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Modelling Evolution of Vascular Disease: Aneurysms
Dr Paul Watton
Department of Computer Science & INSIGNEO Institute of in silico Modelling, University of Sheffield, UK
Department of Engineering Science, University of Oxford, UKSchool of Mathematics & Statistics, University of Glasgow, UK
Cerebral aneurysm evolution
Abdominal aortic aneurysm evolution
ROADMAP AHEADInterdisciplinary collaborative research
APPLICATION: Modelling other cardiovascular disease evolutionatherosclerosis, hypertension, in-stent restenosis, heart disease, tissue engineering
Basic ScienceVascular biologists
TranslationClinicians, engineers
Modelling aneurysm evolution
Illustrative example: Modelling framework for cerebral aneurysm evolution
Modelling Aneurysm Evolution
provide insight into (patho)physiology of disease
predictive models which have clinical application:
• AAA - predict growth rates and guide frequency of monitoring
• IA: identify stable aneurysms
Framework for Modelling Aneurysm Evolution
Mechanical environment
of vascular cells
Structural analysis
synthesis degradation
CFD
• CLINICAL CASE
• GROW ANEURYSM
• TEST HYPOTHESES FOR G&R
Aparicio et al (2014), IJNMBE
DOI: 10.1002/cnm.2620
Modelling developments:
endothelial heterogeneity
adventitial adaption (conceptual mathematical model)
Signalling pathways related to collagen regulation.
Influence of SMC apoptosis on aneurysm evolution.
Cerebral Aneurysms
OCCURRENCE: 3-5% of population.
DETECTION: increasingly diagnosed (improved imaging).
RUPTURE: LOW risk <1% per year.
RUPTURE OUTCOME: 30%-40% fatality.
What is the best treatment ?
1. Clip 2. Coil
3. Coil & Stent
4. Blood Flow divertors
Intervention: risky (1-7% morbidity)
expensive
Low rupture risk (0.1%). Do Nothing ?
6 months
MODELLING ANEURYSM EVOLUTION (& more generally vascular disease):
Model healthy arterial wall (Holzapfel et al 2000)
theoretical framework to describe growth and remodelling of tissue.
Aneurysm evolution: enlargement
Abrupt destruction of medial layer:
fragmentation/loss of elastin fibres
adaption of collagen fabric
Aneurysmal artery(cerebral)
Layers of the Arterial Wall
intima (endothelial cells)
media (elastin, collagen, smooth muscle cells)
adventitia (collagen, fibroblasts)
Healthy artery
Natural reference configurations for constituents
CHALLENGES:
For Biological tissues, the unloaded configurations of constituents/cells:
Modelling Evolution of Vascular Disease:
Mathematical framework needed to keep track of the unloaded geometrical reference configurations of tissue constituents.
Our skin grows with us!
may evolve
may be distinct
wavy collagen fibres
Shu Chien (2007)
Key Concept: Mechanotransduction
Modelling Evolution of Vascular Disease:
• Quantify mechanical environment of vascular cells.
• Understand/Model the influence of the mechanical stimuli on cell functionality.
Mechanical forces stimulate vascular cells…
- leading to modulations gene/protein expression and cellular functions
- influencing growth and remodelling (G&R) of tissue.
Mechanobiology
Structural analysis
(Quasi-static: Systolic/ diastolic deformations)
Mechanical environment
of vascular cells
synthesis degradation
Computational
fluid dynamics
ANSYS CFX
Perl wrapping script; Loosely coupled Fluid-Solid-Growth Framework
Aparicio et al (2014), IJNMBE DOI: 10.1002/cnm.2620
Solid Mechanics: Nonlinear elastic membrane
Principal of Virtual Displacements governs equilibrium displacement field.
0int =Π−Π extδδ
Watton et al (2004) A Mathematical Model for the Growth of the Abdominal Aortic Aneurysm, Biomechanics and Modelling in Mechanobiology, 3:98-113
Functional forms for strain energy density functions of arterial tissue required such that growth and remodelling can be simulated.
Solves steady (large) deformations: as tissue adapts (growth and remodelling)
DevelopmentThick-walled model: Schmid, Watton et al (2010)
Thick-walled FSG framework: Grytsan, Watton & Holzapfel (2013)
Volumetric Growth: Eriksson, Watton et al (submitted)
Thick-walled FSVG: Grytsan, Eriksson, Watton, Gasser (in progress)
+ simple conceptual mathematical models (still many open questions!)
Unloaded configuration
recruitment stretch:
onset of load bearing of collagen fibre
Recruitment Stretch (in 1D):
0Ω
Rλ
LRλ
L
Collagen fibers: - crimped in unloaded artery
- recruited to load bearing at physiological pressures
R
C
λ
λλ =
Elastinous and collagenous constituents have different natural reference configurations at which they begin to bear load.
Theoretical formulation (extended to 3D) enables:
- mechanical response of constituents to be defined relative to configuration they begin to bear load
- remodelling scheme for collagen to be implemented.
1
1
R
C
γ
γγ
λ
λλ =
zλ
λ
zλ
recλ1=C
γλ
γ
Rγλ
1=γλ
Cγλ
γλ
UNLOADED
CONFIGURATION
RECRUITMENT
CONFIGURATION
UNLOADED
RECRUITMENT
CONFIGURATION
LOADED
FIELDS OF RECRUITMENT STRETCHES DEFINED THROUGHOUT MEDIA AND ADVENTITIAL LAYERS
),( tRR
Xγγ λλ =
0Ω
R
γΩ
tΩ
C
γF
R
γ
C
γFFF =
R
γF
F
γa
recruitment stretch fields
normalised mass density elastin
( ) ( )( )0,
,,
=≡
t
ttm
E
EE
X
XX
ρ
ρ
( ) ( )( )γγ
γγ
γ
γγ
λ
λλ
aaC
aaC
×
×=
=
:
:2
2
R
R
C
)3( 1 −=Ψ Ikm E
EE
SEF for Elastin (defined relative to unloaded configuration)
( )( )0,
,
==
t
tm
C
C
C
X
X
γ
γγ
ρ
ρ
SEF for Collagenous constituents (defined relative to fibre stretch)
Strain Energy Density Function of Aneurysmal Tissue
COLLAGEN G&R
( )tR ,Xγλ
normalised mass density collagen
ELASTIN DEGRADATION
(Neo-Hookean SEF)
∑Ψ+Ψ=Ψγ
)()( C
γFF
CER
γ
C
γFFF =
)(trtr1 FFCT
I ==
R
T
RR FFC =
(1-2) IMPLY
collagen remodels to maintain its max stretch to
ASSUMPTIONS(1) Fibroblasts configure the collagen to achieve a maximum stretch during the
cardiac cycle – denoted the attachment stretch:
(2) Attachment stretch is constant during aneurysm evolution.
Cells: secrete collagen molecules and collagenase:
Continual turnover of collagen. New fibres attach in a state of stretch (Alberts et al)
C
ATλ
Collagen Remodelling
C
ATλ
−=
∂
∂C
AT
C
ATMAX
CR t
t
t
λ
λλα
λ γγ),(),(
0
XXevolve collagen reference configurations to restore maximum stretch of collagen fibres in the physiological configuration to the attachment stretch. (RATE-BASED APPROACH)
MAX
Ct),(Xγλ Denotes maximum stretch of collagen during cardiac cycle
C
ATλ
synthesis removal
Growth/Atrophy Collagen
ECM synthesis
Collagenase
INCREASE
STRETCH
ECM synthesis
Collagenase
Collagen mass increases
Collagen mass decreases
Fibroblasts reconfigure integrin attachments to matrix
DECREASE
STRETCH
synthesis removal
HomeostasisHomeostasis
ASSUMPTIONS
• Concentration of fibroblasts α concentration of collagen:
Ref. configuration of fibroblasts = Ref. configuration of collagen fabric: CF
γγ λλ =
[ ] [ ] [ ]
<∂
∂⇒<
>∂
∂⇒>
=∂
∂⇒= 0&0&0
t
m
t
m
t
mC
F
ATMAX
F
C
F
ATMAX
F
C
F
ATMAX
F γγ
γγ
γγ λλλλλλ
−=
∂
∂⇒
C
AT
C
ATMAX
C
C
C
E
EEm
t
m γ
γγ β0
Requirements for functional form:
CFmm γγ β0=
HOMEOSTASISINCREASE COLLAGEN
DECREASE COLLAGEN
)( FF
C
hmt
mγγ
γ λ=∂
∂
2/)1( 2 −= λE
Application of Aneurysm Evolution: Clinical Case
Mechanical environment
of vascular cells
Structural analysis
synthesis degradation
CFD
• CLINICAL CASE
• RECONSTRUCT HEALTHY ARTERY
• GROW ANEURYSM
• TEST HYPOTHESES FOR G&R
Alisa Selimovic, Patient-specific modelling of cerebral aneurysm evolution, PhD Thesis, Dept. of Eng.
Science, University of Oxford, Oct 2009-2012.
Clinical case: courtesy of James Byrne, John Radcliffe Hospital, Oxford
Linking Growth and Remodelling to the
Haemodynamic Environment: WSS
Prescribe aneurysm inception:
degrade elastin in localised region to create perturbation to the geometry
Creates altered haemodynamic environment.
Watton et al (2009) ASME Journal of Biomechanical Engineering
Linking elastin degradation to Haemodynamics
)(τDf
τ
( )
≥
<≤−
<−
=
n)degradatio no(2:0
25.0:
n)degradatio of rate maximum(5.0:
τ
ττ
τE
DD
E
DE
maf
ma
dt
dm
75.0;1)(0:
)(
=≤≤
−=
DD
E
DD
E
af
mafdt
dm
τ
τ
Watton et al (2009) Coupling the Haemodynamic Environment to the Evolution of Cerebral
Aneurysms: Computational Framework and NumericalExamples, JBiomech Eng, 131:101003.
WSS== ττ
WSS Elastin Concentration
Elastin Strain Collagen Strain
MODEL
Patient Aneurysm
MODEL: Elastin degradation linked to low wall shear stress.
Collagen remodels to achieve maximum stretch during cardiac cycle.
Collagen growth linked to magnitude of cyclic deformation of cells during cardiac cycle.
Results consistent with clinical observations: tentative support for modelling hypotheses
≥
<
−−
=
h
h
h
h
hD t
tt
f
ττ
τττ
ττττ
0
),(
),(),(
),(
2
X
XX
)0,(),( == tth XX ττ dttT
t
t
TtL
h
L
∫−
= ),(1
),( XX ττ
Heterogeneous Homeostatic
WSS (non-adaptive)
Heterogeneous Homeostatic
WSS (adaptive)
Spatially Heterogeneous and Temporally Adaptive Homeostatic WSS
WSS distribution is spatially heterogeneous in the arterial tree.
Endothelial Cells are in continual state of turnover.
Are endothelial cells preprogrammed with a homeostatic WSS or is this calibrated to local
haemodynamic environment ?
Assumptions: Elastin degradation driven by deviations of WSS from homeostiatic values.
(i) homeostatic WSS is spatially hetergeneous.
(ii) Is temporally adaptive
E
DhD
E
mafdt
dm),( ττ−=
Time Period = 10 years
Definition of Endothelial cell homeostasis has implications for disease evolution.
Guidance is needed from vascular biology.
Heterogeneous Homeostatic
WSS (non-adaptive)
Heterogeneous Homeostatic WSS
(adaptive)
Aparicio P, Mandaltsi A, Boamah J, Chen H, Selimovic A, Bratby M, Uberoi R, Ventikos Y,
Watton PN (2014) Modelling the Influence of Endothelial Heterogeneity on Progression of
Arterial Disease: Application to Abdominal Aortic Aneurysm Evolution, International
Journal for Numerical Methods in Biomedical Engineering’ DOI: 10.1002/cnm.2620
The cyclic deformation is important too!• Cummins et al (2007) Cyclic strain-mediated matrix
metalloproteinase regulation within the vascular endothelium: a
force to be reckoned with. AJPHCPUniaxial cyclic stretch
Healthy artery 1.1
Equi-biaxial cyclic stretch
Collagenous artery 1.02
Cyclic stretch affects functionality and alignment of vascular cells:
• Need to quantify the cyclic deformation mechanical environment
• Link G&R to cyclic deformation stimuli
BIOLOGY
• Fibroblasts sensitive to magnitude of cyclic deformation.
• Greater magnitude of cyclic deformation greater rate of synthesis
QUANTIFY CYCLIC DEFORMATION:
CYCLIC AREAL STRETCH CYCLIC STRETCH
diastole21
systole21
aa
aa
×
×=CS
A
DIAS
C
SYS
C
CS
γ
γ
γλ
λλ =
( )C
ATMAX
CCCSCS
C
EEmAt
m−=
∂
∂γγγγ
γ λξ ),(
( )
−
−=
==
0,1maxexp0,1maxexp,0;0
0 CS
t
CS
CS
CS
t
CSCS
A
CSCS
A
AA
γ
γγγγ
λ
λξξξλξ
Linking Collagen Growth to Cyclic Deformation
Growth linked to cyclic areal stretch
If G&R is not linked to cyclic deformation – Neck region can develops large cyclic deformation.
MODEL: Linking G&R to cyclic deformation contracts neck region and helps to promote aneurysms with a well-defined necks
Cyclic stretch of aneurysm 2-4% - consistent with clinical studies
Hypothesis: Collagen growth linked to cyclic deformation – a mechanism for neck formation ?
Cyclic Deformations
0=BSIχ
Biaxial Stretch Index (BSI)
MOTIVATION: Cell alignment and functionality is influenced by cyclic deformation.
cylindrical artery – cyclic deformation is 1-dimensional
idealised aneurysm (sphere) – cyclic deformation is equi-biaxial.
How does the cyclic deformation environment evolve and how can this be characterised ?
Watton et al (2011) PROPOSE NEW INDEX: Biaxial Stretch Index (BSI)
Watton et al (2011) Modelling evolution and the evolving mechanical environment of
saccular cerebral aneurysms, Biomechanics and Modeling in Mechanobiology, 10:109-132.
1=BSIχ
cyclic deformation is 1-dimensional
cyclic deformation is equi-biaxial.
( )( )CSCS
CSCSSIB
21
21
,max
,min
εε
εεχ =
1−= CSCS
αα λε magnitude of cyclic (linearised) strain during cardiac cycle
(defined in direction of principle stretches)
Cyclic Areal Stretch Biaxial Stretch Index (Watton, 2011)
cyclic deformation
is 1-dimensional
cyclic deformation
is equi-biaxial.
Watton et al (2011) Modelling evolution and the evolving mechanical environment of
saccular cerebral aneurysms, Biomechanics and Modeling in Mechanobiology, 10:109-132.
Adaption of the functional role of the adventitia.
[4]
[3]
Abrupt destruction of medial layer:
fragmentation/loss of elastin fibres
adaption of collagen fabric
Aneurysmal artery(cerebral)
Healthy young artery: adventitia acts as protective sheath to prevent overdistension
Cerebral aneurysm: Adventitia may become primary load bearer – How ?
Change in definition of homeostasis for collagen fabric
Haoyu Chen(Oxford 2010-14)
Simple mathematical models can guide understanding...
REALITY: distributions of attachment and recruitment stretches
Observation of collagen waviness distribution in the unloaded configuration
Schrauwen, J. T. C.,
Stergiopulos, N., Vosse, F. N.
van de, Bovendeerd, P. H. M.,
Rezakhaniha, R., & Vilanova,
A. (2012). A method for the
quantification of the
pressure dependent 3D
collagen configuration in the
arterial adventitia. Journal of
Structural Biology.
Distribution of collagen attachment stretches in the physiological configuration
Distribution of collagen recruitment stretches
Need for a novel mathematical model…
Distribution of (3 parameters)
Distribution of medial collagen fibre attachment and recruitment stretches
Assumption: medial collagen bears load in physiological configuration.
EQUIVALENTLY: Distribution of
Physiological stretch
∙
∙
∙
stretched crimped
! 1
MODELLING: Use triangular distribution function: 2 additional parameters
Distribution of #
Application: adventitial collagen fibres distributions $ %&'& at ( )
Assumption: adventitial collagen bears no load at physiological stage (* 0). # , 1
Conceptual Model of aneurysm evolution
Watton, P. N., Ventikos, Y., &
Holzapfel, G. A. (2009). Modelling
the growth and stabilization of
cerebral aneurysms. Mathematical
medicine and biology : a journal of
the IMA, 26(2), 133-164.
(i) 1D two-layer non-linear cylindrical membrane subjected to blood pressure and axial stretch
(ii) Media: elastin + collagen, Adventitia: collagen
(iii) Adventitia layer starts load bearing at systolic configuration
0 -./
.0
* -12 12
3 451-
66 7897: 9 4
1-67:#6
Model overview
Mechanical governing equation
6786 ;8<8 1 = 1
1>?
Modelling elastinous constituents (elastin + passive SMC) as Neo-hookean solid
Need to integrate NOVEL collagen model!
Collagen strain energy function (SEF)
7:@∗ :@ <B@2 :@ = 1 >
Distribution of @
Where: D=@, E=@ and F @.
G @ H@
@
Recruited Crimped
G @
0 1 I @2 @ = DF = D E = D D , @ , E2 F = @F = D F = E E I @ , F
0 @ ! F7:@ ;:@ J 7:@∗
@ G @ H@
K
L
Mechanical response of a single fibre
Hill, M. R., Duan, X., Gibson, G. A., Watkins, S., & Robertson, A. M.
(2012). A theoretical and non-destructive experimental approach for
direct inclusion of measured collagen orientation and recruitment
into mechanical models of the artery wall. Journal of Biomechanics.
SEF of a fibre
SEF of all fibre
Stress-stretch response of the collagen constituents
1st P-K stress-stretch response
67:@6 ;:@
H M 7:@∗ @ G @ H@K
LH
A nonlinear mechanical response of collagen can be obtained by assuming a linear response
for each collagen fibre and gradual recruitment of the fibres. (Piecewise analytic function)
67:@6
MODEL: Destruction of Media and growth of adventitia
ASSUMPTION
Prescribe degradation of medial elastinous constituents (elastin/SMC) and medial collagen
;8 ;: NOP0
Medial layer degraded during aneurysm evolution.
Watton, P. N., Ventikos, Y., & Holzapfel, G. A. (2009). Modelling the growth and stabilization of
cerebral aneurysms. Mathematical medicine and biology : a journal of the IMA, 26(2), 133-164.
6;:#6* ;:#
:# = ##
Adventitial collagen growth: simple stretch-based evolution law
Collagen remodelling
GQ@RNew formulation: evolve collagen recruitment stretch distribution GQ@R to restore the
collagen stretch distribution in LOADED configuration towards attachment stretch distribution.
H@H* S ∙ :@
= @Q*R@Q*R
H@H* S ∙ :@
= @Q*R@Q*R
MEDIAL COLLAGEN: Assume attachment stretch distribution fixed during evolution
Q*)=Q*=0) Q*)= Q*=0)
@ * T * , @Q*R U * , @Q*R V *
#Q*R= #Q* 0R
Q*RMODEL: Attachment stretch distribution can adapt but
functional form (triangular distribution) does not change.
MODEL: Evolve recruitment stretch distributions, e.g
Collagen remodelling: adaption of adventitial collagen attachment distribution $Q%&'WRPhysiological motivation
- Change of functional role of the adventitia from a protective sheath in normal
physiological condition to the main load bearer in the aneurysmal condition
H#H* S ∙ :#
= #Q*R#Q*R
H#H* S ∙ :#
= #Q*R#Q*R
Prescribe evolution of attach
stretch distribution
ADVENTITIAL COLLAGEN: Assume attachment stretch distribution adapts during
evolution.
# * T * , #Q*R U * ,
#Q*R V *(NEED TO LINK TO BIOLOGY)
Results illustration: mass density & stretches
Adventitial collagen growthMedial fibre degradation
:# #Media degradation prescribed, adventitial collagen evolves, system expands by a factor of
about 2.5, :# stabilises at the values of #.
Expansion of the arterial radius Evolving maximum and minimum adventitial
collagen stretch
Results illustration: remodelled thickness & thickness/radius ratio
Thinning of the arterial
wall during cerebral
aneurysm evolution
Model with adaption of attachment stretch distributions shows more realistic:
collagen growth, volumetric growth, evolution thickness/radius ratio.
Thickness/radius ratio (new vs old model)
The old model assumes single and
constant values of .
How the adventitia adapts may influence whether an aneurysm stabilises or ruptures:
Guidance from experiments to improve modelling.
Summary and Outlook
Integrated Fluid-Structure-Growth (FSG) computational framework for vasculature adaption:– Physiological geometries
– Model of the arterial wall
– (membrane (F77), thick-wall (FEAP/CMISS), volumetric growth (FEAP))
– G&R linked to cyclic deformation and steady haemodynamics.
Development (ongoing):– Improved representation of fibroblast mechanobiology
– Coupling framework with agent-based models of vascular cells.
– Improved models for adaption of collagen fabric.– (fibril distribution function and fibre reorientation/dispersion remodelling)
Development (outlook/collaborations):– Application to animal models (guidance/validation)
– Experimental guidance of how mechanical stimuli influence functionality
of vascular cells
– Modelling framework a basis for modelling range of vascular diseases.
– Application to vascular construct design.
– Translation (clinicially useful predictive models of aneurysm evolution)
IA: Stable/enlarging; AAA: rate of enlargement (frequency of monitoring)
Theoretical Mechanobiology Group
Hamna
Afaq
www.themebio.org
Chen, Mandaltsi, Boamah, Hornsby, Aparicio, Dickinson, Chan, Afaq
Acknowledgements:
Nick Hill (Glasgow), Yiannis Ventikos (UCL),
Gerhard Holzapfel (Graz), Anne Robertson (Pittsburgh),
www.insigneo.org
INSIGNEO Institute of in silico Medicine
Over 100 academics from engineering, medical physics , computer
science and hospital
Largest institute of its type Europe, Established May 2013
Open questions: How to couple with signalling
pathway models…
Modelling of Signalling Pathways
in Vascular Mechanobiology
[4]
[3]
Pedro AparicioOct 2013-2016
Need for collaborations with vascular biologists and experimental arterial
mechanobiology
Flowchart
update
arterial
layer
radii
update
constitutive
model
update
inner
radius
update
thrombus
radius
P
MECHANICAL EQUILIBRIUM OF MEMBRANE
λzL
λR X Y Z[\]]^_`, Z\b(^_`
L: Media, Adventitia
J: Elastin, Collagen, SMCs
OXYGEN MASS TRANSPORT
O2 diffusion throughL = T, M, A
TROMBUS
LUMEN
WALL
cW Y dW, eWfeWf Y dW, cW
GROWTH AND REMODELLING
ghiWg( Y jiW ,hiW , %iW , %iW&'
g%iWkg( Y liW , %iW , %iW&'
THROMBUS PROPAGATION
WALL
Shear regulated
thrombus
propagation
m' nof
update
O2 regulated
growth rates
jiW YQcWR
( )( ) 073.02/12
=−= C
AT
C
ATE λ
073.0
30,30 −+=Mβ
60,60 −+=Aβ
Adventitial collagen strain
Medial collagen strain
Normalised time
Pressure (Pa)
073.0
Azimuthal elastin strain
Healthy Artery Model
Material, geometric & remodelling
parameters
( )
( )
+
−+=
=
∑≥±== 0:;,
2
22
42
cosexp11
1
)(
C
pJp
ppp
EpAMJR
J
J
C
J
CC
J
CC
JJ
z
E
AA
E
MM
z
EaEakhkHkH
R
pp
λ
γ
λλλ
λ
40 expCollagen
4/ (A)Collagen
KPa2 (M)Collagen
10/ (A)Elastin
KPa135 (M)Elastin
=
=
=
=
=
C
C
M
C
A
C
M
E
M
E
A
E
M
a
kk
k
kk
k
3
years1.0
years 8.0
1-
0
-1
=
=
=
CS
Aβ
β
α
oo
oo
60,60, (A) angles Fibre
30,30, (M) angles Fibre
3/ thickness(A)Adventitia
3/2 thicknessMedia(M)
5/ thickness wallUnloaded
1 radius Unloaded
25.1 (systole)stretch ntialCircumfere
3.1 stretch Axial
0
−=
−=
=
=
=
=
=
=
−+
−+
AA
MM
A
M
z
HH
HH
RH
mmR
γγ
γγ
λ
λ
Pressure-stretch relationship for cylindrical membrane model of artery (derived from variational equation).
GEOMETRIC
MATERIAL
Remodelling
Radius (mm)
Pressure (kPa)
http://www.scsitaly.com/products/AIMA
@neurist projectSemi-automated simulation pipeline: Segmentation
to CFD
Boundary conditions: 1D circulation model
included in AIMA
by
AIMA: a software features a medical imaging pipeline to create computational models of AIMA: a software features a medical imaging pipeline to create computational models of
the haemodynamics of cerebral aneurysms. Integrated with ANSYS ICEM and CFX
The importance of aneurysm research….
2014: Current biomechanical model (UK) used by clinicians for predicting risk of abdominal aortic aneurysm rupture
Need: Improved biomechanical models to aid clinical decisions!!!
economic savings and health benefits
Intervention may be recommended
No intevention
Decision on whether to operate based on aneurysm diameter
Inference of arterial wall thickness during aneurysm evolution
Remodelled wall thickness evolution
Remodelled wall thickness/evolving radius ratio
p8 , p: dryweightmassfractions of fibres in the media
Vtuv * w *w * 0
xyz|~y~∙
4
* ~yzy
zzy|z~
* =ℎtuv *
*=ℎtuv *
- *=
w *
w5 0 + w 0
4
-
1
* >
w *
w * = 0=
p8;8* + p:;:
* w5(0) + ;:#w 0
w5 0 + w 0
Assumptions
1. Constant mass fraction of water in the arterial wall.
2. Volumetric growth of the wall is proportional to the mass growth of the constituents.
Wall thickness evolution is approximated by assuming that the volumetric growth is a
function of the mass growth /atrophy of the constituents.
Is the attachment stretch constant ?
Lzλ
R0λ
Lzλ
Rt)(λ
t
0Ω tΩ
Rr 0λ= Rtr )(λ=
tΩt
0Ω
[ ] [ ] ???0 ⇒=∧→ C
AT
CEm λλ
( ) ( )( )CCCEE
z
mmH
Rp λσλσ
λλλˆˆ
1+
=
( ) ( )( )CCCEE mmH
Rp λσλσ
λλˆˆ
22
+
=
Given functional forms for stress of elastin and collagen.
Assume elastin degrades and collagen adapts to achieve
new homeostasis.
What are the implications of:
(i) Constant attachment stretch
(ii) Collagen stress-strain function independent of time (fibril undulation distribution constant))
Watton PN, Ventikos Y, Holzafpel GA (2009) Modelling the growth and stabilisation of cerebral aneurysms, Mathematical Medicine and Biology, 26:133-164.
( ) ( )( ) )1(ˆˆ1 CCCEE
z
mmH
Rp λσλσ
λλλ+
=
( )( ) ( )C
AT
CE
ECEP
λσλσ
λσ
ˆˆ
ˆ
0
0:
0+
=
( )Z
CCEE Hthhmmth
λλϕϕ =+≈ )(:)(
~
constant1
~
0
0
:
=
−=
r
h
Pr
h
CE
Cϕ
fractions mass dry weight, =CE ϕϕ
Material equilibrium [ ]C
AT
Ct λλ =)(
Volumetric growth (assuming mass fraction water is constant)
( ) )0()( =+= tVmmtV CCEE ϕϕ
Mechanical equilibrium (cylinder)
Wall thickness evolution (loaded configuration)
[ ] [ ] [ ]
−
→⇒=∧→∧
CE
CC
AT
CE
Pmtm
:
0
2
0 1
1)(0(1)
λ
λλλ
Proportion of load borne by elastinous constituents at t=0.
Analytic expression for ratio of remodelled thickness to radius (loaded configuration)
Theoretical prediction of (remodelled wall thickness/radius) ratio
as aneurysm enlarges:(Independent of G&R approach and functional form of stress functions)
LzλRt)(λ
( ) ( )( )CCCEE
z
mmH
Rp λσλσ
λλλˆˆ
1+
=
( )
−=
42
2 11ˆ
λλλλσ
z
E
Ek ( ) ( )
5
2ˆ
=
C
AT
C
C
CCC
E
Ekλλσ
5/01.0)( tEtm =
8.0:
0 =CEP 5.05.0 == CE ϕϕ
KPa165/1/1.1/3.1
1.13.13.1
0
0
===
===
pRHR
C
ATz
λ
λλλ
( )C
AT
CCC
EEmdt
dm−= β
( )C
AT
CR
EEdt
d−= α
λ
0
0
:123.0
~
r
h
Pr
h
CE
C
−=→
ϕ
G&R
Parameters
Verification of theory with
numerical model
constant attachment
stretch implies unrealistic
increases in collagen mass
Conclusion: attachment
stretch not constant.
- Assume:
(i) Distribution of attachment stretches
(ii) Distribution of attachment stretches can adapt
Novel Model for microstructural remodelling of collagen
- Explore consequences using conceptual model of cerebral aneurysm evolution.
Inception: Growth and Remodelling
RadiusElastin ConcentrationCollagen strain
(media, postively wound)
WSS Elastin E22 Collagen concentration
Growth and Remodelling (G&R)
Growth/Atrophy
Remodelling
Changes in mass of the constituents
Reorganization of the constituents (no change in mass), e.g. Constituent reference configurations, orientations.
synthesis degradation
Cells continuously produce ECM and enzymes to degrade the ECM.
Homeostasis
G&R approaches: Integral based (Humphrey and co-workers) and rate based
(Humphrey).