

[K+]S[K+]A

[Na+]S[Na+]A

(

[Cl−]S

[Cl−]A

)2]

, (2.8)

where the subscripts S and A refer to the synaptic and astrocytic spaces, respectively.

JNKCC,max is the scaling factor that determines the amplitude of the pump flux. Unlike

the other channels, the NKCC is electrically neutral (INKCC = 0) because its total uptake

comprises two positive charges (an Na+ and K+ ion) for every two negative charges (two

Cl− ions): a net charge of zero. However, the Na+ intake affects the Na-K pump activity,

27

which is hyperpolarizing. The intracellular Na+ concentration obeys

d[Na+]Adt

= JNaK,Na + JNKCC , (2.9)

where JNaK,Na = −INaK,Na/(Castγ).

Astrocytes in the cortex have homomeric Kir4.1 channels and heteromeric Kir4.1/5.1

channels; both are present in the perisynaptic processes, but the endfeet express only the

heteromer [78]. Also, it is worth noting, astrocytes in thalamus and hippocampus, where

there are abundant synapses, express predominantly the Kir4.1 homomer [78]. The het-

eromer has a higher single channel conductance, but few data exist on the Kir channel

densities along astrocyte bodies except in the retina [105, 92], where astrocyte function

is unique and highly specialized. Therefore, we estimate the relative whole-cell Kir con-

ductances at the endfeet and processes by adjusting for the appropriate K+ fluxes and

astrocyte membrane potential during simulation. The main difference between the Kir4.1

homomer and Kir 4.1/4.5 heteromer is the difference in their response to pH [78]. At the

moment, the model does not include a description of astrocytic pH, but when this is added

at future time, it will be important to consider its nonuniform inhibitory effects on the

process and endfeet Kir channels.

The Kir fluxes at the perisynaptic process and perivascular endfoot, JKir,AS and JKir,AV ,

respectively, are

IKir,AV/S = gKir,AV/S(VA − VKir,AV/S), (2.10)

where AV or AS stands for the Astrocyte Vessel-adjacent endfoot or Synapse-adjacent pro-

cess. The ionic flux, J , is computed from the electrical current, I, as JKir = IKir/(Castγ),

where Cast is the astrocyte cell capacitance, and γ is a scaling factor for relating the net

movement of ion fluxes to the membrane potential [104]. The conductance and reversal

potential, gKir,AV/S and VKir,AV/S are

gKir,AV/S = gKir,V/S

√

[K+]PV/S , and VKir,AV/S = EKir,endfoot/proc log[K+]PV/S

[K+]A,

(2.11)

where [K+]PV/S is the potassium concentration in the perivascular/synaptic space in mM,

28

and gKir,V/S is a proportionality constant. EKir,endfoot and EKir,proc are the Nernst con-

stants for the astrocyte KirAS and KirAV channels, respectively (about 25 mV [149]).

2.2.2 Mechanosensation

Significant work has been done modeling the vascular response to neural and astrocytic

inputs [17, 46, 25, 3, 157, 164] but very little has been done to explore the effect that

vascular activities may have on neurons and astrocytes. The hemo-neural hypothesis, pro-

posed by [128], implies that the cerebral vasculature has pivotal effects on neural function

through a variety of direct and indirect mechanisms. We suggest that one of these indirect

mechanisms is activated by astrocytic mechanosensation of vascular motions. Cerebral

astrocytes have been shown to express the transience receptor potential vanilloid-related

channel 4 (TRPV4), a mechanosensitive cation channel, and these have been observed to

be particularly abundant in astrocytic processes facing blood vessels [12]. There has also

been experimental documentation in vitro and in vivo of astrocytic depolarization and

intracellular calcium increase in response to vessel dilations [26], both of which could be

explained by TRPV4 channel activity.

Figure 2.1 outlines the conceptual model we use for astrocyte mechanosensation via

TRPV4 channels. As the vessel expands, it stretches the perivascular astrocyte endfoot

encircling it, opening stretch-gated Ca2+-permeable TRPV4 channels in the endfoot. As

a result, the astrocyte experiences a membrane depolarization and a rise in intracellular

Ca2+ due to an influx of Ca2+ through the TRPV4 channels. The Ca2+ influx has a

cumulative effect on the BK channels, which prolongs the K+ signal to the arteriole.

Given that the arteriole movements stretch the membrane of the enclosing astrocyte

endfoot, it is worth mentioning the possibility of mechanical feedback to the arteriole, which

is not addressed specifically in this model. Because the parameters in the arteriole model

were calibrated to in vivo data, we can consider the mechanical model for the vascular SMC

as a lumped model that treats the viscoelastic arteriole wall and the enclosing astrocyte

endfeet as a single viscoelastic structure.

In the astrocyte perivascular endfoot, the stretch-activated calcium influx from ex-

tracellular space, JTRPV , is determined by the arteriolar tone at the location where the

29

endfoot encloses the microvessel. The electrical current through the channel is

ITRP = gTRP s(VA − vTRP ), (2.12)

where gTRP is the maximum channel conductance; vTRP is the channel reversal potential,

and VA is the membrane potential (see Eq. (2.4) above). The calcium ion flux through

the channel is given by JTRP = −(1/2)ITRP /(Castrγ), where Castr is the astrocyte cell

capacitance, and γ is a scaling factor for relating the net movement of ion fluxes to the

membrane potential [104]. There is a factor of -1 because JTRP is a flux of positive

ions, whereas electrical current, ITRP , always describes the motion of negative charges (an

outflux of electrons being equivalent to an influx of positive ions). The factor of 1/2 is

there because there are two positive charges for every one calcium ion.

The TRPV4 channel current is activated by mechanical stretches, and, after activation

stops, experiences a slow decay in the absence of extracellular Ca2+, and a fast decay in

the presence of high extracellular Ca2+ [142, 190]. Thus, we model the open probability

as an ODE that decays to its variable steady state, s∞ (Eq. (2.14), below), according to

ds

dt=

1

τCa([Ca2+]P )(s∞ − s), (2.13)

where the Ca2+-dependent time constant τCa([Ca2+]P ) = τTRP /[Ca

2+]P , where [Ca2+]P

is the perivascular Ca2+ concentration (Eq. (1.19), below) expressed in µM, and s∞

is the strain- and Ca2+-dependent steady-state channel open probability: We model the

steady-state TRPV4 channel open probability, s∞, by the Boltzmann equation [73, 104]:

s∞ =

(

1

1 + e−(ǫ−ǫ1/2)/κ

)[

1

1 +HCa

(

HCa + tanh

(

VA − v1,TRPv2,TRP

))]

. (2.14)

The first term 1/(1 + e−(ǫ−ǫ1/2)/κ) describes the material strain gating, adapted from [104].

The strain on the perivascular endfoot, ǫ, is taken to be the same as the local radial strain

on the arteriole ǫ = (r − r0)/r0 (see Eq. (1.47) in 1.3.1), while ǫ1/2 is the strain required

for half-activation. The second term describes the voltage gating and Ca2+ inhibitory

behavior, based on the experimental results from [190] and [143]. The inhibitory term,

30

HCa, is

HCa = ([Ca2+]

γCai+

[Ca2+]PγCae

), (2.15)

where [Ca2+] is the astrocytic intracellular Ca2+ concentration (Eq. (1.5)); [Ca2+]P is

the perivascular Ca2+ concentration (Eq. (1.19), below), and γCai and γCae are constants

associated with intra- and extracellular Ca2+, respectively.

The astrocyte intracellular Ca2+ concentration, modified from Eq. (1.5), must include

influx from the TRPV4 channels:

d[Ca2+]

dt= β(JIP3

− Jpump + Jleak) + JTRP , (2.16)

where the Ca2+ flux through the TRPV4 channels is JTRP = −(1/2)ITRP /(Castrγ) (see

text below Eq. (2.12).

In our simulations of neural-induced astrocyte activation, we represent the total synap-

tic activity in the astrocyte domain as a uniform, continuous smooth pulse of glutamate

([Glu]) and of synaptic potassium ([K+]s).

2.3 Results — Bidirectional signalling

Simulation Procedures

For validation, we simulated two experimental procedures following [26]. In the first,

we simulate astrocyte membrane depolarization in response to purely mechanical vessel

dilations by imposing a time-dependent radial strain which we extracted from [26] and

then interpolated (see Figure 2.2).

In the second procedure, we simulate an in vivo experiment in which myogenic re-

sponse is induced via drug application. Because myogenic constriction requires influx of

Ca2+ into the SMC, the extracellular environment in the perivascular space is affected

by dilations involving myogenic relaxation; namely, the perivascular Ca2+ concentration

increases, which may affect the astrocytic TRPV4 channels. Thus, it is important to study

the astrocyte response to arteriole dilations due to myogenic relaxation in addition to me-

chanical stretching. Experimentally, application of the drug pinacidil stimulates myogenic

31

vessel dilations by activating Kir channels ([26]) in the vascular SMC, leading to a de-

crease in SMC intracellular Ca2+. We simulate the effect of this drug by enforcing a set

Kir channel open population, and we compare the results to the in vivo data from [26].

All differential equations were solved in MATLAB using the solver “ode23tb,” which

implements TR-BDF2, an implicit Runge-Kutta formula with backward differentiation

(BDF).

TRPV4 Model Validation

0 50 100 150 200 250-80

-70

-60

-50

-40

-30

time (sec)

Ast

rocy

te M

emb

ran

e P

ote

nti

al /

mV

Astrocyte Depolarization (Cao (2011))

Astrocyte Depolarization (simulation)

Vessel Dilation (Cao (2011))

0

20

40

Ves

sel

Dil

atio

n /

%

Figure 2.2: Astrocyte response to mechanical stretching of vessel. Thick curves are experimental in vitro

data extracted from [26], Chapter 4, Fig. 2: thick black curve is the vessel radial strain (20-40% of vessel

radius); thick grey curve is the astrocyte membrane potential. Simulation data are shown as the black

dotted line. The vessel dilations used in the simulation were interpolated from the data extracted from

[26].

0 50 100 150 200 250 300 350-0.05

0

0.05

0.1

0.15Arteriole Dilation During Pinacidil Application

time /sec

Art

erio

le R

adia

l S

trai

n

0 50 100 150 200 250 300 3500.14

0.15

0.16

0.17

0.18Astrocyte Perivascular Endfoot Calcium

Endfo

ot

[Ca2

+]

/µM

time /sec

Simulation

In Vivo Data (Cao (2011))

Simulation

In Vivo Data (Cao (2011))

200 2250.0875

0.125

Figure 2.3: Astrocyte response to drug

induced vasodilation. Upper plot shows

vessel radius in response to application of

pinacidil; inset shows zoomed-in view for

better resolution of vasomotion. Lower

plot shows astrocyte intracellular Ca2+

concentration increasing in response to ves-

sel dilation. Grey curves are data inter-

polated from [26], Chapter 4, Figure 7A.

Black curves are simulation results.

Figure 2.2 shows the astrocyte membrane depolarization in response to quick, mechan-

32

ical stretches in the radial direction. The simulation was intended to mimic the in vitro

experiment by [26] in which an arteriole in a brain slice was pressurized in brief bursts

that inflated it between 20-40%. Data extracted from [26] are shown as thick black curves

(vessel dilation) and thick grey curves (astrocyte membrane potential). In the simulation,

the imposed strain on the vessel radius was interpolated from the extracted data. The

black dotted line is the simulated astrocyte depolarization, which is in good agreement

with the experimental results.

The astrocyte response is characterized by a quick rise – ∼2 seconds – to maximum

depolarization, followed by a slow decay: ∼4 seconds to decay to half maximum, and ∼15

seconds to recover to baseline. Note that these simulation results are a product of the

TRPV4 channel equations included in the model perivascular astrocyte endfoot, so the

results in Figure 2.2 support the hypothesis that TRPV4 channels are responsible for the

astrocyte depolarization observed in response to vessel perturbations.

We further validate the model by simulating application of the vasodilatory drug

pinacidil and compute the resulting Ca2+ concentration in the astrocyte, analogous to

the in vivo experiment by [26]. Pinacidil induces vasodilation by opening the Kir channels

in the vascular smooth muscle cell ([26]). Further, data have suggested that pinacidil only

affects the vascular SMC, and has no direct effect on neurons and astrocytes ([27]). We

simulated the effect of pinacidil by imposing the SMC Kir channel open state k (see Eq.

(1.20)), directly. In order to reproduce the same vessel dilation as the analogous in vivo

experiment by [26] (extracted data shown in Figure 2.3, lower plot), we estimated that the

pinacidil induced Kir activation progressed according to

k(t) = 0.01

(

1 + tanht− 270

60

)

, (2.17)

where t is time in seconds. For this simulation, Eq. (2.17) is substituted into Eq. (1.20)

instead of solving for k with the ODE in Eq. (1.23). The results are shown in Figure 2.3;

the lower plot is the arteriole dilation due to pinacidil application, and the upper plot is the

resulting Ca2+ level in the adjacent perivascular astrocyte endfoot. As the vessel dilates,

it activates the TRPV4 channels on the astrocyte perivascular endfoot, initiating a Ca2+

33

influx into the astrocyte. Again, the simulation (black curves) shows good agreement with

experimental results (grey curves, extracted from [26]).

Astrocytic and vascular bidirectional interaction

0 10 20 30 400

0.5

1

1.5

K+/μ

M

time/s

K+, glutamate

A

0 10 20 30 400

0.1

0.2

IP3/μ

M

time/s

K+, glutamate

B

0 10 20 30 40

5

10

15

20

K+/m

M

time/s

K+, glutamate

F

0 10 20 30 40

−70

−60

−50

−40

Vk/m

V

time/s

K+, glutamate

E

0 10 20 30 40

0.2

0.3

0.4

Ca2

+/μ

M

time/s

K+, glutamate

C

0 10 20 30 40

0.1

0.2

0.3

0.4

SM

C C

a2+/μ

M

time/s

K+, glutamate

G

0 10 20 30 40

21

22

23

24

radiu

s/μ

m

time/s

K+, glutamate

H

0 10 20 30 40

1

2

3

EE

T/μ

M

time/s

K+, glutamate

D

Figure 2.4: Astrocytic and vascular bidirectional response during neural stimulation. Horizontal black

bars indicate period of synaptic K+ and glutamate stimulus. Solid lines are results using the equations

above. Dashed grey lines are results when TRPV4 channels are excluded from the model. A K+ concen-

tration in synaptic space. B Astrocytic intracellular IP3 concentration. C Astrocytic intracellular Ca2+

concentration. D Astrocytic intracellular EET concentration. E Astrocyte membrane potential. F Extra-

cellular K+ concentration in the perivascular space. G Arteriolar SMC intracellular Ca2+ concentration.

H Arteriole radius.

The results in the whole system when the astrocyte is given a transient input of synaptic

glutamate and K+ are shown in Figure 2.4. The astrocytic IP3 immediately increases

(Figure 2.4B), while the subsequent rise in intracellular Ca2+ experiences a brief (<1 sec)

delay (Figure 2.4C). The EET rise (Figure 2.4D) follows that of Ca2+. All three reach

steady state within ∼10 sec, while (Figure 2.4E) the astrocyte continues to depolarize due

to influx of K+ from the synaptic space, which is eventually balanced by the K+ outflux

due to Ca2+- and EET-dependent opening of the BK channels in the perivascular endfoot.

The K+ concentration (Figure 2.4F) in the perivascular space activates arteriolar SMC

34

Kir channels, further increasing the perivascular K+, and causing a hyperpolarization of

the SMC membrane (not shown) as well as damping the oscillations in the SMC Ca2+ and

vessel radius. The hyperpolarization closes Ca2+ channels in the SMC, reducing the SMC

Ca2+ concentration (Figure 2.4G), thereby causing vessel dilation (Figure 2.4H). As the

perivascular K+ concentration continues to increase during stimulation, the dilated vessel

contracts slowly due to the [K+]P -dependent shift in the Kir channel reversal potential,

which reverses the direction of the K+ flow in the Kir channels and changes the polarity of

the vessel response. The vessel dilation stretches the enclosing astrocytic endfoot, opening

the stretch-gated TRPV4 channels in the membrane, causing an influx of Ca2+ into the

astrocyte (Figure 2.4C). After the stimulus, the vessel maintains its dilation until the

remaining K+ in the perivascular space decays close to its baseline value.

There are two notable differences that arise in the simulation results due to our addition

of the TRPV4 channel equations. The first is that the vessel response (Figure 2.4G,H) in

the TRPV4 model (black lines) is both prolonged and it rises to maximum dilation faster

than when TRPV4 channels are excluded (grey lines). This is a result of the second differ-

ence, which is that the astrocytic TRPV4-mediated Ca2+ influx maintains the astrocytic

Ca2+ signal and resulting EET production and BK channel activity (Figure 2.4C–E, black

curves) until the vessel reconstricts, well past the end of the neural stimulus. In contrast,

without TRPV4 channels included (grey curves), the astrocyte activity drops to baseline

as soon as the neural stimulus ends, even while the vessel remains dilated. In this way, our

model (with TRPV4 channels) predicts a more physiological result, consistent with the

experimental data from ([56]). Notice that while the grey curves drop to baseline within

seconds of the end of the neural stimulus, the black curves remain steady and drop to

baseline just after 40 seconds, about the time that the vessel has returned to its initial

level of constriction.

Shown in Figure 2.4G and H are the SMC intracellular Ca2+ and vessel radius, re-

spectively. Before the onset of the stimulus, the vessel experiences vasomotion: note the

oscillating behavior in the absence of neural activity. In periods without neural stimuli, the

amplitude and frequency (∼0.5 Hz) of the oscillations are consistent with those observed in

experiment ([56, 89, 129, 113]). During neural stimulation, the vessel dilates, and vasomo-

35

tion is inhibited – note the flat, non-oscillatory response in the SMC Ca2+ and vessel radius

while the vessel is in the dilated state. This phenomenon is consistent with the experimen-

tal results of [56, 22, 168]. Dilation along with suppression of vasomotion occur in response

to the hyperpolarization that the SMC experiences when astrocytic K+ release activates

the SMC KIR channels. This hyperpolarization closes SMC Ca2+ channels, preventing

influx of Ca2+. The reduction in intracellular Ca2+ allows myogenic relaxation, resulting

in vasodilation Figure 2.4H. Further, the hyperpolarization suppresses Ca2+ oscillations

in the SMC, thus suppressing radius oscillations (Figure 2.4G,H). Note that vasomotion

resumes after the termination of the functional hyperemia response, which is also observed

experimentally ([56, 22]).

2.4 Results — Potassium buffering

2.4.1 Effect of Astrocyte K+ Buffering on Neurovascular Coupling

We simulate neural activation of the astrocyte by imposing a smooth pulse of extracellu-

lar glutamate and K+ in the synaptic space to approximate neural stimulation. In this

section, we consider two extracellular regions: (1) the vessel/astrocyte interface (perivas-

cular space), where K+ buffering helps determine the dynamics of functional hyperemia,

and (2) the astrocyte/neural interface, where the astrocyte modulates the extracellular

environment in the synaptic space.

Astrocyte/vessel interaction

In this model, the introduction of the astrocytic Kir channels allows the astrocyte to re-

spond to changes in extracellular and intracellular potassium concentration. To understand

how KirAS and KirAV channels impact the neurovascular interaction, we compare the re-

sults of this model with a lumped version that does not include astrocytic Kir. In the

lumped version, we remove the astrocyte Kir current (JKir,AS = JKir,AV = 0), and instead

describe the total membrane current at the synapse-adjacent side of the astrocyte, IAS , as

a lumped model for the combination of currents from the Na-K pump, KirAS and KirAV

channels: IAS = IKir,AS + INaK,K + INaK,Na ≈ INaK,K (see Eqs. (2.5) – (2.7)), similar

36

to the models in [193, 46]. We adjust the lumped model’s leak current (see Eq. (1.9)) so

that the baseline and maximum astrocyte membrane potential match those of the detailed

buffering model that includes KirAS and KirAV.

lumped (no Kir)lumped (no Kir)

0 20 40 60 80time /s

explicit (Kir)

-90

-80

-70

-60

-50

VA /

mV

[K+] A

/m

M

120

25

0

50

75

ρ,

bo

un

d m

Glu

R /

%ρ

, b

ou

nd

mG

luR

/%

[K+]S

/m

M

0

5

10

15[K+]S

ρ

explicit (Kir)lumped

(no Kir)

0 20 40time /s

[IP

3]

/µM

0

0.4

0.2

[Ca2

+]A

/µ

M

0.1

0.2

0.3

0.4

0

[EE

T]

/µM

1

2

3

4

time /s0 20 40 60

18

radiu

s /µ

m

20

22

24

26

28

[K+]P

/m

M

5

10

15

20

0

Cu

rren

t /n

A

100

50IKir,V

IBK

-25

(a)

(b)

(c)

(e)

(d)

(f)

(h)

(g)

114

116

118

112

110

115

0

Figure 2.5: Astrocyte Kir effect on neurovascular coupling. Black curves – astrocyte model equations

described in this chapter. Grey curves – astrocyte model equations without KirAS or KirAV channels. (a)

Extracellular K+ in the synaptic space. Thin red curve is glutamate transient represented by the ratio

of bound to unbound glutamate receptors, ρ (see Eq. (1.3)). (b) Solid lines – intracellular astrocytic K+

concentration. Dashed lines – Astrocyte membrane potential. (c) Astrocyte intracellular IP3 concentra-

tion. (d) Astrocyte intracellular Ca2+ concentration. (e) Astrocyte intracellular EET concentration. (f)

Astrocyte perivascular endfoot BK (dashed lines) and KirAV (dash-dot lines) currents. (g) Extracellular

K+ concentration in the perivascular space. (h) Arteriole radius.

Figure 2.5 shows the effect of astrocyte Kir channels on the neurovascular unit. Black

curves show the NVU under normal conditions, and the grey curves show the NVU in

which the astrocyte KirAS and KirAV are removed (IKir,S and IKir,V both set to 0). The

system experiences a brief period of neural activity (Figure 2.5a, thick black and grey

curves show synaptic K+; thin red curve shows glutamate transient), triggering astrocyte

membrane depolarization and intracellular K+ increase (Figure 2.5b, dashed and solid

curves, respectively).

The glutamate initiates IP3 production in the astrocyte (Figure 2.5c), leading to release

of Ca2+ from internal stores (Figure 2.5d), causing EET production (Figure 2.5e). The

astrocytic Ca2+ and EET activate BK channels in the astrocyte endfeet (Figure 2.5f,

dashed curves) where K+ is released into the perivascular space (Figure 2.5g). Meanwhile,

the membrane depolarization and the increase in intracellular astrocyte K+ results in an

37

outward K+ flux through the endfoot KirAV channels (Figure 2.5f, dash-dot curve). In the

absence of astrocyte KirAS and KirAV, astrocyte K+ release into the perivascular space is

delayed, causing a delay in the vascular response (Figure 2.5h). According to the simulation

results, this is because the KirAV is responsible for the immediate release of K+, while the

BK current rises later (Figure 2.5f). This may explain why previous generations of this

astrocyte model, without a description of K+ buffering or astrocyte Kir channels [46],

produced a non-physiological delay of ∼25 seconds in the neurovascular response.

In the black curves, there is a short period of arteriole constriction during the neural

stimulation period (Figure 2.5h): at about 25 seconds, the radius stops increasing and

the vessel begins to constrict. This is a phenomenon observed by [65] in which moderate

increases in extracellular K+ cause vasodilation, but increases beyond ∼15 mM will cause

the vessel to constrict. The results are also consistent with the simulations of [46], who

postulated that the change from dilation to constriction during sustained activity was

caused by the arteriole KirSMC channels, which have a reversal potential that experiences

a depolarizing shift with increasing extracellular potassium: when the extracellular K+

rises above 15 mM, the KirSMC reversal potential shifts from below to above the SMC

membrane potential, reversing the direction of the current, which causes a depolarization

that reopens Ca2+ channels, causing in turn constriction. This model is discussed in more

detail in [46].

Astrocyte/neuron interface: extracellular K+ undershoot

[K+] S

(si

mu

lati

on

)

[K+] S

in

viv

o (

see

cap

tio

n)

2 m

M

60 s

(a) (b)

Figure 2.6: K+ undershoot. K+ undershoot in

the extracellular synaptic space following stimu-

lus is more pronounced with increasing length of

activation period. Stimulus period is indicated

by thick black bars. (a) Simulation results. (b)

Experimental results interpolated from Figure 3

in [32].

Figure 2.6 shows the extracellular K+ concentration in the synaptic space over a cycle

38

of stimulation and recovery for several different lengths of stimulus time (simulations in

Figure 2.6a are compared with experimental results from [32], interpolated in Figure 2.6b).

In the post-stimulus recovery period, the extracellular K+ initially displays a fast drop to

below baseline level before returning gradually to resting state equilibrium concentration.

This undershoot is more pronounced as the length of the stimulation period increases:

note that the 60 second stimulus in the top plot results in the greatest undershoot and the

longest period of recovery to baseline. With decreasing length of activation period (top

plot to bottom plot), the undershoot magnitude and recovery time also decrease, a trend

which has been reported from in vivo studies in the mouse hippocampus [32]. The same

experiments also validate the time-dependent characteristics of the undershoot: a fast drop

with a slow return up to baseline.

Our model suggests that the undershoot is a result of the activities of the Na-K pump

and NKCC. The astrocyte Na-K pump flux is an inward movement of K+ from the synaptic

space and an outward flow of Na+ and is activated by high extracellular K+ and high

intracellular Na+. Meanwhile, the NKCC flux is an inward K+ and Na+ flux that increases

with decreasing concentrations of intracellular K+ and Na+. During stimulation, the rise

in K+ in the synaptic space drives the Na-K exchange, and the astrocytic Na+ decreases.

Although the K+ influx and Na+ outflux from the Na-K pump provide competing signals

for the NKCC, the Na-K pump exchanges three Na+ ions for every two K+ ions, so the

result favors an increased NKCC influx.

At the end of the stimulus, the synaptic K+ decreases towards baseline, so the decreased

extracellular K+ and intracellular Na+ result in a decreased Na-K pump flux. At this

time, the NKCC is required to replenish the intracellular Na+, which means that K+

uptake is continued via the cotransporter. At the same time, with rising intracellular

K+ and decreasing extracellular K+, the astrocyte KirAS flux reverses, counteracting the

K+ uptake through the cotransporter. Thus, there is competition at the synaptic space

between K+ uptake by astrocyte NKCC and K+ release by astrocyte KirAS. When the

stimulus period is sufficiently long, the Na+ has enough time to reach a low enough level

that the magnitude of the NKCC flux exceeds the KirAS release, so the K+ uptake continues

beyond the point at which synaptic K+ has reached baseline, resulting in an undershoot in

39

extracellular synaptic K+. The drop below baseline continues until Na+ has risen enough

for the NKCC flux to decrease, and the KirAS outflux returns the extracellular K+ back

up to baseline concentration.

2.4.2 Kir channel blockade

[K+]S

/m

M

0time /s

2

4

6

00time /s

0

[K+]S

/m

M 6

2

4

(c)

52 40

[K+]A

/m

M

65

[K+]A

/m

M

55

6054

56

stim

(a) (d)

(f)

-80

-60V

A /

mV

-90

-70

-90

-80

-60

VA /

mV

-70

stimstimstimstim

stimstim

stimstimstimstim

(b) (e)-50 -50

6030 906030 90

(a)

8 8

Figure 2.7: Astrocyte response

to K+ channel blocker with short

stimulus spike. Black curves –

neural-induced astrocyte stimu-

lation under control conditions.

Grey curves – astrocyte stimu-

lation in presence of K+ chan-

nel blocker. (a) Intracellular as-

trocytic K+ concentration. (b)

Astrocyte membrane potential.

(c) Extracellular K+ concentra-

tion in the synaptic space. (d-

f) show corresponding experi-

mental results interpolated from

Figure 7 in [7].

Figure 2.7 shows the results when the KirAS and KirAV channels in the astrocyte are

blocked. We simulate the effect of the Kir channel blocker Ba2+ [7] by setting the Kir

currents equal to zero (see Eqs. (2.4) and (2.10)). The astrocyte is activated by a tran-

sient “spike” of K+ in the synaptic space (Figure 2.7c). Under control conditions (black

curves), the astrocyte responds with a quick rise in intracellular K+ concentration (Figure

2.7a). In the presence of Ba2+ (grey curves), the astrocyte baseline K+ is higher, and

it rises more slowly to a lower peak concentration. The astrocyte membrane potential

(Figure 2.7b) experiences a hyperpolarization in the presence of Ba2+ during activation,

and has a depolarized equilibrium value compared to the control. These results are all in

good qualitative agreement with the experiments of [7], shown here in Figure 2.7d-f for

comparison.

40

2.5 Sensitivity Analysis

Parametric uncertainty is a major limitation of this model, as well as of previous models

of [193, 46, 17]. The astrocyte component alone contains 55 parameters, many of which

are only crude estimates because not enough experimental data are available. To address

these limitations, we perform a global parameter sensitivity analysis using the ANOVA

functional decomposition and stochastic collocation [174, 175, 75] in which we vary eight

key parameters simultaneously. The eight parameters were identified based on preliminary

sensitivity analysis used to narrow down the 55-parameter set to the subset most critical

to these experiments. Sensitivity indices are computed from the ANOVA representation in

[175]. The sample points are Gauss-Legendre quadrature points that come from a tensor

product of the one-dimensional quadrature rule computed with the code provided in [75].

Undershoot ∆ [K+]A,0

(Kir blockade)

∆ [K+]A,max

(Kir blockade) ∆ VA,max

(Kir blockade)

JNaK,max

KKoa

KNai

JNKCC,max

RdcK+,S

EKir,proc

EKir,endfoot

[Na+]S

JNaK,max

KKoa

KNai

JNKCC,max Rdc

K+,S

EKir,proc

EKir,endfoot

[Na+]S

JNaK,max

KKoa

KNai

JNKCC,max Rdc

K+,S

EKir,proc

EKir,endfoot

[Na+]S

JNaK,max

KKoa

KNai

JNKCC,max Rdc

K+,S

EKir,endfoot

[Na+]S

EKir,proc

Figure 2.8: Sensitivity of K+

undershoot and effects of Kirblockade. Diameters of smallcircles indicate single parame-ter sensitivity; color indicateswhether increasing parametermagnitude will increase (white)or decrease (black) the valueof metric: (top left) synapticK+ undershoot; change in as-trocytic K+ after Kir block-ade at (top right) baseline,∆[K+]A,0, and (bottom left) ac-

tive state, ∆[K+]A,max; (bottomright) maximum astrocyte hy-perpolarization, ∆VA,max, dueto activation during Kir block-ade. Thicknesses of greylines indicate sensitivity of two-parameter interaction pair.

The results for our 8-dimensional global sensitivity analysis are shown in Figure 2.8.

We define “undershoot” as the amount by which the extracellular K+ in the synaptic

region drops below baseline levels following a neural stimulus. To understand the figure

in each quadrant, consider that all eight model parameters of our subset are arranged in

a large ring (to make the diagram easier to see, we have only labeled the parameters we

have determined to be most sensitive). The sensitivity of a single parameter is shown as

41

a small circle, with the diameter equal to the sensitivity of that parameter. For example,

in the top left quadrant, it is shown that the potassium undershoot is most sensitive to

JNaK,max and RdcK+,S , the maximum pump rate of the sodium-potassium exchange and

the decay rate of K+ in the synaptic space, respectively. The fill color – white or black –

of the circles indicates whether the sensitivity is a constructive or destructive, respectively.

In other words, when the value of JNaK,max is increased, the undershoot is increased,

whereas when RdcK+,S is increased, the undershoot effect is diminished. The grey lines

show the interaction of two parameters, where the thickness of the line segment is equal

to the sensitivity of the interaction pair: this means that we are measuring how much the

results will be changed when two parameters are changed at once. For instance, the most

critical interaction pair for the undershoot is JNaK,max and JNKCC,max, the maximum flux

rate of the NKCC pump. Now that we have established how to interpret the figure, we

can discuss the results in more detail.

The parameter sensitivity of the undershoot is shown in the top left quadrant of Figure

2.8. The parameters JNaK,max and JNKCC,max, the maximum flux rates of the Na-K

and NKCC pumps, respectively, have the highest sensitivity (taking into account their

individual sensitivity, the white circles, and their interaction term, thick grey rectangle).

Both of these parameters have a positive impact on the undershoot: when either parameter

is increased, the undershoot also increases. This is consistent with the hypothesis that

the Na-K and NKCC pumps are responsible for the K+ undershoot. Note also the high

sensitivity of the parameter RdcK+,S , the decay rate of K+ in the synaptic space. This

implies that the undershoot phenomenon may be a result of additional factors besides the

astrocyte alone, for example, changes in local synaptic activity following a period of neural

activation.

In the top right quadrant, we show the sensitivity of the shift in baseline astrocyte K+

concentration after a Kir channel blockade is applied, ∆[K+]A,0 = [K+]A,0(Kir blockade)−

[K+]A,0(control conditions). The results demonstrate that the astrocytic KirAS on the

synapse adjacent process are more critical in setting the baseline astrocyte K+, whereas

in the lower left quadrant, it is apparent that the maximum astrocyte K+ level depends

mainly on the endfoot KirAV.

42

The bottom right quadrant shows the sensitivity of the astrocyte hyperpolarization that

occurs when the astrocyte is activated in the presence of a Kir blockade. Under normal

conditions, the activated astrocyte would experience a depolarization due to K+ influx

through the KirAS channels on the synapse adjacent processes. The only other mechanism

present on the astrocyte process in this model is the Na-K exchange, which exchanges two

K+ ions into the cell for three Na+ ions leaving the cell, an overall hyperpolarizing effect

(the NKCC pump is electrically neutral as it pumps in two positive ions, one K+ and one

Na+, along with one Cl2− ion). Thus, it is reasonable that the maximum Na-K pump flux,

JNaK,max, is the most sensitive parameter for the astrocyte hyperpolarization during a Kir

blockade.

[K+]S,0

JNaK,max

KKoa

Cast

γ

EKir,proc

[K+]S,max

JNaK,max

KKoa

VRSA

RdcK+,S

EKir,proc

[K+]P,0

JNaK,max

KKoa

γ

EKir,endfoot

[K+]P,max

JNaK,max

VRSA

RdcK+,S

EKir,proc

EKir,endfoot

baseline, synaptic space

baseline, perivascular space

synaptic space, active state

perivascular space, active state

synaptic space

perivascular space

bas

elin

e

acti

ve

stat

e

EKir,proc

Figure 2.9: Sensitivity of base-

line and maximum extracellu-

lar potassium. Small circles

indicate single parameter sen-

sitivity equal to diameter of

the circle; style indicates that

increased parameter magnitude

will increases (open) or decrease

(solid) the value of the met-

ric: (top left and right) extra-

cellular K+ in synaptic space

at baseline, [K+]S,0, and ac-

tive state, [K+]S,max; (bottom

left and right) extracellular K+

in perivascular space at base-

line, [K+]P,0, and active state,

[K+]P,max, respectively. Thick-

nesses of grey lines indicate sen-

sitivity of two-parameter inter-

action pair.

In order to narrow the set of important parameters, we analyze the system sensitivity

to all 55 parameters using the ANOVA functional decomposition and stochastic collocation

[174, 175, 75] in which only two parameters at a time are varied simultaneously, but all 55

parameters are compared.

Figure 2.9 shows four quadrants, each with sensitivity results for extracellular potassium

43

concentration: The top row is the sensitivity of potassium concentration in the synaptic

space, while the bottom row is the potassium concentration in the perivascular space; the

left column is the baseline level, and the right column is the maximum concentration when

the system is in the active state1.

In each quadrant, consider all 55 the model parameters are arranged in a large ring, but

to make the diagram easier to see, we have only labeled the parameters we have determined

to be most sensitive. The data in this figure are visualized the same way as in Figure 2.8.

In the top left quadrant, the most sensitive parameters for the synaptic space baseline

K+ level are the following: JNaK,max; KKoa, the threshold value of extracellular K+

for the Na-K pump in the astrocyte process; EKir,proc; γ, the conversion factor relating

flux of ionic concentration to electric current in the astrocyte membrane; and Cast, the

capacitance of the astrocytic membrane. The most sensitive parameter is EKir,proc, which

implies that the astrocyte KirAS channels are the most important mechanism setting the

baseline level of synaptic K+. Likewise, in the lower left quadrant, the baseline perivascular

K+ level is most sensitive to the astrocyte endfoot KirAV and process KirAS channel reversal

potentials (EKir,proc and EKir,endfoot, respectively). We note that these findings are limited

to an isolated astrocyte; intercellular inputs from adjacent astrocytes via gap junctions and

diffusible biochemical messengers may have a profound impact impact on the results if they

are included in the model.

It is worthwhile to consider baseline K+ sensitivities (the left-hand side, top and bottom

quadrants) together: note that the synaptic space and perivascular space baseline K+ levels

are both sensitive to several of the same parameters: JNaK,max, KKoa, EKir,proc, and γ.

This reveals an inherent model assumption about the connectivity of the two ends of the

astrocyte, which is a possible limitation of the model. In the future, it may be necessary

to upgrade the model into a multi-compartment model in which some loss exists in the

propagation of ions and electric signals from one end of the cell to the other. Gap junctions

between neighboring astrocytes may also need to be added to the model in the future as

these would contribute to the distribution of ions throughout the glial network. The same

1meaning that there is synaptic activity, which we model as a release of potassium and glutamate in thesynaptic space near the synapse-adjacent astrocyte process

44

is evident in comparing the active state K+ in the synaptic and perivascular regions (right-

hand side, top and bottom quadrants). Again, out of all 55 parameters in the model, the

maximum K+ level in the synaptic and perivascular spaces share four out of five of their

top most sensitive parameters: V RSA, RdcK+,S , JNaK,max, and EKir,proc.

2.6 Model Uncertainty

In this section, we address the uncertainty in the model by testing the effects of either omit-

ting or modifying certain mechanisms of the model, as opposed to parametric uncertainty

or parameter sensitivity analysis.

2.6.1 Astrocytic TRPV4 channels

0 10 20 30 40 508

10

12

14

16

18

time (sec)

[Na+

]A (

mM

)

0 10 20 30 40 50

−0.4

−0.2

0

0.2

time (sec)

Ast

rocy

te

Cu

rren

ts (

pA

)0

5

10

15

[K+

]S (

mM

)

0 10 20 30 40 5052

53

54

55

56

57

58

59

[K+

]A (

mM

)

time (sec)

2

4

6

8

10

12

14

16

[K+

]P (

mM

)

−85

−80

−75

−70

−65

−60

−55

−50

−45

−40

VA

(m

V)

0 10 20 30 40 5018

20

22

24

26

28

30

times (sec)

rad

ius

/µm

−0.5

synaptic space

glutamate

K+

astrocyte K+

Astrocyte Na+

Perivascular

Space K+

Astrocyte

membrane

potential

Vessel radius

INaK,K

IKir,S

IBK

ITRPV

IKir,V

Figure 2.10: Effects of TRPV4 K+ and Na+ effluxes. Blue curves – original model from paper, in whichTRPV4 includes only a Ca2+ permeability. Red curves – modified model in which TRPV4 current includesa Ca2+ influx as well as an outward K+ and Na+ component.

Because TRPV4 is a nonspecific cation channel, it is possible that it has a permeability

to potassium and sodium ions [180]. In this model, we considered only the calcium com-

ponent of the channel current, but we test here whether including potassium and sodium

outward fluxes through astrocyte the TRPV4 channel would impact the neurovascular cou-

pling in the simulation. Figure 2.10 compares the results from our model equations above

45

(blue curves) with the results when we include K+ and Na+ TRPV4 currents. Because

the data are limited on the specific K+ and Na+ permeabilities of astrocyte TRPV4, we

estimate that the TRPV4 K+ and Na+ effluxes have equal magnitude and have a com-

bined magnitude of 75% of the Ca2+ TRPV4 current. Based on our simulation results, the

model is not particularly sensitive to the addition of the TRPV4 K+ and Na+ effluxes, as

the astrocyte Kir channels, particularly at the endfoot, oppose the hyperpolarizing effects

of the TRPV4 effluxes. Also, because the Kir channels are sensitive to extracellular K+,

the TRPV4 potassium efflux into the perivascular space causes a reduction in the endfoot

Kir potassium release, leaving the perivascular potassium almost unchanged after adding

K+ and Na+ effluxes to the TRPV4 model. Because of the limited data on TRPV4 K+

and Na+ currents and the low sensitivity of such currents in our model, we find that it is

preferable to leave these out of the model, although they can be added in the future as

more data are available.

2.6.2 Role of astrocyte BK channels on neurovascular coupling

In this section we consider the effect of astrocytic endfoot BK channels on neurovascular

coupling. Figure 2.11 shows the results from our model when the astrocyte BK channels are

omitted. The top plot shows the perivascular potassium concentration during a 30 second

2

4

6

8

10

12

14

[K+] P

/m

M

0 10 20 30 40 5019

21

23

25

27

time /s

rad

ius

/µm

perivascular

space

Vessel

60

originalBK removed

stim

Figure 2.11: Astrocyte BK chan-nel effect on neurovascular coupling.A 30 second neural stimulus occursstarting at time = 5 seconds (blackhorizontal bar). Blue curves – re-sults under control conditions (withBK channels). Red curves – resultswhen astrocytic BK channels are re-moved by setting the astrocyte BKconductance to 0. Top plot – perivas-cular potassium concentration. Bot-tom plot – arteriole radius.

neural stimulus (indicated by the black, horizontal bar), and the bottom plot shows the

vessel radius. The blue curves are the results when the model equations are unchanged,

and the red curves are the results when we remove the BK channels from the astrocyte

46

model. Although the astrocyte Kir can sustain a moderate potassium efflux, without

the BK channels, the perivascular potassium, and consequentially the vessel dilation, is

significantly reduced according to our model.

These results are reasonable based on the literature. A study by [80] found that the BK

channel blockers paxilline and TEA inhibited astrocyte mGluR-induced outward current,

which was reduced by 87.6%. Similarly, [57] found that BK channels in astrocyte endfeet

produced large-conductance outward potassium currents that were activated by neural

stimulation, and that blocking BK channels or ablating the gene encoding BK channels

actually prevented neuronally induced vasodilation, which is believed to be caused by

astrocyte release of potassium at the gliovascular space. Another study demonstrated that

rSlo KCa (BK) channels are highly concentrated in astrocyte perivascular endfeet [162].

2.6.3 Adult brain astrocyte model

Recent experimental work by [177] demonstrates a fundamental difference between astro-

cyte glutamate receptor expression in young and adult brains, suggesting the possibility

that only astrocytes from young brains respond to synaptic glutamate release with an in-

tracellular calcium rise. The glutamate receptor linked to IP3 production and subsequent

intracellular Ca2+ increase is mGluR5; the results in [177] show that cortical astrocytes

from developing brains expressed mGluR5, but that both adult human and adult mouse

cortical astrocytes did not. The same study found that the primary type of mGluR ex-

pressed in adult cortical astrocytes is mGluR3, which does not trigger Ca2+ increases.

Based on these findings we explore a possible preliminary model of an adult brain

astrocyte which is a modified version of the equations given in Sections 1.2 and 2.2. In the

modified “adult” version of the model, we remove the glutamate triggered IP3/Ca2+/EET

pathway (removing Eq. (1.4) and setting JIP3= 0 in Eq. (2.16)) that is characteristic of

the mGluR5, but not mGluR3.

Results comparing the original young brain model and the modified “adult” brain

model are shown in Figure 2.12. The green curves are from the original model, and

the red curves are the results from the adult brain model. What is interesting is that

the that neural-induced vasodilation is predicted in both the young brain model and the

47

50

52

54

56

58

60

[K+] A

(m

M)

0.1

0.2

0.3

0.4

[IP

3]

(µM

)[C

a2+] A

(µ

M)

0 800

1

2

3

4

[EE

T] A

(µ

M)

time (sec)

−90−80−70−60−50−40−30

VA (

mV

)

5

10

15

20

[K+] P

(m

M)

0

100

200

300

400

[Ca2

+] S

MC (

nM

)

0 10 45 57 8018202224262830

rad

ius

(µm

)

time (sec)

00

0.1

0.2

0.3

0.4

0.5

Astrocyte K+

Astrocyte IP3

Astrocyte calcium

Astrocyte EET

stim

ulus

per

iod

Perivascular K+

Astrocyte membrane

potential

Arteriole SMC calcium

Arteriole radius

young brain

adult brain

10 45 57

Figure 2.12: Compari-son of young and “adultbrain” astrocyte models.Green lines are the youngbrain model; red lines arethe modified “adult” brainmodel. Shaded area indi-cates time period of neuralstimulus

modified “adult” model, suggesting that the astrocyte glutamate receptors may play a

smaller role than expected in functional hyperemia. In fact, the modeling paper by [46]

suggested that the intracellular calcium and EET rises due to astrocyte mGluR activation

were necessary for vasodilation because calcium and EET both activate the BK potassium

outflux at the astrocyte perivascular endfeet. They propose that astrocytic potassium

“is released by endfoot BK channels due to the action of EETs on them, not due to the

depolarising membrane.” Upon experimenting with the equations from their paper, we

found that what they propose is true for the small astrocyte depolarizations simulated in

their paper. However, when we increased the amplitude of the neuronal potassium release,

we found that the resulting increase in astrocyte depolarization was sufficient to activate

the astrocyte BK channels even when glutamate was excluded from the simulation. Our

model prediction that astrocyte-mediated neural induced vasodilation does not depend

on astrocyte mGluR activity could potentially be a unifying property of astrocytes in

developing and mature brains; the functional differences may have more to do with IP3

48

and Ca2+ dependent glutamate release from astrocytes which would occur in developing

brains, but not in mature brains, whose astrocytes lack mGluR5. This difference could

potentially be related to what distinguishes learning in the developing and adult brains.

We also note that the above findings contradict the ideas proposed by [124]. While

we recognize the importance of the work, we are cautious about dismissing astrocyte spa-

tial potassium transfer altogether as a mechanism for neurovascular coupling. First, the

experiments in [124] were performed in the retina, which is a unique and very distinctive

part of the brain, and the types of astrocytes expressed in the retina are primarily fibrous,

a functionally and biochemically different class from the protoplasmic astrocytes typically

found in the cortex. Second, the implications of the study are slightly unclear: the authors

state that astrocyte endfeet release potassium in response to depolarization. The authors

also demonstrate that application of extracellular potassium near an arteriole induces a

dilation. However, the paper does not focus on addressing the reason why depolarizing

an astrocyte in contact with an arteriole might still fail to produce a dilation. In other

words, if depolarized astrocytes release potassium from their vessel-adjacent endfeet, and if

application of potassium alone dilates vessels, then why would it be that astrocyte depolar-

ization does not dilate vessels? It seems the only possible implication is that the astrocyte

does not release sufficient amounts of potassium, but the authors state in the discussion

that the current injection in the experiment was large enough and localized properly on

the astrocyte “ensuring that the experimentally produced depolarizations will evoke sig-

nificant K+ efflux from the endfeet.” One thing that the authors do not discuss is the

initial arteriole tone: if an arteriole is in or near its fully dilated state, neural stimulation

or application of potassium would not cause a dilation at all (see [19]).

2.7 Conclusions and Discussion

While potassium transport is accepted as a primary function of cerebral astrocytes, pre-

vious models of astrocytes omit any description of intracellular K+ dynamics even when

electrical K+ currents are included [132, 161, 46, 193]. Because astrocytes express Kir

channels, which are sensitive to both intra- and extracellular K+, it is necessary to model

49

the intracellular K+ concentration as this effects the dynamics of the astrocyte potassium

release and uptake. Notably, previous generations of this model without intracellular as-

trocyte potassium dynamics and KirAS/KirAV [46, 193] predict a non-physiological delay

(roughly 25 seconds in [46], and 15 seconds in the lumped model in Section 2.4.1) in

the neurovascular response. The KirAS and KirAV included in this model accelerated the

astrocytic K+ release into the perivascular space, which helped correct the delay.

Astrocyte perivascular endfeet have been observed to express both BK channels [162,

196, 65] and KirAV– specifically the Kir4.1 subunit [137, 78, 24, 79], both of which may con-

tribute to neural-induced K+ release into the perivascular space. Our results suggest that

astrocyte endfoot KirAV may account for the initial response due to the faster activation

rate of KirAV compared to BK, while the BK channels are responsible for sustaining the

response as their conductance is much higher than that of KirAV channels [57]. According

to our sensitivity analysis, the astrocyte KirAS and KirAV channels are essential to K+

buffering.

Part of astrocyte potassium buffering is the clearance of extracellular K+ in the synap-

tic region following neural activation. After extended periods of activation, the recovery

to baseline K+ is preceded by a drop below baseline levels due to extra astrocyte uptake,

a phenomenon observed in vivo [32]. The undershoot is most likely a result of the astro-

cyte K+ uptake via NaKCC and Na-K exchange, which temporarily exceeds K+ release

through KirAS [110]; in fact, the undershoot is increased in Kir knockout cases [32, 137].

The astrocyte also has been observed to experience a hyperpolarization during the pe-

riod of extracellular K+ undershoot [188]. Our results are in good agreement with these

experimental findings, supporting the hypothesis that the NKCC and Na-K pumps are

responsible for the undershoot, while the KirAS in the perisynaptic processes behave as a

counterbalance. This is also supported by the results of our sensitivity analysis (Figure

2.8, top left).

It is well established that extracellular potassium affects neural health and behavior

[33, 95, 183, 188]. Thus, astrocytic potassium buffering likely has both protective and

functional implications in the neurovascular unit. While astrocyte controlled K+ clearance

from the synaptic space could be a primarily protective mechanism to prevent potassium

50

accumulation from reaching neurotoxic levels, it is possible that astrocytes may also reg-

ulate extracellular K+ as a means of modulating synaptic activity and overseeing neural

network organization.

Rises in extracellular K+ were observed to result in heightened neural excitability due to

the increase in neural potassium ion channel reversal potential [6, 33]. Also observed were

decreases in inhibitory GABAergic synaptic transmission in the hippocampus [5, 96, 183].

Hippocampal CA3 neurons were found to experience a hyperpolarizing shift in the Cl−

reversal potential, resulting in greater inhibitory activity in the presence of low (below

normal baseline) extracellular K+ [5, 183]. Therefore, the potassium undershoot that

follows long periods of synaptic activity may behave as a balancing mechanism to reduce

excitability and prevent further continued activation.

While our model was able to produce results with a good qualitative match to several

different experiments, we were unable to attain a quantitative match for all of them. In

particular, our model predicts a less pronounced K+ undershoot effect than that seen in

[32]. Our sensitivity analysis offers two possible explanations: (1) because the K+ decay

rate in the synaptic space turned out to be among the most sensitive parameters to the

undershoot, it is possible that other local cellular activity (e.g. changes in neural behavior)

may also contribute to the undershoot; (2) the model may be limited by the fact that it

is a single compartment, meaning that any changes felt at one end of the cell will be

felt immediately and entirely at the other end (see Supporting Material). This is not

physiologically likely. For instance, it is probable that electrical signals will be subject to

significant loss as they propagate down the long, thin astrocyte processes. In addition to

the numerous studies characterizing electrical propagation along neural dendrites, there are

some limited data suggesting similar losses occur for electrical propagation along glial cells

[138]. A single compartment model assumes there is no loss, so a membrane depolarization

that occurs at the endfoot would be grossly overestimated in terms of its effect at the end

of the synapse-adjacent process. Similarly, a multi-compartment model would predict a

more accurate transfer of ion concentration across the cell. In fact, recent studies have

demonstrated that astrocyte intracellular ion diffusion has unique characteristics in the

endfeet and processes, and have revealed that isolated subcellular compartments can occur

51

within the processes and endfeet in which highly localized ion concentration fluctuations

occur without diffusing to or from other parts of the cell [136, 40, 146].

Chapter Three

Neuronal response to

hemodynamics

53

In the previous chapters, we presented a model of an indirect relationship between neurons

and vasculature in which astrocytes mediate communication. In this chapter, we consider a

direct interaction between microvessels and neurons. In the cortex, arterioles and capillaries

are in direct contact with local GABA-ergic interneurons [30, 72, 42]. These include the

fast spiking interneuron (FS cell), the most common type of interneuron in the cortex. FS

cells throughout the brain express the mechanosensitive channel pannexin-1 (Px1), which is

abundant in the cortex of adult and developing brains [186, 195]. Their expression of these

mechanosensitive channels combined with their proximity to arterioles makes FS cells prime

candidates for bidirectional neurovascular coupling. In one direction, the interneurons

directly control vascular tone by providing GABA innervation to the vascular smooth

muscle [30, 72]. In the other, the FS cells’ mechanosensitive Px1 channels would allow them

to respond to local hemodynamic changes including vascular constriction and dilation. In

this chapter, we describe a model of FS cells that focuses on direct neuronal response

to vascular movement. We exclude the FS cell control of vascular tone through GABA

innervation because this model is intended to explore specialized experimental conditions

in which microvessel dilation and constriction is explicitly fixed with optogenetic controls.

In this context, the vessel is restricted in its ability to respond to local neuronal (and

astrocytic) inputs.

3.1 Introduction

Based on preliminary experimental observations from our collaborators Tyler Brown and

Chris Moore, we intended to create a model of vessel-adjacent interneurons that would

demonstrate two specific behaviors: (1) they should increase monotonically in baseline

spiking rate with increasing vessel dilation. (2) They should decrease in their sensitivity

to brief sensory stimuli while the adjacent vessel is dilated. This means that during a

dilation, the sensory-driven spiking response should decrease even though the baseline

spiking is increased. An example of this type of behavior is illustrated in Figure 3.1A. The

figure shows a graphical depiction (not real data) of the intended FS cell spiking behavior.

Short (<< 1 sec) sensory stimuli (black arrows) induce a large spiking output from the FS

54

cell. During a dilation, the spiking rate during these input stimuli is decreased compared

to the undilated state. At the same time, baseline spiking is higher during dilation. An

important distinction to make is that the decrease in sensory driven spiking refers to the

total number of times the FS cell fires during the sensory event, and not the increase in

spike rate compared to baseline. Figure 3.1B depicts behavior that is not an example of

behavior observed by our collaborators. In this example, the increase in spiking during

a sensory event is diminished during dilation, but the total sensory-driven spiking is not

decreased.

1 sec

(1)

(2)

1 sec(1)

(2)incorrect

A

B

dilation onno dilation

Figure 3.1: Illustration of perivascular

FS spiking behavior An example of the ex-

pected spiking response of an FS cell in

contact with a dilating microvessel. Ar-

rows indicate short input stimuli that drive

large spiking increase in the neuron. A

Correct behavior — when the microvessel

is dilated, the neuron shows (1) an increase

in baseline spiking and (2) a decrease in

stimulus driven spiking. B Incorrect exam-

ple behavior (marked in red) — although

the dilated state in this example correctly

shows an increase in baseline firing (1), the

spiking rate during sensory events (2) is not

decreased compared to the undilated state.

We aimed to identify the simplest model that could reproduce these phenomena. In

doing so, we formulated a preliminary hypothesis explaining the mechanisms behind neu-

ronal response to dilation, which we will discuss in detail below. Predictions made by

the mininimalistic model would be useful in providing better discerning experiments that

would shed light on the problem and help refine the hypothesis.

In an unpublished experiment, our collaborators observed cortical neurons during op-

togenetic controlled vasodilation and constrictions of local microvessels. We speculated

that the the spiking response observed in the interneurons is linked to any combination

of four possible causes or mechanisms: (1) mechanical stimuli from the adjacent dilating

vessel, (2) synaptic input from other neurons responding either directly or indirectly to the

dilation, (3) biochemical reaction to diffusible messengers (either release into the extracel-

55

lular environment or uptake that would result in a significant change in the extracellular

environment) derived from vascular cells reacting to membrane stress caused by dilation,

or (4) diffusible messengers from other (non-vascular) local cells responding to the dilation.

We will briefly discuss the merits of the above four mechanisms. The close proximity

between cortical interneurons and microvasculature [72, 42], is compatible with the hypoth-

esis that there is a direct link between a dilating microvessel and the response observed in

local FS cells. All mechanisms being otherwise equally likely, we use proximity to rule out

(2) and (4) and develop a preliminary model using only mechanisms of direct interaction

(1 and 3). We first evaluate the feasibility of (3), direct biochemical reaction to vascu-

lar derived diffusibles. In this case, there would need to be a mechanosensitive response

from vascular smooth muscle or endothelial cells that diffuses outside the vessel wall (not

just into the blood stream) and that produces a spiking response in FS cells. Vascular

endothelial and smooth muscle cells do express mechanosensitive calcium channels [67],

but it is unclear what kind of chemical release or uptake — and in what magnitude —

may occur. Whatever possible changes to the extracellular environment, one would need

to attempt to identify various possible mechanisms by which the FS cells would respond.

To our knowledge, there are no clear biochemical pathways that stand out as providing

a likely explanation for FS cell response to dilation. On the other hand, mechanism (1)

is a direct mechanical link. In this case, the FS cells would not only need to be in close

proximity to the vessel, but would have to express some mechanosensitive mechanism. In

fact, as was discussed in the introduction, FS interneurons have been observed to possess

the mechanosensitive channel Pannexin1 (Px1) in the neocortex and other regions of the

brain [186, 195]. Given that FS cells express mechanosensitive channels and are observed in

close proximity with microvessels in the cortex, it would not be surprising if they showed

some response to direct mechanical stimulation from an adjacent vessel. We infer from

this that (1) is the simplest explanation for FS response to vasodilation: that the FS cells

were mechanosensing the vascular event. To our knowledge, the only reported channels in

cortical FS cells known to be mechanosensitive are Px1. Therefore, we propose a model of

vessel-to-interneuron communication in which vessel movement activates mechanosensitive

Px1 channels in the membranes of adjacent FS cells.

56

With limited data on pannexin channels in cortical interneurons and their specific

mechanosensitive properties, a complicated model is liable to include inaccurate mecha-

nisms that are difficult to validate. Instead, we made a series of attempts to model pannexin

in FS cells, beginning with the simplest model and gradually adding complexity as needed

until the model reproduced the two behaviors we initially sought to explain. In all versions

of the model, we do not model the vessel explicitly. Instead, we define an arbitrary dilation

level, L, (that may or may not vary with time) as an input to the FS cell, as we will de-

scribe below in Section 3.2. In a simple model, this is sufficient to simulate optogenetically

controlled vessel tone. An outline of the model development stages, which we discuss in

detail in the sections below, is as follows: Model 1 comprises only a single perivascular FS

cell in contact with a dilating microvessel. We demonstrate that this model is not capable

of reproducing the two target behaviors. Model 2 comprises two synaptically connected

perivascular FS cells both in contact with the same microvessel. This model is capable of

reproducing the two target behaviors. Model 3 comprises the system in Model 2 but with

additional FS cells not in contact with the microvessel. We explored Model 3 to test the

feasibility of Model 2 and to determine whether the same behavior demonstrated by the

perivascular FS cells would be translated to neighboring cells in the network that were not

in contact with the vessel.

3.2 Theoretical framework

We model the FS cell as a single compartment (point neuron) with Hodgkin-Huxley type

sodium (INa), potassium (IK), and leak (IL) currents, to which we add a pannexin channel

current, IPx1, described by

IPx1 = gPx1(v − EPx1), (3.1)

with

gPx1 = (gon − goff )1

2tanh(

L− L0

kPx1+ 1) + goff , (3.2)

where the amount of vessel dilation, L, determines the channel conductance. L0 is

the amount of dilation required for half activation of the channel. gon is the maximum

57

Px1 conductance (when the membrane is fully stretched), and goff is the conductance in

the absence of any mechanical perturbation. Experimentally obtained parameters from

literature for Px1 conductance (see Table 3.1) have been recorded using mechanical per-

turbation techniques such as cell swelling [194] or by applying suction pressure to the patch

pipette. However, limited data are available indicating the channel response over a range

of stretch or stress values, which leaves uncertainty for the parameter values of L0 and

kPx1. In this case, we reduce the model to a linear relationship where we eliminate these

two parameters:

gPx1 = (gon − goff )L+ goff . (3.3)

Here L varies from 0 to 1, where the cell experiences no mechanical perturbation at 0,

while the vessel is fully dilated at L = 1.

Table 3.1: Px1 physical constants

parameter/description value patch clamp recording method source

gon active Px1 conductance 1250 pS whole cell, isolated retinalfig 4D in [194]

goff inactive Px1 conductance 250 pS ganglion neuron

EPx1 Px1 reversal potential mVsingle channel and whole cell for

fig 2c,e in [118]rat CA1 hippocampus interneurons

3.3 Results

Simulations were performed with a parallel NEURON/python code written by Stephanie

Jones, Shane Lee, and Maxwell Sherman [98, 97, 112], which we modified slightly to include

pannexin channels using Eqs. (3.1) and (3.3). Parameters used for Px1 are shown in Table

3.1. All other parameters, including Hodgkin-Huxley channels were the same as those used

in [112, 97] for layer 2/3 cortical FS interneurons. In all simulations, FS cells were given

a stochastic Poisson input that produced a baseline firing rate of approximately 10 Hz,

the typical value for FS cells [189, 107]. To account for variations caused by stochastic

background noise, all simulations were repeated for 10 trials and the results were averaged.

58

Sensory stimuli were simulated by injecting short (50 ms) 0.4 mA current pulses into the

FS cells.

3.3.1 Model 1 — single perivascular FS cell

We began testing the model by simulating a single FS cell as it responded to a gradual

dilation that increased steadily over nine seconds. Figure 3.2 shows the spiking response

in a perivascular FS cell (blue curve). The time course of the adjacent vessel dilation is

indicated by the black dashed line. The dilation starts at 5 seconds into the simulation and

reaches maximum (L = 1) at 14 seconds. The neuron does not appear to respond until

about a quarter of the way into the dilation, at which point quickly doubles. This model

predicts a threshold exists that prevents small dilations from inducing a baseline spiking

response in the FS cell.

0 5 10 150

1

2

3

4

5

time (sec)

spik

es p

er 5

0 m

s bi

n(t

rial a

vera

ge)

dilation

Figure 3.2: Single FS cell re-

sponse to gradual dilation. Blue

curve shows spike count for

perivascular FS cell. Time

course of vessel dilation is shown

in black dashed line.

Figure 3.3: Effect of dilation

on single FS cell response to

sensory input Blue curve shows

spike count for perivascular FS

cell. Sensory inputs occur ev-

ery 1 second and last 5 ms (the

length of one time bin). The

large jumps in spiking are the

cell’s response to these short

stimuli. Time course of ves-

sel dilation is shown in black

dashed line.

0 5 10 150

1

2

3

4

5

6

time (sec)

spik

es p

er 5

0 m

s bi

n(t

rial a

vera

ge)

dilation

We next tested how a dilation would affect the FS cell response to short input stimuli.

59

We simulated brief sensory inputs as direct 0.4 mA current injections into the cell lasting

50 ms each. The current injections were applied once every second (or every 1000 ms)

throughout the simulation. The time course of vessel dilation was the same as in Figure

3.2. Results are shown in Figure 3.3. The tall narrow peaks in the spiking pattern occur

when the 50 ms sensory inputs are applied. Time binning for the spike count was chosen

at 50 ms so that the response to sensory stimuli would be visible after the time binning.

In this trial, although the stimulus-driven spiking increases less dramatically during the

dilation, we did not get the effect we were expecting: the total spikes in a bin during a 50

ms sensory input did not decrease at any point in the dilation. While Figure 3.3 shows only

one example, we repeated this simulation procedure over a wide range of magnitudes and

durations for sensory input currents as well as a wide range of values for Px1 parameters

and found that the results were not improved.

The results suggest that Model 1 was not sufficient to explain the FS cell decreased

response to short input stimuli during dilation.

3.3.2 Model 2 — perivascular FS interneuron network

inhibitorysynapses

dilation

dilation

FS

FS

px1

px1

vessel

Figure 3.4: Schematic of perivascular FS

cell network Two FS cells (red circles) are

mutually inhibitive, each projecting an in-

hibitory synaptic input to the other. Both

FS cells are in contact with the same mi-

crovessel. Dilations activate Px1 channels

on both the FS cells.

We next considered whether network effects could be partially responsible for the phe-

nomenon, as cortical FS cells are connected via inhibitory synapses. To test this, we

simulated two FS cells that were both in contact with the same microvessel. A schematic

is shown in Figure 3.4. Two perivascular FS cells (red circles) were located adjacent to each

other along the microvessel which they both contacted. The dilation (thick black arrows),

which was uniform along the length of the vessel, activated the Px1 channels in both of

60

the FS cells according to Eq. (3.3). Each FS cell received an inhibitory synaptic input

from the other. The synaptic parameters and connectivity followed that of [97]. Both FS

cells received independent stochastic background noise, but they both received the same

50 ms sensory input stimuli; the timing of these inputs is indicated by short black bars at

the top of the plot. Figure 3.5 (upper plot) shows the results when the FS cells received

sensory inputs in the absence of any dilation for 7.5 seconds, followed by 7.5 seconds during

which the vessel was instantaneously set at full dilation while the sensory inputs continued

every 1 second. In the top plot, the spiking patterns of each FS cell are shown as dark

and light blue curves. These results show both of the behaviors our model was intended to

reproduce: dilation causes both an increase in baseline spiking and a decrease in sensory

stimulus driven spiking. The findings suggest that FS cell response to vasodilation may be

explained by the electrical activity of mechanically activated Px1 channels (as opposed to

the purinergic effects of Px1 ATP release) combined with the mutually inhibitory synaptic

connections between FS cells. This hypothesis would also require that there be at least two

FS cells in contact with the same dilating microvessel or that the dilation of one section of

microvessel would induce a dilation on several nearby branches.

This model also predicts that vessel dilation may cause gamma rhythm in perivascular

cortical FS networks. Figure 3.5B,C shows correlation histograms for baseline spiking be-

fore and after dilation. In these plots, two synaptically connected FS cells are in contact

with the same microvessel, as in Figure 3.5A, but they receive only stochastic background

noise, with no input stimulus pulses. In the absence of dilation, (Figure 3.5B) the his-

togram indicates that there is no correlation between the two interneurons at baseline.

At full dilation, (Figure 3.5C) the correlation histogram shows strong peaks located at

roughly 28 ms intervals, which is in the range of gamma frequencies. It is not yet clear

whether this effect is physiologically accurate or whether it is an artifact of the model. The

effect of dilation on gamma oscillations has not been tested experimentally. However, the

model prediction offers insight towards further experimental investigation that could lead

to interesting results in addition to either validating or negating the model as a hypothesis.

61

0

1

2

3

4

spik

es

pe

r b

in (

tria

l ave

rag

e)

−200 −100 0 100 2000

1

2

x10−4 Cross−correlation at rest

(no dilation)

time (ms)−200 −100 0 100 200

0

1

2

3

4

x10−3 Cross−correlation during dilation

time (ms)

cell 1

cell 21 sec

dilation (t=7.5 sec)

A

B C

Figure 3.5: Perivascular FS cells response to dilation in network A Spiking behavior of two synaptically

connected, perivascular FS cells before and after a dilation. The spike patterns of each cell is distinguished

with light and dark blue. The cells are identical, but differences in their spiking patterns are due to the

random differences in their stochastic background inputs. 50 ms sensory inputs occur every 1 sec; the

time period of the input pulses are indicated by the short black bars above the blue curves. The vessel is

undilated (L = 0) from 0–7.5 sec. At 7.5 sec, the dilation is set to maximum (L = 1) for the remainder of the

simulation. B,C Cross-correlation histogram of the two FS cells baseline spiking patterns. Both histograms

are computed using baseline spike trains, without sensory stimuli. B Cross-correlation histogram of baseline

spike trains for the two FS cells when the vessel is undilated. C Cross-correlation histogram of baseline

spike trains for the two FS cells when the vesse is dilated.

3.3.3 Model 3 — networks of perivascular and peripheral FS cells

In the two-neuron simulation, we assume that the entire network is in contact with a

dilating microvessel. While it is true that microvessels in the cortex are very often in contact

with local interneurons, FS cells are dense in the cortex, and not all make contact with

vasculature. We decided to test whether the same behavior would occur when the network

was extended beyond the immediate vicinity of the dilation, such that some interneurons

in the network were not in direct contact with the vessel.

In the next example we used a two-by-three grid of FS cells as illustrated in Figure

3.6A. Following [98, 97, 112], synaptic connections in the network were all-to-all, but the

synaptic strengths were scaled according to distance between the cells. Synaptic weights

and distance scaling parameters are the same as from [97, 112]. The middle row was in

62

0

1

2

4

spik

es

pe

r b

in (

tria

l ave

rag

e)

0 5 10 15time (sec)

0 5 10 15time (sec)

A B

0

1

2

4

spik

es

pe

r b

in (

tria

l ave

rag

e)

C

vess

el

3 3

Figure 3.6: 2x3 FS network response to dilation A Model schematic. Two-by-three grid of FS cells (red

circles) with the two middle cells in contact with the vessel segment (grey). The two middle FS cells shown

on the vessel are the perivascular cells; all others are distal and their Px1 channels are unaffected by any

vessel dilation. Synaptic connectivity (not shown) is all-to-all. B Average spiking for both FS populations.

Time course of dilation is shown in black dashed line. Green curves — perivascular FS spiking. Black

dotted curves — distal FS spiking. C Average spiking across all FS cells in the network (solid black curve).

Time course of dilation is shown in black dashed line.

contact with the vessel. Thus, the dilation would only have a direct effect on the middle

row, while the two rows on either side would be affected only indirectly through their

synaptic connections.

The network is given the same inputs as in Figure 3.3. The vessel dilation begins at 5

seconds and reaches full dilation at 10 seconds. Figure 3.6B shows the average spiking for

the perivascular and distal FS populations. The thick green curve shows the average spiking

for the two perivascular interneurons, and the dotted black curve shows the average spiking

for the more distal interneurons. The time course of the dilation is indicated by the black

dashed line. The two perivascular neurons show the same response to dilation as before,

with an increase in baseline spiking and a decrease in sensory-driven spiking. The distal

interneurons also show a decrease in sensory-driven spiking during the dilation, but they

actually show a slight decrease in baseline spiking due to their inhibitory connections with

the two perivascular interneurons. However, the average spiking across the entire network,

Figure 3.6C, shows both the increased baseline spiking and the decreased stimulus-induced

spiking in response to dilation.

We next extended the network even further beyond the vessel, adding an additional row

of interneurons on either side, as in Figure 3.7A. The results are shown in Figure 3.7B,C.

The average spiking for the two perivascular interneurons are shown in the thick green

curve, and shows the same behavior as before. The average distal interneuron spiking,

63

0

1

2

3

spik

es

pe

r b

in (

tria

l ave

rag

e)

0 5 10 15time (sec)

0 5 10 15time (sec)

A B

0

1

2

3

spik

es

pe

r b

in (

tria

l ave

rag

e)

C

vess

el

Figure 3.7: 2x5 FS network response to dilation A Model schematic. Two-by-five grid of FS cells (red

circles) with the two middle cells in contact with the vessel segment (grey). The two middle FS cells shown

on the vessel are the perivascular cells; all others are distal and their Px1 channels are unaffected by any

vessel dilation. Synaptic connectivity is all-to-all. B Average spiking for both FS populations. Time course

of dilation is shown in black dashed line. Green curves — perivascular FS spiking. Black dotted curves —

distal FS spiking. C Average spiking across all FS cells in the network (solid balck curve). Time course of

dilation is shown in black dashed line.

dotted black curve, no longer shows a decrease in stimulus-evoked spiking during the

dilation. The overall average network spiking, shown in Figure 3.7C, also no longer shows

a decrease in stimulus-evoked spiking, although there is a slight increase in baseline spiking.

3.4 Conclusions

We formulated a minimalistic model of cortical perivascular interneurons capable of re-

sponding to local vessel dilation with both an increase in baseline spiking and a decrease

in stimulus-driven spiking. The results from Models 1 and 2 support the hypothesis that

the behavior is a combined effect of inhibitory synaptic connections between interneurons

and the depolarizing current through mechanically activated Px1 channels expressed on

the interneurons. The model is accurate at predicting the activity of interneurons in direct

contact to the dilated vessel and for small populations in the immediate vicinity of the

vessel (e.g. the two-by-three grid in Model 3). As the network extends farther from the

vessel in Model 3 (two-by-five grid), the model is incapable of predicting the correct spiking

response to dilation. The results from Model 3 reveal the limitations of the model, sug-

gesting that our hypothesis is only a partial explanation. Additional mechanisms must be

involved, possibly including ATP release from active Px1 channels, which would transmit

signals across the network. The interactions between FS cells and excitatory pyramidal

64

neurons is also likely to have an impact in the neurovascular response.

Chapter Four

Discrete particle model of arteriole

66

In the first generation of our NVU model we have employed the lumped arteriole smooth

muscle cell (SMC) space model first developed in [67]. Here, we propose to replace this

rather empirical model with a microvascular model that we will construct from first prin-

ciples using an atomistic approach. In this chapter, we present a discrete particle model

of an arteriole that follows the true microstructure of the arteriole wall. We begin with a

discussion of arteriole structure and composition to motivate the conceptual model for the

arteriole.

4.1 Background on arteriole structure

This section will provide a background on the physical structure of arterioles, paying partic-

ular attention to parenchymal (also called intracerebral or penetrating) arterioles, as these

are primarily the type around which astrocyte endfeet are found [29, 57]. Because these

are located deep within the brain tissue, there are more data on pial arterioles, which occur

near the outer layers of the brain and are more easily accessible for experimental analysis.

It is important to acknowledge that there are likely to be structural and functional differ-

ences between parenchymal and pial arterioles in the brain due to the differences in their

local environments (for instance, mechanisms involved in parenchymal arteriole response

to astrocytes may not be present in pial arterioles). However, data from pial cerebral arte-

rioles, as well as arterioles in different regions of the body, are still useful as they are likely

to have several qualitative and quantitative similarities with parenchymal arterioles, and

as a group, they provide a general range of characteristics in which parenchymal arterioles

may fit.

Parenchymal arterioles

The authors of [34] characterized parenchymal arterioles from the rat brain as having one

layer of endothelial cells covered by a single layer of smooth muscle cells (SMC) surrounded

by a basal lamina, with a thin adventitial (exterior connective tissue) layer consisting of a

leptomeningeal sheath. The nuclei of the endothelial cells were observed to be parallel to

the vessel axis, while the SMC nuclei were oriented perpendicular. A diagram illustrating

67

the layers is given in Figure 4.1. This orientation allows the smooth muscle contractile

mechanisms to have efficient control of the vessel radius. The structure of the endothelial

and SMC layers also contributes to the anisotropic mechanical properties of the arteriole,

which is stiffer in the axial direction than the circumferential.

smooth muscle

endothelial cells

basal lamina

adventitia

Figure 4.1: Parenchymal arteriole structure

Axially aligned endothelial cells are surrounded

by a single layer of smooth muscle cells oriented

in the circumferential direction. The basal lam-

ina encloses the smooth muscle layer. The outer

layer, adventitia, is a thin layer of connective tis-

sue consisting of a leptomeningeal sheath [34].

The same study also reported a mean vessel diameter of 36.7 µm for the passive (uncon-

stricted) arterioles pressurized at 60 mmHg; increasing the arteriole pressure from 10-120

mmHg dilated the vessel from a diameter of ∼34 µm to ∼38 µm [34].

Pial cerebral and non-cerebral arterioles

Pial arterioles in the rat brain stem and cerebrum have an inner layer, the intima, composed

of endothelial cells and elastin; the middle layer, media, is composed of one to two layers of

SMC with collagen between the individual cells (the pial SMC layer was found to be thicker

in the brain stem than cerebrum), and the arachnoid layer, comprising basement membrane

and collagen, sits outside the arteriole wall [10]. The collagen distribution was observed

to be irregular in the arachnoid layer, with several gaps that contained no collagen, and

basement membrane was also found to line endothelial and smooth muscle cells [10].

The composition of arterioles depends on their location in the body. For instance, in

the hamster cheek pouch, SMC accounts for 45% of the arteriole wall by volume, compared

to 89% in the in rat cerebral pial arterioles; in arterioles in general, SMC content is high

compared to large arteries (typically around 30%) [11]. Arteriole structure is also adjusted

under pathological conditions. With chronic hypertension, increased SMC content (by

volume) was observed in rat pial arterioles, while the endothelium decreased. The wall

thickness also increased due to chronic hypertension, specifically in the elastin and SMC

68

layers [11].

In the frog mesentery, [192] observed arterioles composed of an endothelium, SMC, and

collagen and elastic fibers. Collagen fibers with a high elastic modulus were circumferen-

tially oriented around the SMC, while axially oriented collagen fibers with a low elastic

modulus occurred in the exterior layers [192].

For reference, we include a list of arteriole mechanical properties for various arteriole

types in Table 4.1

Table 4.1: Arteriole mechanical properties

Radius Wall thickness Elastic modulus Location Source

10 µm 5.3 µm 0.074 MPa cat omentum arteriole [91]

27 µm 4.4 µm 0.1-0.8 MPa rat pial arteriole [11]

12 µm 4.2 µm 0.23 MPa active hamster cheek pouch arteriole calculated from [36]

20 µm 4.2 µm 0.033 MPa active hamster cheek pouch arteriole calculated from [36]

4.2 Dissipative Particle Dynamics model of flexible arteriole

— single layer

In the first generation of our NVU model we have employed the lumped arteriole smooth

muscle cell (SMC) intracellular space model first developed in [67]. Here, we propose to

replace this rather empirical model with a microvascular model that we construct from

first principles using an atomistic approach. In this regime, we will be able to study blood

flow in the microvessel at the cellular level. To this end, we can incorporate into the flow

simulations a Dissipative Particle Dynamics (DPD) red blood cell model (see [160, 48, 163])

that can accurately simulate the properties and dynamic behavior of healthy red blood

cells (RBCs) as well as diseased RBCs. The multiscale model can represent a RBC at the

molecular (spectrin) level with 30,000 points [116] or at a coarser level with 500 points

by proper scaling of the physiologically correct parameters [160, 48]. In order to take

advantage of these red blood cell models, it is necessary to convert the arteriole model

from the continuum to DPD.

69

In this section we will first briefly review the DPD method and then we will discuss

two different approaches to constructing a DPD model of a flexible arteriole that takes

into account the anisotropic nature of the arteriole wall. The first attempt, which we

describe in Section 4.2.2 was to construct a single layer spring network that could be made

anisotropic by using different bond parameters for different bond orientations within the

mesh. This approach proved to be unsuccessful as adjusting the spring parameters alone

was not sufficient to tune the anisotropic properties of the material; the structure of the

mesh itself – the orientations of the springs – was a constraint on the mechanical properties.

We briefly considered a different mesh, which we will describe below in Section 4.2.3, before

changing the approach altogether and using a two-layer model.

In Section 4.3 we describe the second attempt, a multilayer model that mimics the

actual microstructure of the vascular tissue. This model consists of an isotropic elastin

layer attached to a layer of stiff fibers oriented at arbitrary angles. This model is two-

dimensional in that there is no variation in stress or strain along the thickness of the

material, but the model does take into account changes in the material thickness based

on the assumption of incompressibility. We compare stretching simulations performed on

this model with an analogous continuum level model. While the multilayer DPD model

performs well in uniaxial stretching tests, it presents issues in biaxial stretch tests, possibly

due to inherent biases related to the connection points of the two layers. We will discuss

these issues and the possible sources of error as well as potential ways to address them in

future work.

4.2.1 Dissipative Particle Dynamics (DPD) method

DPD is a coarse-grained discrete particle simulation method in which DPD particles repre-

sent molecular clusters, first developed by [84]. The system consists of N DPD particles of

massmi, position ri, and velocity vi. The particles interact interact through a conservative

70

force (FCij), a dissipative force (FDij ), and a random force (FRij) [44]:

FCij =

aij(1− rij/rc)rij , for rij ≤ rc

0, for rij > rc

FDij = −γωD(rij)(vij · rij)rij ,

FRij = σωR(rij)ξij√dt

rij ,

(4.1)

where the vector pointing from DPD particle i to particle j is rij , with rij = rij/rij ,

and the relative velocity between particles i and j is vij = vi − vj . The conservative

force coefficient between particles i and j is aij . The coefficients γ and σ, along with

the corresponding weight functions ωD and ωR define the strength of the dissipative and

random forces, respectively. The dissipative force includes the random variable ξij , which

is normally distributed with zero mean, unit variance, and ξij = ξji. The cutoff radius,

rc defines the length scale in the DPD system, and all forces are truncated beyond this

length.

The random and dissipative forces obey the fluctuation-dissipation theorem for DPD

so that [44]

ωR(rij) =√

ωD(rij),

σ =√

2kBTγ,

(4.2)

where T is the temperature, and kB is the Boltzmann constant. The weight function is

given by

ωR(rij) =

(1− rij/rc)k, for rij ≤ rc

0, for rij > rc,

(4.3)

where k = 1, as in the original DPD method, although lower values of k have been used

in order to increase the viscosity [45, 50].

4.2.2 Single layer DPD arteriole model with triangular mesh

The arteriole wall is defined by a two-dimensional triangulated network in which the DPD

wall particles are positioned at the triangle vertices, and the triangle edges are viscoelastic

71

Figure 4.2: Triangulated arteriole wall Circles

are DPD particles; dashed red lines are DPD

bonds. Sizes and line weights were adjusted to

emphasize perspective, but all bonds and parti-

cles in this figure are identical in the DPD sim-

ulation.

springs connecting the vertices. The triangulated mesh for a cylinder is illustrated in

Figure 4.2. The total energy in the system is given by [48, 49]

V (xi) = Vin−plane + Varea, (4.4)

where xi is the set of coordinate points of all N vertices (DPD particles) in the system.

Vin−plane is the spring energy, and Varea comes from the area constraints; both are defined

below.

Spring forces in the arteriole wall come from conservative elastic forces of the bonds

and are expressed in terms of an energy potential Us as a function of lj , the length of spring

j of a total Ns number of springs. The potential energy of the viscoelastic wall is given by

Vin−plane = Σj∈1...Ns [Us(lj)]. (4.5)

For the bonds connecting the arteriole wall particles in the passive (non-myogenic) model,

we use a nonlinear spring model, the wormlike chain (WLC) combined with a repulsive

force in the form of a power function (POW) based on the one formulated by [48], which

has the energy potential Us = UWLC +UPOW (for simplicity in notation, we will leave out

the particle subscripts, j, from here on):

UWLC =kBT lmax

4p

3x2 − 2x3

1− x, (4.6)

72

and

UPOW =

kp(m−1)lm−1 , for m > 0,m 6= 1

−kp log(l), for m = 1,

(4.7)

where x = l/lmax; lmax is the maximum spring extension length (equilibrium length is l0),

and p is the persistence length. The POW force coefficient is kp andm is the exponent. The

persistence length and kp are computed from balancing the in plane forces at equilibrium

(f =∂Vinplane

∂l |l=l0 = 0, Eq. (4.5)) and from their relation to the shear modulus µ0 (see

derivation in supporting material for Fedosov et al. 2010 [48]):

µ0 =

√3kBT

4plmaxx0

(

x02(1− x0)3

− 1

4(1− x0)2+

1

4

)

+

√3kp(m+ 1)

4lm+10

, (4.8)

where x0 = l0/lmax.

Area constraints developed by Fedosov 2010 [48, 49] are defined as

Varea = Σj∈1...Ntka(Aj −A0)

2

2A0, (4.9)

where Nt is the number of triangles, and karea is the area constraint coefficient. Aj is the

current area of triangle j, and A0 is the equilibrium value of the triangle area.

Simulation procedures

The anisotropic structure of the arteriole wall (see Section 4.1) gives the material anisotropic

mechanical properties such that its stiffness is greater in the axial direction than the circum-

ferential direction. In an attempt to reproduce anisotropic mechanical properties observed

experimentally, we used a non-uniform set of bond parameters such that the circumferen-

tially oriented bonds (see Figure 4.2) would be tuned with one set of parameters, while

the diagonal bonds would be given a different unique set of parameters. The results were

compared to experimental results from an isolated rat cremaster arteriole in [68].

To calibrate the model, we combined two simulation test procedures. First, we simu-

lated an axial stretch on a vessel segment with a uniform bond type and calibrated the

parameters to experimental results. Next, we simulated internal pressurization of another

73

vessel segment, again with uniform bond type, and calibrated those parameters to experi-

ment. The calibrated bonds from the pressurization test then replaced the circumferentially

oriented bonds in the calibrated vessel from the axial stretch test. As will be described

below, we then attempted to recalibrate the model until the same set of nonuniform bond

parameters produced the correct axial and circumferential stiffnesses. Unfortunately, the

triangulated structure was not able to achieve sufficient anisotropic properties. It was not

possible to make a material that was sufficiently stiff in the axial direction while at the

same time being sufficiently flexible in the circumferential direction.

Axial stretch of a uniform bond mesh

Figure 4.3: Passive arteri-

ole with axial stretch Results

are shown for different values of

lmax (see Eqs. (4.7) – (4.8)):

Thick solid lines – lmax = 2.0;

thin solid lines – lmax = 1.7.

Shear modulus, µ0 of the bonds

is also varied (see color legend

in figure). Grey dashed lines

are experimental results using

rat cremaster arterioles in [68]

(see Figure 4A and B): – myo-

genic (active) arteriole; © – pas-

sive arteriole.

0 30 60 901.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

axial stress (kPa)

axia

l st

rain

µ0 = 5.0 kPa

µ0 = 9.1 kPa

µ0 = 20 kPa

expt., active

expt., passive lmax = 2.0

lmax = 1.7

1.8

Figure 4.3 shows results of an axial stretch test for an arteriole with a 12 µm radius, 40

µm length. The wall thickness was 5 µm. The ends of the cylinder were held fixed in the

radial and circumferential directions to mimic experimental stretching conditions in which

the ends of the arteriole would be tied to a cannula. Axial stress from 10–75 kPa was

distributed uniformly to the ends of the cylinder. In this and all other simulations in this

chapter, we exclude pairwise interactions in Eqs. (4.1) – (4.3) because there are no fluid

particles in the system. Because this simulation was a preliminary attempt to determine

the approximate bond parameters, we also left out area constraints. Simulation parameters

74

are given in Table 4.2. In the table, Nv is the total number of DPD particles (or number

of triangle vertices) in the mesh. The other parameters are the bond parameters in Eqs.

(4.6) – (4.8).

Table 4.2: Parameters for axial stretch of triangular mesh cylinder

Nv l0 kBT lmax µ0 m

1320 1.64 µm 4.28× 10−21m2kg s−2K−1 2.0, 1.7 (see text) 20, 10, 5 kPa (see text) 2

To calibrate the model, we varied two of the bond parameters: lmax, the maximum

bond length (see Eqs. (4.7) – (4.8)); and the shear modulus µ0 as indicated in the figure

legends. Also shown (grey dashed lines) are experimental results from [68] for a passive

arteriole (open circles) and an active arteriole (filled circles) from the rat cremaster. We

chose to vary the shear modulus close to that estimated for the cat omentum arteriole

(∼0.074 MPa [91]), as this was likely to be comparable to the rat cremaster arteriole.

At higher values of lmax (thick solid lines), the stress-strain curves tended to bend more

gradually than for lower values. Using a value of lmax = 2.0 and µ0 = 0.01 MPa (thick

light green line), we were able to achieve results for the passive vessel that had a good

match with experimental data. Because the calibration processes is an iterative procedure,

it is not necessary to achieve a perfect match at this point.

Radial stretch of a nonuniform bond mesh

Figure 4.4 shows the results of pressurization of a vessel with different bond parameters

used for the circumferential and diagonal bonds. The vessel is held fixed in the axial

direction for comparison with experimental results of pressurization of a cannulated vessel

in [68]. As before, the vessel ends are also fixed in the radial and circumferential directions,

simulating the experimental conditions in which the ends of the vessel are tied to the tips

of the cannulae. Simulation parameters and mesh are the same as those given in Table 4.2

except that the maximum spring extension, lmax and shear modulus, µ0, are given different

values depending on the spring orientation, as described below.

As the vessel is more flexible in the circumferential direction than the axial direction,

75

Figure 4.4: Pressurization

of passive arteriole Results are

shown for different values of

lmax (see Eqs. (4.7) – (4.8)):

Thick solid lines – lmax = 2.0;

thin solid lines – lmax = 1.7.

Shear modulus, µ0 of the bonds

is also varied (see color legend in

figure). Grey dashed curve is ex-

perimental result from pressur-

ization of passive rat cremaster

arterioles interpolated from Fig-

ure 2A in [68]

1.20 40 60 80 100 120

1.4

1.6

1.8

2.2

2.4

2.6

2.8

3

3.2

Pressure (mmHg)

radia

l st

rain

(r/

r 0)

2

20

µ0,C = 0.0018 kPa, lmax,C = 4.0

µ0,V = 0.91 kPa, lmax,V = 2.0

expt., passive (Guo et al 2007)

µ0,C = 0.009 kPa

µ0,C = 0.045 kPa

µ0,C = 0.18 kPa

lmax,C = 4.0

µ0,V = 9.1 kPa

lmax,V = 2.0

the bond parameters for the diagonal bonds need to be stiffer than those for the circum-

ferential bonds. Experimental results for pressurization interpolated from Figure 2A in

[68] are shown in open circles. The dashed green, solid green, and black curves are the

results obtained using the diagonal bond parameters obtained above in the axial stretch

calibration above, with circumferential bond stiffnesses 0.18 kPa, 0.045 kPa, and 0.009

kPa, respectively, each roughly a quarter the stiffness of the one before it. In the plot

legends, the parameters µ0,C and lmax,C are the stiffness and maximum spring extension

for the circumferential bonds, while the diagonal bond parameters are indicated with the

subscript V . The results demonstrate that the model converges to a maximum radial strain

vs. pressure response well below experimental values when the diagonal bond parameters

used are those taken from the axial stretch calibrations above. In order to match exper-

iment for pressurization, the axial bond parameters needed to be reduced. The orange

curve shows the results when the vertical bond stiffness is reduced one order of magnitude

(µ0,V = 0.91 kPa, one tenth the value of the light green curves in Figure 4.3). Although

we would be able to match the experimental results for pressurization by further tuning

the parameters in the orange curve, the results demonstrate that no match can be attained

76

unless the vertical bond stiffness (µ0,V or lmax,V ) is reduced to a value below 9.1 kPa. In

this case, the axial stretch would no longer match experiment. Thus we concluded that the

single layer triangular mesh is not sufficiently tunable to be implemented as an anisotropic

vascular model.

4.2.3 Single layer DPD arteriole with square mesh

While the triangular mesh proved to be inappropriate for an anisotropic model, we consid-

ered an alternative mesh in which the DPD particles were arranged in a square grid with

diagonal bonds connecting the opposite corners. The structure is illustrated in Figure

4.5. In the figure, there are nine DPD particles indicated by black circles located at the

square vertices. For the derivation of macroscopic properties, below, the bonds connected

to the central DPD particle are highlighted in light blue, orange, and navy blue to help

distinguish the diagonal, vertical, and horizontal bonds, which each have unique spring

parameters. This structure was eventually abandoned in favor of a multilayer structure

that could take into account the true microstructure of vascular tissue (Section 4.3, below),

but we discuss here the preliminary work done with the single layer square model so that

it can be revisited in the future.

(bx,by)

x (θ)

y (axial direction)

(-bx,-by)

(ax,ay)(-ax,-ay)

(cx,cy)

(-cx,-cy)

(dx,dy)

(-dx,-dy)

S

Figure 4.5: Representative area elementof square DPD grid DPD particles indi-cated with filled black circles are locatedat intersections of vertical and horizontallines. Colored lines indicate the bonds act-ing directly on the central DPD particle,and separate bond parameters are used fordiagonal, horizontal, and vertically alignedbonds. Dashed border indicates the repre-sentative area element S surrounding thecentral particle.

Below, we derive the macroscopic properties for the DPD structure in Figure 4.5 us-

ing the virial theorem as in [48]. We begin with the generalized stiffness tensor for an

orthotropic material in continuum. An orthotropic material is a special type of anisotropic

material which has two or three orthogonal planes of symmetry. Arterial walls are generally

accepted to be cylindrically orthotropic [83], which allows us to simplify the continuum

77

stiffness tensor by using the orthotropic case. The continuum and DPD models are equated

by deriving their stress/strain relationships in terms of the continuum stiffness tensor and

DPD spring parameters. We start with the continuum model and then we derive the cor-

responding stress/strain relationships in DPD by applying three deformations from which

we can calculate the stress due to the spring forces.

Orthotropic stiffness tensor in continuum model

The stress strain relation is written as

ǫ = D σ, (4.10)

where ǫ is the displacement vector; σ is the stress vector, and D is the stiffness tensor.

For an orthotropic material, D can be written as

D =

C11 C12 0

C12 C22 0

0 0 C33

, (4.11)

The components of D can then be written in terms of the components of the stress and

strain vectors, where

σ =

σxx

σyy

σxy

, ǫ =

ǫxx

ǫyy

ǫxy

. (4.12)

Taking the displacements

ǫ1 =

ǫxx

0

0

, ǫ2 =

0

ǫyy

0

, ǫ3 =

0

0

ǫxy

, (4.13)

78

gives the following relations for the components of D in terms of the stress and strain:

C11 =σxxǫxx

, C12 =σyyǫxx

,

C21 =σxxǫyy

, C22 =σyyǫyy

,

C33 =σxyǫxy

.

(4.14)

Derivation of Elasticity Tensor from DPD Equations

In this section, we derive expressions for the components of the elasticity tensor above in

terms of the parameters for the DPD wormlike chain bonds for the analogous DPD model

for orthotropic elastic material.

The Cauchy stress comes from the Virial theorem:

ταβ = − 1

A

∑

rk∈pair

f(rk)

rkrαk r

βk , (4.15)

for pair-interactions, where A is the area of the representative area element (RAE) as the

area enclosed in the dotted black line in Figure 4.5, and k goes over all interacting pairs

in the RAE, each pair with radial distance rk, and directional normal rk.

The bond forces are a combination of wormlike chain (WLC) and power (POW), defined

as fWLC-POW = fWLC + fPOW, where

fWLC1(L1) = −kBTp1

(

1

4(1− x)2− 1

4+ x

)

, x = L/Lmax,1,

fWLC2(L2) = −kBTp2

(

1

4(1− x)2− 1

4+ x

)

, x = L/Lmax,2,

fWLC3(L3) = −kBTp3

(

1

4(1− x)2− 1

4+ x

)

, x = L/Lmax,3,

(4.16)

and

fPOW1(L1) =kp,1Lm1

,

fPOW2(L2) =kp,2Lm2

,

fPOW3(L3) =kp,3Lm3

,

(4.17)

where p1,p2 and p3 are the persistent lengths, kB is Boltzmann constant and T is the

79

temperature. L is the length of the spring, and Lmax1, Lmax2, Lmax3 are the contour lengths

of the springs; in the power term, kp is the force coefficient, and m is the exponent.

Combining these with Eq. (4.15) and including an area constraint (for incompressibil-

ity), the total in-plane stress is

ταβ = − 1

A

[

fWLC,POW1

aaαaβ +

fWLC,POW3

bbαbβ +

fWLC,POW2

ccαcβ +

fWLC,POW3

ddαdβ

]

+(ka + kd)(A0 −A)

A0δαβ ,

(4.18)

where the last term is the area constraint: ka and kd are the local and global area con-

straints; A0 is the initial area of the RAE, and δαβ is the Kronecker delta.

From here, one can calculate the stress that results from applying a small strain ǫ and

then compute ∆ταβ/ǫij = σαβ/ǫij to get the expressions in Eq. (4.14).

Taking the differential of Eq. (4.18) gives

∆ταβ = − 1

A[∆(fWLC-POW1(a)/a)(aαaβ)0 + fWLC-POW1(a0)/a0∆(aαaβ)

+∆(fWLC-POW3(b)/b)(bαbβ)0 + fWLC-POW3(b0)/b0∆(bαbβ)

+∆(fWLC-POW2(c)/c)(cαcβ)0 + fWLC-POW2(c0)/c0∆(cαcβ)

+∆(fWLC-POW3(d)/d)(dαdβ)0 + fWLC-POW3(d0)/d0∆(dαdβ)]

−(ka + kd)∆A

A0δαβ .

(4.19)

From Eqs. (4.16) and (4.17), we find

∆fWLC(l)

l=kBT

pL20

1

4(1− x0)2− x0

2(1− x0)3− 1

4

∆l, and (4.20)

∆fPOW(l)

l= − 1

L20

kp(m+ 1)

Lm0∆l, x = l/Lmax, x0 = L0/Lmax, (4.21)

The derivative of the area ∆A comes from

A = |a× c| = axcy − aycx

∆A = cy∆ax + ax∆cy − cx∆ay − ay∆cx.

(4.22)

80

In order to calculate the components ofD we impose on the lattice the three incremental

engineering strains (one at a time) from Eq. (4.13) which correspond to the following three

deformation tensors:

J1 =

√1 + 2γ 0

0 1

,J2 =

1 0

0√1 + 2γ

,J3 =

√1− γ

√γ

√γ

√1− γ

, (4.23)

r = r0J1, r = r0J2 or r = r0J3, where we use γ << 1 for the small strain approximation.

Note that

ǫxx =1

2(J1

TJ1 − I) = γ;

ǫyy =1

2(J2

TJ2 − I) = γ;

ǫxy =1

2(J3

TJ3 − I) =√

γ(1− γ).

(4.24)

For the square configuration, we let L0 define the equilibrum length of the edge, so

that a0 = L0 = x0Lmax1, c = L0 = x0Lmax2 and b0 = d0 =√2L0 = x0Lmax3, where

Lmax1 = Lmax2 = Lmax3/√2.

J1 (pure shear in x direction)

By imposing r = r0J1, the coordinates of the lattice and their variations are given as

a = (L0

√

1 + 2γ, 0), ∆a = L0γ +O(γ2), (4.25)

b = (L0

√

1 + 2γ, L0), ∆b =L0γ√

2+O(γ2), (4.26)

c = (0, L0), ∆c = 0, (4.27)

d = (−L0

√

1 + 2γ, L0), ∆d =L0γ√

2+O(γ2). (4.28)

To compute ∆τxx1, we start with

∆(axax) = 2L20γ +O(γ2), (4.29)

∆(bxbx) = 2L20γ +O(γ2), (4.30)

∆(cxcx) = 0, (4.31)

∆(dxdx) = 2L20γ +O(γ2), (4.32)

81

∆τxx1 = − 1A

fWLC-POW1(a0)/a0∆(axax) +∆(fWLC-POW1(a)/a)(axax)0

+fWLC-POW3(b0)/b0∆(bxbx) +∆(fWLC-POW3(b)/b)(bxbx)0

+fWLC-POW2(c0)/c0∆(cxcx) +∆(fWLC-POW2(c)/c)(cxcx)0

+fWLC-POW3(d0)/d0∆(dxdx) +∆(fWLC-POW3(d)/d)(dxdx)0

−(ka + kd)∆A

= L0

E0(

2√2

p3+ 2

p1

)

− E1(

1√2p3

+ 1p1

)

+ 1Lm0

(

kp,1(m− 1) + kp,3m−3√2m+1

)

−(ka + kd)L0

γ

(4.33)

where we have substituted the expressions E0 ≡ kBT

14(1−x0)2

− 14 + x0

and E1 ≡

kBT

14(1−x0)2

− x02(1−x0)3

− 14

for simplicity.

We get the expression for C11 (Eq. 4.14) from C11 = σxx/ǫxx = dτ/dγ:

C11 =E0L0

(

2p1

+ 2√2

p3

)

− E1L0

(

1p1

+ 1√2p3

)

+ 1Lm+10

(

kp,1(m− 1) + kp,3m−3√2m+1

)

−(ka + kd).(4.34)

Similarly, we can derive the expression for C12 using the yy component of the stress for

the same deformation:

C12 = − 1

L0

E1√2p3

+1

Lm+10

m+ 1√2m+1kp,3 − (ka + kd). (4.35)

Below, we show the expressions for the rest of the components of D which can be

82

derived similar to the above method:

C21 = − 1

L0

E1√2p3

+1

Lm+10

m+ 1√2m+1kp,3 − (ka + kd),

C22 =E0

L0

(

2

p2+

2√2

p3

)

− E1

L0

(

1

p2+

1√2p3

)

+1

Lm+10

(

kp,2(m− 1) + kp,3m− 3√2m+1

)

− (ka + kd),

C33 =E0

L0

(

1

p1+

1

p2+

2√2

p3

)

− E1

L0

(

2√2

p3

)

− 1

Lm+10

(

kp,1 + kp,2 − kp,3m√21−m

)

.

(4.36)

Anisotropic stretch test

1

1.2

1.4

1.6

1.8

vertical stretch

ε y

0 20 40 60 800.7

0.8

0.9

1

stress (kPa)

ε x

1

1.2

1.4

1.6

1.8

horizontal stretch

ε x

0 20 40 60 800.7

0.8

0.9

1

stress (kPa)

ε y

n=16

n=100

n=256

expt, passive (Guo et al. 2007)

Figure 4.6: Uniaxial stretch of anisotropic square mesh Blue curves – coarse grained mesh using 16 DPD

particles. Black blue curves – mesh with 100 DPD particles. Red curves – fine grained mesh with 256 DPD

particles. Grey dashed curves – experimental results for axial stretch of passive rat cremaster arterioles

interpolated from Figure 4A in [68]. Top, left – strain in the x-direction (σx=length/initial length) due to

horizontal (x-direction) stretch. Grey dashed curve shows results for circumferential stress vs. strain in

rat cremaster arteriole interpolated from Figure 3A in [68]. Bottom, left – strain in the y-direction due to

horizontal stretch. Top, right – strain in the y-direction due to vertical (y-direction) stretch. Bottom, right

– strain in the x-direction due to vertical stretch.

We performed two uniaxial stretching tests – one in the horizontal and one in the vertical

direction – for a square sheet of material using the mesh in Figure 4.5. We used a 9 µm by

83

9 µm square. The bond parameters were m = 2.0, Lmax,1 = Lmax,2 = Lmax,3/√2 = 2.0,

and C11 = 0.016, C22 = 0.05MPa, C33 = 0.024MPa. Area constraints were excluded

(ka = kd = 0). Table 4.3 gives the meshes for the three levels of coarse graining used in

the simulation.

Table 4.3: Mesh values used for square sheet with square mesh

Nv L0

16 3.0 µm

100 1.0 µm

256 0.6 µm

The results for three different levels of coarse graining are shown in Figure 4.6. Blue

curves show results for a highly coarse grained mesh with 16 DPD particles (4 by 4). The

results for 100 and 256 DPD particles (black and red curves) demonstrate good convergence.

The grey dashed curves in the top plots are experimental results for stretching tests with

a passive rat cremaster arteriole interpolated from [68]. The top left plot compares DPD

results for horizontal stretch with experimental results for circumferential stress vs. strain

(interpolated from Figure 3A in [68]). The top right plot shows results for stretch along

the vertical direction (y-axis) compared with experimental axial stress vs. strain data

interpolated from Figure 4A in [68].

4.3 Multilayer arteriole model in DPD

In this section, we present a multilayer model of an arteriole that mimics the true mi-

crostructure of the vascular tissue. The DPD model is derived from the continuum level

models of fiber-reinforced anisotropic arterial tissues. The first model, by Holzapfel and

coworkers [62, 83] is a thick-walled cylinder composed of a neo-Hookean elastic material

reinforced by stiff fibers, in which the neo-Hookean component represents elastin, and the

fibers represent collagen present in the vascular wall. The fibers in the model are uniformly

distributed. The model uses two identical fiber families oriented symmetrically to the axis

of the cylinder, making the material orthotropic. The authors Ferruzzi, Humphrey, and

coworkers later introduce a four fiber family version of the model in which they also use a

84

circumferentially aligned fiber family and an axially aligned fiber family [54, 53].

In DPD, the elastin matrix layer of the arteriole model is an isotropic layer composed of

a set of DPD particles forming a triangulated network. The fiber layer is orthotropic and

is built by a parallelogram network in which the DPD particles are located at the crossings

between the two symmetrical fibers. The DPD particles comprising the fiber layer we will

refer to as fiber particles, while those in elastin layer will be referred to as matrix particles.

Figure 4.7 shows the structure for an example DPD arterial segment. The fiber and elastin

layers are bound at the contact points between the fiber particles and the matrix triangle

faces according to an adhesion relationship described below.

The vessel wall is treated as a two-dimensional material, in that the material variation

is constrained to only the axial and azimuthal directions, while the material is uniform

across the wall thickness (in the radial direction). However, we do consider variations in

wall thickness based on the assumption that the material is incompressible. In this way,

the wall thickness at any location along the wall is calculated based on the change in area

of the triangular faces in the elastin layer. Thus, ht = ht,0At,0/At, where ht is the triangle

thickness; At is the triangle area, and the subscript 0 denotes the unloaded value.

coll

ag

en

b

ers

ela

stin

ma

trix

Figure 4.7: Structural schematic of

two-layer fiber-reinforced arterial wall The

fiber-reinforced vascular wall structure in

DPD (left) is made of two attached layers

shown separately on the right. Collagen

fibers are indicated in red; elastin matrix

bonds are indicated in blue. DPD parti-

cles in each layer are located at the bond

intersections.

For the two-layer model, the total energy in the system (Eq. (4.4)) is adjusted to

include the interaction between the two layers, so that the system energy is now [158]

V (xi) = Vin−plane + Varea + Vint, (4.37)

where Vint comes from Peng et al. 2013 [158]:

85

Vint =∑

j∈1...Nmf

kmf (dj − d2j0)

2, (4.38)

where Nmf is the number of connection points between the matrix and fiber layers; in

this case, Nmf is equal to the number of fiber particles because the layers are connected

at the intersections points of the fiber particles and the triangle faces on the matrix layer.

dj is the distance between the vertex j in the fiber layer and its corresponding projection

point j′ on the matrix layer; dj0 is the initial distance (in the unloaded material) between

j and j′. kmf is the spring constant of the interaction potential. For our purposes, we

set kmf to a large value such that the two layers are permanently attached and no sliding

between the two layers is permitted. This mimics the continuum model, which assumes

there is no deatchment between the layers and does not include any sliding of the collagen

fibers within the material. To do this, we set the value of kmf four orders of magnitude

higher than the shear modulus of the elastin layer, µ0, which was more than sufficient to

prevent the layers from sliding or separating. We also found that at this value, further

increases in kmf did not change the results of simulations of stretch tests (whereas much

lower values of kmf gave a decrease in the overall material stiffness). In future models,

it will be important to consider weaker interaction forces between the layers in order to

simulate mechanical remodeling of the wall in pathological conditions. Here we focus on

developing a working constitutive model of a fiber-reinforced arteriole in DPD that can

match continuum model results.

Elastin layer

The elastin layer is composed of wormlike chain bonds that follow the same equations as

in Section 4.2.2 with a slight modification to account for variations in thickness which we

describe here.

For thin-walled problems, such as modeling a red blood cell membrane, it is a reasonable

approximation to ignore variations in the thickness of the material. Arterioles, however,

have thick walls, so the thin walled approximation is not applicable. In DPD, we address

this problem by offering a “2.5-dimensional” solution, in which we still ignore variations

86

in stress across the thickness of the wall (along the radial direction), but we compute

and account for changes in the wall thickness due to deformations of the material. The

thickness of the material is computed using the incompressibility constraint and thus local

volume conservation: the volume of any triangle (triangle area times thickness) is constant,

so each triangle i has thickness hi = A0h0/Ai where Ai is the triangle area; A0 and h0 are

the unloaded triangle area and thickness, respectively. Each matrix particle is assigned

a thickness equal to the average thickness of all triangular faces associated with it (for a

matrix particle in the center of the cylinder, there will be six associated triangles of which

it is located at a vertex, while matrix particles at the ends of the tube have three associated

triangles). Fiber particles are attached to the matrix layer at the location within a matrix

triangle where they intersect in the unloaded configuration. Because they will not always

fall directly in the middle of a triangle, they are not assigned the same thickness of the

triangle at which they attach, but instead they are assigned a weighted average of the

thicknesses of the three matrix particles located at the triangle vertices. The weighted

average is based on proximity to the fiber particle. Thicknesses of bonds are determined

by the averages of the two DPD particles they connect.

In the DPD red blood cell model [47], the physical shear modulus of the red blood cell

membrane is scaled to a one-dimensional value in DPD with units of force/length rather

than force/length2 by multiplying the shear modulus by the membrane thickness. Here,

we preserve the true dimensions of the physical shear modulus of the vascular elastin layer,

but when computing the DPD bond parameters p from Eq. (4.8), we modify the equation

by multiplying µ0 by the current value of the bond thickness. Thus changes in thickness

will result in changes in bond stiffness. In this way, we can approximate a thick-walled

problem while only considering a two-dimensional material. Thus, the persistence length,

p of the WLC springs is recalculated every timestep to account for the changing thickness,

and Eq. (4.8) becomes

hjµ0 =

√3kBT

4pjlmaxx0

(

x02(1− x0)3

− 1

4(1− x0)2+

1

4

)

+

√3kp(m+ 1)

4lm+10

, (4.39)

where hj and pj are the current thickness and persistence length, respectively, of matrix

87

particle j. As described above, hj is computed as the average thickness of the triangles

associated with particle j:

hj =∑

i∈1...Nt,j

hi, (4.40)

where Nt,j is the number of triangles associated with particle j, and hi is the area of

triangle i.

Fiber layer

To derive a DPD bond type equivalent to the stress function in the continuum model of

Gasser et al [62], consider stretching a 2D sheet consisting of nf parallel fibers in the x

direction from initial length l0 to l with an effective continuum cross section area of Af .

The fiber orientation vector is a0 = [1 0] and the deformed fiber orientation vector is

a0 = [l/l0 0].

For the no dispersion case (κ = 0), the deformed structure tensor [62] is given as

h = a0 ⊗ a0 =

l2

l200

0 0

, (4.41)

and the Green-Lagrange strain-like quantity is given as

E = tr(h)− 1 =l2

l20− 1. (4.42)

The stress function in continuum (See Table 2 in [62]) is given as

ψ′f = k1E exp(k2E

2) = k1(l2

l20− 1) exp

[

k2(l2

l20− 1)2

]

, (4.43)

where k1 and k2 are two material parameters. k1 determines the initial stiffness and k2

determines the hardening behavior.

Thus, the projected Kirchhoff stress tensor for the fibers (Eq. 4.8 in [62]) is

τ f = 2ψ′f h =

2k1(l2

l20− 1) l

2

l20exp

[

k2(l2

l20− 1)2

]

0

0 0

, (4.44)

88

where the Kirchhoff stress tensor is related to τ f through τ f = p : τ f . Here p is the

fourth-order projection tensor defined as I− 12I⊗ I. After some algebra, we find τ f = 1

2 τ f .

The Cauchy stress in x direction is given by

σ11 = τf,11/J =2k1l

2

Jl20(l2

l20− 1) exp

[

k2(l2

l20− 1)2

]

, (4.45)

where J is the Jacobian, which presents the volume change. If volume incompressible,

J = 1.

Since the summation of fiber forces (in DPD or reality) equals to the stress times the

cross section area (in effective continuum media), i.e.

nfFf = σ11Af , (4.46)

the final exponential form of the DPD bond force is

Ff = σ11Af/nf =2k1Af l

2

Jnf l20

(l2

l20− 1) exp

[

k2(l2

l20− 1)2

]

, (4.47)

where l and l0 are the current and unloaded bond lengths, respectively; J = 1 for volume

incompressibility. The above formulas were derived by Zhangli Peng.

Also, it may be noted that Af/nf is the cross-sectional area (thickness, h times the

width of the sheet) divided by the number of fibers; thus Af/nf = dfh, where df is the

distance between two parallel fibers along the normal direction, which can be calculated

from the initial fiber configuration along with volume incompressibility, as we describe here.

First, we assume local volume incompressibility such that the local volume is preserved.

Local volume is equal to the fiber bond length l times the distance to the nearest parallel

fiber, df , times the thickness h. Thus,

V0 = l0df0h0 = V = ldfh, (4.48)

where the subscript 0 denotes the initial (unloaded) value. From this, we get an expression

89

for dfh:

dfh = (l0df0h0)/l. (4.49)

The values l0 and h0 are known, and df0 can be computed from the initial configuration

of the diamond grid:

df0 = l0 sin(2α), (4.50)

where α is the fiber angle and l0 sin(2α) is the distance along the orthogonal direction

between two parallel edges of an equilateral diamond with sides having length l0 and

angles 2α and 180 − 2α. The final expression for dfh is

dfh = (l20 sin(2α)h0)/l. (4.51)

The thickness h of the fiber bond is equal to the average thickness of the two fiber par-

ticles it connects. As described above, fiber particle thickness is computed as the weighted

average of the thicknesses of matrix particles located at the vertices of the triangular face

that the fiber particle is attached to. Figure 4.8 shows a diagram for reference. The filled

circles are matrix particles, and the open circle is a fiber particle.

d1 d2

d3

1 2

3

f

Figure 4.8: Adhesion between fiber and matrix parti-

cles Open circle — fiber particle. Filled circles — ma-

trix particle. The fiber particle adheres to the elastin

matrix layer at the location of its intersection point

with the triangular face in the elastin layer. The thick-

ness of the fiber particle is computed as the weighted

average of the thicknesses of the matrix particles lo-

cated at the three vertices of the triangle to which it

is attached.

The thickness, hf of fiber particle f , is computed as

hf =∑

i∈[1,2,3]

hiwi, (4.52)

where hf is the thickness of the fiber particle, and hi is the thickness of the matrix

particle located at one of the three triangle vertices. The weights, wi depend on the

90

distance between the matrix particles and the fiber particle and are defined as

wi =D − di2D

,with

D =∑

j∈[1,2,3]

dj ,(4.53)

where di is the distance between the fiber particle and matrix particle i, and D is the

sum of the three distances.

4.3.1 Results and verification – uniaxial stretch

Uniaxial stretch of thick rectangular sheet

We follow the same tensile test for a rectangular sheet as in [62]. Uniaxial loads were

applied to two rectangular strips with complementary fiber angles; these represented slices

from the adventitial layer of an arteriole cut along the axial and circumferential directions.

The unloaded dimensions of the sheets were length L = 10.0mm by widthW = 3.0mm, and

unloaded thickness T = 0.5mm. The fiber angle of the adventitial layer is γ = 40.02 with

respect to the axis; thus the circumferential strip had fiber angle γ = 49.98. Following

[62], we apply a rigid boundary at the short edges of the sheets where the pulling force is

applied, mimicking the conditions in an experimental testing machine in which the mounted

specimen would be constrained on each end. Table 4.4 gives the mesh and the parameters

used in the simulation.

Table 4.4: Parameters for uniaxial stretch of fiber-reinforced sheet

layer Nv l0 lmax µ0 m k1 k2

matrix 825 0.216506 mm 5.0 7.64 kPa 2

fiber (axial) 725 0.212056 mm 996.6 kPa 524.6

fiber (circumferential) 694 0.217632 mm 996.6 kPa 524.6

Figure 4.9 shows the thickness map for a deformed axial and circumferential specimen

with an applied uniaxial load of 1 N. The deformed shapes for both specimens is thinner

near the ends where the material is held rigid. In the center, where the loading force causes

a decrease in the width, the thickness of the specimen is increased due to the material

91

thickness (mm)

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

5 mm

circumferential

axial

Figure 4.9: Thickness map for uniaxial stretch of axial and circumferential sheets Deformed axial and

circumferential sheets are shown for an applied load of 1 N. Triangulated mesh lines show the deformed

structure of the matrix layer (fiber mesh not shown). Color map indicates thickness across the surface of

the axial and circumferential strips. Scale bar is 5 mm.

incompressibility. The deformed structures and the pattern of the thickness contour maps

are in agreement with the continuum results for three-dimensional material in [62].

Figure 4.10 shows lengthwise strain due to force for the circumferential and axial spec-

imens. DPD results are shown for two different values of area constraint (indicated in

the legend in DPD units). Interpolated results from Figure 11 in [62] are shown in black

open circles; spectral element results performed by Alireza Yazdani are shown in yellow.

At low applied force (<0.2 N), the curves show large initial deformation, followed by very

small increases in strain at higher forces. The bend in the curves occurs when the stiff

fibers are rotated nearly in the direction of the applied force, at which point they carry

92

most of the load. Initially, most of the load is carried by the weaker elastin matrix, so the

force causes a large deformation, which causes the rotation of the collagen fibers. Because

the fiber angle is 40.02 with respect to the vessel axis, the fibers in the axial specimen

are slightly more aligned with the applied force in their undeformed state, compared with

the circumferential specimen. This means that in the axial specimen, the collagen fibers

support a larger portion of the applied load than they do in the circumferential specimen,

making the axial specimen stiffer overall.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

strain

axi

al f

orc

e (

N)

Gasser, Holzapfel, et al. 2006Gasser, Holzapfel, et al. 2006

spectral elementspectral element

DPD, ka = 80DPD, ka = 80

DPD, ka = 120DPD, ka = 120

axi

al s

pe

cim

en

circ

um

fere

ntia

lsp

eci

me

n Figure 4.10: Uniaxial stretch

of axial and circumferential

sheets Blue (or red) – DPD

model results. Black open cir-

cles – interpolated results from

Figure 11 in [62]. Yellow – spec-

tral element results performed

by Alireza Yazdani

Pressurization of a thin-walled tube for two fiber model

0.4 0.6 0.8 10

5

10

15

axial strain

pres

sure

(kP

a)

1 1.2 1.4 1.6 1.8 20

5

10

15

circumferential strain

pres

sure

(kP

a)

Gasser et al. 2006 DPD

α=50.02 α=40.02 α=30.02

α=30.02 α=40.02 α=50.02

B

A

Figure 4.11: Pressurization

of cylinder with two fibers Solid

grey curves – interpolated re-

sults from Figure 7 in [62].

Dashed black curves – DPD

model results. Fiber angle val-

ues (α) displayed in degrees.

A axial strain due to pressure.

B circumferential strain due to

pressure.

93

We perform pressurization of a discrete particle tube embedded with two diagonal fibers

symmetric across the axis for verification against the continuum results in Figure 7 from

[62]. As in the example from [62], we use a cylinder with a mean radius R = 4.745mm and

wall thickness H = 0.43mm in the stress-free configuration. We compute the axial and

circumferential stretches, λz and λθ, respectively, due to internal pressurization using three

tubes with different fiber angles, α, where α is the reference angle between each diagonal

fiber and the axis. Table 4.5 gives the mesh and the parameters used in the simulation.

Table 4.5: Parameters for pressurization of thin-walled tube

layer Nv l0 lmax µ0 m k1 k2

matrix 260 1.484563 mm 5.0 7.64 kPa 2

fiber (α = 50.02) 221 1.481925 mm 996.6 kPa 524.6

fiber (α = 40.02) 240 1.439540 mm 996.6 kPa 524.6

fiber (α = 30.02) 260 1.483666 mm 996.6 kPa 524.6

Results for the three tubes are shown in Figure 4.11 along with the interpolated results

from [62]. The top plot shows the axial strain due to pressure, while the low plot shows

circumferential strain. As the tube is pressurized, causing the circumference to expand,

the axial length of the tube decreases due to the incompressibility constraint. The tube

with the 50.02 fiber angle experiences the least circumferential strain because these fibers

are closest to being aligned with the circumferential direction, and thus they support

more of the circumferential load than the 40.02 or 30.02 fibers. The 50.02 fiber angle

also experiences the least amount of axial shortening (top plot) due to the fact that the

circumferential deformation is smaller. The DPD model is an excellent match with the

continuum for the circumferential strain, while there is slight discrepancy for the axial

strain, which may be explained by slight differences in surface area constraints and bending

rigidity, which are modeled differently in the discrete framework than the continuum.

4.3.2 Results and verification – biaxial stretch

While the DPD model was successful in matching results for uniaxial stretch with the

continuum model, there were some issues in biaxial stretch experiments. We will discuss

those issues in this section and some possible solutions.

94

Biaxial stretch of square sheet

circumferential

triangle alignment

axial triangle

alignment

α=40.02˚

Figure 4.12: Alignment of fibers and matrix triangles

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

strain (along width)

axi

al f

orc

e (

N)

−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 00

0.2

0.4

0.6

0.8

1

strain (lengthwise)

axi

al f

orc

e (

N) L0=0.206, circ

L0=0.206, axial

L0=0.4 circ

L0=0.4, axial

spectral element

Figure 4.13: Biaxial stretch of square sheet with α = 40.02 Top plot — lengthwise strain vs. applied

force. Lower plot — strain along width vs. applied force. Yellow curves show spectral element results

performed by Alireza Yazdani. Open triangles — DPD results for axial matrix orientation. Filled triangles

— DPD results for circumferential matrix orientation. Fine graining of the mesh is indicated in the plot

legend, which gives the bond length for the matrix layer.

We begin with some examples of biaxial stretch of a square sheet. In the first example,

we use the same parameters as the axial sheets from Figure 4.10 except that the sheet is a

10 mm by 10 mm square instead of 10 by 3 mm. The sheet thickness was 0.5 mm, as it was

in the uniaxial tests. Force was applied in the vertical direction to the horizontal edges

and in the horizontal direction to the vertical edges. The boundary conditions allowed

movement in the x− and y− directions throughout the material so that no edges were held

95

rigid.

We applied an equal amount of force in each direction such that for an isotropic ma-

terial, the square would have transformed into a larger square. Because this material was

orthotropic, the square was stiffer in the vertical direction (fiber angles were at a 40.02

with respect to the vertical axis; see text surrounding Figure 4.10 for details). Thus, the

deformed structure was expected to be rectangular with the width being greater than the

length.

In this biaxial stretch test, we used two different alignments for the matrix layer, as

illustrated in Figure 4.12. In the first configuration, the triangulated matrix layer was

aligned such that the flat edges of the triangular sheet were oriented in the circumferential

direction; in the second, the flat edges of the matrix layer were oriented in the axial

direction. We used two different levels of coarse graining for each configuration. Table 4.6

gives the meshes used in the simulation.

Table 4.6: Mesh values for square sheet in Figure 4.13

layer Nv l0 lmax µ0 m k1 k2

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 2503 0.204 mm 996.6 kPa 524.6

fiber (axial) 2503 0.205 mm 996.6 kPa 524.6

matrix 769 0.4124 mm 5.0 7.64 kPa 2

fiber (circumferential) 644 0.4081 mm 996.6 kPa 524.6

fiber (axial) 644 0.4092 mm 996.6 kPa 524.6

The results for biaxial stretch of the square sheet are shown in Figure 4.13. The top

plot shows the strain along the length of the sheet, and the lower plot shows the strain

along the width. The circumferential matrix alignment is shown in filled triangles, while

axial alignment is shown in open triangles. Spectral element results performed by Alireza

Yazdani are shown in yellow curves. Because of the 40.02 fiber alignment, the fibers

support a greater proportion of the lengthwise loading than the horizontal (width-wise)

load, so the material is stretched in the horizontal direction, aligned with the width. This

deformation also rotates the fibers closer to the horizontal alignment, which causes the

decease in the length of the sheet. Because the matrix layer is isotropic, we expected that

96

the two matrix layer alignments (open and filled triangles) would have identical results.

However, this was not the case. The axial configuration (open circles) overpredicted the

horizontal stiffness of the material (bottom plot), which resulted in a corresponding un-

derprediction of the length decrease (upper plot). While the circumferential results (filled

triangles) were much closer to the spectral element results, they slightly overpredict the

strain along the width, such that the yellow curves lie somewhere in the middle of the

curves for the two DPD configurations. We show that the results match for two different

levels of coarse graining (light and dark blue curves) to demonstrate that the error between

the two matrix alignments is not corrected with fine graining.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

strain (along width)

axi

al f

orc

e (

N)

−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 00

0.2

0.4

0.6

0.8

1

strain (lengthwise)

axi

al f

orc

e (

N) L0=0.206, circ

L0=0.206, axial

spectral element

α=40.02˚

α=40.02˚

α=30˚

α=30˚

Figure 4.14: Biaxial stretch of square sheet with α = 30 Top plot — lengthwise strain vs. applied force.

Lower plot — strain along width vs. applied force. Yellow curves show spectral element results performed

by Alireza Yazdani. Open triangles — DPD results for axial matrix orientation. Filled triangles — DPD

results for circumferential matrix orientation. Fine graining of the mesh is indicated in the plot legend,

which gives the bond length for the matrix layer. Results for α = 30 and α = 40.02 fiber angle are shown

as labeled on the plot.

To determine whether the fiber angle had an impact on the sensitivity of the results to

the matrix layer orientation (i.e. how much the results for axial and circumferential matrix

alignments disagreed), we repeated the same biaxial test keeping all parameters the same

except that the fiber angle was changed to 30. Table 4.7 gives the mesh values for the

axial and circumferential sheets with a 30 fiber angle.

Figure 4.14 shows the results when the fiber angle is 30. The results from Figure 4.13

97

Table 4.7: Mesh values for square sheets with fiber angle 30 in Figure 4.14

layer Nv l0 lmax µ0 m k1 k2

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 2822 0.206197 mm 996.6 kPa 524.6

fiber (axial) 2765 0.208333 mm 996.6 kPa 524.6

for the 40.02 fiber angle are also shown for comparison. Again, the top plot shows the

strain along the length, and the lower plot shows the strain along the width. Yellow curves

are the spectral element results by Alireza Yazdani, and black curves with triangles are

DPD results. Open triangles show the results for the axial matrix alignment, and filled

triangles show DPD results for the circumferential matrix alignment. Compared to the

40.02 fiber angle, the 30 fibers are closer to being aligned with the vertical (lengthwise)

direction, so they support a smaller portion of the horizontal load than the 40.02 fibers.

For this reason, the sheet with the 30 fiber angle experiences significantly higher strain

along the width (bottom plot) resulting in increased shortening along the length (upper

plot). The difference between the two matrix orientations is much more pronounced for

the 30 fiber angle, which is expected given that 40.02 is much closer to 45, which is

identical for both orientations.

Because the coarse graining is not related to the error, we induced that the diamond grid

structure of the fiber layer must introduce some biasing due to the anisotropic connectivity

pattern between the matrix and fiber layers. We decided to test whether fine graining the

fiber layer without changing the matrix layer would make the solution converge such that

both matrix orientations gave the same results. We repeated the biaxial stretch for the

same sheet as in Figure 4.13 (fiber angle equal to 40). We held the equilibrium spring

length constant for the matrix layer, L0,m and adjusted the length for the fibers L0,f using

the ratios L0,m : L0,f =1:1, 2:1, 3:1, 4:1, where previously, the ratio had been held at 1:1.

Table 4.8 gives the meshes used for the various levels of fine graining of the fiber layer.

The results of biaxial stretch at different levels of fine graining in the fiber layer are

shown in Figure 4.15. Lengthwise strain is shown in the top plot, and strain along the

width is given in the lower plot. As discussed above, the 40.02 oriented fibers hold a

98

Table 4.8: Mesh values for square sheets in Figure 4.15

layer Nv l0 lmax µ0 m k1 k2

ratio 1:1

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 2503 0.204 mm 996.6 kPa 524.6

fiber (axial) 2503 0.205 mm 996.6 kPa 524.6

ratio 2:1

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 9589 0.10363 mm 996.6 kPa 524.6

fiber (axial) 9589 0.10367 mm 996.6 kPa 524.6

ratio 3:1

matrix 769 0.4124 mm 5.0 7.64 kPa 2

fiber (circumferential) 5463 0.138914 mm 996.6 kPa 524.6

fiber (axial) 5693 0.136410 mm 996.6 kPa 524.6

ratio 4:1

matrix 769 0.4124 mm 5.0 7.64 kPa 2

fiber (circumferential) 9843 0.103634 mm 996.6 kPa 524.6

fiber (axial) 9740 0.103672 mm 996.6 kPa 524.6

higher proportion of the lengthwise load than the horizontal load, so the sheet is stretched

along the width (upper plot), and it shortens along the length (lower plot) due to the in-

compressibility constraint. Yellow curves again show spectral element results from Alireza

Yazdani. Grey curves are for the 1:1 ratio of matrix bond equilibrium length to fiber bond

equilibrium length, and are the same as the blue curves in Figure 4.13. The figure legend

indicates the level of fine graining for the fiber layer by giving the ratio L0,m : L0,f (as

described above, the matrix bond equilibrium length L0,m is held constant, while L0,f is

decreased. As the fine graining of the fiber layer is increased (light blue curves to dark

blue, to black), the difference in the results between the two matrix orientations gets more

pronounced. In fine graining the fiber layer, we increase the number of fiber particles in

the mesh, which means that we increased the number of adhesion points between the two

layers. The increase in adhesion points altered the competition between biaxial loads in

a way that amplified difference between the two matrix orientations. In other words, for

the circumferential matrix orientation in the original mesh (L0,m : L0,f=1:1, grey filled

triangles), the sheet experienced more strain along the width than in the spectral element

results (yellow curve) or the axial matrix orientation (grey open triangles), meaning that

99

the circumferential sheet was more resistant to lengthwise stretching and less resistant to

horizontal stretch. With increased fine graining (dark blue and black filled triangles), the

horizontal load competed even more favorably against the lengthwise loading, such that the

widthwise strain is increased compared to the grey curves, and the lengthwise shortening

(or negative strain, upper plot) is also increased. The opposite is true for the axial matrix

orientation (open triangles): the increased number of adhesion points from fine graining

the fiber layer caused a decrease in the competition between the vertical and horizontal

loads, such that the fine grained sheets (dark blue and black open triangles) experienced

less horizontal strain and less shortening along the length than the 1:1 ratio mesh (grey

open triangles).

1:12:1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

strain (along width)

axi

al f

orc

e (

N)

−0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 00

0.2

0.4

0.6

0.8

1

strain (lengthwise)

axi

al f

orc

e (

N)

spectral element

ratio 1:1, circ

ratio 1:1, axial

ratio 4:1, circ

ratio 4:1, axial

ratio 3:1, circ

ratio 3:1, axial

ratio 2:1, circ

ratio 2:1, axial

3:1

4:1

Figure 4.15: Biaxial stretch of square sheet with fine grained fiber layer Top plot — lengthwise strain vs.

applied force. Lower plot — strain along width vs. applied force. Yellow curves show spectral element

results performed by Alireza Yazdani. Open triangles — DPD results for axial matrix orientation. Filled

triangles — DPD results for circumferential matrix orientation. Fine graining of the fiber layer is indicated

in the plot legend, which gives the ratio of matrix bond equilibrium length to fiber bond equilibrium length,

L0,m : L0,f , where L0,m is held fixed.

We next considered whether we would achieve the opposite effect by coarse graining

the fiber layer. Whereas in the previous example, increasing the fine graining of the fiber

layer resulted in increased sensitivity to the matrix layer orientation, we wanted to test

whether decreasing the fine graining of the fiber layer (starting from L0,m : L0,f=1:1)

100

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

strain (along width)

axi

al f

orc

e (

N)

−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 00

0.2

0.4

0.6

0.8

1

strain (lengthwise)

axi

al f

orc

e (

N)

spectral element

ratio 1:1, circ

ratio 1:1, axial

ratio 1:2, circ

ratio 1:2, axial

ratio 1:3, circ

ratio 1:3, axial

Figure 4.16: Biaxial stretch of square sheet with coarse grained fiber layer Top plot — lengthwise strain

vs. applied force. Lower plot — strain along width vs. applied force. Yellow curves show spectral element

results performed by Alireza Yazdani. Open triangles — DPD results for axial matrix orientation. Filled

triangles — DPD results for circumferential matrix orientation. Coarse graining of the fiber layer is indicated

in the plot legend, which gives the ratio of matrix bond equilibrium length to fiber bond equilibrium length,

L0,m : L0,f , where L0,m is held fixed.

would continue to decrease the sensitivity of the matrix orientation. We used the ratios

L0,m : L0,f =1:1, 1:2, 1:3, where the L0,m was again held fixed, and the fiber equilibrium

spring length was changed. Table 4.9 gives the meshes used for each level of coarse graining

of the fiber layer.

Table 4.9: Mesh values for square sheets in Figure 4.16

layer Nv l0 lmax µ0 m k1 k2

ratio 1:1

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 2503 0.204 mm 996.6 kPa 524.6

fiber (axial) 2503 0.205 mm 996.6 kPa 524.6

ratio 1:2

matrix 2879 0.206 mm 5.0 7.64 kPa 2

fiber (circumferential) 644 0.408059 mm 996.6 kPa 524.6

fiber (axial) 644 0.409231 mm 996.6 kPa 524.6

ratio 1:3

matrix 4106 0.169809 mm 5.0 7.64 kPa 2

fiber (circumferential) 419 0.502227 mm 996.6 kPa 524.6

fiber (axial) 388 0.518359 mm 996.6 kPa 524.6

101

Figure 4.16 shows the results for the biaxial stretching at various levels of coarse graining

in the fiber layer. The upper plot gives the strain along the length, and the lower plot gives

the strain along the width. The spectral element results from Alireza Yazdani are shown

again in yellow curves. Open triangles show DPD results for axial matrix orientation, and

filled triangles give results for circumferential matrix orientation. Grey curves show the

results for L0,m : L0,f =1:1 and are the same as in the previous figure. Dark blue curves

give the results for the first level of coarse graining L0,m : L0,f =1:2, and light blue curves

give the results for the second level of coarse graining L0,m : L0,f =1:3. The results for

all three levels are almost identical. The circumferential matrix orientation results (filled

triangles) match very closely for all three ratios, as do the axial results (open triangles).

Coarse graining the fiber layer had virtually effect on the biaxial stretching results.

l 0 c

os

α

l0 cos α

α

l0

B

A

Figure 4.17: Structural schematic of two-

layer fiber mesh A Mesh structure for ma-

trix layer with two fiber layers each com-

prising parallel fiber bonds. Grey lines

show matrix mesh; blue lines show paral-

lel fibers in one direction; orange lines show

the parallel fibers in the opposite direction.

Shaded region shows square area within the

mesh. B Parallel fibers from one layer that

occupy the square shaded region in A.

We considered whether the diamond grid structure of the fiber layer could be responsible

for the mismatching of results for the two matrix layer orientations. We made one attempt

to restructure the fiber layer to test whether we could make the results insensitive to the

matrix orientation. Figure 4.17 shows the mesh structure for the fibers. In this mesh, the

symmetrical fibers do not share particles. Instead, the fibers form two distinct layers of

parallel bonds. This allows us to set an arbitrary spacing between parallel fibers without

102

changing the fiber angle. Figure 4.17A shows the three-layer structure, where the matrix

layer mesh is shown in grey lines, and the fibers are shown in blue and orange lines. The

bond equilibrium length for the fibers, l0, is equal to the bond equilibrium length for the

matrix. The shaded region covers a square subsection of the mesh. Figure 4.17B shows

one layer of parallel fibers that occupy the square shaded region. The fibers in each layer

are spaced such that the density of fiber particles (and thus the density of adhesion points

between the matrix and each fiber layer) will be equal along the horizontal and vertical

directions.

We repeated the biaxial test for a 10 mm by 10 mm square sheet (thickness was 0.5

mm) using the mesh given in Figure 4.17. The fiber angle was 30, and the mesh details

are given in Table 4.10. The square was given an equal amount of force in each direction

from 0–0.5 N. The results for biaxial stretch are given in Figure 4.18. The top plot shows

the lengthwise strain, and the lower plot shows the strain along the width. The yellow

curves show the spectral element results performed by Alireza Yazdani. The open triangles

show the results for the axial matrix orientation, and the filled triangles show the results

for the circumferential matrix orientation. The results for the new fiber mesh give very

similar behavior as the previous, diamond mesh, as in Figure 4.14. The circumferential

alignment (filled triangles) overpredicts the strain along the width compared to the spectral

element results, and correspondingly, it overpredicts the lengthwise shortening. Also, the

axial matrix orientation (open triangles) underpredicts the strain along the width and

underpredicts the lengthwise shortening.

Table 4.10: Mesh values for square sheets with fiber angle 30 in Figure 4.18

layer Nv l0 lmax µ0 m k1 k2

matrix 1240 0.3207 mm 5.0 7.64 kPa 2

circumferential matrix alignment

fiber (layer 1) 1333 0.3207 mm 996.6 kPa 524.6

fiber (layer 2) 1333 0.3207 mm 996.6 kPa 524.6

axial matrix alignment

fiber (layer 1) 1341 0.3207 mm 996.6 kPa 524.6

fiber (layer 2) 1340 0.3207 mm 996.6 kPa 524.6

103

−0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 00

0.2

0.4

strain (lengthwise)

axi

al f

orc

e (

N)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

strain (along width)

axi

al f

orc

e (

N)

DPD, circ

DPD, axial

spectral element

Figure 4.18: Biaxial stretch of square sheet with α = 30 for two-layer fiber mesh Top plot — lengthwise

strain vs. applied force. Lower plot — strain along width vs. applied force. Yellow curves show spectral

element results performed by Alireza Yazdani. Open triangles — DPD results for axial matrix orientation.

Filled triangles — DPD results for circumferential matrix orientation.

4.3.3 Results and verification – biaxial stretch of four-fiber model

colla

ge

n

be

rse

last

in m

atr

ix

Figure 4.19: Structural schematic of two-

layer fiber-reinforced arterial wall with four

fibers The fiber-reinforced vascular wall

structure in DPD (left) is made of two

attached layers shown separately on the

right. Elastin matrix bonds are indicated

in blue. Collagen fibers are indicated in red

(diagonal fiber family), black (circumferen-

tial fibers), and grey (axial fibers); DPD

particle locations are shown on the right.

DPD particles (black circles) in the matrix

layer are located at the triangle vertices;

fiber particles are located at the diamond

vertices.

A limitation of the two-fiber model discussed above is that it cannot adequately capture

the mechanical behavior of true arterial tissue without somehow accounting for dispersion

in the fiber orientation [62]. Gasser et al. [62] successfully addressed the problem of

dispersion in their continuum model using a parameter κ to characterize the distribution

104

of fiber orientation angles. Another approach by Ferruzzi, Vorp, and Humphrey [54] was to

introduce two additional fiber families aligned axially and circumferentially in a “four fiber

family” model. Without considering dispersion explicitly, the mixture of the four fiber

orientations provided a good approximation of the macroscopic effects of collagen fiber

dispersion [54]. The circumferential fibers also account for the passive elastic properties of

the smooth muscle cells, which are aligned along the circumferential direction (see Section

4.1, above). Of the two models, the latter is more readily adaptable to DPD because it

defines each fiber orientation explicitly, whereas the dispersion model is more ambiguous

in a discrete framework.

In this section we introduce the four-fiber model in DPD following the continuum model

of Ferruzzi, Vorp, and Humphrey [54]. This model uses the same constitutive equations for

the fiber stress energies as in the two-fiber model discussed above, but with two additional

fiber families oriented in the axial and circumferential directions. The DPD structure of

the four-fiber model is given in Figure 4.19. The elastin matrix is shown in blue, and the

original two diagonal fiber families are shown in red. The axial fibers are shown in grey,

and the circumferential fibers are shown in black. The axial and circumferential fibers

share DPD particles with the diagonal fibers. The connectivity for the DPD particles is

shown on the right. DPD particles are indicated in black circles. In the fiber layer, the

particles lie at the vertices of the diamonds formed by the diagonal fibers. The axial and

circumferential vertices connect the opposite vertices.

Biaxial stretch of thin square sheet

Using the four-fiber mesh in Figure 4.19, we perform the same biaxial tests as in the

previous section, again on a 10 mm by 10 mm square sheet with thickness of 0.5 mm. The

angle of the diagonal fibers was 30. The parameters and details of the mesh are given in

Table 4.11.

Results for biaxial stretch of the four-fiber sheet are given in Figure 4.20. Top plot

shows the lengthwise strain, and lower plot shows the strain along the width. Spectral

element results performed by Alireza Yazdani are shown in yellow curves. DPD results

using the circumferential orientation of the matrix layer are shown in filled triangles, and

105

Table 4.11: Parameters and mesh values for square sheets in Figure 4.20

layer Nv l0 lmax µ0 m k1 k2

circumferential matrix orientation

matrix 769 0.4124 mm 5.0 7.64 kPa 2

diagonal fibers 740 0.4124 mm 996.6 kPa 524.6

circumferential fibers 0.4124 mm 800.0 kPa 300.0

axial fibers 0.714286 mm 925.0 kPa 250.0

axial matrix orientation

matrix 769 0.4124 mm 5.0 7.64 kPa 2

diagonal fibers 711 0.416667 mm 996.6 kPa 524.6

circumferential fibers 0.416667 mm 800.0 kPa 300.0

axial fibers 0.721688 mm 925.0 kPa 250.0

open triangles show the results for the axially oriented matrix layer. The results for the

four fiber model show an important difference from the two fiber results: in the two-

fiber results, the square was stretched along its weakest direction (along the width) and

compressed along its strongest direction. In the four-fiber results, the square is stretched

along both directions, and the incompressibility constraint causes the thickness to decrease

(not shown) instead of the length. In this example, the square experiences more strain along

horizontal direction (width) than the length because the 30 diagonal fibers support more

load in the lengthwise direction, whereas the axially and circumferentially aligned fibers

(Table 4.11) have close to the same stiffnesses. (Here, the axial direction is oriented along

the length of the sheet, and the circumferential direction is oriented along the width of

the sheet, as shown in Figure 4.12). The presence of the axial and circumferential fibers

changes the qualitative effects of the competing loads in the biaxial stretch. These fiber

families are each oriented exactly in the direction of the two orthogonal loads, and because

they are able to carry these loads throughout the deformation, they prevent some of the

rotation of the diagonal fibers, which allows the sheet to resist shortening along the length.

Thus, the sheet is stretched along both the length and the width. As with the two-fiber

examples, the results for the four-fiber stretch demonstrate that there is sensitivity to the

orientation of the matrix layer, such that the circumferentially aligned matrix overpredicts

strain along the width (filled triangles, lower plot) and underpredicts lengthwise strain

(upper plot), while the reverse is true for the axially aligned matrix (open triangles). In

106

contrast with the two-fiber results above, the axial curves (open triangles) are much closer

to the spectral element results (yellow), but it is not clear whether this is characteristic of

the four-fiber grid structure, or whether it would change depending on the fiber parameters.

In this example, the values for the parameters of all four fibers (Table 4.11) were very close.

However, as will be shown below, when the four fiber parameters are fit to biaxial stretch

data from real arterioles, the parameters for the diagonal fibers are two orders of magnitude

off from the values for the orthogonal fibers.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.2

0.4

0.6

0.8

1

strain (lengthwise)

axi

al f

orc

e (

N)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.2

0.4

0.6

0.8

1

strain (along width)

axi

al f

orc

e (

N)

DPD, circ

DPD, axial

spectral element Figure 4.20: Biaxial stretch of four-fiber

family square sheet with α = 30 Top plot

— lengthwise strain vs. applied force.

Lower plot — strain along width vs. ap-

plied force. Yellow curves show spectral

element results performed by Alireza Yaz-

dani. Open triangles — DPD results for

axial matrix orientation. Filled triangles

— DPD results for circumferential matrix

orientation.

Biaxial stretch of thick-walled cylinder

The advantage of performing biaxial stretching tests to calibrate a model arteriole is that

in their in vivo state, blood vessels experience both luminal pressure and significant axial

stretch. Therefore, the biaxial tests say more about the mechanical behavior of a vessel

in its in vivo state than uniaxial tests can. A unique mechanical property that has been

demonstrated in vessels is that when extended to their in vivo level of axial stretch, they

maintain a constant level of axial force that does not change with pressure [184, 191, 21].

When extended beyond the in vivo axial stretch, increased pressure causes an increase in

the axial force, while vessels held below the in vivo axial stretch experience a decrease in

axial force as pressure is increased [53].

107

While the two-fiber model cannot reproduce this mechanical behavior, our collabora-

tors, Jay Humphrey and Chiara Bellini, successfully fit their experimental in vitro data

to the four-fiber continuum model. In this small subsection, we simulate biaxial stretch

of a left common carotid artery (lCCA) and compare our results with experimental data

provided by our collaborators. We use a four fiber DPD tube, for which our collaborators

provide us parameters they obtained through a best-fit estimation for the four fiber con-

tinuum model following the procedures outlined in [53]. In their in vitro experiment, they

use an artery with the following dimensions in the unloaded (stress-free) state: length =

5.57 mm; outer diameter = 447 µm; wall thickness = 65 µm.

Following the experiment of our collaborators, we simulate pressurization of the DPD

vessel over a range of 0-20 kPa while the vessel is held at the in vivo axial stretch, and

once while the vessel is held slightly above and slightly below the in vivo axial stretch.

These stretch levels (λz = length/unloaded length) are provided to us by our collaborators:

λz = 1.64, 1.73, 1.81. In DPD, we compute the axial force by summing the material forces

on each DPD particle at the end of the tube. In the experiment, each end of the artery

is attached to a cannula so that it can be pressurized. We avoid the non-uniformities at

the ends of the artery by simulating only a short segment at the center of the vessel (1

mm length) where the radius is roughly uniform. Thus, we simulate the vessel with free

boundary conditions in the radial and circumferential directions such that the entire length

of the vessel (including the ends) is free to move radially. In the simulation, the tube is first

extended axially until it reaches one of the three values of λz, at which point it is restricted

from moving axially for the rest of the simulation so that it can maintain a constant axial

stretch. After the axial stretch, pressure is applied in small steps and the tube thickness,

diameter, and axial force are recorded.

Although our DPD model computes the thickness change in the wall, it is still only a

two-dimensional approximation of, in this case, a thick-walled problem. This means that

there is some ambiguity about where the DPD particles should be placed in terms of a

reference radius. Because the pressure in DPD was computed based on the locations of

the matrix particles and the surface area they inhabit, placing the DPD particles at the

inner radius, rin, would give a correct estimate of the force they should experience due

108

to pressure. On the other hand, placing the DPD particles at the middle radius, rmid

would more accurately predict the circumferential extension, as inner radius placement

may cause an underestimate. Therefore, we performed the biaxial pressurization tests

using both reference radius values rmid and rin; parameters and meshes are given in Table

4.12.

Table 4.12: Parameters and mesh values for thick walled tube in Figure 4.21

layer Nv l0 lmax µ0 m karea k1 k2

reference radius = rmid

matrix 868 42.77 µm 2.0 21.22 kPa 2 0

diagonal fibers 812 43.1097 µm 0.04 kPa 1.16

circumferential fibers 42.77 µm 7.025 kPa 0.09

axial fibers 74.8245 µm 9.25 kPa 0.07

reference radius = rin

matrix 744 41.38 µm 2.0 21.22 kPa 2 0

diagonal fibers 696 41.7050 µm 0.04 kPa 1.16

circumferential fibers 41.38 µm 7.025 kPa 0.09

axial fibers 72.3729 µm 9.25 kPa 0.07

Figure 4.21 shows the results for pressurization tests. The thick solid lines are the

experimental data obtained by Chiara Bellini and Jay Humphrey. Yellow curves show the

DPD results when rin is used as the reference radius, and the grey open circles show DPD

results when rmid is the reference radius. The top plot (Figure 4.21A), shows the outer

diameter vs. pressure, and the bottom plot (Figure 4.21B) shows the axial force due to

pressure. The DPD results using rmid (grey open circles) show an excellent match both

qualitatively and quantitatively. In the bottom plot, the in vivo axial stretch curve is

relatively flat, so that as the pressure increases, the axial force remains constant. When

the vessel is held at an axial stretch above the in vivo value, increased pressure causes an

increase in the axial force. When the vessel is held slightly below the in vivo axial stretch,

the axial force decreases with increased pressure. When rin is used as the reference radius

(yellow curves), the DPD results still show a qualitative match with the experiment, but

there is a quantitative disagreement. In the top plot they significantly overestimate the

circumferential stiffness (or underestimate the circumferential extension due to pressure).

In the lower plot, they underestimate the axial force by several mN.

109

300 400 500 600 700 8000

5

10

15

20

outer diameter (µm)

pre

ssu

re (

kPa

)

0 5 10 15 200

5

10

15

20

pressure (kPa)

axi

al f

orc

e (

mN

)

λz = 1.81

λz = 1.73

λz = 1.64

λz = 1.81> in vivo

λz = 1.73in vivo

λz = 1.64< in vivo

A

B

DPD, rin

DPD, rout

expt, Bellini & HumphreyFigure 4.21: Pressure vs. diameter

at three axial stretch levels Vessels were

held at constant axial stretch during ap-

plications of incremental internal pressure.

Thick solid lines — experimental data by

Chiara Bellini and Jay Humphrey. Yellow

filled circles — DPD results using inner ra-

dius, rin, as reference radius. Grey open

circles — DPD results using outer radius,

rout, as reference radius. A Outer diameter

vs. pressure. B axial force vs. pressure.

Because the DPD results had a better match when using the middle radius as the

reference point, we used this mesh for an additional biaxial test in which the pressure was

held constant and the axial force was gradually increased. In this simulation, we began by

pressurizing the tube to one of four levels of internal pressure (10, 60, 100, or 140 mmHg).

Next, an axial force of 0–18 mN was applied to the ends of the tube (distributed uniformly

over the circumference). The tube had free boundary conditions throughout the simulation

so that the DPD particles were free to move axially and radially.

One thing that needs to be pointed out is how we treated the axial force due to pres-

surization to match the experiment of our collaborators. In the experiment, the luminal

pressure in the cannulated vessel was acting on a close-ended vessel, which meant that the

axial force experienced by the vessel was equal to ft, the axial force applied to the vessel in

order to stretch it axially, plus fp,z = πr2inP , or the pressure times the inner cross-sectional

area of the tube. In DPD, we applied pressure to an open-ended tube, so we had to correct

for the axial force after the simulation. To do this, when we plot the axial force, we first

110

Figure 4.22: Force vs. length tests at

four levels of internal pressure Vessels were

held at constant pressure during applica-

tions of incremental axial force. Thick

solid lines — experimental data by Chiara

Bellini and Jay Humphrey. Grey open cir-

cles — DPD results using middle radius,

rmid, as reference radius. A Outer diam-

eter vs. axial strain, λz B Axial force vs.

axial strain.

0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600

700

800

strain (λz)

ou

ter

dia

me

ter

(µm

)

0.8 1 1.2 1.4 1.6 1.8 2strain (λz)

axi

al f

orc

e (

mN

)

0

5

10

15

20

10 mmHg 60 mmHg 100 mmHg 140 mmHg

10 mmHg

60 mmHg

100 mmHg

140 mmHg

DPD

expt, Bellini & Humphrey

A

B

subtract πr2inP , where P is the pressure used in the simulation (in units of Pascals), and

rin is the current inner radius.

The results for axial force tests are given in Figure 4.22. The thick grey curves show

the experimental results from Bellini and Humphrey, and the open circles show the DPD

results using the first mesh in Table 4.12 (rmid). The upper plot shows the outer diameter of

the tube with respect to axial strain (λz). For low pressure (10 mmHg), the DPD matches

fairly well with the experiment until the axial stretch exceeds 1.7 times the unloaded

length. At higher pressures, the matching is fairly worse. At 60 mmHg, the DPD results

predict a larger diameter than experiment, and at 100 and 140 mmHg, the DPD results

underestimate the diameter. However, in Figure 4.22B (lower plot), the DPD matches

experimental results for axial force vs. length slightly better, especially at 100 and 140

mmHg. The point where all for curves cross (roughly 1.73) is the in vivo axial stretch

value. The force correction, mentioned above, is what allows the first data point in each

111

curve to have a negative value for the axial force (this is most noticeable in the 100 mmHg

curve).

To address the error in the DPD results for the axial force tests, we repeated both of the

above biaxial test with a slightly modified set of parameters for the matrix layer, without

changing the mesh. Table 4.13 shows the original and adjusted parameter sets. In the new

parameters, we increased the maximum spring extension from 2.0 to 2.1 to try and allow

the diameter to extend further at high pressure and correct the error shown Figure 4.22A.

In an initial trial (not shown) this adjustment alone only provided minor improvement for

the diameter estimates, while at the same time, it introduced additional error in the axial

force vs. length results, so we added a small area constraint (karea, see Eq. (4.9), above)

to prevent the tube from extending too far from axial force.

Table 4.13: Original and adjusted matrix layer parameters for Figure 4.23

layer Nv l0 lmax µ0 m karea

matrix (original parameters, rmid) 868 42.77 µm 2.0 21.22 kPa 2 0

matrix (adjusted parameters rmid) 868 42.77 µm 2.1 21.22 kPa 2 10.0

Figure 4.23 gives the results for the pressure and axial force tests for the adjusted

parameter set. The DPD results for the new parameter set are shown in open circles, with

the experimental results shown in thick solid curves. For comparison, we show the results

from the original DPD parameter set in thin red curves. The left column (A and B) shows

the results for the pressure tests. Figure 4.23A shows the diameter vs. pressure when the

tube is held at three levels of axial stretch, and Figure 4.23B shows the axial force vs.

pressure for the same test. The adjusted parameter set gives a slightly better match with

experiment for the pressure vs. diameter (Figure 4.23A), but the results for axial force

vs. pressure (Figure 4.23B) are significantly worse. The parameter adjustment gives a

considerable underestimate for the axial pressure. In addition, there is a slight qualitative

difference at the in vivo axial stretch, as this curve is not flat, but decreases with respect

to pressure. This difference is explained in Figure 4.23D, which shows the axial force vs.

pressure during the axial stretch tests, in which the vessel is held at a constant pressure.

The DPD curves for the adjust parameters cross at an axial strain slightly above 1.73,

112

300 400 500 600 700 8000

5

10

15

20

outer diameter (µm)

pre

ssu

re (

kPa

)

0 5 10 15 200

5

10

15

20

pressure (kPa)

axi

al f

orc

e (

mN

)

0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600

700

800

strain (λz)

ou

ter

dia

me

ter

(µm

)

0.8 1 1.2 1.4 1.6 1.8 2strain (λz)

axi

al f

orc

e (

mN

)

λz = 1.81

λz = 1.73

λz = 1.64

λz = 1.81> in vivo

λz = 1.73in vivo

λz = 1.64< in vivo 0

5

10

15

20

10 mmHg 60 mmHg 100 mmHg 140 mmHg

10 mmHg

60 mmHg

100 mmHg

140 mmHg

DPD

DPD, alt. parameters (see text)

expt, Bellini & Humphrey

A

B

C

D

Figure 4.23: Biaxial stretch tests of four-fiber vessel Thick solid lines — experimental data by Chiara

Bellini and Jay Humphrey. Red curves — DPD results using original parameter set with rmid as reference

radius. Grey open circles — DPD results using middle radius rmid as reference radius with adjusted

parameter set in Table 4.13. A,B Pressure vs. diameter tests — Vessels were held at constant axial stretch

during applications of incremental internal pressure. A Outer diameter vs. pressure. B axial force vs.

pressure. C,D Force vs. length tests — Vessels were held at constant pressure during applications of

incremental axial force. C Outer diameter vs. axial strain, λz D Axial force vs. axial strain.

meaning that with these parameters, the model predicts an in vivo axial stretch value

above 1.73. Figure 4.23C shows the axial strain vs. diameter when the vessel is held at

constant pressure. There is some improvement in the DPD results for higher pressures,

but at lower levels of axial strain, the diameter predicted by DPD at 100 and 140 mmHg

is still below the experimental value. However, the most important part of the data to

match is near the in vivo range of pressure and stretch; thus, it is better for the curves to

match at axial stretch values in the range of 1.6–1.8 than 1-1.4.

Chapter Five

Multiphysics Neurovascular

Coupling and Future Directions

114

This chapter provides a discussion of the multiscale and multiphysics problems related to

neurovascular coupling. Future applications for neurovascular modeling are discussed and

motivated. We first demonstrate an example of a simplistic multiphysics neurovascular cou-

pling modeling model that couples astrocyte ODEs from Chapter 2 with a fiber-reinforced

DPD vessel described in Section 4.3 using modifications described below (Section 5.1.1)

that convert the model from a passive, purely mechanical, model to an active myogenic

model that can respond to pressure and astrocytic inputs.

5.1 Example of Neurovascular Coupling with Multiphysics

DPD Vessel

We present an example of a simplistic neurovascular coupling model using DPD and an

astrocyte. The astrocyte model is the ODE system described in Chapter 2. The DPD vessel

responds to extracellular potassium (due to astrocytic release) with active dilation and

constriction. As we will discuss in detail below, the model describes the interaction between

extracellular potassium and the myogenic response, a phenomenon in small arterioles in

which internal pressure increases induce constriction. Increases in extracellular potassium

induce dilation by reversing the myogenic constriction. The astrocyte responds to changes

in the DPD vessel radius via mechanosensitive TRPV4 channels (Eqs. (2.12) – (2.14),

except that in Eq. (2.14), ǫ = (r − r0)/r0 is now the length change of the DPD arteriole

radius).

We use a four fiber family version of the two-layer DPD vessel described in Section

4.3. The diagonally oriented fibers represent collagen fibers, while the axially oriented

fibers represent the combination of axially oriented wall components which include the

endothelial cells and axially aligned collagen fibers. The circumferentially oriented fibers

represent the smooth muscle cells, which are aligned circumferentially, in combination with

circumferentially aligned collagen.

Precise quantitative biaxial stretching data are available for some larger arterioles (in

the range of 500 µm diameter), and these data have been fit to the four fiber model

[53]. However, at the penetrating arteriole level (∼50 µm diameter), where astrocytes are

115

present and neurovascular coupling is most critical, few data are available, so it is efficient

to begin with a more lumped model rather than considering several layers separately.

Before we discuss how the model relates vessel diameter to extracellular K+, it is

necessary to start by explaining the necessary initial conditions – namely, myogenic tone

– that allow an arteriole to dilate when exposed to K+. As demonstrated experimentally

in vitro by [19], astrocyte-induced vasodilation depends on the initial level of arteriole

constriction, or tone. Their results are in agreement with the hypothesis that potassium

behaves as a vasodilator by releasing the constriction in the smooth muscle cells. This

hypothesis is also supported by the mechanistic model proposed by [46] which takes into

account the interactions of potassium and calcium ion channels in the smooth muscle (see a

discussion of this model in Chapter 1. We adapt this model into our own model in Chapter

2).

A DPD arteriole model that responds to extracellular K+ should include some descrip-

tion of baseline arteriole tone. In experimental settings, isolated vessels or vessel in vitro

can achieve physiological baseline tone through cannulation, pressurization, or the applica-

tion of vasoconstrictors like thromboxane [19, 22]. Under normal physiological conditions

in vivo, baseline arteriole tone develops in response to transmural pressure from blood flow.

Thus, we propose a model in DPD that includes a description of pressure-induced arteriole

constriction, or myogenic response.

The myogenic response is a phenomenon occurring predominantly in smaller arterioles

in which pressure increases cause an active constriction, rather than inflating the arteriole

like a passive elastic tube. The constriction is a result of myogenic contraction of smooth

muscle cells which express stretch-activated calcium channels. Internal pressure or luminal

flow produces an inward calcium flow through these channels, and the increased intracellu-

lar Ca2+ enables the binding of myosin-actin chains in the smooth muscle, which contract

as the chains bind and slide together.

As discussed in Chapter 1, extracellular potassium is a vasodilator that functions by

inducing a depolarizing outward K+ flux through smooth muscle Kir channels, thus clos-

ing voltage-gated Ca2+ channels, so that the reduction in intracellular Ca2+ concentration

causes the detachment of myosin-actin crossbridges, allowing the filaments to slide apart.

116

Thus, vasodilation and constriction occurs through a literal length change of smooth mus-

cle cells. In penetrating arterioles, the myogenic response describes a pressure and flow-

induced constriction that at in vivo baseline state is close to ∼40% decrease in diameter

compared to a passive (unconstricted) arteriole. Therefore, for a simplistic example of

neurovascular coupling in DPD, we consider the vessel response to both astrocyte derived

potassium and internal pressure in the vessel. Below, we define a basic relationship be-

tween DPD smooth muscle cell length and pressure and extracellular potassium. In this

example, while the DPD vessel is three dimensional, we do not consider diffusion in ex-

tracellular space, and assume that the extracellular potassium is uniform in space. This

is a reasonable assumption given the tight spacing between the astrocyte endfoot and the

arteriole wall. Also, we note that while each astrocyte endfoot typically covers a length of

20µm along the arteriole it encircles, we assume in this example that the entire length of

the DPD arteriole segment is encircled by endfeet from a uniform population of astrocytes,

so that extracellular potassium release is uniform, but it is a trivial matter to convert this

model into a modular array of non-uniform astrocytes in the future.

5.1.1 Modeling Framework and Constitutive Equations

As discussed above, the length of penetrating arteriole smooth muscle cells depends on

internal pressure and extracellular potassium. It should be specified that while the length

of the SMC decreases during constriction, the tissue is still elastic and can be stretched.

Therefore, in DPD, we describe myogenic response as a relationship between pressure and

L0,SMC , the equilibrium length of the circumferentially oriented DPD fiber bonds that

represent smooth muscle:

L0,SMC

L0= 1− 1− λmin

2

(

1 + tanh∆P − Pmid

kP

)

, (5.1)

where L0 is the unloaded, passive SMC bond length. The length change due to pressure is a

sigmoid curve that decreases monotonically from 1 to λmin, where λmin is chosen to prevent

an equilibrium length equal to zero or at a value below the physical limit. The internal

pressure is ∆P , while Pmid defines the pressure value that gives 1/2 maximum constriction,

117

and kP defines the slope of the relationship. These last two parameters are chosen so that

there is virtually no constriction at zero pressure (L0,SMC/L0 ≈ 1 at ∆P = 0), while the

maximum constriction (L0,SMC/L0 ≈ λmin) occurs by ∆P = 200 mmHg.

For this model, we choose Pmid = 90 mmHg, kp = 72 mmHg, λmin = 0.2. The first two

values were chosen based on experimental data for myogenic response for small (∼40 µm

diameter) arterioles in [38, 35]. In these studies, pressurized myogenic arterioles reached

their maximum level of constriction at just over 140–160 mmHg pressure and were about

half-way to their maximum constriction around 90 mmHg. Taking Pmid to be 90 mmHg,

we get L0,SMC/L0 to approach its minimum asymptotic value (representing maximum

myogenic constriction) around 190–200 mmHg when we use a value of 72 mmHg for kp,

while still maintaining L0,SMC/L0 ≈ 1 at 0 mmHg of pressure.

To inform our value for λmin = 0.2, we looked at an experimental study of rat intracere-

bral arterioles by Dacey and Duling [34] on the mechanical properties of passive arterioles

and “maximally active” arterioles; for the latter, they induce as much constriction as pos-

sible in the smooth muscle cells with the application of extremely high extracellular K+

(140 mM). At the lowest pressure levels (close to 0 internal pressure), the vessel radius

for the passive arteriole is approximately 19 µm, while the radius for the maximally con-

stricted arteriole was close to 5 µm, or close to 0.25 times the passive radius. While it is

not clear from these results whether high levels of pressurization would be able to induce

the maximum arteriole tone, we use their data to set a reasonable lower limit on λmin,

which we set λmin = 0.2, slightly below 0.25 to account for uncertainty (as there were no

data shown for exactly 0 mmHg).

0 50 100 150 2000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pressure (mmHg)

Sm

oo

th m

usc

le e

qu

ilib

riu

m le

ng

th (

L0

,SM

C/L

0)

Kshift=0, [K+]=3 mM

Kshift=20, [K+]=10 mM

Figure 5.1: SMC equilibrium length vs pressure in

DPD Relation given in Eq. (5.2). K+ dependence

comes from Eq. (5.3). Black curve — K+ = 3 mM;

blue curve — K+ = 10 mM.

118

We can include the dilatory contribution of extracellular potassium as a shift to the

right in the length-pressure relationship described by

L0,SMC

L0= 1− 1− λmin

2

(

1 + tanh∆P − Pmid −Kshift

kP −Kshift

)

, (5.2)

where Kshift has units of pressure in mmHg and adjusts the SMC equilibrium length

according to

Kshift = kshift,m

(

1− |[K+]P − 10 mM|7 mM

)

, (5.3)

where [K+]P is the extracellular potassium in the perivascular space due to astrocytic

release (Eq. (2.3)) and kshift,m is the maximum value of Kshift (when [K+]P=10 mM). The

vertical bars around |[K+]P −10 mM| denote absolute value. Figure 5.1 shows the relation

between pressure and SMC length from Eq. (5.2) at 3 and 10 mM K+, corresponding to

baseline and maximum values of Kshift, respectively. We use a kshift,m = 20 mmHg as

a best guess so that the shift in the pressure vs. SMC length curve (Eq. (5.2)) will be

noticeable but without eliminating the myogenic response altogether, which is diminished

but still present with elevated K+ [122]. The other parameters here are not calibrated

to the model but come directly from well established experimental studies: the baseline

extracellular potassium in the perivascular region is 3 mM [85, 106, 57, 46], so at [K+]P = 3

mM, Kshift will be zero. Below 10 mM, increased extracellular K+ dilates arterioles to

anywhere between 20% to 60% of their diameter at baseline K+ [102, 85, 134, 57]. However,

at higher extracellular concentrations, increases in K+ beyond 10 mM [102] or sometimes

15 mM [85] are observed to have the opposite effect and cause constriction. Thus, we use

an absolute value in Eq. (5.3) so that when extracellular potassium increases above 10

mM, Kshift will decrease and cause SMC constriction, while K+ increase between baseline

(3 mM) up to 10 mM Kshift increases, giving a dilatory shift in Eq. (5.2). The 7 mM in

the last term in Eq. (5.3) is simply the difference 10 mM - 3 mM.

119

0 20 40 60 80 100 120 140 160 18015

20

25

30

35

40

45

rad

ius

(um

)

pressure (mmHg)

0 20 40 60 80 100 120 140 160 18020

25

30

35

40

pressure (mmHg)

rad

ius

(um

)

myogenic, 10 mM K+

passive

passive

myogenic, 10 mM K+

myogenic

myogenic

A

B

C

D

Figure 5.2: Myogenic response in DPD. Results are compared with a dynamical smooth muscle cell model

and two examples from experimental studies. A Arteriole radius vs. pressure relation for myogenic DPD

model. Orange curve - passive model (purely mechanical, with L0,SMC/L0 = 1 at all pressures). Yellow

curve - myogenic model from Eqs. (5.2) and (5.3) with K+ at baseline value (3 mM) so Kshift = 0. At low

pressure, myogenic response is minor and the vessel expands with increased pressure. Closer to typical in

vivo pressure (50 to 60 mmHg), the vessel constricts with increasing pressure until constriction maximizes

at which point further increases in pressure cause the vessel to expand. Blue curve - myogenic model

with 10 mM extracellular K+. Extracellular potassium produces a vasodilatory response in an arteriole

depending on its initial level of preconstriction. B Simulation of myogenic response for dynamical model

of smooth muscle from [67] discussed in Chapter 1. Orange curve - passive model (Eq. (1.34) modified to

v1=-17.4, to eliminate calcium current dependence on pressure. Intracellular Ca2+ remains below 1 nM at

all pressures, compared with 100-400 nM under normal physiological conditions). Yellow curve - myogenic

response under normal baseline conditions. Blue curve - myogenic model with 10 mM extracellular K+. C

Figure 1B from [171]: Experimental data taken from rat tail small artery. Open circles (upper curve) are

for a vessel in a calcium-free solution (thus, a passive vessel, as calcium is required for constriction). Filled

circles are for a vessel in physiological saline (1.6 mM calcium). D Experimental data from rat cerebral

arteries from Figure 6B in [102]. Vessels were immersed in physiological salt solution (PSS) that mimics

the extracellular environment of cerebral arteries. Myogenic response is evident in the vessels in regular

PSS (filled circles). Passive vessel behavior occurs for calcium-free PSS (open circles) and PSS with 10 nM

nisoldipine (triangles), an inhibitor of voltage-dependent calcium channels.

5.1.2 Results

Myogenic response

We simulate the DPD arteriole response to varying pressure from 0–180 mmHg to demon-

strate the model’s ability to reproduce myogenic response, in which pressure increases

120

induce constriction of the vessel. In these simulations, we pressurize a penetrating arte-

riole with an unloaded outer radius of 25 µm and unloaded wall thickness of 5 µm. The

ends of the arteriole are held fixed in the axial direction so that the axial length is constant

but the entire vessel is free to move radially. In this way, we approximate an experimental

setup in which a vessel is cannulated and pressurized, but in our simulation, we only con-

sider a section of the tube near the center, far from the cannulated ends where the radius

is tapered and nonuniform. Because of the highly limited data available for mechanical

properties of penetrating arterioles, we chose parameters for the four fibers based on typical

values fit for slightly larger vessels (∼200 µm unloaded radius) as in [53]. The angle of the

diagonal fibers is 30 deg with respect to the cylinder axis. The parameters and mesh for

the DPD vessel are given in Table 5.1(a). The parameters for the four fibers are best fit

parameters for a left common carotid artery (lCCA) provided to us by our collaborators

Chiara Bellini and Jay Humphrey who performed experimental biaxial stretch experiments

and fit the data to the four fiber model according to the procedures outlined in [53].

Figure 5.2A shows the myogenic response in a DPD arteriole. The orange curve is a

passive arteriole model, where Eq. (5.2) is replaced with L0,SMC/L0 = 1 so that there

is no pressure (or K+) dependence of bond equilibrium length, and the model is purely

mechanical. This model is also equivalent to an experimental setup in which a penetrating

arteriole is pressurized in a Ca2+-free bath, preventing SMC constriction.

The yellow curve is the myogenic model from Eq. (5.2) with Kshift set equal to 0

in the absence of potassium. The myogenic response shown is a great qualitative match

with experimental results and is in good quantitative agreement for the pressure values

and ranges of constrictions; compare for example with Figure 1B in [171] or Figure 6B

from [102] shown here in Figure 5.2C and D. From 0 to ∼ 50mmHg, the pressure com-

petes with myogenic constriction enough to expand the vessel. At pressures in the range

50 − 150mmHg is where the myogenic response occurs: active constriction becomes more

significant, and the vessel constricts rather than expands due to increased pressure. As

the constriction begins to maximize near 160mmHg pressure, further increased pressure

overpowers the constriction and the vessel expands due to mechanical stretching of the con-

stricted smooth muscle cells. The blue curve shows the myogenic response when the vessel

121

is exposed to 10 mM extracellular K+. The potassium produces a vasodilatory response

depending on the level of constriction. At lower pressures, the blue and yellow curves are

closer as there is very little constriction due to pressure for the extracellular potassium to

reverse. However, at values where the myogenic response is more pronounced, extracel-

lular potassium induces dilations of ∼ 20%, which is comparable with dilations typically

observed during neural/astrocytic induced vasodilation [65, 178].

Because the DPD model of myogenic response is relatively simplistic and empirical,

it is worthwhile to compare myogenic response in the more detailed, mechanistic model

of smooth muscle cells from [67] detailed in Section 1.3.1, which we use with the astro-

cyte model in Chapter 2. While this model is purely dynamical, it explicitly includes the

smooth muscle potassium and calcium ion channels and their complex gating mechanisms;

it also includes a mechanistic model of intracellular calcium dynamics taking into account

the calcium ion current and intracellular buffering dynamics. The dynamical equations

also detail the calcium dependence of SMC contraction and relaxation. The myogenic re-

sponse simulation results from this model are shown in Figure 5.2B. The two models were

not designed or calibrated to match exactly, but the results are shown to demonstrate the

qualitative agreement. The parameters for the ODE model are given in Table A.5 in the

appendix, except for a few modified parameters which are given here in Table 5.1(b). In

both models, the myogenic vessel (yellow curves) expands slightly at low pressures and con-

stricts with pressure increases above 40 mmHg, and then expands again at high pressures

beyond 150 mmHg or 120 mmHg. This trend is also visible in the experimental examples

from literature shown in Figure 5.2C and D. The effect of potassium on myogenic response

(blue curves Figure 5.2A and B) shows good qualitative agreement across models. When

exposed to extracellular K+, both models predict that the minimum pressure needed to

cause myogenic constriction (or the location of the maximum radius) is increased, meaning

that the myogenic response still occurs but is slightly inhibited. This idea is supported

by experimental results in [122] (see Figure 2B,C) which demonstrate that the level of

constriction caused by a pressure increase from 30 to 70 mmHg is reduced by roughly one

half for a vessel with high extracellular K+. Similar results have also been observed in

various other types of arterioles (see review in Table 1 in [37] and Figures 2 and 3 in [35]).

122

Table 5.1: Parameters for myogenic arteriole

(a) myogenic DPD arteriole

layer mesh parameters

matrix Nv l0 µ0 lmax m

630 4.181 µm 10.0 kPa 2.0 2

fibers Nv l0 k1 k2 kshift,m

diagonal 630 4.181 µm 0.04 kPa 1.16

circ. 4.181 µm 7.025 kPa 0.09 20 mmHg

axial 7.239 µm 9.25 kPa 0.07

(b) arteriole ODE from [67]

parameter/value

Caref 400 nM

Cai,m 350 nM

x0 205 µm

A 1025.0 µm2

S 50000.0 µm2

v1 (-18-0.075∆P ) mV

At this point, two things that should be noted: first, the results in DPD were obtained

without any calibration of the model; the parameters in Eqs. (5.2) and (5.3) were chosen

based on literature. Second, as we used a sigmoid function in Eq. (5.2), it could seem

possible that the behavior of the myogenic arteriole could be highly sensitive to the slope

of the L0,SMC/L0 relation, which flattens at high and low pressures. For example, it is

reasonable to imagine be that the myogenic vessel expands at low pressures only because

the slope of the L0,SMC/L0 curve is still nearly flat and the pressure is increasing more

than the SMC is contracting. In this way, the myogenic response in this model would be an

artifact of the precise shape of the sigmoid function. However, we found that when applying

a purely linear relationship in place of Eq. (5.2), such as L0,SMC/L0 = 1− 0.0038∆P , the

myogenic response maintained the same features and ranges of values as in the sigmoid

relation.

Neurovascular coupling in DPD

Here we demonstrate a basic example of neurovascular coupling in a DPD arteriole. Using

Eqs. (5.2) and (5.3), we can couple a DPD vessel with the dynamical astrocyte model from

Chapter 2 to simulate astrocyte-induced vasodilation. As described earlier in the chapter,

the DPD vessel responds to perivascular potassium from the dynamical astrocyte model,

while the astrocyte responds to the DPD vessel radius through its mechanosensitive TRPV4

channels (Eqs. (2.12) – (2.14)). In this example, extracellular potassium concentration is

assumed to be uniform along the length of the arteriole. Prior to the start of the simulation,

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we bring the system to equilibrium by pressurizing the DPD arteriole to 50mmHg, inducing

a myogenic constriction. The vessel is restricted from moving in the axial direction, but

can expand and contract radially.

0.1

0.2

0.3

0.4

[Ca

2+] A

(µ

M)

0

5

10

15

[K+] P

(m

M)

0 10 20 30 40 50 6022

24

26

28

30

time (sec)

rad

ius

(µm

)

A

B

C

dynamical arterioleDPD arteriole

stim

Figure 5.3: Dynamical astro-

cyte simulation with DPD vessel

Simulation of neural/astrocyte

induced vasodilation using a

DPD vessel (blue curves) com-

pared with dynamical arteriole

model (red curves) from [67].

Black horizontal bars indicate

period of neural stimulus. A

Astrocyte intracellular calcium

concentration. B Perivascular

potassium concentration. C Ar-

teriole radius.

To combine the DPD model with the astrocyte model, we need to couple two mutually

incompatible numerical solvers. DPD is a three-dimensional PDE model built into a large

scale parallel code, while the astrocyte model is a dynamical ODE system that runs in a

serial code. To combine these, we use a coupling method and software package developed

by Yu-Hang Tang et al. [179] called Multiscale Universal Interface (MUI). MUI acts as

an interface between the solvers and passes data back and forth between them. MUI also

interpolates the data in space and time so that different time and length scales can be

used in the two solvers. In this basic example, the astrocyte solver and DPD vessel are

coupled through two variables: the first is the extracellular potassium from the astrocyte

solver, which governs the smooth muscle bond equilibrium length in DPD; the second is

the DPD vessel radius, which, as it dilates, affects the mechanosensitive TRPV4 channels

on the astrocyte endfoot. In the example we present here, we are assuming the astrocyte

endfoot covers the entire length of the vessel uniformly, and that the potassium released by

the endfoot is uniform in space. Diffusion of potassium is also ignored, but this is unlikely

to have a significant role in the interaction, as the space between the astrocyte endfoot

and the vessel is very tight, so there is little room for the potassium to diffuse. Therefore,

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MUI only passes two scalar values between the astrocyte ODE and DPD solvers. Each

time step, the DPD solver receives a global value for the extracellular potassium, sent from

the astrocyte solver. At the same time, the astrocyte solver receives a scalar value for the

vessel radius, sent from the DPD solver. The radius is computed in DPD each time step,

and it is computed as the average radius along the length of the vessel that the astrocyte

covers (in this case, the entire vessel).

Figure 5.3 shows simulation results for a basic example of a neurovascular unit using a

DPD vessel (blue curves) compared with the results for a purely dynamical vessel model

(red curves). In this example, we tune the parameters for the ODE model arteriole The

parameters are the same as in Table A.5 in the appendix, except adjusted so that the

radius is 23 µm at baseline wth 50 mmHg of pressure. Because the model was originally

intended for larger arterioles [67], it was never optimized for small penetrating arterioles.

Had we used the same parameters for this example as we had used to generate the myogenic

response curve in Figure 5.2, the dilation here during neurovascular coupling would have

been significantly diminished. The DPD vessel used here was also slightly different than

the one used in the myogenic response curves. The unloaded outer radius of this DPD

vessel was 20 µm, and the unloaded wall thickness was 5 µm; the length was 54 µm. The

mesh and parameters are given in Table 5.2.

layer mesh parameters

matrix Nv l0 µ0 lmax m

630 3.136 µm 5.0 kPa 4.0 2

fibers Nv l0 k1 k2 kshift,m

diagonal 630 3.136 µm 0.04 kPa 1.16

circ. 3.136 µm 7.025 kPa 0.09 40

axial 5.429 µm 9.25 kPa 0.07

Table 5.2: Parameters for DPD arteriole used in neurovascular coupling

Figure 5.3A shows the astrocyte intracellular calcium concentration. The solid black

bar indicates the period of neural stimulus. Figure 5.3B shows extracellular potassium,

and Figure 5.3C shows the vessel radius. The red curve is the dynamical arteriole model

we use from [67] along with the modifications Eqs. (1.20) – (1.26) adopted from [46].

Ignoring the oscillations present in the red curves (as the DPD model does not include

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vasomotion), the two arteriole models are in good agreement both in onset of response and

amplitude of dilation. At approximately 20 seconds, there is a drop in the radius due to the

extracellular potassium increasing beyond 10 mM. The dynamical model of the arteriole

sustains dilation for longer than the DPDmodel, although the DPDmodel drops to baseline

radius at a slower rate. This can be explained by the simplistic nature of the DPD model.

As the extracellular potassium decays slower than it rises, the DPD arteriole dilates and

reconstricts at roughly the same rate as the extracellular potassium. The more complex

dynamical arteriole radius depends on the ion currents through a capacitive membrane that

carry calcium in and out of the smooth muscle cells. In this model, there are different time

constants for the separate ion channel gating variables and for the intracellular calcium

buffering, all of which affect the dynamics of the dilation response during a change in

perivascular potassium. The difference in the astrocyte response to the DPD and dynamical

vessel models is evident during the period directly after the neural stimulus, when the

extracellular potassium begins to decay to baseline at 30 seconds. In the red curves,

in Figure 5.3A, the astrocytic calcium response is sustained until about 55 seconds and

then drops sharply, while in the blue curve, the calcium begins to fall to baseline around

45 seconds but drops at a slow rate. This causes a slight difference in the extracellular

potassium around 45-60 seconds (Figure 5.3B).

Figure 5.4 shows some sensitivity analysis of the model. We repeat the simulation

above in Figure 5.3 while varying two of the model parameters: µ0, the shear modulus of

the elastin layer, and kshift. For these results, we return to the same parameter sets as in

the myogenic response curves in Figure 5.2A,B (see Tables 5.1(a,b) ) except for the curves

for which µ0 and kshift are varied, as indicated in the figure legend. The top plots show the

astrocyte intracellular calcium; the middle plots show the extracellular potassium in the

perivascular space, and the lower plots show the vessel radius. The thick yellow curves show

the results from the dynamical model. In the left column, we vary the parameter kshift,m

(see Eq. (5.2) above). The blue dotted curve is when kshift,m = 20 mmHg, the same curve

shown in Figure 5.3. The dark blue curve shows the result when we increase the value of

kshift,m to 30 mmHg, and the light blue curve shows the result when we decrease kshift,m

to 10 mmHg. kshift,m determines how much the extracellular potassium will release the

126

0.1

0.2

0.3

0.4

[Ca

2+] A

(µ

M)

2

16

4

6

8

10

12

14

[K+] P

(m

M)

36

600 10 20 30 40 5028

30

32

34

ou

ter

rad

ius

(µm

)

time (sec)

ODE

Kshift,m = 20 mmHg

Kshift,m = 10 mmHg

Kshift,m = 30 mmHg

µ0 = 10

600 10 20 30 40 50time (sec)

Kshift,m = 20

µ0 = 10 kPa

µ0 = 5 kPa

µ0 = 15 kPa

ODE

Figure 5.4: Sensitivity analysis of neurovascular coupling in DPD Top plots – astrocyte intracellular

calcium. Middle plots – perivascular potassium. Lower plots – arteriole radius. Left column – sensitivity of

results when kshift,m is varied from 10 mmHg (dark blue curves) to 20 mmHg (blue dashed curves) to 30

mmHg (light blue curves). Results from dynamical ODE model are shown in yellow curves (same in both

columns.) Right column – sensitivity of results when shear modulus of elastin matrix, µ0, is varied from 5

kPa (light grey curves) to 10 kPa (dashed blue curves; dashed blue curves are the same in both columns)

to 15 kPa (black curves).

smooth muscle constriction, so as the value increases, the vessel will experience a large

dilation during periods of high extracellular K+. The increased dilation also increases

the astrocyte calcium (top left plot) via increased activation of astrocyte endfoot TRPV4

channels.

In the right column, we vary the stiffness of the elastin layer by changing its shear

modulus, µ0. The blue dotted line is the same as the one in the left column: kshift,m = 20

mmHg and µ0 = 10 kPa. We show the results when the elastin stiffness is decreased (µ0 = 5

kPa, grey curves) and increased (µ0 = 15 kPa). As the stiffness increases, the radius (lower

right plot) decreases both at baseline and at the active state. Unlike kshift,m, which only

affects the vessel’s response to non-baseline extracellular potassium, the material stiffness

127

determines how much the vessel will expand in response to pressure, which is present at

baseline. In these simulations, the vessel is held at an internal pressure of 50 mmHg, typical

for cerebral penetrating arterioles, which makes the result very sensitive to mechanical

parameters. The astrocyte calcium level (top right plot) is also slightly more sensitive to

µ0 than kshift,m, due to the large variance in vessel radius when µ0 is altered. There is also

some sensitivity in the extracellular potassium (middle plot). This is because astrocytic

potassium release is partly calcium dependent, as the endfoot BK channels are activated

by intracellular calcium. In addition, when the arteriole dilates and actives the astrocyte

endfoot TRPV4 channels, the influx of positive calcium ions into the astrocyte depolarizes

the membrane, which also affects the endfoot BK and Kir channels, increasing the outflux

of potassium ions from both channels.

5.2 Future Directions for Multiscale and Multiphysics Neu-

rovascular Models

Dynamical DPD model of myogenic arteriole

The present DPD arteriole model is a static model. It does not take into account the true

viscosity of the material or the time dependent mechanical response properties. There

are also time dependent biochemical factors, namely the myogenic response which will be

important to consider. In [38], authors Davis and Sikes demonstrate the rate sensitivity

of the myogenic response in isolated hamster cheek pouch arterioles (∼40 µm internal

diameter at 60 mmHg). Their results show a drastically different response for arterioles

subjected to instantaneous pressure changes compared with gradual increases or decreases

(over several minute). Increasing pressure from 29.4-103 mmHg over 210 seconds produced

a gradual constriction in which the internal diameter decreased monotonically from 85-55%

of its value at 60 mmHg internal pressure (which they refer to as L0, so that typical values

of L0 were ∼40 µm). The same pressure increase applied over 1 second produced a rapid

initial dilation from 82-94% of L0 over approximately 15 seconds, followed by a ∼1 minute

constriction to 47% of L0. This was followed by a secondary relaxation over ∼3 minutes

128

in which the internal diameter increased back to 55% of L0, the same final constriction

level as the vessel subjected to gradual pressure (see Figure 4 in [38]). In the same study,

they found that in response to small instantaneous pressure changes (30-45 mmHg or 30-60

mmHg), constrictions were monophasic and did not show a secondary relaxation, but in all

cases, arteriole response lasted 2-4 minutes before reaching equilibrium constriction. The

same trend was observed in pressure increases starting at 45 or 60 mmHg rather than 30

mmHg (see Figures 5, 6 and 7 in [38]).

In another study, McCarron and coworkers study time dependent myogenic response

in isolated rate cerebral arteries (∼145 µm diameter at 30 mmHg pressure) in [122]. Their

results were similar to [38] except with a slightly longer relaxation time: when increasing

cerebral arteries in a fast step from 30 to 70 mmHg, the vessels initially increased their

diameter by 25% and took roughly 5 minutes to constrict to their equilibrium myogenic

tone.

While the exact nature of the relationship between rate of pressure changes and myo-

genic response dynamics is not clear, both studies demonstrate that a static or equilibrium

model will be inaccurate for short term pressure changes. The time scale of myogenic

response may be on the order of minutes. This will be critical to consider when modeling

downstream effects of vasodilation, which occurs on the timescale of seconds and may easily

last less than 45 seconds depending on the stimulus time. In this case a simulation would

grossly overestimate downstream myogenic effects due to pressure changes from upstream

dilation unless the time dependent characteristics of myogenic response were added to the

model.

Another potentially important time dependent process to consider in neurovascular

coupling is pulsatile flow. Oscillations in flow are present at the arteriole level and are

significantly larger in arterioles than in venules [60]. Table 5.3 shows some measurements

for pulsatile flow in different arteriole types. At these time scales (0.5-1.25 seconds), the

small pressure oscillations (∼10 mmHg in cat pial arterioles) are unlikely to have an effect

on myogenic constriction, but they will be relevant during astrocyte-invoked vasodilation

especially when modeling the behavior of individual red blood cells.

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Table 5.3: Pulsatile flow measurements in various arterioles

Flow metric wavelength (sec) vessel/animal source

Pressure (mmHg)peak trough60 57 1.25 25 µm rabbit omentum arteriole [90]11 5 1.25 119 µm frog length arteriole [119]49 51 0.5 25 µm cat pial arteriole

[172]55 69 0.5 85 µm cat pial arteriole55 65 0.5 195 µm cat pial arteriole55 69 0.5 260 µm cat pial arteriole60 75 0.5 285 µm cat pial arterioleRBC velocity (mm/sec)12 25 0.5 22.5 µm cat mesenteric arteriole [60]

Astrocyte release of neurotransmitters

Astrocytes are known to regulate neural activity through the calcium-dependent release

of glutamate and other neurotransmitters onto synapses [159, 55, 51, 87, 153, 2, 152, 76].

This lead to the notion of the “tripartite synapse” [2] which consists of a pre- and post-

synaptic neuron terminal along with the astrocyte perisynaptic process that wraps around

the neural synapse. Transmitters such as ATP and glutamate are released from vesicles in

the astrocyte membrane [4, 127, 126] similar to the synaptic vesicles from which neurons

release neurotransmitters at a synapse. While the exact roles of these astrocyte-derived

transmitters is not well known, it is likely that they have a wide variety of complex func-

tions. For instance, astrocytic glutamate release modulates both pre- and post-synaptic

terminals and can be inhibitory (suppressing neural spiking) and excitatory (inducing neu-

ral spiking) [152, 2, 159, 55, 51].

There is evidence that astrocyte neurotransmitter release plays a major role in synaptic

plasticity [152, 159, 76, 185, 55]. In a recent experimental study using rat hippocampal

slices, Henneberger et al. [76] observed that astrocytic release of the neurotransmitter D-

serine onto postsynaptic terminals enables long term potentiation (LTP), which is a form

of long-term plasticity in which an increase in the normal synaptic response is sustained

for hours or longer. Short term plasticity in the hippocampus was observed in response

to astrocyte glutamate release [159]. In this case, the increased synaptic strength began

within 20 seconds of astrocytic stimulation and lasted roughly 1 minute [159].

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Astrocytes also modulate the fast dynamics of neural function. A recent study by Lee

et al. [111] demonstrated in vivo that inhibition of astrocytic vesicle release (the main

pathway of glutamate) caused suppression of cortical gamma oscillations. Impaired per-

formance in object recognition was also observed. The study demonstrates the importance

of astrocyte vesicular release on multiple scales: both at the cognitive level, and at the

sub-cellular level, where the release targets specific synaptic terminals.

Modular network modeling

In the example in Section 5.1, above, the entire vessel segment is assumed to be encircled

by a single astrocyte endfoot. Physiologically, a single astrocyte endfoot only covers a

length of approximately 20 µm along an arteriole so that the vessel is encircled by a row

of several endfeet from different astrocytes [101, 135, 173, 130, 178, 120]. The example we

presented essentially assumes the vessel is lined by a uniform community of astrocytes all

reacting to the same inputs. Here we will discuss how this can be improved by developing

the model into a modular network in which short length segments on the vessel are each

connected to distinct astrocytes that also communicate with each other.

Figure 5.5: Immunolabeling image of

cortical perivascular astrocytes from [173]

GFAP immunolabeling of astrocytes in cor-

tex copied from Figure 1A in [173]. GFAP

is an intermediate filament expressed ex-

clusively by astrocytes in brain. Astro-

cytic processes terminating in perivascular

endfeet are indicated by red arrows. Inset

shows astrocyte with two perivascular pro-

cesses. Scale bar is 10 µm in the figure and

40 µm in the inset.

In the cortex, arterioles are essentially sheathed by contiguous but non-overlapping

astrocyte endfeet [101, 135, 173, 130, 178, 120]. While a single astrocyte endfoot covers

roughly 20 or 30 µm of length along a vessel, they often have two or three endfeet contacting

a single vessel [173, 130, 178]. Also, it is typical for cortical arterioles to be in contact

with astrocyte endfeet from opposite sides [173, 150, 101] so that we can approximate

131

that each astrocyte covers a length of roughly 75 µm along the vessel and covers half its

circumference. An example of an immunolabeling image of perivascular astrocytes in the

cortex is shown in Figure 1A from Simard et al. 2003 [173] which we reproduce here in

Figure 5.5. Red arrowheads indicate some of the processes that terminate in perivascular

endfeet, which are identifiable because they are straight, unbranched, and have a much

wider diameter than the other astrocytic processes. There are three microvessels in the

image but it is easiest to see the arteriole at the center which has several thick, straight

astrocyte processes extending to it from both the left and the right. The inset shows a

single astrocyte contacting the vessel with two of its endfeet on the left, while the right

half of the arteriole is covered in endfeet from another astrocyte. The scale bar shows 10

µm in the figure and 40 µm in the inset.

Also important when considering the morphology of cortical astrocytes is that they are

nonoverlapping but contact neighboring astrocytes at the edges [71, 135]. Each cortical

astrocyte occupies a volume of 23000 µm3 and contacts 23,000-35,000 neural synapses

[71]. Figure 2.1 in Chapter 2 outlines the detailed model describing astrocytic response

to synaptic activity in its domain. If the modular model were developed to include the

neurons explicitly, then each astrocyte would respond to the average of all neural glutamate

and potassium across all the synapses in its domain.

Figure 5.6 illustrates the morphology and astrocyte-vessel signaling mechanisms for a

potential modular network model. We start with the connectivity between an astrocyte

and an arteriole, shown in Figure 5.6A. Each astrocyte inhabits a unique, nonoverlapping

domain and responds to the synaptic activity in that region. As an example of a neural

stimulation pattern, astrocyte domains with high neural activity are indicated with yellow

stars and labeled with the synaptic glutamate (glut) and potassium (K+) inputs that

activate the astrocyte. These astrocytes release potassium through their endfeet onto the

arteriole segment they cover (roughly 75 µm along the length, and half way around the

circumference) initiating a dilation by releasing smooth muscle cell contraction in that

segment. The dilation stretches the adjacent astrocyte endfoot, activating Ca2+ influx

through mechanosensitive TRPV4 channels (see details in Chapter 2).

Figure 5.6B demonstrates how this modular structure of two astrocytes and a 75 µm

132

length of vessel can be repeated into a larger arteriole tree. The model is a homogenized

approximation based on the structure of cortical penetrating arterioles which branch off of

the larger pial arterioles. Pial arterioles are located at the surface of the brain, while the

penetrating arterioles branch perpendicularly downward into the brain, creating a layer of

parallel arterioles perpendicular to the brain surface [43, 167]. Based on in vivo mapping

of the rat cortex, the mean distance between neighboring pairs of penetrating arterioles

is 130 µm [145], and about the same distance is seen in the human cerebral cortex [167],

which would allow enough space for two astrocytes between two parallel arterioles, the

same structure proposed by [135] and visible in Figure 5.5. These arterioles have typical

diameters of 30-60 µm and lengths of 1.2-2 mm where they form terminal branches [167].

The first branches occur at 300-500 µm, with the second branch occurring roughly 100 µm

later. Figure 5.6 is a summary of these measurements assuming each astrocyte domain

takes up 75 µm in each direction.

As discussed above, the advantage of a multicellular NVU network model is that it can

account for spatial variations in activity. We will now discuss intercellular communications

among astrocytes and how this can be included in the model to simulate propagation of

signals across the network. Intercellular communication between neighboring astrocytes

produces a rise in intracellular Ca2+ that propagates across the network. In a single

astrocyte, the intracellular calcium rises on a timescale of roughly 5 seconds, with a slow

decay on the order of 15-40 seconds [15, 117]. Typically, calcium waves travel at speeds

of roughly 10-25 µm/s [13, 169, 170, 63, 20]. Calcium waves in networks of cultured

astrocytes have been observed to propagate distances of 200 µm up to 600 µm [63, 20, 13]

and transmission can occur between two astrocytes separated by distances of up to ∼40-80

µm [13, 69]. In retinal slices, astrocytic calcium waves have been reported to travel a radius

of 85 µm [139]. In vivo recordings from the mouse hippocampus showed calcium waves

propagating across 100-150 astrocytes at mean speed 61 µm/s [108].

Figure 5.7 shows the two mechanisms of communication between astrocytes that cause

the propagation of intercellular calcium waves. In the spinal cord and archicortex, the dom-

inant communication pathway between astrocytes is purinergic transmission (signaling via

extracellular ATP release and receptor activation), while gap junctions are predominant for

133

cortical astrocyte networks [115, 16]. There are a few models of purinergic transmission in

astrocytes [15, 16, 117]. Purinergic transmission begins with the neurotransmitter adeno-

sine triphosphate (ATP) binding to P2Y receptors on the astrocyte membrane [15, 16, 41].

ATP can be released by neurons or astrocytes. The P2Y receptors initiate a G-protein

cascade leading to IP3 production inducing the release of Ca2+ from internal stores into

the intracellular space, similar to the astrocyte glutamate receptors (see detailed descrip-

tion and model in Section 1.2). IP3 also leads to release of ATP, which initiates the same

response in neighboring astrocytes [15, 16, 117].

The right panel in Figure 5.7 illustrates the gap-junction transmission pathway by

which calcium waves propagate through astrocytes in the cortex. Although ATP signal-

ing is also present in cortical astrocytes, the predominant mechanism is the transfer of

cytosolic IP3 through gap junctions [115, 16]. When IP3 is transferred from a neighboring

astrocyte through a gap junction, it initiates a rise in intracellular calcium through release

of internal stores. A possible additional mechanism of IP3 production within the cell is

Ca2+-dependent PLCδ-mediated synthesis [16, 66, 82]. The calcium signal is then trans-

ferred to the next neighboring astrocytes via IP3 diffusion out through gap junctions. The

illustration on the right in Figure 5.7 is a summary of the quantitative models presented

in [82, 16, 66, 115]. These models could be integrated into our astrocyte model by adding

a gap-junction term to the existing IP3 dynamical equations.

Randomness and distributed modeling

In computational neuroscience, stochastic modeling is a standard practice that allows mod-

elers to simulate seemingly random background noise and to account for probabilistic be-

havior of synapses. Stochastic models, along with statistical tools, allow researchers to

analyze neural coding and information processing, essentially linking statistical firing pat-

terns in the neural network with specific cognitive functions or sensory inputs.

It is likely that stochastic modeling may be valuable in the study of astrocytes too.

Astrocytes are known to have spontaneous, irregular fluctuations in intracellular calcium,

a behavior referred to as calcium oscillations [155, 156, 136, 141, 28, 39, 66]. Unlike

intercellular calcium waves discussed above, calcium oscillations are confined within a the

134

cell and are not transmitted. At resting conditions, a single spontaneous calcium transient

lasts roughly 10–30 seconds, usually with 1–5 minutes between oscillations although on

some occasions the calcium peaks occur consecutively for several minutes [136, 141].

There is some recent evidence that astrocytes may actually encode information in these

calcium oscillations, as their dynamics are affected by various inputs including synaptic

activity and mechanical stimuli [39, 28, 155, 156, 136, 141]. Some evidence suggests the

existence of both frequency and amplitude modulated encoding of inputs [18, 39, 28]. In a

computational study, Goldberg, de Pitta, and coworkers [66] demonstrated that when as-

trocytic calcium oscillation dynamics exhibit frequency modulated encoding, long-distance

regenerative signaling (e.g. intercellular calcium wave propagation) is supported, but that

the range of intercellular propagation is restricted when calcium oscillations patterns show

amplitude modulation encoding. There is even evidence of plasticity in astrocyte cal-

cium oscillations. Pasti and coworkers [155, 156] observed long term changes in astrocytic

response to glutamate and mGluR agonists in the form of increased Ca2+ oscillation fre-

quency.

Another modeling technique that could benefit future generations of our neurovascular

model is to employ full diffusion-reaction equations in a large-scale, multicellular network.

Diffusion-reaction equations are already used to model cortical spreading depression (CSD)

in a neurovascular context [16, 115, 31]. Cortical spreading depression is a pathological

phenomenon linked to migraine, epileptic seizure, hypoxia and traumatic brain injury. It

is characterized by a self-propagating wave of depolarization in neurons and astrocytes

that travels through the cortex. Large changes in vascular constriction levels also occur.

At any specific location in the cortex, CSD lasts approximately 1 minute [176, 16], and

it travels at 40–50 µm/sec [187, 16]. Unusually high levels of extracellular potassium are

also an important factor in sustaining and propagating CSD. For this reason, diffusing

of extracellular K+ has been a critical part of various models of CSD [16, 115, 31]. The

model developed by Chang and coworkers [31] pays particular attention to the effects

of extracellular potassium on arterioles in CSD, as these elevations have a strong and

immediate impact on vascular tone. In addition to potassium, diffusion of glutamate may

be an important factor in CSD, which most likely propagates via a combined effect of

135

glutamate and K+ diffusion with neuronal and glial buffering through gap junctions [176].

Summary of future modeling directions

We summarize the length and time scales of the mechanisms discussed above in Table 5.4,

below. The importance of each mechanism in the context of a large neurovascular model

is also ranked. The two most critical mechanisms in neurovascular coupling are already

included in our model: arteriole response to astrocyte-derived extracellular potassium, and

astrocytic response to synaptic activity. We rank the dynamics of myogenic response as

having low importance because fast and drastic pressure changes are not typical under

normal physiological conditions. Pulsatile flow is also ranked as having moderate impor-

tance. As outlined in Table 5.3, the mechanical effects on the arterioles are minimal, so

it is unclear how and under what timescales pulsatility affects neurons and astrocytes,

although it is more relevant to fluid dynamics when modeling blood flow in DPD.

Because the astrocyte and neurovascular models we have developed were informed

by experimental data from the cortex and neocortex, we place high importance on gap

junctions, the dominant means of astrocytic communication in the cortex, compared to

purinergic transmission, which is more critical in other brain regions, as discussed above.

The next step in the future directions of our astrocyte model will be to add astrocytic

release of neurotransmitters. The current model focuses heavily on the astrocyte-vascular

interface and mostly ignores the neuroglial interface. Although our model includes astro-

cytic regulation of extracellular potassium near the synaptic space, astrocyte neurotrans-

mitter release is a more significant and more versatile modulator of neural function. Recent

mathematical models of the tripartite synapse [133, 182] provide an explicit description of

the bidirectional interaction between neurons and astrocytes in which the astrocyte mod-

ulates synaptic function via calcium-dependent glutamate release. As our model already

includes the dynamics of synaptically-evoked astrocytic calcium, these tripartite models

can be adapted by adding their equations for Ca2+-dependent astrocytic glutamate release

along with their explicit descriptions for the pre- and post-synaptic neurons.

In order to model neurovascular coupling in DPD using an arteriole tree or just a long

segment of an arteriole, multicellular modular modeling of astrocytes will be critical. This

136

Table 5.4: Summary of neurovascular coupling mechanisms

mechanism/phenomenon time scale(s) length scale(s) importance

dynamic arteriole behavior

dynamic myogenic response ∼3 min [38, 122] low

pulsatile flow 1 sec [90, 119, 172, 60] 1 m low

astrocyte communication

purinergic (single cell) 5 s (rise) 15-40 s (decay) 40-80 µm (extra-

low in cortex[15, 117] cellular) [13, 69]

purinergic (network) 30-60 s [13, 169, 170] 200-600 µm [63, 20, 13]

gap junctions (single cell) ∼10-20 sec [16, 58]high in cortex

gap junctions (network) 10-30 sec [58] 100-350 µm [63, 58]

Cortical spreading depression 1 min (single millimeters [77, 187, 16] moderate

location) [176, 16]

astrocyte-to-neuron signaling

neurotransmitter release 20 sec [159] adjacent cells high

synaptic plasticity 1 min (short-term) [159];

hours (LTP) [76]

astrocyte encoding

calcium oscillations 10-30 sec; 0-5 min moderate

btwn peaks [136, 141]

is another important short-term goal in the future directions of the model. Each astrocyte

only covers roughly 75 µm along the length of a ∼1000 µm arteriole (see Figure 5.6 and

surrounding text, above). To build this modular structure correctly, astrocyte gap junc-

tions should be added to the model, as this is the main pathway of astrocyte-astrocyte

communication in the cortex. Simultaneously, it will be important at this point to add

an explicit neural network that covers the same region as the astrocyte network. Synap-

tic activity is transmitted across neighboring astrocyte domains directly through neural

network connections. Unlike astrocytes, neurons overlap and extend across several astro-

cyte domains, so it will be inaccurate to model synaptic activity isolated within separate

astrocyte domains.

137

A

dilation

Ca2+ Ca2+TRP TRP

K+ K+

high synaptic

activity

K+ ,glut

glutglut

glut, K+

arterioleastrocytes astrocytes

pial arteriole

terminal branches

pe

ne

tra

tin

g a

rte

rio

le

75

µm

60

0 µ

m1

50

µm

ast

roc

yte

s

Bdilation

low synapticactivity

Ca2+ TRP TRP

K+

glut

glut, K+

Ca2+

glut

Figure 5.6: Modular astrocyte model in arteriole tree A Multicellular model of astrocyte-vessel interaction

along a single arteriole segment. Astrocytes occupy nonoverlapping domains in which they respond to local

synaptic activity. Astrocyte domains with high synaptic activity are indicated by yellow stars. Synaptic

glutamate (glut) and potassium (K+) released by active neurons trigger astrocytic release of potassium at

the perivascular region they cover (outlined in blue). The resulting arteriole dilation activates astrocyte

endfoot TRP channels producing a Ca2+ influx into the astrocyte. Rise in intracellular Ca2+ causes

astrocytic release of glutamate (or other neurotransmitters) onto neural synapses, altering synaptic activity

and modulating short- and long-term synaptic plasticity. B Astrocyte network in arteriole tree. Pial

arteriole sits at the surface of the brain and branches into penetrating arterioles that extend downward

into the brain. See text under Modular network modeling for references motivating structure and

measurements in the figure.

138

IP3/Ca2+

IP3/Ca2+

IP3/Ca2+

IP3/Ca2+

ATP

ATP

P2Y

glutamate

IP3

GJ

Ca2+

GJ

IP3

IP3

Ca2+

IP3

Ca2+

GJ

GJ

GJ

IP3

IP3

Ca2+

Purinergic Transmission

(spinal cord, archicortex)

Gap Junctional Transmission

(cortex, neocortex)

Figure 5.7: Models of astrocyte intercellular communication Astrocytes are represented by grey circles.

Left — purinergic transmission is the primary astrocytic communication in the spinal cord and archicortex.

ATP (blue circles) binds to P2Y receptors stimulating IP3 production, causing a release of internal stores

of Ca2+ into the intracellular space. The rise in intracellular Ca2+ triggers the release of ATP which

diffuses through the extracellular space and binds to P2Y receptors on neighboring astrocytes. Right – gap

junctional transmission is the primary means of astrocytic intracellular communication in the cortex and

neocortex. A stimulated astrocyte (top left) receives an input such as glutamate (orange circles) which

stimulates IP3 production, or IP3 from a neighboring astrocyte diffuses into the membrane through a gap

junction (GJ). Intracellular IP3 causes release of internal stores of Ca2+ into the intracellular space. Then

IP3 diffuses to adjacent astrocytes through gap junctions.

Appendix A

Simulation Parameters

140

Parameters used in the model described in Chapters 1 and 2 are given on the following

page in Tables A.1 – A.5. See Sections 1.2.1–1.2.3 and 2.2 for parameter descriptions; for

descriptions of parameters used in the vascular SMC equations, see Section 1.3.1.

Table A.1: Ωs Synaptic Space

Ωs Synaptic Space Description Source

RdcK+,S 0.07 s−1 K+ decay rate in synaptic space estimation

V Rsa 3 volume ratio of synaptic space to astrocyte intracellular space estimate – [46]

[Na+]S 169 mM synaptic space Na+ concentration estimate – [147]

Table A.2: Ωastr Astrocytic Intracellular Space

Ωastr Astrocytic Intracellular Space Description Source

astrocyte perisynaptic process

EKir,proc 26.797 mV Nernst constant for KirAS channels estimate – [149]

gKir,S 144 pS proportionality constant for KirAS conductance [78]

KNai 1 mM intracellular Na+ threshold for Na-K pump estimation

KKoa 16 mM estimation

JNaKmax1.4593 mM/sec maximum Na-K pump rate estimation

JNKCC,max 0.07557 mM/sec NKCC pump rate estimation

δ 0.001235 [46]

KG 8.82 [46]

astrocyte soma

rh 4.8 µM [46]

kdeg 1.25 s−1 [46]

βcyt 0.0244 [46]

Kinh 0.1 µM [17]

kon 2 µM−1s−1 [17]

Jmax 2880 µMs−1 [17]

KI 0.03 µM [17]

Kact 0.17 µM [17]

Vmax 20 µMs−1 [17]

kpump 0.192 µM estimation

141

PL 5.2 µMs−1 [46]

[Ca2+]min 0.1 µM minimum Ca2+ required for EET production [46]

VEET 72 s−1 [46]

kEET 7.1 s−1 [46]

Cast 40 pF astrocyte membrane capacitance [46]

gleak 3.7 pS leak channel conductance estimation

vleak -40 mV leak channel reversal potential estimation

γ 834.3 mVµM−1 ion flux factor [154]

RdcK+,A 0.15 s−1 astrocyte intracellular K+ decay rate estimation

perivascular endfoot

gKir,V 25 pS KirAV conductance factor [78]

EKir,endfoot 31.147 mV Nernst constant for endfoot KirAV channels estimate – [149]

gBK 200 pS BK channel conductance [57]

vBK -80 mV BK channel reversal potential [57]

EETshift 2 mVµM−1 [46]

v4,BK 14.5 mV [46]

v5,BK 8 mV [46]

v6,BK -13 mV [46]

ψn 2.664 s−1 [46]

Ca3,BK 400 nM [46]

Ca4,BK 150 nM [46]

gTRPV 50 pS TRPV4 channel conductance [109]

vTRPV 6 mV TRPV4 channel reversal potential [12]

κ 0.1 fit to [26]

v1,TRPV 120 mV fit to [142, 190]

v2,TRPV 13 mV fit to [142, 190]

ǫ1/2 0.1 fit to [26]

γCai 0.01 µM fit to [190]

γCae 0.2 mM fit to [142]

τTRPV 0.9 s−1 fit to [26, 190]

Table A.3: ΩP Perivascular Space

ΩP Perivascular Space Description Source

142

[Ca2+]P,0 5 µM estimation

V Rpa 0.04 estimate – [46]

V Rps 0.1 estimate – [46]

Rdc 0.2 s−1 K+ decay rate in perivascular space estimate – [46]

[K+]P,0 1 mM minimum K+ concentration in perivascular space estimate – [46]

Table A.4: ΩSMC KirSMC Channels in Vascular Smooth Muscle Cell

ΩSMC Vascular Smooth Muscle Cell Space Description Source

vKIR,1 48.445 mV factor for KirSMC reversal potential estimate – [193, 46]

vKIR,2 116.09 mV minimum KirSMC reversal potential estimate – [193, 46]

gKIR,0 120 pS KirSMC conductance factor estimate – [46]

αKIR 1020 sec parameter for KirSMC opening rate, αk [46]

av1 18 mV parameter for KirSMC opening rate, αk [46]

av2 6.8 mV parameter for KirSMC opening rate, αk [46]

βKIR 26.9 s parameter for KirSMC closing rate, βk [46]

bv1 18 mV parameter for KirSMC closing rate, βk [46]

bv2 0.06 mV parameter for KirSMC closing rate, βk [46]

143

Table A.5: ΩSMC Vascular Smooth Muscle Cell Space

Parameter Source

∆P 60 mmHg [85]v1 -23.265 mV [67]v2 25 mV [67]v4 14.5 mV [67]v5 8 mV [67]v6 -15 mV [67]Ca3 400 nM [67]Ca4 150 nM [67]φn 2.664 [67]vL -70 mV [67]vK -85 mV [67]vCa 80 mV [67]C 19.635 pF [67]gL 63.617 pS [67]gK 314.16 pS [67]gCa 157 pS [67]Kd 1000 nM [67]BT 10000 nM [67]α 4.3987e15 nM C−1 [67]kCa 135.68 s−1 [67]Cam 170 nM [67]q 3 [67]Caref 285 nM [67]kψ 3.3 [67]

σ#y0 2.6e6 dyne cm−2 [67]

σ#0 3e6 dyne cm−2 [67]ψm 0.3 [67]

Parameter Source

νref 0.24 [67]a′ 0.28125 [67]b′ 5 [67]c′ 0.03 [67]d′ 1.3 [67]x′1 1.2 [67]x′2 0.13 [67]x′3 2.2443 [67]x′4 0.71182 [67]x′5 0.8 [67]x′6 0.01 [67]x′7 0.32134 [67]x′8 0.88977 [67]x′9 0.0090463 [67]u′

1 41.76 [67]u′

2 0.047396 [67]u′

3 0.0584 [67]y′0 0.928 [67]y′1 0.639 [67]y′2 0.35 [67]y′3 0.78847 [67]y′4 0.8 [67]x0 188.5 µm [140]a 753.98 µm2 [34]S 40000 µm2 [34]we 0.9 [67]wm 0.7 [67]τ 0.2 dyne cm−1 [67]

Appendix B

Manual for LAMMPS code

145

atom style dpd/full/thick

Computes varying thickness of 2D solid based on local area changes combined with the

assumption of incompressibility (constant volume).

This atom style is required for LAMMPS simulations of thick-walled DPD solids. For

single layer materials, the atom style assumes that the mesh is equaleral triangles. For

multilayer materials, at exactly one layer must be an equilateral triangle mesh. The trian-

gulated layer we can call the matrix layer, and the other layers we can call fiber layers. For

the matrix layer, each triangle is indexed as an angle and the thickness of the triangle is

computed using the angle style area/volume/thick (see below). The thickness of any

atom in the matrix layer is computed as the average thickness of all angles associated with

it. The thickness of a triangluar face ht in the matrix layer is

ht = ht,0At,0/At, (B.1)

where At is the triangle area, and the subscript 0 denotes the unloaded value.

The additional layers (fiber layers) must be attached to the matrix layer using the

improper style fiber/thick (see documentation below), which is where the thicknesses

of these atoms are computed. For details, please see Eqs. (4.52) and (4.53) in Section 4.3,

above.

Specifications for input data file

See read data command in original LAMMPS manual. The structure of the data file

follows that shown in the original LAMMPS manual except for the modifications shown

here:

Atoms section:

•one line per atom

•line syntax as follows

atom-ID molecule-ID type-ID angles-per-atom thickness x y z

146

The keywords have these meanings:

•atom-ID = integer ID of atom

•molecule-ID = integer ID of molecule the atom belongs to

•type-ID = type of atom (1–Ntype)

•angles-per-atom = number of angles (triangular faces) associated with the atom

•thickness = thickness of atom (distance units)

•x,y,z = coordinates of atom

Developer notes

Changes to verlet.cpp

This file needs to be modified from the original LAMMPS version to handle the thickness

computations in parallel. These computations do not work the same as any of the original

pack comm or pack border computations. Instead, the thickness calculation using MPI

needs to be done similar to the force computation, which is done in verlet.cpp. LAMMPS

needs to compute the area of each triangle face in a solid in order to determine the thickness

of that triangular face. Then it assigns a thickness to each atom that is equal to the

average thickness of all the triangles associated with it. (For a fiber-reinforced solid, the

fiber particles are attached to the triangulated matrix layer via fiber style impropers. The

improper style fiber/thick assigns each fiber particle a thickness equal to the weighted

average of each matrix particle associated with it in the improper relationship.)

It is not trivial to compute the (matrix) atom thickness as the average thickness of

its associated angles, not all associated angles are local, and there is no built in DPD

variable that stores the number of non-local DPD angles for each atom. For this reason,

the total (global) number of angles associated with each atom is read in via the initial data

file as described above. Then the angle style area/volume/thick computes the triangle

thickness and adds this value to the variable atom->th comp[i], where i=i1,i2,i3 the atom

index of each of the three vertices of the triangle. Thus, each timestep, the atom thickness

needs to be computed in parallel by summing local and nonlocal angle thickness for each

147

atom (via AtomVecDPDFullThick::un/pack comm/border()) so that the thickness of the

matrix atom i equals atom->th comp[i] divided by global number of angles per atom,

which equals the average thickness of the associated triangles.

However, this method alone only computes the current atom thickness by the end of

each timestep, but the thickness also needs to be used for computations of bond, angle,

dihedral, and improper forces during the timestep. This means that two variables must

exist: one to store the thickness of each atom computed from the previous timestep, and

one to store the sums of local angle thicknesses for each atom (th comp[]). The for-

mer, atom->th[] receives the unpacked values from atom->th comp[] after each timestep.

It is also necessary to clear th comp[] to zero at the beginning of each timestep via

verlet::thickness clear (similar to verlet::force clear())

bond style wlc/pow all visc/thick command

Syntax:

bond style wlc/pow all visc/thick

Examples:

bond style wlc/pow all visc/thick

bond coeff type-ID kBT lmax µ0 m 0 0

Description:

The wlc/pow all visc/thick bond style uses the potential

U =kBT lmax

4p

3x2 − 2x3

1− x−

kp(m−1)lm−1 , for m > 0,m 6= 1

−kp log(l), for m = 1,

(B.2)

148

where the persistence length, p is computed from the shear modulus µ0 as

hjµ0 =

√3kBT

4pjlmaxx0

(

x02(1− x0)3

− 1

4(1− x0)2+

1

4

)

+

√3kp(m+ 1)

4lm+10

, (B.3)

where x = l/lmax; lmax is the maximum spring extension length (equilibrium length is

l0). The following coefficients must be defined for each bond type via bond coeff as in the

example above or in the data file or restart files by read data or read restart commands:

•kBT = Boltzmann constant * temperature

•lmax = maximum spring extension

•µ0 = macroscopic shear modulus

•m = power term

•0 (dummy argument)

•0 (dummy argument)

bond style fiber/diamond/thick command

This bond style assumes a 2- or 4-fiber layer with a diamond grid mesh as described in

Section 4.3.3.

Syntax:

bond style fiber/diamond/thick

Examples:

bond style fiber/diamond/thick

bond coeff type-ID α k1 k2 J h0 αdiag axis

Bond force is given in Eq. (4.47), using the following parameters:

•α fiber angle

•k1 = macroscopic fiber stiffness parameter for small strain

•k2 = macroscopic fiber stiffness parameter for large strain

149

•J = Jacobian (equals 1 for incompressible material)

•h0 = initial (unloaded) thickness

•αdiag = angle of diagonal fibers in same layer

•axis = alignment of axis that defines fiber angles (x=0, y=1, z=2)

angle style area/volume/thick command

Syntax:

angle style area/volume/thick

Examples:

angle style area/volume/thick

angle coeff type-ID 0 1 0 1 ka A0 0 0 h0

Description:

The angle style uses the following potential for surface area constraint, as described above

in the text surrounding Eq. (4.9):

Varea = Σj∈1...Nthjka(Aj −A0)

2

2A0, (B.4)

where Nt is the number of triangles, and ka is the area constraint coefficient. Aj and hj

are the current area and thickness of triangle j, and A0 is the equilibrium value of the

triangle area.

The following coefficients must be defined via angle coeff as in the example above (where

0 and 1 coefficients are dummy arguments) or in the data file or restart files by read data

or read restart commands:

•ka = area constraint coefficient

•A0 = equilibrium triangle area

•h0 = initial (unloaded) triangle thickness

150

improper style fiber/thick command

Syntax:

improper style fiber/thick

Examples:

improper style fiber/thick

improper coeff type-ID I1 I2 f1 f2 0 0

Description:

Improper style binds one fiber atom to three matrix atoms located at the vertices of the

matrix triangle that the fiber atom intersects. This style was written by Zhangli Peng. For

discussion of the improper style and the forces, please see the supporting material for Peng

et al. 2013 [158]. In this version of the improper style, we added the small modification

that allows the calculation of the fiber particle thickness as the weighted average of the

three matrix atom thicknesses it is bound to (For details, please see Eqs. (4.52) and (4.53)

in Section 4.3, above).

The following coefficients must be defined via improper coeff as in the example above

(where the 0 coefficients are dummy arguments) or in the data file or restart files by

read data or read restart commands:

•I1=I2 = strength of improper force

•f1=f2 = friction parameter

Please see the supporting material for Peng et al. 2013 [158] for details on the coeffi-

cients.

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