Hemodynamic Model of the
Cardiovascular System during
Valsalva Maneuver and
Orthostatic Changes
Niklas Moberg
November 23, 2011
Master’s Thesis in Engineering Physics, 30 creditsSupervisors at Med-Uni Wien: Univ.-Prof. Dr. techn. H. Schima, Dipl.-Ing.
Dr. techn F. MoscatoSupervisors at TU Wien: Univ.-Prof. Dr. techn. A. Kugi, Dr.-Ing. W.
Kemmetmüller, K. SpeicherExaminer: Urban Wiklund
Umeå University
Department of Engineering Physics
SE-901 87 UMEÅSWEDEN
Abstract
The goal of the Master’s Thesis was to extend an existing cardiovascular model to includethe mechanics of the lungs, thus allowing to simulate breathing maneuvers such as theValsalva maneuver and the Forced Vital Capacity maneuver. This included a remodeling ofthe pulmonary capillaries and of the existing interactions of the model with the intrathoracicpressure. The existing description of the vascular compartments was found to be insufficientto describe the hemodynamic response to orthostatic changes and was extended to includea compartment representing the upper body. Stress relaxation was included into all thelarger vascular compartments. The results showed an improved accuracy of the extendedmodel when subjected to large intrathoracic pressure changes and during orthostatic stress.The internal responses of the newly modeled pulmonary capillaries were studied and verifiedagainst literature with satisfying results.
Sammanfattning
Målet med detta examensarbete var att utvidga en existerande kardiovaskulär modell till attinkludera lungmekanik, för att därigenom möjliggöra simulering av andningsmanövrar somValsalva-manövern och Forced Vital Capacity-manövern. Detta inkluderade en omformningav de pulmonära kapillärerna samt hur modellen som helhet påverkades av det intratho-rakala trycket. Den existerande vaskulära modellen ansågs vara otillräcklig för beskriva dehemodynamiska responsen under ortostatisk stress och utökades därför till att inkludera ettfack som representerade överkroppen. Stress relaxation inkluderas in i all större vaskulärakärlrum. Resultatet visade på en förbättrad noggrannhet hos den utökade modellen vidstörre intrathorakala tryckändringar och även vid ortostatisk stress. De interna svaren hosden utökade pulmonära kapillära modellen studerades och verifierades gentemot litteraturmed positiva resultat.
Contents
1 Problem Description 2
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 The Human Cardiovascular System 4
2.1 Basic Structure of the Cardiovascular System . . . . . . . . . . . . . . . . . . 4
2.2 Physiology of the Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 The Circulatory System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Arterioles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.4 Venules and Veins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.5 Systemic Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.6 Pulmonary Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Regulatory Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Arterial Baroreflex Controller . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Pathology of the Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1 Hypertension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.2 Systolic Dysfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.3 Cardiac Arrhythmia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Mechanical Circulatory Assist Devices . . . . . . . . . . . . . . . . . . . . . . 16
2.6.1 Pulsatile Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.2 Continuous Blood Pumps . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Model Theory 19
3.1 Conservation of Mass/Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Modeling of Blood Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Heart Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Active and Passive Pressure Functions . . . . . . . . . . . . . . . . . . 22
i
CONTENTS ii
3.3.2 Contractility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 Atria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.4 Ventricles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.5 Intraventricular Septum . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.6 Heart Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Baroreflex Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Systemic Arterial Resistance Controller . . . . . . . . . . . . . . . . . 29
3.4.2 Heart Rate Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.3 Unstressed Volume Controller . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Systemic Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 Arterial Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.2 Venous Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Lungs and Airways Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Pulmonary Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7.1 Pulmonary Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7.2 Pulmonary Capillary Model . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7.3 Pulmonary Veins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 LVAD Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Valsalva Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.10 FVC Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Matlab Simulink Implementation 55
4.1 Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Matlab-code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 System Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Results and Conclusions 57
5.1 Normal Physiological Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1 Breathing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Left Heart Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Orthostatic Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Valsava Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.1 Pathological Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 FVC-Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Review 73
6.1 Achieved Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7 Acknowledgements 76
CONTENTS iii
A Parameters and Variables 80
B Full Model Schematic 81
Introduction
This Master’s Thesis is divided into six main parts: a problem description, a basic intro-duction to the human cardiovascular system, the theory behind the model development, themodel implementation, the results achieved and a discussion reviewing the results.
The first chapter describes the aims of this Master’s Thesis and the methods used toaccomplish them. The second chapter describes the working physiology of the human heart,the vessels, the body’s own control mechanisms and gives some basic information on thepathology of the heart. The third chapter describes the equations ruling the physical system,how they were derived and their physiological meaning. The fourth chapter explains howthe model was implemented in Matlab-Simulink and the basic function calls within the code.The simulated results of the model are then presented and compared to experimental resultsfrom articles and books. The thesis then reviews the results including limitations, futurework and includes a summary of the whole project.
The main goal of this work is to build a comprehensive mathematical model that givesan accurate representation of the flows, pressures and blood volumes in the human car-diovascular system and its interaction with a Left Ventricular Assistant Device (LVAD).This model shall include the cardiopulmonary interaction with the lung mechanics and theresponse to orthostatic position changes. It is important that the hemodynamic responsesof orthostatic position changes and breathing maneuvers, such as the Valsalva maneuver,are modeled correctly. This since the hemodynamic responses can be used to diagnose leftheart failure in patients and also test reflex controller reactions.
The Master’s Thesis is based on the Master’s Thesis done by Michael Baumann incooperation with the Research Group in Cardiovascular Dynamics and Artificial Organsat the Medical University of Vienna situated at the General Hospital of Vienna and theAutomation and Control Institute (ACIN) at Vienna University of Technology [1].
1
Chapter 1
Problem Description
1.1 Problem Statement
As mentioned in the introduction, this is a continuation from another Master’s Thesis doneby Michael Baumann [1]. He modified and extended an existing mathematical model of thehuman cardiovascular system and its interactions with a cardiac support system developedby the Medical University of Vienna [30]. The goal of this project is to extend the modelmaking it a more accurate representation of the human cardiovascular system.
1.2 Goals
I. Extend the existing mathematical model, found in [1], by including the interactions ofthe lungs and pulmonary capillaries.
II. Evaluate the new interactions by simulating changes in lung volume and pressure, andverifying the results against literature.
III. Extend the existing systemic venous model to better reproduce the hemodynamicresponses associated with orthostatic stress.
1.3 Purpose
The purpose of including the cardiopulmonary interactions of the lungs is to be able toaccurately simulate the influence of breathing maneuvers such as the Valsalva maneuver.Breathing maneuvers are non-invasive to the patients, the expected cardiovascular responseis known and can therefore be used for diagnosis.
1.4 Methods
The Master’s Thesis starts with an introductory study of the human cardiovascular systemand heart mechanics. This is needed to get a basic understanding of the hemodynamics ofthe body. The study is followed by a review of the equations describing the mathematicalcardiovascular model and their physiological meaning as described in Baumann [1].
2
1.4. Methods 3
In order to extend the existing model to include the interactions of the lungs, a math-ematical description needs to be created. This is done by searching for already existingmodels describing the cardiopulmonary interactions of the lungs and choosing a suitableone to implement. When a model is found, an in-depth study needs to be made to de-termine the important parts of the model, which simplifications can be made and how toinclude the new cardiopulmonary model into the entire cardiovascular system model.
The Master’s Thesis then continues with extending the existing systemic venous circu-lation making it a more physiologically meaningful model. Again a literature review and anin-depth study needs to be made for existing models to find a suitable model for implemen-tation.
Chapter 2
The Human Cardiovascular
System
This chapter describes the basic structure of the cardiovascular system, the mechanics ofthe heart and common pathologies involving left heart failure.
2.1 Basic Structure of the Cardiovascular System
The human cardiovascular system is basically a complex array of valves and tubes repre-senting the blood vessels. The heart is the pump that keeps the blood flowing through thesystem and the nervous system is the computer controlling the system. The heart can befound almost in the middle of the chest. There are blood vessels directly connected to theheart in which blood flows from the heart out into the rest of the body and through whichblood returns back into the heart.
Figure 2.1 shows a schematic of the human cardiovascular system. The figure showshow oxygenated blood flows from the left heart through arteries, capillaries, venules, veins,through the upper and lower vena cava into the right heart. This systemic circulation sup-plies the organs in the body with minerals and oxygen as is represented by the transition fromoxygenated blood to oxygen-deprived blood in the systemic capillaries. Oxygen-deprivedblood then flows from the right heart through the pulmonary circulation, also containingarteries, capillaries, venules and veins, and returns to the left heart. It is in the pulmonarycirculation, more specifically in the capillaries, that blood is supplied with oxygen beforereturning to the left heart [13].
2.2 Physiology of the Heart
The heart can be divided into four parts: left and right ventricle and left and right atrium.The left and right ventricle are the two larger pumps, mentioned in the last section, thatkeep the blood flowing through the body and the lungs. The left and right atrium can beseen as two primer pumps that fill the ventricles with blood. The right and the left ventriclework basically in the same way, just under different pressures. The left heart pumps theblood out into the high pressure systemic arterial system and therefore needs to be able tocreate pressures higher than 100 mmHg. The right heart pumps the blood out into the lowpressure pulmonary arterial system and just needs to create pressures up to 20 mmHg [13].
4
2.2. Physiology of the Heart 5
Systemicvessels Arteries–13%
Arteriolesandcapillaries–7%
Heart–7%
Aorta
Pulmonary circulation–9%
Veins, venules,and venous
sinuses–64%
Inferiorvena cava
Superiorvena cava
Figure 2.1: Schematic of the human cardiovascular system with the direction of blood flowgoing from the left heart (red) to the right heart (blue). Also marked is the percentage ofblood volume stored in each part of the circulation [13].
Aorta
Pulmonary artery
Inferiorvena cava
Superiorvena cava
Right ventricle
Tricuspidvalve
Pulmonaryvalve
Right atriumPulmonaryvein
Left atrium
Mitral valve
Aortic valve
Leftventricle
Lungs
HEAD AND UPPER EXTREMITY
TRUNK AND LOWER EXTREMITY
Figure 2.2: Cross section of the heart and their respective inlets and outlets [13].
2.2. Physiology of the Heart 6
Each ventricle can be thought of as an elastic chamber that allows for filling and emptyingof the chamber through two one-way valves, one for input and one for output. Figure 2.2shows a cross section of the heart detailing the different chambers, the heart valves and theblood vessels directly connected to the heart [13].
The heart’s pumping of the blood can be divided into two phases, a filling phase calleddiastole where blood from the atrium fills the ventricle and a contraction phase called systolewhere the ventricle contracts and pumps the blood out into the vessels.
The amount of blood being pumped out into the body during a time unit is called thecardiac output qco. The cardiac output is determined by multiplying the heart rate fhr withthe stroke volume Vsv. The heart rate is the number of beats the heart makes in one minute.The stroke volume is the amount of blood ejected by the ventricle during one contraction.An average person has normally a heart rate of 70 min−1 and a stroke volume of 70 mlwhich results in a cardiac output (qco = fhrVsv) of 4.9 l/min [17].
Pressure-Volume (PV) Diagram
A important tool for studying the interactions of the heart and the circulatory system isthe so called pressure-volume (PV) diagram. To better understand what the PV diagramis, first take a look at the time dependent pressures for one heart cycle of the left ventricle,which are depicted in figure 2.3. The PV diagram is created by plotting the pressure as afunction of the volume, seen in figure 2.4.
120
100
Pre
ssu
re (
mm
Hg
)
80
60
40
20
0130
90
Vo
lum
e (
ml)
50
R
P
a c v
Ventricular volume
Ventricular pressure
Atrial pressure
Aortic pressure
A-V valveopens
A-V valvecloses
Aortic valvecloses
Aorticvalveopens
Isovolumiccontraction
Ejection
Isovolumicrelaxation
Rapid inflow
DiastasisAtrial systole
Figure 2.3: Time-dependent pressure and volume of the left ventricle [13].
The following explanation of the phases describes the working of the left ventricle cor-responding to the illustrations shown in figures 2.3 and 2.4. However, it is important toremember that the right ventricle and the atria work in the same manner, the differencesbeing that the pressures are lower and the names of the inlet/outlet valves of the heartchambers change.
The points A, B, C and D represent important phases of the heart cycle. Point A infigure 2.4 indicates the so called end-diastolic-volume (EDV) and the end-diastolic-pressure(EDP). A is the point in time when the diastole ends and the systole begins. During thefollowing period of the heart cycle, represented by moving from point A to B, the mitralvalve (i.e. the left ventricle inlet valve) closes and the ventricle rapidly builds up pressurealmost without changing its volume. This is called an isovolumic contraction (II).
2.2. Physiology of the Heart 7
25000
Left ventricular volume (ml)
Intr
aven
tric
ula
r p
ressu
re (
mm
Hg
)250
200
150
100
50
50 100 150 200
300
Isovolumicrelaxation
Isovolumiccontraction
Systolic pressure
EW
III
IV
I
II Diastolicpressure
Period of ejection
Period of filling
A
C
D
B
ESPVR
EDPVR
Figure 2.4: PV diagram of the left ventricle [13].
The point B is reached when the ventricular pressure exceeds the aortic blood pressurecausing the opening of the aortic valve (i.e. the left ventricle outlet valve) and the ventricleejects the blood into the blood stream. This phase is the so called ejection phase (III) andis represented by the segment B −C in the PV diagram. When the point C is reached, theaortic valve closes and the ejection phase is over.
At point C the diastole begins with an isovolumic relaxation (IV) of the ventricle rep-resented by the segment C −D. As the ventricular pressure falls below the pressure in theatrium the mitral valve opens and the ventricle starts to fill (I) along the curve segmentD − A. When filling is complete, the PV diagram is back to its starting point A making aPV-loop. The stroke volume Vsv can easily be obtained from figure 2.4 by measuring thevertical distance B − C.
Other things to note from the PV diagram in figure 2.4 are the two limiting functionsfor the PV-loop, which are the End-Systolic-Pressure-Volume-Relationship (ESPVR) andthe End-Diastolic-Pressure-Volume-Relationship (EDPVR). The ESPVR is the top curve inthe PV diagram and always coincides with the upper left shoulders of the PV-loop, pointC. The ESPVR represents the contractility1 of the heart. The EDPVR is the bottom curveand always connects to the end-diastolic point A of the PV-loop. The EDPVR representsthe passive stiffness of the heart and is a measure of the pressure-volume relation in a fullyrelaxed chamber.
1The maximum pressure that can be developed for a certain volume during an isovolumic contraction.
2.3. The Circulatory System 8
2.3 The Circulatory System
The human circulatory system is, as previously mentioned, divided into a systemic and apulmonary circulation. When looking at the distribution of blood in the body it is said thatabout 84 % of the entire blood volume can be found in the systemic circulation and 16 %in the heart and lungs. Of the 84 % contained in the systemic circulation, 64 % can befound in the veins, 13 % in the arteries, and 7 % in the systemic arterioles and capillaries,see figure 2.1. Of the 16 % contained in the heart and lungs, the heart contains about 7 %and the pulmonary vessels 9 % of the blood [13].
Pre
ssu
re (
mm
Hg
)
0 Systemic Pulmonary
60
80
100
120
40
20
0
Aort
a
Larg
e a
rteries
Sm
all
art
eries
Art
eriole
s
Capill
aries
Venule
s
Sm
all
vein
s
Larg
e v
ein
s
Venae c
avae
Pulm
onary
art
eries
Art
eriole
s
Capill
aries
Venule
s
Pulm
onary
vein
s
Figure 2.5: Typical pressure profiles in the different vessels [13].
A schematic of the typical pressure profiles in the different vessels can be seen in figure2.5. The reason for the oscillations observable in the circulation is the pulsatile nature ofthe heart which causes the systemic arterial pressure to fluctuates between a systolic (120mmHg) and a diastolic (80 mmHg) pressure with an average pressure of 100 mmHg. Thereason for the small increase in the amplitude of the pressure oscillations in the large arteriesis due to wave reflection in the arterial tree2, for more information see Frans [8] and Milnor[29]. The pressure in the pulmonary circulation fluctuate for the same reason but at muchlower values, between 25 and 8 mmHg with a mean of 16 mmHg [30]. The central venouspressure (CVP), measured in the inferior vena cava, has a mean value of about 4 mmHg. Anillustration of the branching of the vessels of the systemic circulation can be seen in figure2.6. It is illustrated how from the aorta, the arteries branch out into the arterioles, whichfurther branches out into the smallest vessels, the capillaries. The capillaries then convergeinto larger vessels, the venules, which finally converge into the veins and the venae cavae.
2.3.1 Arteries
Arteries are thick-walled vessels and have a large amount of elastin and collagen fibers.These fibers allow for the expansion of the arteries and can thus provide an additionalstorage of blood. This expansion property is especially needed during systole when blood israpidly pumped out into the arterial system. During diastole the diameter of the arteries is
2The pulsatile blood flow propagates as a wave through the arterial branches. Wave reflections are causedby the elasticity of the vessels and changes in the arterial geometry, stiffness and bifurcations [8].
2.3. The Circulatory System 9
Aorta
Large
Artery
Arteriole
Capillaries
Venule
Vena
Cava
Vein
Small
Artery
Figure 2.6: Schematic of the branching of the vessels in the systemic circulation [21].
in turn reduced, which contributes to the stabilization of the arterial pressure. The largestartery in the human body is the aorta with a diameter of about 25mm [17].
2.3.2 Arterioles
As described in Heller [17], "the arterioles are narrower and structured differently than thearteries. In relation to their diameter, the arterioles have a greater wall thickness and agreater proportion of smooth muscle than the arteries". This allows the blood flow to beregulated throughout the organs. The arterioles are referred to as resistance vessels becauseof their high and tunable resistance [17].
2.3.3 Capillaries
The capillaries are the most narrow vessels in the entire cardiovascular system. The capil-laries have a diameter of only 3 µm. In order to get a better understanding of this dimensionremember that the blood cells in the human body have a diameter of 7 µm. Thus, the bloodcells have to deform in order to travel through the capillaries. The capillaries, in contrastto the arteries and arterioles, have no smooth muscles and can therefore not change theirdiameter [17].
2.3.4 Venules and Veins
The venules and veins have very thin walls when compared to their diameter. This makesthem very sensitive to small changes in transmural pressure, i.e. the difference betweeninternal and external pressure of the vessel. As the arteries, the venules and veins alsohave smooth muscles in the walls which allow a change of their diameter. Compared to thearteries and arterioles, the venules and veins have a much higher compliance, which meansthat they can expand a lot more. This in turn, allows to store much blood in these vessels,which is the reason why so much of the blood in the body (64 %) can be found in the veins.Most of the larger venous vessels also have one-way valves that prevent reverse flow. This isof great importance to the workings of the cardiovascular system when standing up or lyingdown [17].
A summary of the number, the cross-sectional area and some physical characteristics ofall vessels can be found in table 2.1.
2.3. The Circulatory System 10
Table 2.1: Summary of the number, the cross-sectional area and some physical characteris-tics of the circulatory vessels [17].
Arteries Arterioles Capillaries Venules VeinsInternal diam. 2.5 cm 0.4 cm 30 µm 5 µm 70 µm 0.5 cm 3 cm
Wall thick. 2 mm 1 mm 20 µm 1 µm 7 µm 0.5 mm 1.5 mmNumber 1 160 5 · 107 1010 108 200 2
Cross-area 4.9 cm2 20 cm2 400 cm2 4500 cm2 4000 cm2 40 cm2 14 cm2
2.3.5 Systemic Circulation
The systemic circulation starts at the left ventricle where oxygenated blood is ejected intothe aorta. The blood is then distributed over the systemic arteries and arterioles into thesystemic capillaries supplying blood to the organs. The oxygen-deprived blood is thentransported through the systemic venules and veins into the right heart.
Orthostatic Stress
The pressures in the vessels are highly dependent on the body’s position. Orthostatic stressis something a body is subjected to when changing body position, e.g. from lying downto standing up. In this case a redistribution of blood volume will occur due to hydrostaticpressure changes. This pressure comes from the weight of the blood in the vessels. Whenlying down, all the organs and limbs are on the same level when compared to the ground.When standing up this will no longer be the case. In the lower part of the body an increasein pressure will occur, as opposed to the upper body where a decrease will occur. Thehydrostatic reference point that does not experience any effects lies in the throat and neck.The systemic vessels supplying blood to the lower body experience a pressure rise of up to90 mmHg as can be seen in figure 2.7. This pressure rise will cause the vessels to distend,especially the more compliant veins, which will cause an accumulation of blood. Since theopposite is true for the upper body, a displacement of blood will occur throughout thesystemic vessels [13]. This is the reason why sometimes people get light headed and evenfaint when standing up too quickly. The one-way valves in the larger venules and veinshelp to redistribute the blood after standing up by preventing back flow when the blood isexpelled [17].
2.3.6 Pulmonary Circulation
The pulmonary circulation carries the blood from the right heart via the pulmonary arteries,through the capillaries in the lungs and through the pulmonary veins into the left side ofthe heart. When the blood passes the lungs it is supplied with oxygen and delivered to theorgans through the systemic circulation.
The pressures in the pulmonary circulation are much smaller than the ones in the sys-temic circulation because of the fact that the structure of the pulmonary circulation isdifferent from the systemic circulation. The pulmonary circulation only feeds blood to onesingle organ, the lungs. This, in combination with the fact that the dimensions of the vesselsin the pulmonary circulation are larger than those in the systemic circulation, causes theoverall resistance to be much less thereby lowering the pressures [17].
2.3. The Circulatory System 11
Sagittal sinus-10 mm
0 mm0 mm
+ 6 mm
+ 8 mm
+ 22 mm
+ 35 mm
+ 40 mm
+ 90 mm
Figure 2.7: Schematic of orthostatic effects present in a fully upright person, where mmstands for [mmHg] [13].
2.3. The Circulatory System 12
Cardiac Output Dependence of Pulmonary Arterial Pressure
When the cardiac output increases, pressures in the blood vessels rise due to an increase inthe blood volume flowing through the vessels. The rise of pulmonary arterial pressure papis, however, limited because of the fact that a rise in cardiac output qco causes a decrease inthe pulmonary capillary resistance. This is possible since, during normal conditions, somecapillary vessels are collapsed and not yet "open" to blood flow. When the cardiac outputincreases more blood is forced through the pulmonary capillaries opening more capillariesand expanding them. This is the cause of a decrease in pulmonary resistance which coun-teracts the increase of the pulmonary arterial pressure [13]. The dependence of pap on qcocan be seen in figure 2.8.
0
30
20
10
0
Normal value
4 8 12 16 20 24
Pu
lmo
nary
art
eri
al
pre
ssu
re (
mm
Hg
)
Cardiac output (L/min)
Figure 2.8: The relation between pulmonary arterial pressure and cardiac output [13].
Lungs
Breathing happens because the human body can control the pressure surrounding the lungs.The intra-thoracic pressure (pith) or also called the pleural pressure is the pressure differencebetween the pressure in the thoracic cage and the pressure in the lungs [25].
When no strain is put on the lung, it is collapsed and is nearby empty of air. A personcan never fully empty the lungs even when one forcibly exhales to the maximum. The lungswill still retain some volume and this volume is called the residual volume. When a personwants to inhale, the lung needs to expand. As a result, in order to inflate the lung, a negativepressure in the thorax relative to the pressure in the lungs is needed. During inspiration thepressure in the thorax decreases and the intra-thoracic pressure becomes more negative andpulls on the lung from the outside, thus expanding it and drawing in air. During expirationthe intra-thoracic pressure increases and the lung contracts, thus emptying it of air [39].During inspiration the intra-thoracic pressure pith is at rest at about −3 mmHg and duringinspiration at about −6 mmHg [39].
Figure 2.9 shows the working volume of the lungs. The tidal volume is the normalworking volume during normal breathing. The tidal volume is normally around 500 ml [13].As can be seen from the figure, the total lung capacity is the maximum lung volume thatcan enter the lungs, the residual volume is the volume that the lungs retain even after fullyexhaling and the vital capacity is the total capacity minus the residiual volume. The residual
2.4. Regulatory Mechanisms 13
volume of an average person is about 1200 ml [13]. Figure 2.10 shows the interactions of
Lu
ng
vo
lum
e (
ml)
5000
6000
1000
2000
3000
4000
Time
Inspiration
Inspiratorycapacity
Inspiratoryreservevolume
Expiratoryreserve volume
Vitalcapacity
Expiration
Total lungcapacity
Tidalvolume
Functionalresidualcapacity
Residualvolume
Figure 2.9: Illustration of the working area of the lungs [13].
the lungs with the pulmonary circulation and how the blood is supplied with oxygen (O2).It also illustrates how the oxygenated blood is transported through the heart and carriesoxygen to the rest of the body.
2.4 Regulatory Mechanisms
The human circulatory system has several regulatory mechanism that respond to and coun-teract physiological changes. These changes include the maintenance of a sufficient cardiacoutput and perfusion pressure for the organs [39]. The regulatory mechanisms can be sepa-rated into three different kinds of mechanisms: short-, medium- and longterm mechanisms.As the name implies, the shortterm mechanisms respond to rapid changes in local bloodflow and react within seconds. An example of a shortterm regulatory mechanism is the pres-sure sensitive baroreflex mechanism. As for the medium- and longterm mechanisms, theyrespond to the local blood flow changes very slowly. The setting of the optimum operatingpoint can take, for the mediumterm mechanism, from minutes up to several hours and forthe longterm mechanisms from days, weeks up to several months [13]. An example of aslower reacting mechanism is the stress relaxation of the systemic veins. Figure 2.11 showshow the regulatory mechanisms are divided according to their response time.
2.4.1 Arterial Baroreflex Controller
As is stated in Heller [17], "appropriate systemic arterial pressure is perhaps the singlemost important requirement for proper operation of the cardiovascular system." The aorticpressure is regulated through many different coupled systems, the most important one beingthe baroreflex controller [17]. The baroreflex is a short-term pressure sensitive controller thatadjusts the aortic pressure through changes in the systemic arterial resistance, the unstressedvolume of the systemic venous compartments, the heart rate and the heart contractility
2.4. Regulatory Mechanisms 14
expired air
alveolar air
tissue fluid
O2-poor blood O2-rich blood
inspired air
respiring tissues
internal respiration
gas transport
external respiration
O2
CO2
O2
CO2
O2
CO2
O2
CO2O2 CO2
O2 CO2
O2
CO2
O2
CO2
Figure 2.10: The process of gas exchange between the lungs and the cardiovascular system[27].
2.5. Pathology of the Heart 15
Maxim
um
feed
back g
ain
at
op
tim
al
pre
ssu
re
Time after sudden change in pressure
•0 1530 1 2 4 8 1632 1 1 2 4 8 162 4 8160
1
2
3
4
5
6
7
8
9
10
11
Acute
change in p
ressure
at
this
tim
e
Seconds Minutes
Renin-angiotensin-vasoconstriction
Hours Days
Baroreceptors
Capillary
Fluid
shift
• • !!
CNSischem
icresponse
Chemoreceptors
Stress relaxationAldosterone
Rena
l–blo
od
volu
me
pre
ssure
contr
ol
Figure 2.11: The different regulatory mechanisms in the human body and their individualresponse times [13].
accordingly. The sensors, so called baroreflex receptors, are located in the walls of theaortic arch, the carotid sinus and in the major thoracic arteries [39].
In order to better understand how the regulation of arterial pressure works with thebaroreflex controller, the regulatory responses to a sudden drop in arterial pressure is ex-plained. When the arterial pressure drops, the baroreflex receptors in the arterial wall tellthe controller to increase sympathetic nerve activity. This in turn increases the heart rate,increases the contractility of the heart, increases venous tone and constricts the arterialvessels. These adjustments increase the arterial pressure towards the optimal value [17].
2.5 Pathology of the Heart
When a patient is suffering from heart failure it simply means that there is a failure of theheart to pump blood as well as it normally does. There are a lot of causes for and typesof heart failures and only a few will be presented in this section. The intention is mainlyto give the reader an introduction to the most common causes of heart failure, for moreinformation see Guyton [13] and Heller [17].
2.5.1 Hypertension
Hypertension is defined as a chronic increase of the arterial blood pressure above 140/90mmHg (systole/diastole values). It is not actually a pathology of the heart but it greatlyincreases the risk of a coronary artery disease, heart failure, stroke and other cardiovascular
2.6. Mechanical Circulatory Assist Devices 16
diseases. Hypertension is one of the most common cardiovascular problems in the world,affecting over 20 % of the adult population in the western world. In most cases (up to 90%), the primary cause of hypertension is unknown and the only thing that can be treatedis the symptom of high blood pressure [17].
2.5.2 Systolic Dysfunction
Systolic heart failure is defined as a "lower than normal cardiac function curve" [17], whichmeans that at any given filling pressure a reduction, compared to normal, in cardiac outputcan be observed. The reduction in cardiac output causes a lowered arterial pressure whichin turn causes reflex activation of the sympathetic nerves (baroreflex). This increased reflexactivation counteracts the heart failure and returns the cardiac output near to its normalworking value. But in the long term, the cardiovascular operation cannot remain at thisincreased reflex activation but will cause an accumulation of cellular fluid in the body, anincrease in the end-diastolic volume (EDV) and a dilation of the heart muscles. Excessivecardiac dilation will impair the functionality of the heart and thus put more strain on thecardiac muscles [17].
When the heart becomes even more severely damaged, the reflex mechanisms will notbe able to reestablish a sufficient cardiac output. In this case the accumulation of fluid hasreached a level where any further retention will not help in cardiac output recovery butinstead cause increasingly severe edema3 [15]. In this situation the heart normally can nolonger recover and a heart transplant is needed. When an immediate transplant is impos-sible, Ventricular Assist Devices (VADs) can be used to establish hemodynamic stabilityand the opportunity for rehabilitation before transplant. For some patients with incurablediseases as cancer or severe infections, who are therefore not viable for transplant, VADsupport is the only chance for survival [30]. For more information on systolic dysfunctionand cardiac failure see Guyton [13] and Heller [17].
2.5.3 Cardiac Arrhythmia
Cardiac arrhythmia is defined as a disturbance in the heart rhythm [39]. Most cardiacarrhythmias are benign4, e.g. those where the heart sometimes skips a beat or has an extrabeat. Cardiac arrhythmia come from the disturbance of the electrical signals that are sentto the heart, some might be delayed or even blocked completely. The disturbances havethe effect that the heart does not contract rhythmically and therefore cause the heart topump less efficiently [13]. An example of cardiac arrhythmia is ventricular tachycardia whichoccurs when the ventricles are driven at unusually high rates as a compensatory mechanismfor heart failure.
2.6 Mechanical Circulatory Assist Devices
A mechanical circulatory assist device is a cardiac support system that is intended to help afailing heart to pump the blood out into the limbs and organs. This section will concentrateon the so-called left ventricular assist devices (LVADs). Cardiac support systems have asa main goal the improvement of the survival rate of transplant patients by bridging themto the transplantation. Other aspects of usage are for the improvement in "quality of life",
3Swelling in the hands, arms, legs and feet caused by fluid retention [17].4Of no danger to health; not recurrent or progressive; not malignant, [44]
2.6. Mechanical Circulatory Assist Devices 17
for ventricular recovery and maybe in the future can be used as a longterm or permanentsupport system for cardiac patients [30].
The research on mechanical circulatory assist devices started as late as in the 1960s andhas evolved into a standard therapy supporting patients with end-stage heart disease. Thefirst left ventricular assist devices were intended for short-term usage in weaning patientsfrom heart-lung machines. The first LVAD used as a bridge to transplantation was a pneu-matic LVAD in 1986. The first electrical powered LVAD used was the HeartMate LVADimplanted in the U.S. in 1991. Since then, the research on electrically powered LVAD hascontinued to this day [9].
Based on their flow characteristics, LVADs are generally divided into two categories,pulsatile and continuous flow pumps [38]. The pulsatile flow pumps mimic the pulsatilenature of the heart by pumping out the blood in pulses while the continuous flow pumpssupply a continuous blood flow.
2.6.1 Pulsatile Devices
Pulsatile pump systems consist of a chamber that is filled during the cardiac cycle andemptied during the ejection phase. The pulsatile devices are built to follow the beating ofthe human heart. A lot of devices use a bag made of polyurethane or a shell/membranecombination. A schematic of an implanted pulsatile LVAD can be seen in figure 2.12.
Pulsatile-flowLVAD
Actuatorbearing
Pusherplate
Pumphousing
Motor
Bloodflow
One-way inflowvalve (open)
One-way outflowvalve (closed)
Flexiblediaphragm
A Pulsatile-Flow LVAD
Leftventricle
Skinentrysite
Externalsystem
controller
Externalbatterypack
Aorta
Percutaneouslead
Percutaneouslead
Blood-pumpingchamber
Figure 2.12: Schematic of an implanted pulsatile LVAD and a cross section of the device[38].
The pulsatile LVAD works by alternately filling and emptying the bag, thereby pumpingthe blood out into the body. The biggest advantage of using a pulsatile LVAD in comparisonto a continuous flow LVAD is the generation of pulsatile blood flow, which generates a flowprofile more similar to the human heart blood flow. The biggest disadvantage lies in thelarge dimensions and in the complex mechanism involved [30]. Moreover, they are usuallymore expensive than the continuous flow LVADs [1].
2.6.2 Continuous Blood Pumps
Continuous blood pumps or rotary blood pumps, as they are also called, work, as the namesuggests, with an impeller that rotates at high speeds and creates a continuous blood flow.They are characterized by their compact design and simple functionality. They have only
2.6. Mechanical Circulatory Assist Devices 18
one moving part, the impeller, which makes them very reliable and it is the compact designthat produces less bodily stress on implant patients than in the case of the larger pulsatileLVADs. Rotary LVADs are connected to the left ventricle and pump the blood via an inflowcannula through a rotor and out through an outflow cannula into the aorta. Figure 2.13shows a schematic of an implanted rotary LVAD.
Continuous-flow LVAD Rotor Inlet stator and
blood-flowstraightener
Bloodflow
From left
ventricleTo aorta
B Continuous-Flow LVAD
Percutaneouslead
Motor Pumphousing
Outlet statorand diffuser
Figure 2.13: Schematic of an implanted rotary LVAD and a cross section of the pump [38].
Rotary pumps can be divided into three categories: radial (centrifugal), diagonal andaxial rotary pumps, see figure 2.14.
CENTRIFUGAL DIAGONAL AXIAL
Figure 2.14: A schematic of the direction of flows through the three types of rotary LVADs,radial, diagonal and axial LVAD [30].
As can be seen in the schematic shown in figure 2.14, radial rotary pumps have an inflowpath that is parallel to the rotation axis while the outflow path is orthogonal to it, e.g. theHeartWare R Ventricular Assist System, Heartware International Inc., Framingham, Mas-sachusetts, USA. Diagonal rotary pumps have an inflow that is parallel to the rotational axiswhile the outflow path is diagonal to it, e.g. the Deltastream, Medos Medizintechnik, Stol-berg, Germany. Axial rotary pumps have both inflow and outflow parallel to the rotationalaxis, e.g. the DeBakey, Micromed Cardiovascular, Houston, Texas, USA.
Chapter 3
Model Theory
This chapter presents the mathematical model of the cardiovascular system. The modeldescribed in this chapter is based on an existing model [1]. Therefore, parts of the theoryin this chapter have been taken from the work of Baumann [1]. The major contributions ofthis Master’s Thesis are the pulmonary circulation model and the systemic venous model.A schematic of the model of the full cardiovascular system can be seen in the Appendix infigure B.1. This can be compared with the old schematic created by Baumann [1] in figureB.2.
3.1 Conservation of Mass/Volume
For all chambers and vessels the law of mass conservation can be applied. When assumingthat the fluid flowing within the chambers and vessels, in this case blood, is an incom-pressible fluid, volume conservation also applies. This means that, for a given point in thecardiovascular circulation, the difference between the volume flow of blood coming in qinand going out qout has to account for the change in volume V . This can be expressed as
dV
dt= qin − qout. (3.1)
3.2 Modeling of Blood Vessels
A segment of any blood vessel can be represented as a tube with blood flowing mostly ina laminar way, according to figure 3.1 [17]. Blood flow through a vessel segment can onlyoccur when there is a pressure difference between p1 and p2. This difference is the drivingforce that moves the blood through the segment. When the blood moves through thesegment, friction will develop between the blood and the stationary walls, thereby resistingthe movement. This is expressed as a so-called vascular resistance R. The stationary flow qthrough a tube can be calculated as the pressure drop ∆p over the tube divided by R [17].
q =∆p
R=
p1 − p2R
. (3.2)
According to the equation of Poiseuille the resistance R can, for a fluid of viscosity η, beexpressed as
R =8ηL
πr4. (3.3)
19
3.2. Modeling of Blood Vessels 20
A
p1 p2r
qinpi
pa
S
qout
Figure 3.1: Schematic of a cardiovascular vessel with length S, cross-section area A, radiusr, flow q and pressure difference p1 − p2 > 0. The pressure inside the vessel is described aspi and the external pressure as pa [17].
This means that a change in the radius r of the vessel greatly affects the resistance R. Thisis an important principle e.g. in baroreflex control [17].
It is important to note that blood does not have constant viscosity but the viscosity ofblood is dependent on the shear rate and the blood composition [1]. About 40 % of bloodis made up of blood cells. These cells are suspended in a plasma1 that accounts for theremaining 60 % [17].
Blood vessels are elastic and there exists a relationship between the distending pressure2
and the blood volume contained within the vessel. The distending pressure is assumed tobe linearly correlated with the active volume, as described in [20]
V − V0 = C(pi − pa), (3.4)
where V is the volume in the vessel, V0 is the dead volume at pi − pa = 0 and C is theso-called compliance of the vessel.
In order to better understand and describe the modeling of the cardiovascular system, thevessels can be expressed using an electrical equivalent. The electrical description, equivalentof figure 3.1, can be seen in figure 3.2. In figure 3.2 the flow q through the vessel is equivalent
C
p1 p2pi
pa
0mmHg
qin R1 L R2qout
Figure 3.2: Electrical schematic of a cardiovascular vessel segment where R1 +R2 = R.
1The plasma has approximately the same viscosity as water [17].2Pressure difference between internal pressure pi and external pressure pa of the vessel.
3.3. Heart Model 21
to the current I, the pressures p to the voltage V , the resistance to blood flow R to electricalresistance R and the inductance L caused by blood acceleration to electrical inductance L.The resistance R of the tube has, in figure 3.2, been divided into two parts R1 and R2,where R1+R2 = R. The values of R1 and R2 can be set as to model the flow characteristicsthe vessel should demonstrate. The internal pressure of the vessel pi can be calculated fromequation (3.4).
pi =V − V0
C+ pa. (3.5)
The outflow qout of the vessel can be calculated from equation (3.2)
qout =pi − p2R2
. (3.6)
To calculate the inflow into the vessel one must consider the effect of inertia for bloodacceleration. This is represented as the inductance L in the figure 3.2, which is importantto consider at points where large accelerations in blood flow takes place, such as duringopening of the heart valves. Assuming a flat velocity profile the inductance L can, accordingto [34], be expressed as
L =∆p
q, (3.7)
where ∆p is the pressure drop over the inductance and q is the acceleration in blood flow.(3.7) can be used to express the change in inflow to the vessel as a differential equation
qin =p1 − pi − qinR1
L. (3.8)
Equation (3.4) can be used to set up the differential equation describing the pressure changewithin the vessel due to blood flow. By taking the time derivative and assuming that the
change in external pressure is negligibledpadt
= 0 and C is constant, the differential equation
can be expressed asV = Cpi. (3.9)
Using equation (3.1), the differential equation for pressure change within a vessel becomes
pi =qin − qout
C. (3.10)
3.3 Heart Model
The equations for the modeling of the heart have been taken from Baumann [1]. Theimportant effects that need to be considered when modeling the human heart are:
I. The pressure and volume changes in the atria and ventricles and the contractility ofthe heart.
II. The intraventricular septum which is an elastic wall that separates the left and rightventricle.
III. The inflow and outflow valves of the ventricles and atria.
In the following subsections, these points will be considered in detail.
3.3. Heart Model 22
3.3.1 Active and Passive Pressure Functions
According to Campbell [3] the ventricles and atria can be modeled using a non-linear time-varying elasticity and an internal resistance. The relation between the pressure and thevolume inside one of the chambers can be described by
p(t) = pe (V (t) , t)−R · q, (3.11)
where t represents the time, V (t) the blood volume inside the chamber, p (t) the instanta-neous blood pressure inside the chamber, R the internal resistance and q the flow out of thechamber. The internal resistance R represents the drop in pressure in the ventricle duringthe ejection phase and comes from friction effects. The resistances for the left and rightventricle have been taken from Ursino [40]. The friction effect for the atria is negligibleand thus the resistance term is set to zero. The pressure function pe (V (t) , t) depends onthe non-linear time-varying elastance of the chambers. The mathematical description ofpe (V (t) , t) can be given in the form
pe (V (t) , t) = pp (V (t)) + pa (V (t))Fiso (t) , (3.12)
where pp (V (t)) and pa (V (t)) represent the passive and the active pressure-volume relation-ship, respectively, and Fiso (t) represents the normalized ventricular contraction function.pp is also the end-diastolic pressure-volume relation (EDPVR) and pp + pa the end-systolicpressure-volume relation (ESPVR). For the left and right atria, pp is described as a linearfunction of V
pp,a (Va) = (Va − V0,a)Emin,a + p0,a, (3.13)
where Va is the volume in the atrium, V0,a is the dead volume, p0,a an offset pressureand Emin,a the elasticity of the atrium. For the left and right ventricle respectively, thenon-linear passive pressure-volume relationship pp is approximated by a hyperbolic function
pp,v (Vv) =λ
Vsat − Vv+ kVv + pλ, (3.14)
where λ is a weighting factor, Vsat the maximum blood volume in the ventricle, Vv theactual blood volume and pλ is a pressure offset. The linear coefficient k is determinedthrough the least squares method as to minimize the square error between the pressure inthe ventricle pp,v(Vv) and the line Emin,v(V − V0) + p0 in the low volume region. Emin,v isthe linear elastance of the ventricles in the low volume region, see figure 3.3. The additionalhyperbolic term is added to describe the limited volume of the ventricle. When blood isfilling the ventricle, Vv increases and the ventricle expands. But the ventricle can not expandto an infinite size. As filling continues and Vv → Vsat the pressure pp,v will rise drasticallyeventually preventing any further expansion of the ventricle.
The active pressure-volume relation pa is, for all chambers, modeled by using a parabolicrelation as described in [2].
pa (V (t)) =
[
1−
(
V ∗ − V (t)
V ∗ − V0
)2]
p∗. (3.15)
This relation is characterized as a downward concavity with a maximum value at (V ∗, p∗)and passing through the point (V0, 0), where V0 is the so called ventricular dead volume, seefigure 3.4. The point (V ∗, p∗) describes the maximum contractility3 of the respective ventri-cle or atrium. Figure 3.4 shows the pressure-volume relationship for the active pressure part
3.3. Heart Model 23
Emin,v(V − V0) + p0
Pre
ssure
[mm
Hg]
Volume [ml]
(V0, p0)
EDPVR
0 50 100 150 200 2500
5
10
15
20
25
Figure 3.3: An example of the passive pressure (pp)volume relationship (EDPVR) of theventricle [23, 30].
(V0,0)
Volume [ml]
Pre
ssure
[mm
Hg]
(V ∗,p∗)
(V ∗,p∗)
0 100 200 300 400 500 600−50
0
50
100
150
200
250
300
Figure 3.4: An example of the active pressure (pa) volume relation of the left ventricle, bothin the physiological case (solid line) and in the pathological case (dashed line) [1].
of the left ventricle. The scope of the curve shown in figure 3.4 is in the physiological casefrom V0 to 150 ml and in the pathological case from V0 to 250 ml [1]. The parameter valuesdescribing the ventricles and atria can be found in Colacino [5] and [30] for the physiologicalcase and in [23] for the pathological case simulating left heart failure.
3The point at which the hearts generates its maximum pressure in an isovolumic contraction.
3.3. Heart Model 24
For more information on the derivation of the passive and active pressure volume relationsof the ventricles and atria see Baumann [1].
3.3.2 Contractility Function
The normalized contractility function Fiso describes the contraction of the ventricles andthe atria. Fiso is basically a time-varying function and has the same shape for the ventriclesas for the atria. The only difference is that the contractility function for the atria has a timedisplacement due to the fact that the atria contract before the ventricles. In Baumann [1]and Moscato [30] Fiso has been determined from data given in Senzaki [36] and approximatedby polynomial interpolation. For detailed information on the derivation of the contractilityfunction Fiso see Baumann et al. [1, 30, 36] and the works cited therein.
(
Tsys
T, 0
)
(t2, 1)
(t1, Fiso(t1))
Fiso(t)
[1]
t/T [1]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
Figure 3.5: The approximated contractility function Fiso [30].
Figure 3.5 shows the approximated contraction function Fiso according to [30]. Threepolynomials are used in the normalized time intervals [0, t1], [t1, t2] and [t2, Tsys/T ]. Theheart period T is set as the sum of Tsys and Tdia [1]
T = Tsys + Tdia, (3.16)
where Tsys is the time of the systolic phase of the heart and Tdia is the time of the diastolicphase of the heart. Both Tsys and Tdia depend on the heart rate fhr. The relationshipbetween the heart rate and the diastolic part Tdia was calculated as in Vollkron [41]
Tdia =1
2
(
e−0.01207(fhr−40) + e−0.038(fhr−40))
. (3.17)
3.3. Heart Model 25
The parameters, t1, t2 and Fiso (t1) are set to the values given in Moscato [30] The time t2
corresponds to the value2
3Tsys/T . The point (t1, Fiso (t1)) is set to the value (0.32t2, 0.42).
At the point (t1, Fiso (t1)), the slope is set to fs = 1/t2. Using piecewise defined cubicpolynomials, Fiso can then be expressed as
Fiso =
a13τ3 + a12τ
2 0 < τ < t1a23τ
3 + a22τ2 + a21τ + a20 t1 < τ < t2
a33τ3 + a32τ
2 + a31τ + a30 t2 < τ < Tsys/T0 Tsys/T < τ < 1
, (3.18)
where τ is the normalized time τ = t/T . The coefficients a12 to a33 are determined fromcontinuity conditions at the three intersecting points, Fiso(t1), Fiso(t2), Fiso(Tsys/T ) ofequation (3.18) and the given slope requirement (fs = 1/t2) at point (t1, Fiso (t1)) and canbe found in Baumann [1].
The final equations describing the atria and ventricles can, with help from (3.1), (3.12),(3.13), (3.14) and (3.15), be established.
3.3.3 Atria
It is known from the electrocardiogram4 (ECG) analysis of the P-wave5 that the contractionof the atria occurs before the contraction of the ventricles. For more information on ECGand on the P-wave see [17]. The time lag of the atrial contraction to the contraction ofthe ventricles is about 20 % of the cardiac cycle. Equation (3.1) can be used to set up thedifferential equation for the atria. This gives for the left atrium:
Vla = qvp − qin,lv with Vla(0) = Vla,0, (3.19)
where qvp is the flow coming from the pulmonary system into the left atrium, qin,lv is theflow out of the atrium into the left ventricle and Vla,0 is the dead volume in the atrium. Thedifferential equation for the right atrium is described in the same manner.
Vra = qvs − qin,rv with Vra(0) = Vra,0, (3.20)
where qvs is the flow coming from the systemic venous system into the right atrium. Byusing (3.12), (3.13), (3.15) and, since the heart is within the thoracic cage, adding the intra-thoracic pressure pith, the expression for the pressure-volume relationship of the left andright atrium takes the form
pia = (Via − V0,ia)Emin,ia + p0,ia +
[
1−
(
V ∗
ia − Via (t)
V ∗
ia − V0,ia
)2]
p∗iaFiso,a(t) + pith, (3.21)
where i = {l, r} represent the left and right atrium [30].
3.3.4 Ventricles
Equation (3.1) is used to set up the differential equation for the ventricles. This gives forthe left ventricle
Vlv = qin,lv − qout,lv with Vlv(0) = Vlv,0, (3.22)
4ECG is a diagnostic tool that measures and records the electrical activity of the heart [17].5The P-wave is a pressure-wave that can travel through the blood [17].
3.3. Heart Model 26
where qin,lv is the flow coming from the left atrium into the left ventricle, qout,lv is the flowout of the left ventricle into the aortic system and Vlv,0 is the dead volume in the ventricle.The equation for the right ventricle can again be described in the same manner
Vrv = qin,rv − qout,rv with Vrv(0) = Vrv,0. (3.23)
By using (3.12), (3.14), (3.15), adding the intrathoracic pressure pith and remembering tosubtract the friction force in (3.12), the pressure-volume relationship of the left and rightventricle takes the form
piv =λ
Vsat,iv − Viv+ kivViv + pλ,iv +
[
1−
(
V ∗
iv − Viv (t)
V ∗
iv − V0,iv
)2]
p∗ivFiso,v(t)+
pith −Rivqout,iv,
(3.24)
where Riv is the internal resistance of the ventricles, i ∈ {l, r} [30].
3.3.5 Intraventricular Septum
The following sections discuss the interaction of the ventricles with the intraventricularseptum. The intraventricular septum is an elastic wall separating the left and right ventricle.Since the pressures in the left and right ventricle differ, the intraventricular septum willbe displaced depending on this difference and this, in turn, influences the volumes in theventricles [1]. When not taking the effect of the intraventricular septum into account thepressures in the ventricles can be expressed according to equation (3.11) and (3.24)
piv (Viv) = pp,iv (Viv) + pa,iv (Viv)Fiso,v (t)−Rivqout,iv + pith, i = {l, r}. (3.25)
When including the intraventricular septum, the volumes Viv in the ventricles need to beadjusted in the form
Vlv = Vlvf + Vspt (3.26a)
Vrv = Vrvf − Vspt, (3.26b)
where Vspt is the volume displacement in the ventricles caused by the displaced intraventri-cular septum and Vivf the non-adjusted ventricular volumes, see figure 3.6. As can be seenfrom figure 3.6, Vspt is actually a fictitious volume with the initial value Vspt,0 = 0. In realitythere are just the left ventricular Vlv and right ventricular Vrv volumes.
The intraventricular septum is displaced due to the pressure difference in the ventricles
pspt = plv − prv. (3.27)
The dynamics of the intraventricular system is described in a similar way as for the ventricles.The pressure pspt is divided into an active and a passive part, where the active part is affectedby the contractility function for the ventricles Fiso,v
pspt = pp,spt + pa,sptFiso,v. (3.28)
Assuming that the intraventricular septum has a linear pressure-volume relationship forboth the active and the passive part, pp,spt and pa,spt can be described by
pp,spt = Espt,minVspt (3.29a)
pa,spt = Espt,maxVspt, (3.29b)
3.3. Heart Model 27
Vrvf Vlvf
prv plv
Vspt
Figure 3.6: Schematic cross-section of the ventricles illustrating the displacement of theintraventricular septum [1].
Vspt
[ml]
pspt [mmHg]0 20 40 60 80 100 120
0
2
4
6
8
10
Figure 3.7: Simulated VP loop of the intraventricular septum [1].
where Espt,min is the minimum elastance of the intraventricular septum and Espt,max themaximum elastance. The parameters Espt,min and Espt,max have been optimized to fit theVolume-Pressure loop of the intraventricular septum observed in literature [1, 4] and seenin figure 3.7.
Adjusting the new volumes in the ventricles Vivf and using (3.11), (3.29a), (3.29b),equation (3.27) can be expressed as
Espt,minVspt + Fiso,vEspt,maxVspt = plv (Vlvf , Fiso,v)− prv (Vrvf , Fiso,v) (3.30)
3.4. Baroreflex Controller 28
Since plv and plv are both non-linear functions of Vspt, (3.30) is a non-linear equation. Thecalculation of the septum volume Vspt is therefore done using the Newton-Raphson method.For more information on the derivation of the equations for the intraventricular septum, seeBaumann [1].
3.3.6 Heart Valves
Each heart valve is modeled using an inductance and two different resistances. One smallvalue resistance in the normal direction of blood flow and one inverse resistance, which hasa much higher value, in the other. A schematic of the electrical equivalent of the heartvalves can be seen in figure 3.8, where L is the inductance of the valve, Rd is the directflow resistance and Ri is the inverse flow resistance, D1 and D2 are diodes only letting onedirection of flow through. The index y = {m, a, t, p} represent the different heart valves,
Rd,yv
Lyv
qin
p1 p2
D1
Ri,yv D2
Figure 3.8: Schematic of the heart valves [30].
mv = mitral valve, av = aortic valve, tv = tricuspid valve, pv = pulmonary valve. Thevalues for the inductance Lyv were taken from Baumann [1] and Moscato [30]. For moredetails on the derivation of the inductance values see Baumann [1]. The differential equationsfor the heart valves can be expressed by using (3.8)
qin =
p1 − p2 − qinRd,yv
Lyvfor qin > 0
p1 − p2 − qinRi,yv
Lyvfor qin < 0
, (3.31)
where qin is the flow through the heart valve. By using (3.31), the differential equation forthe mitral valve can be expressed in the form
qin,lv =
pla − plv − qin,lvRd,mv
Lmvqin,lv > 0, mitral valve open
pla − plv − qin,lvRi,mv
Lmvqin,lv < 0, mitral valve closed
. (3.32)
The differential equations for the flows through the other heart valves can be expressed inthe same manner [30].
3.4 Baroreflex Controller
As mentioned in chapter 2, the baroreflex controller is a pressure controller sensitive tochanges in aortic pressure pao. In order to correctly model the regulation of the aorticpressure, three different controllers are implemented in the model: a controller that adjuststhe total systemic arterial resistance, a controller that changes the heart rate and a controllerthat controls the unstressed volume in the systemic veins.
3.4. Baroreflex Controller 29
3.4.1 Systemic Arterial Resistance Controller
The structure of the systemic arterial resistance controller has been taken from Ursino [40].The controller reacts to changes in the mean transmural aortic pressure from the referencevalue and counteracts these changes by altering the systemic arterial resistance Ras. Thetransmural pressure is the pressure difference between pressure within and outside the vessel.This is important since vessels within the thoracic cage, as the aorta, are subject to an outerpressure of pith. Therefore the mean transmural aortic pressure is expressed as
pao,tm = pao − pith. (3.33)
The model of the control loop for the systemic arterial resistance controller can be seen infigure 3.9.
+GRas
1 + τRass
1Delay
pao,ref
pao,tm++
-DRas
Ras,const
∆RasxRas σRase
Ras
Figure 3.9: Schematic of systemic arterial resistance controller [40].
The error e in the transmural aortic pressure is multiplied by a factor GRas
xRas = GRas (pao,tm − pao,ref ) , (3.34)
where pao,ref is the reference value for the transmural aortic pressure. The limiting functionσRas described by the saturation block in figure 3.9 is expressed as
σRas =Ras,min +Ras,maxexp
−
xRas
kRas
1 + exp−
xRas
kRas
with kRas =Ras,max −Ras,min
4SRas, (3.35)
where Ras,max/Ras,min is the max/min change in the systemic arterial resistance caused bythe controller, xRas is the weighted error, SRas is the slope of the limiting function at e = 0[40]. The limiting function σRas with Ras,min = −0.6, Ras,max = 0.6 and SRas = 1 can beseen in figure 3.10.
The differential equation for the change in systemic arterial resistance ∆Ras can finallybe expressed as
∆Ras =1
τRas(σRas (t−DRas)−∆Ras) , (3.36)
where τRas is the time constant for the transfer function in figure 3.9 and DRas is the timedelay. The total systemic arterial resistance can be expressed as a constant mean value plusthe change caused by the controller
Ras = ∆Ras +Ras,const, (3.37)
where Ras,const is the mean systemic arterial resistance as described by Guyton [13] andMoscato [30]. The parameter values Ras,min, Ras,max, SRas for the systemic arterial resis-tance controller have been taken from Ursino [40].
3.4. Baroreflex Controller 30
σRas
[mm
Hgs
/ml]
e [mmHg]
Normalworking point
Ras,max
Ras,min
−100 0 100−1
−0.5
0
0.5
1
Figure 3.10: Saturation function σRas with pao,ref = 100− pith,mean = 104.5 mmHg [1].
3.4.2 Heart Rate Controller
The structure of the heart rate controller has been taken from Ursino [40]. The heart ratecontroller works in a similar way to the systemic arterial resistance controller, it is only alittle more complex. The heart rate controller model the changes in heart rate fhr presentin patients who are subjected to changes in the aortic pressure. The controller works byaltering the heart rate when changes in the mean transmural aortic pressure pao,tm from thereference value pao,ref are present. The controller is made out of two parts, a sympatheticand a vagal one. The vagal part is fast reacting and the sympathetic part needs a couple ofseconds to react [13, 40]. A schematic of the model of the heart rate controller can be seenin figure 3.11.
+GaTv
1 + τTvs
1Delay
pao,ref
pao,tm ++-
DTv
vT
GaTs DTs 1 + τTss
1
T
sT
b
Delay
xTv
xTs
xT
Figure 3.11: Schematic of heart rate controller [40].
Each part xTi, i = {v, s} is calculated in the same way as for the baroreflex controller.The weighted error of the mean transmural aortic pressure from the reference value is ex-pressed as
iT = GaTi (pao,tm − pao,ref ) , i = {v, s}, (3.38)
where GaTi is the weighting factor for the vagal and sympathetic parts. The differential
3.4. Baroreflex Controller 31
equation for the two parts is then
xTi =1
τTi(iT (t−DTi)− xTi) , i = {v, s}. (3.39)
The total heart period T can be expressed with the variable xT , where
xT = xTs + xTv, (3.40)
and with the limiting function
T =Tmin + Tmaxexp
−
xT
kT
1 + exp−
xT
kT
, with kT =Tmax − Tmin
4ST0. (3.41)
The parameters for the heart controller have been taken from Ursino [40].
3.4.3 Unstressed Volume Controller
The structure of the unstressed volume controller has been taken from Heldt [16]. Theunstressed volume controller regulates the vascular tone in venous compartments thus mod-ifying the pressure-volume relationships in the compartment and consequently the amountof blood stored. A schematic of the unstressed volume controller can be seen in figure 3.12.
GusvDelayDusv
xusvσusv
1
1 + τusvs
khead
kkid
ksp
kleg
∆Vhead,usv
∆Vkid,usv
∆Vsp,usv
∆Vleg,usv
∆Vusv,tot
+
—
pao,tm
pao,ref
Figure 3.12: Schematic of unstressed volume controller [40].
The control loop has been derived from Ursino [40]. The control loop is made up by aweighting factor Gusv, a time delay Dusv, a linear saturation function σusv and a transferfunction of the first order with the time constant τusv. The weighted error xusv of the meantransmural aortic pressure from the reference value is expressed as
xusv = Gusv (pao,tm − pao,ref ) . (3.42)
The implemented limiting function σusv is a simple linear function with a minimum and amaximum value
σusv =
Lusv xusv(t−Dusv) < Lusv
xusv(t−Dusv) Lusv < xusv(t−Dusv) < Uusv
Uusv xusv(t−Dusv) > Uusv,(3.43)
where xusv(t − Dusv) is the weighted error with the included time delay Dusv, Uusv themaximum value and Lusv the minimum value for the limiting function σusv. The value for
3.4. Baroreflex Controller 32
Dusv has been taken from Ursino [40] and τusv, Uusv, Lusv were set as to minimize theovershoot and for the rapid decay of any oscillations.
In comparison to Baumann [1] the structure of the controller stayed the same but all pa-rameters related to the controller were recalculated since the systemic venous compartmentshave been changed and an additional compartment representing the head and upper bodyhas been added to the previous model configuration. The factors khead, ksp, kkid, kleg arethe volume percentiles that give the change in unstressed volume in each compartment [16].The weighting constant Gusv has been recalculated, when multiplied with the factors khead,ksp, kkid, kleg, to correspond to the gain values for each compartment given in Heldt [16].The final change in unstressed volume is given as ∆Vhead,usv for the head, ∆Vkid,usv for thekidneys, ∆Vsp,usv for the splanchnic and ∆Vleg,usv for the legs. The differential equationfor the total change in unstressed volume in the venous compartments is
∆Vusv,tot =1
τusv(σusv (t−Dusv)−∆Vusv,tot) . (3.44)
Figure 3.13 shows the pressure-volume relation in the systemic veins according to Guyton[13] and what happens when the controller adjust the unstressed volume. pi is the pres-
Vi
∆Vusv,i
Vact,iVusv0,i
pi
pi,2
pi,1
Figure 3.13: Pressure-volume relation in the systemic veins.
sure in a vascular compartment, ∆Vusv,i is the change in unstressed volume caused by thecontroller, Vusv0,i is the initial unstressed volume and Vact,i is the active volume in the ve-nous compartment. The active volume Vact,i is the volume in the compartment that, dueto streching of the compartment vessels according to (3.4), give rise to a linear increase inpressure pi. The change ∆Vusv,i in unstressed volume, following the control loop in figure3.12, eventually increases the pressure from pi,1 to pi,2.
3.5. Systemic Circulation 33
3.5 Systemic Circulation
3.5.1 Arterial Circulation
The model describing the systemic arterial circulation has been taken from Baumann [1].The electrical schematic of the arterial circulation can be seen in figure 3.14. Rao represent
Rao
qCao
Cao
pCao
pith
qo,lv Lao
qLao
Cas
qCas
pCas
pith
0mmHg0mmHg
Ras
pvpaoqRas
Figure 3.14: Electrical schematic of the arterial systemic circulation model [1].
the resistance to flow of the aorta and the branched off arteries and Cao the respectivecompliance. The inductance Lao represents the inertia of the blood in the aorta. Ras
represents the total resistance and Cas the total compliance of the systemic arterial vessels.The systemic arterial resistance Ras is expressed as a sum of a constant part Ras,const andan adjustment ∆Ras controlled by the baroreflex controller (3.37). As explained earlier, achange in Ras is equivalent to a change in the diameter of the vessels. The flow over theinductance Lao can be described by
qLao =pao − pCas
Lao, (3.45)
where pao is the pressure in the aorta and pCas the pressure in the arteries. From figure3.14 it can be seen that the pressure in the aorta can be described as
pao = pCao + (qo,lv − qLao)Rao, (3.46)
where pCao is the transmural pressure over the aorta. The pressure pCao can be describedby the differential equation
pCao =pao − pCao
CaoRao, with pCao(0) = pCao,0, (3.47)
where pCao,0 is the initial pressure over the aorta. The pressure pCas in the arterial systemcan be described through
pCas =qLao − qRas
Cas, with pCas(0) = pCas,0, (3.48)
3.5. Systemic Circulation 34
where Cas is the total compliance of the arterial vessels and pCas,0 is the initial pressure inthe arterial system. qRas is the flow over the systemic arterial resistance Ras and can becalculated from
qRas =pCas − pv
Ras, (3.49)
where pv is the pressure in the veins. The component values of the equations above can becalculated from the physiological conditions of the human cardiovascular system at rest to
Ras,const = 1.066 mmHgs/ml
Cas = 1.6 ml/mmHg
Rao = 0.048 mmHgs/ml
Cao = 1.48 ml/mmHg
Lao = 0.0099 mmHg s2/ml,
where Ras,const is the estimated average value for the systemic arterial resistance, Cas thesystemic arterial compliance, Rao the resistance and Lao the inductance of the aorta [1, 28,30, 32]. For more information on the derivation of the parameters of the systemic arterialcirculation see Baumann et al. [1, 30, 28, 32].
3.5.2 Venous Circulation
The important effects that have been considered in modeling the systemic venous circulationare:
I. The effect of the orthostatic stress on the more compliant veins. Due to hydrostaticeffects blood displacement will occur and the veins will expand/contract.
II. The effect of stress relaxation where according to Heusden [18] "stress relaxation refersto the ability of the veins to stretch slowly when the pressure rises and to contractslowly when the pressure falls".
III. The effect of changes in abdominal pressure pabd on the kidneys and intestines.
In the following subsections, these points will be considered in detail. The basis for themodel of the systemic venous circulation has been taken from Baumann [1] but has beenextended to include an extra venous compartment representing the head and upper body,the effect of stress relaxation and the effect of external abdominal pressure pabd.
Simple Model
To first get a general idea of the resistances and blood flows that make up the systemicvenous system and to estimate some parameters, a simple model of the system is studied.The schematic of the simple model can be seen in figure 3.15. qRas is the flow coming fromthe system arterial circulation, qvsin is the eventual flow coming from an infusion, qvs is theflow out of the systemic venous circulation into the right atrium, Rvs is the total systemicvenous resistance and Cvs the total systemic venous compliance. According to Coleman etal. [14, 30] Rvs can be calculated by
Rvs =0.164
pv, (3.50)
3.5. Systemic Circulation 35
Cvs
pv
pith
0mmHg
qRasRvs qvs
pra
qvsin
Figure 3.15: Simple model of the systemic venous circulation [14, 30].
where pv is the pressure in the vena cava. By using equation (3.50) and estimating pv to 4mmHg as in Baumann [1], Rvs is calculated to
Rvs =0.164
4= 0.041 mmHgs/ml. (3.51)
The value for Cvs has been taken from earlier estimations in Guyton [13, 30].
Cvs = 50 ml/mmHg. (3.52)
When extending the model to include all the venous compartments the idea is to keep thetotal systemic venous compliance Cvs and resistance Rvs constant. This to keep the totalinflow, outflow and blood volume stored in the veins the same as in the simple model.
Extended Model
In order to correctly simulate orthostatic effects, such as hydrostatic pressure changes andblood volume displacement an extended model of the one seen in figure 3.15 needs to bederived. The model divides the systemic venous system into several compartments modelingdifferent parts of the body. This makes it possible to simulate the different hydrostaticpressure changes and blood volume displacements in the different parts of the body. Theextended model is based on the model presented in Heldt [16] with the venous systempartitioned into four compartments: the head, the kidneys, the splanchnic and the legs.The structure of the extended model can be seen in figure 3.16. Blood from the systemicarterial system qRas and fluid from an infusion qvsin enter the systemic venous system andis distributed into the four major compartments of the human body. The blood then flowsfrom the head into the superior vena cava and from the lower compartments into the inferiorvena cava . Both the inferior and superior vena cava are vessels that lie within the thoraxand are therefore subjected to the pressure pith due to breathing. The inferior and superiorvena cava finally transport the blood into the right atrium.
3.5. Systemic Circulation 36
Rleg,1qleg,1
Cleg
portho,leg
Cv
0mmHg
Rleg,2qleg,2
0mmHg
pleg
qRaspv
qvs pra
Rsp,1qsp,1
Csp
portho,sp
Rsp,2qsp,2
pabd
0mmHg
psp
qvsin
Rkid,1qkid,1
Ckid
portho,kid
Rkid,2qkid,2
pabd
0mmHg
pkid
Rhead,1qhead,1
Chead
portho,head
Rhead,2qhead,2
0mmHg
phead
psvc
Rsvc
qsvc
Csvcpith
0mmHg
qv,low
pivc
Rivc
qivc
Civc
pith
0mmHg
Figure 3.16: Schematic of the model of the extended systemic venous system [16].
3.5. Systemic Circulation 37
Stress Relaxation
The pressure pi in each compartment is calculated from the sum of an artificial pressurepstr,i, simulating the effect of stress relaxation and the pressure change caused by a changein unstressed volume ∆Vusv,i. Stress relaxation of the systemic veins has mainly been addedin order to simulate the hemodynamics during orthostatic stress more accurately.
As mentioned in section 3.4.3, a pressure rise can also occur when the controller changesthe unstressed volume as seen in 3.13. Because a change in unstressed volume is causedby the unstressed volume controller and not due to a displacement of blood volume, it isassumed that the only volume under the effect of stress relaxation is the active volume Vact,i.
According to the model described in Heldt [16] only the compartments representing thekidneys and splanchnic are thought to be within the abdomen and are therefore subjectedto an external pressure pabd. The pressure in each vascular compartment can therefore beexpressed as
pi = pstr,i −1
CiVusv,i, i = {leg, head} (3.53)
pj = pstr,j −1
CjVusv,j + pabd, j = {kid, sp} , (3.54)
where Ci is the compliance value for each of the compartments. The equation describingthe stress relaxation variable pstr,i were taken from Heusden [18]. The equation describesthe step response in pressure pstr,i(t) to a change in volume V0,i occuring at time t = 0 ina compartment
pstr,i(t) =V0,i
Ci
(
1 +5
3e−t/τ
)
i = {kid, sp, leg, head}, (3.55)
where τ = 30 s is the time constant. (3.55) is a step response to a volume change whichmeans the active time-dependent volume Vact,i(t) in a compartment can be expressed as
Vact,i(t) = V0,iu(t)
{
u(t) = 0 t ≤ 0u(t) = 1 t > 0
. (3.56)
The related linear differential equation describing pstr,i can be determined by making aLaplace transformation L(pstr,i(t))(s) and inserting the Laplace transform of the step func-tion Vact,i(s) = V0,i/s
pstr,i(s) =Vact,i(s)
Ci
(8/3s+ 1/τ)
s+ 1/τ, i = {kid, sp, leg, head}. (3.57)
The linear differential equation of pstr,i(t) for a time-dependent volume Vact,i(t) representing(3.57) is expressed as
pstr,i = −pstr,iτ
+Vact,i
Ciτ+
8
3
(qi,1 − qi,2)
Ci, i = {kid, sp, leg, head}, (3.58)
where Vact,i is the active volume in each systemic venous compartment. The inflow qi,1 andoutflow qi,2 of the vascular compartments are calculated by
qi,1 =pv − (pi − portho,i)
Ri,1, i = {kid, sp, leg, head} (3.59a)
3.5. Systemic Circulation 38
qi,2 =(pi − portho,i)− px
Ri,2, i = {kid, sp, leg, head}, x = {ivc, svc}, (3.59b)
where px = psvc when i = {head} and px = pivc when i = {kid, sp, leg}. The differentialequations describing changes in pivc and psvc are calculated using (3.10)
psvc =qhead,2 − qsvc
Csvc(3.60a)
pivc =qv,low − qivc
Civc, (3.60b)
where Csvc is the compliance of the superior vena cava, Civc is the compliance of the inferiorvena cava, qsvc the flow out of the superior vena cava and qivc the flow out of the inferiorvena cava. qivc and qsvc are calculated using (3.2)
qivc =pivc − pra
Rivc(3.61a)
qsvc =psvc − pra
Rsvc, (3.61b)
where Rivc is the resistance of the inferior vena cava and Rsvc is the resistance of thesuperior vena cava. qv,low is the summarized flow of the lower compartments that flows intothe inferior vena cava
qv,low = qkid,2 + qsp,2 + qleg,2. (3.62)
The total inflow into the right atrium is expressed as
qvs = qivc + qsvc. (3.63)
Orthostatic Pressure
The pressure portho,i represents the hydrostatic pressure change in a venous compartmentdue to a change in body position. A change in body position could be, e.g. standing up.A way to measure the effect of orthostatic stress is often done through the so-called Head-Up-Tilt (HUT) test. During the test a patient lies on a tilt table that gradually increasesits angle of tilt compared to the ground, slowly putting the patient in an upright positionand then returning them back to the initial lying position. The hydrostatic pressure changeportho,i for each compartment is calculated by
portho,i = pbias,i sin(α), (3.64)
where portho,i is dependent on the angle of tilt α of one’s body compared to the ground, seefigure 3.17. pbias,i represents the hydrostatic effects that are present when a person is fullyupright, i.e. α = 90o. The values for pbias,i have been taken from Guyton [13], see figure2.7 in section 2.3. For pbias,leg an estimated value, lying between the hydrostatic pressurearound the hips and the pressure in the feet in figure 2.7, was used
pbias,kid = 15.0 mmHg (3.65a)
pbias,sp = 22.0 mmHg (3.65b)
pbias,leg = 70.0 mmHg (3.65c)
pbias,head = −10.0 mmHg. (3.65d)
3.5. Systemic Circulation 39
α
Tilt Plate
Ground
Figure 3.17: Schematic of a tilt table, tilted at angle α compared to ground.
Estimation of Parameters
In the estimation of the parameters Ri,1, Ri,2, Ci, Rivc, Rsvc, Civc and Csvc for the systemicvenous circulation as seen in figure 3.16, the values given in Heldt [16] were used as a basis.There is one important difference between the model described in Heldt [16] and the modelin this Master’s Thesis. The calculated parameter values presented in Heldt [16] havebeen calculated to represent both the systemic arterial and the systemic venous part ofthe circulation. The systemic venous model has divided the systemic arterial and systemicvenous circulation into two separate parts. In order to obtain the same flow patterns andvolume ratios as presented in Heldt [16], the same parameter ratio is used but the parametervalues themselves need to be recalculated.
Rkid,tot,lit = Rkid,1,lit +Rkid,2,lit = 4.4 mmHgs/ml (3.66a)
Rsp,tot,lit = Rsp,1,lit +Rsp,2,lit = 3.18 mmHgs/ml (3.66b)
Rleg,tot,lit = Rleg,1,lit +Rleg,2,lit = 3.19 mmHgs/ml (3.66c)
Rup,tot,lit = Rhead,1,lit +Rhead,2,lit +Rsvc,lit = 4.19 mmHgs/ml, (3.66d)
where Ri,tot,lit, i = {kid, sp, leg} is the total resistance of each lower body vascular com-partment and Rj,1,lit is the inflow resistance and Rj,2,lit, j = {kid, sp, leg, head} the outflowresistance of each compartment. Rup,tot,lit is the sum of the resistances in the vascularcompartment representing the head and the resistance of the superior vena cava Rsvc,lit.All resistances with index lit are given according to the values in Heldt [16]. The totalresistance Rv,low,lit of the lower body compartments can be calculated using
1
Rv,low,lit=
1
Rkid,tot,lit+
1
Rsp,tot,lit+
1
Rleg,tot,lit⇒ Rv,low,lit = 1.253 mmHgs/ml (3.67)
The final resistance values that need to be calculated from Heldt [16] are
Rlow,tot,lit = Rv,low,lit +Rivc,lit = 1.278 mmHgs/ml (3.68a)
Rvs,lit = 0.979 mmHgs/ml, (3.68b)
where Rlow,tot,lit is the total resistance of the lower body venous compartments and theresistance of the inferior vena cava Rivc,lit. Rvs,lit is the total venous resistance accordingto Heldt [16]. It is, from the calculated values, clear that the model in this report and
3.5. Systemic Circulation 40
the one in [16] describe two different systems since according to 3.51 Rvs was calculated tobe 0.041 mmHgs/ml which clearly differs from Rvs,lit = 0.979 mmHgs/ml. The importantratios between the resistances in the upper and lower body are
Rup,tot,lit/Rvs,lit = 4.2788 = r1 (3.69a)
Rlow,tot,lit/Rvs,lit = 1.30499 = r3 (3.69b)
Rkid,tot,lit/Rv,low,lit = 3.512 = r5 (3.69c)
Rsp,tot,lit/Rv,low,lit = 2.538 = r7 (3.69d)
Rleg,tot,lit/Rv,low,lit = 3.1128 = r9. (3.69e)
The internal ratios of each compartment are
Rhead,2,lit/Rsvc,lit = 3.8333 = r2 (3.70a)
Rhead,1,lit/Rsvc,lit = 65 = r4 (3.70b)
Rkid,1,lit/Rkid,2,lit = 13.667 = r6 (3.70c)
Rsp,1,lit/Rsp,2,lit = 16.667 = r8 (3.70d)
Rleg,1,lit/Rleg,2,lit = 12 = r10 (3.70e)
Rv,low,lit/Rivc,lit = 50.12 = r12. (3.70f)
The ratios r1, . . . , r12 are important because in order to keep the flow pattern the samein the implemented model as the one described in Heldt [16] these ratios need to stay thesame. By using r1, . . . , r12 and the total systemic resistance Rvs = 0.041 mmHgs/ml theresistances for the extended model can be calculated.
The calculation of the resistances in the upper body compartment and the superior venacava is done by using the fact that the ratio between the total resistance of the upper bodyRup,tot and Rvs should be as described in 3.69a (Rup,tot/Rvs = r1) where
Rup,tot = Rhead,1 +Rhead,2 +Rsvc. (3.71)
Next by using the calculated values for the ratios between Rhead,1/Rsvc = r4 and Rhead,2/Rsvc =r2, where Rhead,1 is the inflow resistance, Rhead,2 the outflow resistance of the upper bodycompartment and Rsvc the resistance of the superior vena cava, the individual resistancecan be calculated. The ratios for the upper body can be rewritten as
Rup,tot = r1Rvs (3.72a)
Rhead,1 = r4Rsvc (3.72b)
Rhead,2 = r2Rsvc. (3.72c)
By using the relations above and putting them into (3.71) the resistance values in the upperbody are calculated to
Rup,tot = Rsvc + r2Rsvc + r4Rsvc = r1Rvs =⇒
Rsvc = 0.0025 mmHgs/mlRhead,1 = 0.1633 mmHgs/mlRhead,2 = 0.0096 mmHgs/ml.
(3.73)
3.6. Lungs and Airways Model 41
The resistances for the remaining compartments are calculated in the same manner by usingthe ratios from Heldt [16] and the fact that Rvs = 0.041 mmHgs/ml
Rivc,2 + r12Rivc,2 = r3Rvs =⇒
{
Rivc,1 = 0.0525 mmHgs/mlRivc,2 = 0.0010 mmHgs/ml
(3.74)
Rkid,2 + r6Rkid,2 = r5Rvs =⇒
{
Rkid,1 = 0.1717 mmHgs/mlRkid,2 = 0.0126 mmHgs/ml
(3.75)
Rsp,2 + r8Rsp,2 = r7Rvs =⇒
{
Rsp,1 = 0.1256 mmHgs/mlRsp,2 = 0.0075 mmHgs/ml
(3.76)
Rleg,2 + r10Rleg,2 = r9Rvs =⇒
{
Rleg,1 = 0.1507 mmHgs/mlRleg,2 = 0.0126 mmHgs/ml
. (3.77)
The calculations of the capacitances are done in the same way
Cvs,lit = Chead,lit + Csvc,lit + Cv + Ckid,lit + Csp,lit + Cleg,lit + Civc,lit = 141 ml/mmHg,
where Cvs,lit is the total systemic venous capacitance, Ci,lit i = {kid, sp, leg, head} is thecapacitance for each compartment, Csvc,lit is the capacitance of the superior vena cava andCivc,lit is the capacitance of the inferior vena cava. Cv = 2 ml/mmHg is a small artificialcapacitance, taken from Baumann [1], representing the capacitance of the vena cava but isnot present in Heldt [16]. The values indexed lit are all values taken from Heldt [16]. Thenew values for the capacitances are a bit simpler to calculate, just using the ratios and thefact that according to Moscato [30] Cvs = 50 ml/mmHg.
Chead,lit/Cvs,lit = 0.0567 = c1 ⇒ Chead = c1Cvs = 2.835 ml/mmHg (3.78a)
Csvc,lit/Cvs,lit = 0.1067 = c2 ⇒ Csvc = c2Cvs = 5.320 ml/mmHg (3.78b)
Cv,lit/Cvs,lit = 0.0142 = c3 ⇒ Cv = c3Cvs = 0.710 ml/mmHg (3.78c)
Ckid,lit/Cvs,lit = 0.1064 = c4 ⇒ Ckid = c4Cvs = 5.320 ml/mmHg (3.78d)
Csp,lit/Cvs,lit = 0.3901 = c5 ⇒ Csp = c5Cvs = 19.505 ml/mmHg (3.78e)
Cleg,lit/Cvs,lit = 0.1348 = c6 ⇒ Cleg = c6Cvs = 6.740 ml/mmHg (3.78f)
Civc,lit/Cvs,lit = 0.1915 = c7 ⇒ Civc = c7Cvs = 9.575 ml/mmHg (3.78g)
3.6 Lungs and Airways Model
When modeling the lungs and airways it is important to consider parameter dimensions.In contrast to the cardiovascular modeling, where dimensions such as [mmHg] and [ml] arestandard, airway models as Liu et al. [25, 26] often use dimensions such as [cmH20] and [l].The relation between [cmH20] and [mmHg] is defined by
1 mmHg = 1.36 cmH20. (3.79)
In order to use equations given in literature [25] the intrathoracic pressure pith in thissection only is expressed in [cmH20]. The goal of the lung model is to accurately depictthe interactions between the lungs and the cardiopulmonary system. The lung model inthis Master’s Thesis is based on an earlier work by Liu [25]. The model described by Liu
3.6. Lungs and Airways Model 42
[25] is a three part model: the first part describing the airway mechanics of the lungs andairways, the second part describing the intra-alveolar pulmonary capillaries and the thirdpart describing the gas exchange taking place between the lungs and the surrounding tissue.Only the first two parts of the model are relevant for the purposes of this Master’s Thesis.The aim is to create a model that is able to reproduce the effect of normal respiration andthe dynamics of cardiopulmonary interaction with particular focus on the hemodynamicresponse of the Valsalva maneuver which is a forced expiration against a closed glottis. Aschematic of the airway mechanics model from Liu [25] can be seen on the right in figure3.18 together with a schematic cross-section of the lungs on the left in figure 3.18. Figure
Lungspithpalv
Intra-alveolar capillaries
Mouth
CollapsibleAirways
Upper
Airways
Thorax
ptmb
pel
ptm
0mmHg
pith
pel(Valv, pith)
ptm(VC)
RLti
palv
Ralv(Valv, pith)
qair,2
RC(VC , qair,1)
qair,1
pC
pref
Figure 3.18: Schematic cross-section of the lungs, redrawn and partially modified from Liu[25] (left) . Schematic of the model in Liu [25] describing the airway mechanics (right).
3.18 shows how air enters through the mouth at reference pressure pref , flows throughthe upper and collapsible airways at pressure pC and into the lungs at pressure palv. Thelungs are characterized by a variable resistance Ralv(Valv, pith) and by a transmural pressurepel(Valv, pith). The collapsible and upper airways are characterized in terms of a volume andflow dependent resistance Rcoll(VC , qair,1) and by the transmural pressure ptm(VC) [25].
The equations describing the transmural pressures pel and ptm were taken from Liu [25].pel is characterized by two parts, one inspiration function pel,I and one expiration functionpel,E representing the outer boundries of the pel-loop depicted in figure 3.19.
pel,E = pel,max
(
Valv − VRV + 0.001
V maxl − VRV + 0.001
)3
(3.80)
pel,I = ξpel,max + nd ln (na/nb)
ξ + 0.001, (3.81)
where V maxl is the total lung capacity and VRV is the minimum lung volume, also called the
residual volume. According to Liu [25] the pel,E and pel,I curves were generated by assumingthat pith is held constant at pith,max and pith,min during the expiration and inspiration,respectively. The mean of these two curves refers to the equilibrium curve correspondingto pith = pith,mean = 4.5 mmHg = 4.5 · 1.36 cmH20. The dependence of pel on pith was
3.6. Lungs and Airways Model 43
calculated using linear interpolation through grading by effort between the equilibrium curveand the pel,E and pel,I curves respectively. na, nb, nc and nd are then calculated accordingto
na =V maxl − VRV + 0.1
V maxl − VRV + 0.001
− 0.99 (3.82)
nb =V maxl − VRV + 0.1
Valv − VRV + 0.001− 0.99 (3.83)
nc =V maxl − VRV + 0.1
0.001− 0.99 (3.84)
nd =pel,max + 25
ln (nc/na), (3.85)
where pel,max is the maximum value of pel and has been taken from measurements in Liu[25]. pel is finally expressed as a sum of pel,E and pel,I
pel =
{
(0.5 + ne)pel,E + (0.5− ne)pel,I for pith(t) ≥ 0(0.5− ne)pel,E + (0.5 + ne)pel,I for pith(t) < 0.
(3.86)
ne is calculated by
ne =
0.5
pith,mean − pith,max(pith,mean − pith(t)) for pith(t) ≥ 0
0.5
pith,mean − pith,min(pith,mean − pith(t)) for pith(t) < 0.
(3.87)
In order for the pel equations to create a full loop during maximum effort, a further conditionon pel needs to be set
pel ≥ 0 ∀ Valv, pith (3.88)
In [25] inspiration is defined as pith ≤ 0 and expiration as pith > 0. During normal con-ditions, when pith = pith,mean, pel is the same during inspiration and expiration. This canbe seen by setting pith = pith,mean in (3.87). Figure 3.19 shows pel as a function of theactive lung volume Valv and graded by effort (different pith). The parameters in figure 3.19have been set according to [25] as V max
l = 5.3 l, VRV = 1.24 l and pel,max = 35 cmH20.According to Liu [25] and Olender [33], the transmural pressure ptm is expressed as
ptm =
5.6− lbptm ln
(
VC,max
VC− 0.999
)
forVC
VC,max> 0.5
saptm − sbptm
(
VC
VC,max− 0.7
)2
forVC
VC,max≤ 0.5
, (3.89)
where VC,max is the maximum volume of the airways. The constant lbptm is calculated by
lbptm =ptm,max − 5.6
6.908. (3.90)
The constants saptm and sbptm are calculated using the conditions of continuity atVC
VC,max=
0.5
ptm
∣
∣
∣
∣
∣
VCVC,max
=0.5
= 5.6− lbptm ln (2− 0.999) = saptm − sbptm (0.5− 0.7)2. (3.91)
3.6. Lungs and Airways Model 44
pel [cmH20]
Va
lv -
VR
V [
l]
pith = pith,min
pith = pith,mean (equilibrium curve)
pith = pith,max
(pel,max,Vmax)l
Figure 3.19: Transmural pressure pel plotted against lung volume Valv and for differentvalues of intrathoracic pressure [25].
dptmdVC
∣
∣
∣
∣
∣
VCVC,max
=0.5
=4lbptm
1.001VC,max=
0.4sbptmVC,max
. (3.92)
Combining (3.91) and (3.92), sbptm and saptm can be expressed by
sbptm =4lbptm0.4004
(3.93)
saptm = 5.6− lbptm ln (1.001) + 0.04sbptm. (3.94)
The transmural pressure ptm as a function of the volume VC in the airways can be seenin figure 3.20. By assuming the air in the lungs and airways is incompressible and using(3.1) the differential equations for the lung volume Valv and the airway volume VC can beexpressed by
Valv = qair,2 (3.95)
VC = qair,1 − qair,2. (3.96)
The flows qair,1 and qair,2 are calculated using (3.2)
qair,1 =pref − pC
RC(3.97)
qair,2 =pC − palv
Ralv, (3.98)
where pC is the pressure in the airways and palv is the pressure in the lungs. The resistancesRC and Ralv are calculated according to Liu [25]. The total airway resistance RC is expressedby
RC = K3
(
VC,max
VC
)2
+K1
2+
√
(
K1
2
)2
+ | pref − pC |. (3.99)
3.6. Lungs and Airways Model 45
ptm
[mm
H20]
VC [l]0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−15
−10
−5
0
5
10
15
20
25
30
35
Figure 3.20: Transmural pressure ptm plotted against the airway volume VC [25].
K1 and K3 are constants taken from [25]. The airway resistance Ralv is expressed by
Ralv = RSme
RSa
Valv − VRV
V maxl − VRV
+RSc, (3.100)
where RSc is calculated by
RSc =
{
RSc,max−0.02pith,max
(pith − pith,max) +RSc,max pith ≥ 0
0.02 pith < 0.(3.101)
RSm is the magnitude of Ralv −RSc at Valv = VRV , RSa is a parameter characterizing thecurvature of Ralv and RSc,max is RSc at the instant of pith = pith,max. The parameter valuesfor RSm, RSa and RSc,max have been taken from Liu [25]. The pressure in the airways pCis calculated by
pC = ptm + pith. (3.102)
The pressure in the lungs palv can, according to [25], be expressed by
palv = pel + pith +RLtiqair,2, (3.103)
where the term RLtiqair,2 represents the pressure drop over the resistance RLti. Solving(3.103) for palv yields
palv =pel +
RLti
RalvpC + pith
1 +RLti/Ralv. (3.104)
The pressures in the lungs and airways are all relative to the atmospheric pressure and thereference pressure pref has therefore been set to zero in all equations.
3.7. Pulmonary Circulation 46
3.7 Pulmonary Circulation
3.7.1 Pulmonary Arteries
The pulmonary arteries are represented by a single resistance Rpul and a single complianceCpul as can be seen in figure 3.21. The model is based on the pulmonary artery model de-scribed in [1]. The differential equations describing the system are calculated using equation
Cpul
pap pcap,a
pith
0mmHg
qout,rv Rpul
qlp
Figure 3.21: Model representing the pulmonary arteries and pre-capillaries [1].
(3.10)
pap =qout,rv − qlp
Cpulwith pap(0) = pap,0 (3.105)
where qout,rv is the outflow of the right ventricle, qlp is the flow into the pre-capillaries andpap,0 is the initial pressure in the pulmonary arteries.
3.7.2 Pulmonary Capillary Model
The pulmonary capillaries can be divided into two parts: the first part describes the pre-capillaries representing the capillary vessels outside the alveolar region, the second partdescribes the intra-alveolar capillaries. The alveolar region is the region where the gas ex-change between the air in the lungs and the blood takes place. The model describing thepulmonary capillaries has been completely modified from the one described in Baumann [1]to include the interactions of the lungs. The model is based on the pulmonary capillarymodel presented in Liu [25], which describes the pulmonary capillary vessels and the inter-actions of the lungs with the pulmonary capillary resistances. A schematic of the modelcan be seen in figure 3.22, where Vpc is the blood volume and ppc the pressure in the intra-alveolar capillaries, ptmb is the transmural blood pressure over the capillary wall, pcap,a is thepulmonary pre-intra-alveolar capillary pressures and pvp is the pulmonary venous pressure.
As can be seen in figure 3.22, the pulmonary capillary resistance is divided into threeparts: a variable pre-intra-alveolar capillary resistance Rcap,a, a variable pulmonary capillaryresistance Rpc and a variable post-intra-alveolar capillary venous resistance Rcap,v. Thepulmonary capillary resistance is divided into two parts, an inflow resistance Rpc/2 and anoutflow resistance Rpc/2. The capacitance of the pulmonary capillaries is divided into twoparts, an intra-alveolar volume-dependent capacitance represented by a transmural pressureptmb(Vpc) and an extra-alveolar capacitance Ccap.
As the intra-alveolar capillaries lie within the alveolar region, the external pressure ef-fecting them is assumed to be the alveolar pressure palv that was calculated in (3.104). The
3.7. Pulmonary Circulation 47
0mmHg
palv
ptmb(Vpc)
pith
Ccap
0mmHg
qlp qin,pul qout,pulppc
Rcap,a(Valv, pith, qout,rv)
Rpc(Vpc, qout,rv)
2
Rpc(Vpc, qout,rv)
2
Rcap,v(Valv, pith, qout,rv)
pcap,a pvp
Figure 3.22: Electrical schematic of the pulmonary capillary circulation [25].
idea is that due to the direct proximity of the capillary vessels to the lungs it is the pressurewithin the lungs palv, not the pressure within the thoracic cage pith, that interacts withthe intra-alveolar capillaries. It is important to remember that the alveolar pressure palv isdifferent from the intrathoracic pressure pith assumed to affect the other vessels within thethoracic cage. The intra-alveolar pulmonary capillaries are modeled as a single tube with avariable volume. Rcap,a and Rcap,v are assumed to be inversely proportional to the alveolivolume Valv but proportional to the intrathoracic pressure pith. Rpc is assumed to be onlydependent on Vpc [25].
Another effect of significance is the effect of opening of pulmonary capillaries and theexpansion of the vessels, described in section 2.3.6, [13]. This is modeled by setting thepulmonary capillary resistances dependent on cardiac output. Which means that when thecardiac output of the right ventricle rises, more capillary vessels will expand and open in par-allel thereby reducing the total pulmonary resistance. The differential equation describingthe pressure in the pre-capillaries pcap,a can be calculated using equation (3.10)
pcap,a =qlp − qin,pul
Ccapwith pcap,a(0) = pcap,a,0, (3.106)
where qin,pul is the flow into the capillaries and pcap,a,0 is the initial pressure in the pre-capillaries. Using (3.1), the state equation for the volume in the intra-alveolar pulmonarycapillaries can be expressed as
Vpc = qin,pul − qout,pul, (3.107)
where qout,pul is the flow out of the capillaries. qin,pul can be expressed in the form
qin,pul =pcap,a − ppc
Rcap,a(Valv, ppl) +Rpc(Vpc)
2
. (3.108)
qout,pul can, in a similar manner, be expressed in the form
qout,pul =ppc − pvp
Rcap,v(Valv, ppl) +Rpc(Vpc)
2
. (3.109)
3.7. Pulmonary Circulation 48
Rcap,a,lit, Rcap,v,lit and Rpc,lit are dictated by observations, showing that the extra-alveolarresistances decrease while the intra-alveolar resistances increase as Valv rises and are de-scribed by Liu [25]
Rcap,i,lit = (kv,φ (Valv − V maxl ) +R0,i)
(
1 +pithkp,φ
)
with i = {a, v} (3.110)
Rpc,lit = Rpc,max
(
Vpc,max
Vpc
)2
, (3.111)
where kv,φ is a parameter characterizing the volume dependence of Rcap,i,lit and kp,φ is aparameter characterizing the pressure dependence of Rcap,i,lit. V
maxl is the maximum alveoli
volume and R0,i is the estimated mean of Rcap,i,lit during normal breathing. Rpc,max is themaximum capillary resistance and Vpc,max is the maximum capillary volume. The calculatedpulmonary capillary resistances above need to be adjusted to include the effect of openingof pulmonary capillaries and the expansion of the vessels, as described in section 2.3.6. Thiseffect is modeled by changing the pulmonary capillary resistances according to the meancardiac output of the right ventricle qout,rv. The capillary resistances are expressed by
Rcap,i =Rcap,i,lit
α(qout,rv)i = {a, v} (3.112)
Rpc =Rpc,lit
α(qout,rv), (3.113)
where α(qout,rv) is a cardiac output dependent parameter. α(qout,rv) is described by
α(qout,rv) =
1 for qout,rv < 5 l/min
1 + kaqout,rv − 5
16− 5for 5 ≤ qout,rv < 16
1 + ka for qout,rv ≥ 16
, (3.114)
where qout,rv is the mean cardiac output for one heart cycle of the right ventricle, ka is aconstant characterizing the relationship of qout,rv with the pulmonary capillary resistances.
In the case of left heart failure the pressure in the pulmonary circulation is already higherthan normal implying that most capillary vessels are already open and somewhat distended.This means that the relation between the pulmonary capillary resistance and the cardiacoutput α(qout,rv) will change. It is assumed that this change can be described by changingka.
In the physiological case ka = 2 has been optimized, through simulation, as to fit thepap — qout,rv curve as close as possible to the one seen in figure 2.8 in section 2.3.6. Inthe pathological case ka = 0.5 has been optimized, through simulations, as to fit the pap— qout,rv curve to the observations made by Janicki [19]. The equation describing thetransmural pressure ptmb(Vpc)
ptmb = mc −mb ln
(
Vpc,max − 0.001
Vpc − 0.001− 0.999
)
, (3.115)
was taken from Liu [25]. The constants ma, mb and mc are calculated according to
ma =Vpc,max − 0.001
Vpc,max − 13.6Cpc − 0.001(3.116)
mb =13.6
6.908 + ln (ma − 0.999)(3.117)
mc = 20.4− 6.908mb, (3.118)
3.7. Pulmonary Circulation 49
ptm
b[m
mH
g]
Vpc [ml]
Normal working point
0 10 20 30 40 50 60 70−15
−10
−5
0
5
10
15
20
25
Figure 3.23: Transmural pressure ptmb from [25] used in modeling the pulmonary capillaries.
where Cpc = 6.9 · 10−4 [l/mmHg] is the mean compliance of the intra-alveolar pulmonarycapillary tube during normal respiration [25]. The transmural pressure ptmb as a functionof the volume Vpc can be seen in figure 3.23.
3.7.3 Pulmonary Veins
A schematic of the pulmonary veins can be seen in figure 3.24. The model for the pulmonaryveins has been taken from Baumann [1].
Cvp
pvp
0mmHg
pith
qout,pul Rvpqvp
pla
Figure 3.24: Schematic of the pulmonary veins [1].
The differential equations describing the system are calculated in the same way as forthe pulmonary arteries, using equation (3.10)
pvp =qout,pul − qvp
Cvp, with pvp(0) = pvp,0, (3.119)
where qout,pul is the outflow from the pulmonary capillaries, qvp is the flow into the leftatrium and Cvp is the compliance of the pulmonary veins. pvp,0 is the initial pressure in thepulmonary veins.
3.8. LVAD Modeling 50
3.8 LVAD Modeling
The LVAD modeled in this Master’s Thesis is a DeBakey axial flow pump from MicroMed(MicroMed Cardiovascular Inc., Houston, TX, USA). The model for the LVAD has beentaken from Baumann et al. [1, 31, 41]. The pump is connected to the left ventricle andthe aorta. The blood is pumped from the ventricle through the LVAD and into the aorta.Figure 3.25 shows a schematic of the flows and pressures in the LVAD.
plv Rican Lican qi pi
Cican
qci
pith
0mmHg
qvad Axial flowpump
i ω
PI-controller
ωset
paoRocanLocanqopo
Cocan
qco
pith
0mmHg
qvad
Figure 3.25: Schematic of the internal flow mechanics of the Debakey LVAD [1]
The pressures pi and po from figure 3.25 are the pressures at the inlet respectively theoutlet of the rotational pump. Two cannulas carry the flow of blood to and from the LVAD.qi is the flow of blood through the inflow cannula that connects the LVAD to the leftventricle. qo is the flow of blood through the outflow cannula and into the aorta. The inflowcannula is very stiff and therefore has a very small compliance Cican. The outflow cannulais made up of a 23 centimeter flexible tube which has a larger compliance Cocan than theinflow cannula. The hydraulic resistances Rican and Rocan and the inductances Lican andLocan are determined from geometrical dimensions [30].
A PI-controller is used to set the rotational speed ω of the pump to maintain a constantvalue ωset using the current i as a control variable. The rotational speed of the LVAD causesin turn a flow qvad through the pump. The differential equations describing the model canbe determined by looking at figure 3.25. For the inflow
pi =qi − qvadCican
with pi(0) = pi,0 (3.120)
qi =plv − pi − qiRican
Licanwith qi(0) = qi,0, (3.121)
where pi,0 is the initial inlet pressure and qi,0 is the initial inlet flow. In the same way theequations for the outflow are
po =qvad − qoCocan
with po(0) = po,0 (3.122)
qo =po − pao − qoRocan
Locanwith qo(0) = qo,0, (3.123)
3.9. Valsalva Maneuver 51
Table 3.1: Working range for the DeBakey axial flow pump [31].Parameter Working Range
qvad 0 — 180 ml/spo — pi 0 — 120 mmHgω 7500 — 12500 min−1
where po,0 is the initial inlet pressure and qo,0 is the initial outlet flow. According toBaumann [1] and Schima [31] the change in flow through the LVAD qvad can be expressedas
qvad =1
b1
(
−b0qvad + b2ω2 + pi − po
)
, (3.124)
where b0, . . . , b2 are constants taken from Schima [31]. For information on the internalmechanics of the axial pump and the derivations of the mechanical equations see Schima[31]. Table 3.1 show the working range of the axial flow pump.
3.9 Valsalva Maneuver
The Valsalva maneuver is a test which is used to evaluate the overall hemodynamic responseof the cardiovascaular system to fast and large changes in intrathoracic pressure pith. Thismaneuver is used to test the model and its overall hemodynamic response. During theValsalva maneuver, a person forcibly exhales against a closed glottis. This elevates theintrathoracic pressure pith to about 40 mmHg and is kept for about 15 seconds. The expectedhemodynamic response of a healthy person performing the Valsalva maneuver can be seenin figure 3.26. The figure shows the changes of the systemic arterial pressure pap and theheart rate fhr. The oscillations visible in pap are from the pulsatile nature of the heart, asmentioned in chapter 2.
Figure 3.26: Arterial pressure response of a healthy person performing the Valsalva maneu-ver [26].
As can be seen in figure 3.26, the expected hemodynamic response to the Valsalva ma-neuver can be divided into four phases. Phase 1 starts at the beginning of the maneuver
3.10. FVC Maneuver 52
Table 3.2: Summary of the expected hemodynamic changes, for a healthy person, duringthe Valsalva maneuver [26].
Measured Data during Valsalva Maneuver
Normal value Phase 1 Phase 2 Phase 3 Phase 4Cardiac Output [l/min] 4.2 4.07 1.64 1.51 4.49
Heart Rate [min−1] 72 65 97 105 57Pulmonary arterial pressure
23/12 53/46 50/44 22/11 28/15Systole/Diastole [mmHg]
with a rapid increase in intrathoracic pressure due to forced exhalation against a closedglottis. This is accompanied by a rapid rise in arterial pressure, due to the increase inintrathoracic pressure, followed by a drop in heart rate. Phase 2 is the maintaining of theraised intrathoracic pressure. This starts with an initial reduction in arterial pressure andan initial rise in heart rate. It continues with a small recovery of the arterial pressure anda continued rise in heart rate. Phase 3 is the opening of the glottis and the rapid fall ofthe intrathoracic pressure. This is followed by a rapid fall in arterial pressure and a furtherrise in heart rate. Phase 4 is the recovery phase after the termination of the maneuver.This starts with an initial overshoot in arterial pressure accompanied by a rapid drop inthe heart rate. It ends with a recovery of the arterial pressure and the heart rate to normalworking values. Further hemodynamic changes, that can be expected during the maneuver,are presented in table 3.2
The expected hemodynamic response of the Valsalva maneuver is for a healthy person,[7, 11, 24, 26, 37], and in the pathological case [43], well documented.
In figure 3.27 the hemodynamic responses to the Valsalva maneuver for different sever-ities of left heart failure can be seen. Figure 3.27 clearly shows the differences betweenthe hemodynamic response to the Valsalva maneuver of a healthy person and of a personsuffering from left heart failure. For patients with severe left heart failure there is no arte-rial pressure decrease during phase 2 of the Valsalva maneuver as in the healthy situation.There is also no apparent overshoot in arterial pressure after the termination of the ma-neuver. This is the so-called square wave arterial pressure response and is indicative of leftheart failure [43].
3.10 FVC Maneuver
The Forced Vital Capacity maneuver (FVC maneuver) is used to evaluate the cardiovascularresponse of the intra-alveolar pulmonary capillaries to rapid changes in intrathoracic pres-sure. This maneuver is also used to test the model and in particular the cardiopulmonarycoupling. The maneuver is performed by first fully exhaling to the residual volume, then in-haling to the total lung capacity V max
l followed immediately by a full exhalation to residualvolume once again and holding this for about 15 seconds [26]. The changes to intrathoracicpressure pith during the maneuver can be seen in figure 3.28.
The expected internal responses of the intra-alveolar capillaries can be seen in figure3.29. The responses from figure 3.29 are hereby explained. As one inhales i, starting fromresidual lung volume VRV , the pulmonary pre-intra-alveolar capillary resistance Rcap,a andpost-intra-alveolar capillary resistance Rcap,v decrease due to inflation of the lungs whichis followed by an increase in pulmonary capillary inflow qin,pul and outflow qout,pul. Thedifference between inflow and outflow cause a drop in capillary volume Vpc which in turn
3.10. FVC Maneuver 53
Figure 3.27: Arterial pressure responses during Valsalva maneuver. (A) Arterial pressureresponse, healthy patients; (B) absent overshoot arterial pressure response, light to mod-erate left ventricular dysfunction; (C) square wave arterial pressure response, severe leftventricular dysfunction [43].
pit
h [cm
H 20
]
Time [s]
Figure 3.28: Measurement of changes in intrathoracic pressure, taken from a healthy person,when performing the FVC maneuver. The dashed lines mark the start of the first forcedexpiration e, the inhalation i and the final full expiration e∗ [25].
3.10. FVC Maneuver 54
q [l
/min
]Vpc
[m
l]R
[mm
Hgs
/l]
Time [s]
qin,pulqout,pul
RpcRcap,a
Rcap,v
Figure 3.29: Simulated pulmonary intra-capillary responses to the FVC-maneuver [25].
increases the pulmonary capillary resistance Rpc. When one starts the final forced expiratione∗, the pressure in the intra-alveolar pulmonary capillaries ppc is relatively low which causea greater inflow than outflow and Vpc increases. The increase in pith causes an increase inRcap,a and Rcap,v which in turn lowers inflow qin,pul and outflow qout,pul. The flow ratesstabilize at around 50 % of normal value for the rest of the maneuver and recover whennormal breathing resumes [25].
Chapter 4
Matlab Simulink Implementation
This chapter describes the implementation that is done in the computational program Mat-lab Simulink. The system requirements are also shortly explained.
4.1 Simulink
The full cardiovascular model is implemented in Simulink. Most of the equations describingthe model lie within a Simulink model block and are written in Matlab within S-functions.Some of the equations describing the control loops with inherent delays and complex functionhave instead been implemented with Simulink blocks. These control loops include:
– Unstressed volume controller
– PI-controller for axial flow pump
4.2 Matlab-code
Most of the equations detailed in this report were implemented in Matlab and written inC as to increase the computational speed. The equations were implemented as first orderdifferential equations. The Matlab-program is made up by the following functions
– mdlInitializeSizes: Initialization of arrays.
– mdlInitializeSampleTimesInitialize: Initialization of the sample time arrays.
– mdlInitializeConditions: Setting of initial states.
– mdlOutputs: Calculation of outputs.
– mdlUpdate: Computation of states.
– mdlDerivatives: Calculation of differential equations.
– mdlTerminateClean: Clean up memory after end of simulation.
The sequence of the function calls can be seen in figure 4.1.
55
4.3. System Requirements 56
Simulink Engine
mdlCheckParameters
Called when
parameters change
mdlProcessParameters
mdlUpdate
mdlGetTimeOfNextVarHit
mdlInitalizeConditions
mdlOutputs
mdlDerivatives
Called when
parameters change
Called if sample time
of this S-function varies
Major Time Step
Minor Time Step
mdlDerivatives
mdlOutputs
mdlOutputs
mdlZeroCrossings
Called if this S-function
detects zero crossings
Zero Crossing Detection
Integration
Called if this S-function
has continuous states
END SIMULATION
START SIMULATION
Figure 4.1: Function call of the Matlab-program during simulation.
4.3 System Requirements
To run the model one will need to have Matlab with the Simulink package installed on onescomputer.
Chapter 5
Results and Conclusions
In this chapter the simulation results will be presented. The results will include simulationsof:
1. Physiological and pathological hemodynamic responses with and without LVAD sup-port.
2. Hemodynamic responses to orthostatic stress.
3. Hemodynamic responses to the Valsalva maneuver, both in physiological and patho-logical situations.
4. Hemodynamic responses of the intra-alveolar pulmonary capillaries to the FVC-maneuver.
The results will mostly focus on the new parts of the extended model.
5.1 Normal Physiological Conditions
This section displays the simulation results during physiological conditions which correspondto a healthy person. Figure 5.1 shows, on the left, the pressure of the left ventricle duringone beat. The pressures seen in the figure are the left ventricular pressure, which risesand falls rapidly during contraction and relaxation of the ventricle respectively, the aorticpressure which rises when blood is ejected from the ventricle and decreases slowly duringthe filling phase, and the mean aortic pressure averaged over one heart beat. On the rightin figure 5.1, the associated PV-loop of the left ventricular pressure is drawn. The pointsA, B, C and D correspond to the same points in the figures and are the phases described insection 2.2. The stroke volume, Vsv ≈ 90 ml, can be read from the picture as the horizontaldistance between point B and C in the PV-loop which agrees with literature [17]. Thesimulated heart rate during physiological conditions is 62 min−1, which together with thestroke volume would indicate a cardiac output of 5.6 l/min. Figure 5.2 shows the sameresponses for the right ventricle. The results show that pao has a systolic pressure of 140mmHg and diastolic of 80 mmHg with an average of 100 mmHg. pap has a systole/diastolepressures of 32/13 mmHg with an average of 20 mmHg. These values are slightly higher thanwhat was described in section 2.3. This is probably because of the adjustments that havebeen made to the systemic venous model. To optimize the systolic and diastolic pressures ofthe left and right ventricle a review of the hemodynamic parameters governing the systemic
57
5.1. Normal Physiological Conditions 58
pao
pao,mean
plv
D
C
B
A D
C
B
A
Pre
ssure
[mm
Hg]
Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
150
D
C
B
AEDPVR
plv
[mm
Hg]
Vlv [ml]
ESPVRPV-loop
0 20 40 60 80 100 120 140 1600
20
40
60
80
100
120
140
160
Figure 5.1: Pressure profiles of the left ventricle (left) and the associated PV-loop (right).
pap
pap,mean
prv
D
C
BA
Pre
ssure
[mm
Hg]
Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
25
30
35
40
D
C
B
AEDPVR
prv
[mm
Hg]
Vrv [ml]
ESPVR
PV-loop
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
20
25
30
35
Figure 5.2: Pressure profiles of the right ventricle (left) and the associated PV-loop (right).
5.1. Normal Physiological Conditions 59
circulation is recommended. Figure 5.3 shows, on the left, a Pressure-Volume (PV) loop ofthe intraventricular septum and the corresponding change in intraventricular septum volumeduring one second of simulation.
Systole
Diastole
Vspt
[ml]
pspt [mmHg]0 20 40 60 80 100 120
0
1
2
3
4
5
6
7
8
Systole Diastole
Sep
tum
vol
um
eVspt
[ml]
Time [s]0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
1
2
3
4
5
6
7
8
Figure 5.3: PV-loop of the intraventricular septum (left). Changes in Vspt during simulation(right).
5.1.1 Breathing
To simulate normal breathing, pith was chosen as a sinusoidal function pith = −4.5 +
sin
(
2π
τbt
)
, where τb is the time period for one breath and was set to 5.5 seconds. Figure
5.4 shows the change in alveolar volume Valv during normal breathing calculated from themodel. The tidal volume is approximately 500 ml which agrees with the results found inliterature [13]. Figure 5.5 shows the effect of breathing on the PV-loop for the left and rightventricle.
The observable change in the PV-loops originates from the intrathoracic pressure pithand the effect it has on the vessels and chambers that lie within the thoracic cage. Duringnormal breathing, pith changes and thereby the ventricle pressures change as according toequation (3.24).
5.1. Normal Physiological Conditions 60
Lung
volu
meValv
[l]
Time [s]0 2 4 6 8 0 2 4 6 8 0
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Figure 5.4: Changes in lung volume Valv during normal breathing.
D
C
B
AEDPVR
plv
[mm
Hg]
Vlv [ml]
ESPVRPV-loop
0 20 40 60 80 100 120 140 1600
20
40
60
80
100
120
140
160
D
C
B
AEDPVR
prv
[mm
Hg]
Vrv [ml]
ESPVR
PV-loop
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
20
25
30
35
Figure 5.5: Effect of breathing on the PV-loop for the left (left) and right (right) ventricle.
5.2. Left Heart Failure 61
5.2 Left Heart Failure
This section describes the simulated hemodynamic responses of the cardiovascular system ofa person suffering from left heart failure. The results also include the expected hemodynamicchanges for a patient with LVAD support when the LVAD rotor is working at a turning speedof 9600 rpm.
With LVAD
Without LVAD
EDPVR
plv
[mm
Hg]
Vlv [ml]
ESPVR
0 50 100 150 200 250 3000
20
40
60
80
100
120
Figure 5.6: PV-loop of left ventricle under left heart failure with and without LVAD support.
Figure 5.6 shows the PV-loop for the left ventricle of a patient suffering from left heartfailure with and without LVAD support. The results show how the systolic pressure inthe ventricle plv ≈ 100 mmHg are lower than normal and, compared to figure 5.1, howthe working volume of the left ventricle greatly increases 200 ml < Vlv < 250 ml which isindicative of left heart failure [17, 13]. With LVAD support the working volume of the leftventricle is reduced 170 ml < Vlv < 220 ml which would reduce cardiac muscle tension andthereby put less strain on the heart. The average heart rate is reduced from 87 to 74 min−1
with LVAD support.
With LVAD
Without LVAD
EDPVR
prv
[mm
Hg]
Vrv [ml]
ESPVR
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
20
25
30
35
40
45
Figure 5.7: PV-loop of right ventricle under left heart failure with and without LVADsupport.
5.3. Orthostatic Stress 62
Figure 5.7 shows the same responses but for the right ventricle. Notice here the increasedstroke volume and size of PV-loop under LVAD support thereby increasing the cardiacoutput towards a more normal working range. The changes in cardiac output with andwithout LVAD support are summarized in table 5.1
Table 5.1: Changes in cardiac output for the left and right ventricle when under LVADsupport.
Left Ventricle [l/min] Right Ventricle [l/min] LVAD [l/min]
Without LVAD 4.0 4.0 0.0With LVAD 1.5 5.5 4.0Healthy 5.6 5.6 -
Table 5.1 shows the simulated cardiac output for the left and right ventricle for a patientsuffering of left heart failure without LVAD support and the changes with LVAD support.An increase, back to the normal working range of > 5 l/min, in cardiac output of the rightventricle can be observed in the table. On the other hand the cardiac output of the leftventricle drops drastically to about 1.5 l/min and instead the blood flows through the LVADat 4 l/min. This results in a total cardiac output from the left ventricle of 5.5 l/min.
The change in intraventricular septum volume Vspt with and without LVAD support canbe seen in figure 5.8. The reason for this change can be attributed to the effect the LVADhas on the ventricle pressures as observed in figures 5.6 and 5.7.
With LVAD
Without LVAD
Vspt
[ml]
pspt [mmHg]0 10 20 30 40 50 60 70 80
0
2
4
6
8
10
12
14
16
18
20
With LVAD
Without LVAD
Vspt
[ml]
Time [s]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
14
16
18
20
Figure 5.8: VP-loop of the intraventricular septum with and without LVAD support (left).Changes in Vspt during simulation with and without LVAD support (right).
5.3 Orthostatic Stress
This section describes the hemodynamic response of a healthy person when under orthostaticstress. This is done by simulating a patient performing a Head Up Tilt (HUT) test. A HUTtest is done by having a patient lying on a tilt table, as described in section 3.5, and graduallyincreasing the tilt angle α = from 0 to 75◦ and then back to zero again. The response ofthe systemic venous system is simulated and displayed in order to evaluate the displacedvolumes in the vascular compartments.
5.3. Orthostatic Stress 63
fhr
α
pao,mean
DownUp
pao,m
ean
[mm
Hg]
,α
[◦],f h
r[m
in−1]
Time [s]100 150 200 250 300 350 400 450 500 550 6000
20
40
60
80
100
120
140
160
Figure 5.9: Mean aortic pressure pao,mean response when subjected to orthostatic stressthrough Head Up Tilt (HUT) test. The tilt angle α and heart rate fhr response is alsodisplayed.
Figure 5.9 shows the simulated mean aortic pressure pao,mean response to the HUT test.The dashed black lines mark the start of tilt up to 75◦ and the start of tilt down back to levelposition. pao,mean is taken as the averaged aortic pressure of one heart beat. Also plotted inthe same figure is the heart rate fhr and the tilt angle α to give an easy overview of wherethe test starts and ends. This does not agree to that described in literature [16, 18]. Thesimulated results shows how pao,mean initially drops and then recovers. However pao,mean
does not fully recover to normal working value of 100 mmHg, as described in literature[16, 18], but stagnates at around 80 mmHg. This is probably because the modeled reflexmechanisms is not enough to fully compensate for the orthostatic pressure changes andvolume distributions occuring during orthostatic stress. This displacement of pao,mean fromthe reference value causes the simulated heart rate to stay elevated during the test, eventhough it should drop to about 70 beats/min according to literature [16, 18].
qo,rv
qo,lv
Car
dia
cou
tput
[l/m
in]
Time [s]100 150 200 250 300 350 400 450 500 550 6001
2
3
4
5
6
7
8
Vsv,r
Vsv,l
DownUp
Str
oke
Vol
um
e[m
l]
Time [s]100 150 200 250 300 350 400 450 500 550 6000
20
40
60
80
100
120
140
Figure 5.10: Cardiac output (left) and stroke volume (right), averaged over one heart beat,for the left Vsv,l and right ventricle Vsv,r when subjected to orthostatic stress (right).
5.4. Valsava Maneuver 64
In figure 5.10, on the left, the cardiac output of the left and right ventricle during theHUT test and on the right, the stroke volume of the left and right ventricle are displayed.When compared to the tabulated values in [18] it can be seen that the drop in both Vsv,i andcardiac output, during the HUT test, is more than what would be expected. In Heusden[18] cardiac output decrease with about 16 — 27 % of original values and stroke volumeabout 30 — 45 % during the maneuver. In figure 5.10 the drop in cardiac output can bemeasured to about 50 % and for stroke volume to about 70 %. This measurement is takenas the difference to the stabilized value that is reached during the maneuver and not theinitial undershoot that can be observed in the figure. These values are much higher thanwhat has been observed in literature [18] and is probably because the current control reflexesimplemented are not enough to fully counteract the hemodynamical changes of orthostaicstress. This would suggest that an improvement in the control loops is needed or an additionof a heart contractility controller.
Vleg
Vsp
Vkid
Vhead
DownUp
Vol
um
e[m
l]
Time [s]100 150 200 250 300 350 400 450 500 550 6000
500
1000
1500
2000 Vtot
DownUpB
lood
Vol
um
e[m
l]
Time [s]
Vheart
Vsys,v
Vsys,a
Vpul
100 150 200 250 300 350 400 450 500 550 6000
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Figure 5.11: Volume displacements in the venous compartments (left) and blood volumesin different parts of the body (right) when subjected to orthostatic stress.
Figure 5.11 shows the simulated volume displacements across the vascular compartmentsduring the HUT test. The largest volume displacement occurs in the legs, about +400 ml.The displaced volume in the head is about -150 ml, in the splanchnic about +150 ml and inthe kidneys +50 ml. When adding these volumes together it is clear that the total volumechange in the systemic veins do not add up to zero. To compensate blood is redestributedfrom other parts of the cardiovascular circulation, which can be seen on the right in thefigure. Some blood from the systemic arteries Vsys,a, the pulmonary circulation Vpul andthe heart Vheart goes into the systemic veins Vsys,v during orthostatic stress. The totalblood volume Vtot stays constant during the simulation which confirms the conservation ofblood in the body.
5.4 Valsava Maneuver
This section shows the simulated hemodynamics of the cardiovascular system when perform-ing the Valsalva maneuver. Figure 5.12 shows the driving force that performs the Valsalvamaneuver, the intrathoracic pressure pith. When pith rises, the maneuver begins and whenit drops the glottis is opened. The figure also shows the mean aortic pressure response
5.4. Valsava Maneuver 65
pao,mean
pith
Pre
ssure
[mm
Hg]
Time [s]130 140 150 160 170 180 190 200
0
20
40
60
80
100
120
140
160
Figure 5.12: Mean aortic pressure pao,mean response during Valsalva maneuver plottedtogether with the changes in intrathoracic pressure pith.
pao,mean during the maneuver. As can be seen, all the four phases described in paragraph3.9 are present and agree with literature [7, 11, 26, 37].
The heart rate response during the maneuver can be seen in figure 5.13. According toliterature [24, 26] there should be an initial decrease in heart rate followed by a steady riseto a value of about 150 % of normal. The simulated results show a very similar behaviouronly that the rise in heart rate is up to 175 % of normal value. The reason for this differencecould be due to slight differences in the heart rate controllers between literature and thisMaster’s Thesis. At the end of the maneuver there is a drop followed by a stabilization ofthe heart rate at the normal working value also agreeing with literature [26, 24]. Examiningthe results of the simulations more closely, the heart dynamics of the simulated results seemto be more fast reacting than that of the experimental results shown in Liang et al. [24, 26].This conclusion is drawn from the fact that the initial drop and following increase in heartrate at the start of the maneuver is much steeper in the simulated results.
f hr
[min
−1]
Time [s]130 140 150 160 170 180 190 200
40
50
60
70
80
90
100
110
120
Figure 5.13: Heart rate fhr response during Valsalva maneuver.
Figure 5.14 shows the changes of cardiac output of the ventricles and the changes ofsystemic arterial resistance Ras during the maneuver. The resulting cardiac output, 60 % of
5.4. Valsava Maneuver 66
normal value, is higher during the maneuver than reported in Lu [26] where cardiac outputdrop to about 40 % of normal value. The increase in Ras during phase 2 of the maneuveris probably because of the drop in transmural aortic pressure pao,tm and is also a reason forthe slow recovery in aortic pressure during the later stages of phase 2.
qo,rv
qo,lv
Car
dia
cou
tput
[l/m
in]
Time [s]130 140 150 160 170 180 190 2000
2
4
6
8
10
12
Ras
[mm
Hgs
/ml]
Time [s]130 140 150 160 170 180 190 200
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Figure 5.14: Cardiac output (left) for the left and right ventricle during the Valsalva ma-neuver; Baroreflex changes of systemic arterial resistance Ras during the maneuver (right).
5.4.1 Pathological Conditions
This section describes the simulated results of the Valsalva maneuver of patients sufferingfrom left heart failure. Figure 5.15 shows the changes in pao,mean during the maneuver.Notice the large difference between the pathological case here and the healthy case in figure
pao,mean
pith
Pre
ssure
[mm
Hg]
Time [s]130 140 150 160 170 180 190 200
0
20
40
60
80
100
120
140
160
pao [
mm
Hg]
Figure 5.15: Mean aortic pressure pao,mean response during Valsalva maneuver and intratho-racic pressure pith (left). Expected pathological aortic pressure response during the Valsalvamaneuver [43] (right).
5.12. As described in literature [43] there is no visible reduction in pao,mean during phase
5.4. Valsava Maneuver 67
2 of the maneuver and no overshoot after the release of the maneuver all indicative of leftheart failure. Figure 5.16 shows the simulated heart rate during the Valsalva maneuver inthe pathological case.
f hr
[min
−1]
Time [s]130 140 150 160 170 180 190 200
40
60
80
100
120
140
160
Figure 5.16: Heart rate fhr response during Valsalva maneuver when suffering from leftheart failure.
The cardiac output and the changes to systemic arterial resistance Ras in the pathologicalcase can be seen in figure 5.17. Since the patient is suffering from left heart failure thecardiac output is lower than normal, only 4.2 l/min. Notice the relatively constant cardiac
qo,rv
qo,lv
Car
dia
cou
tput
[l/m
in]
Time [s]130 140 150 160 170 180 190 2000
2
4
6
8
10
12
Ras
[mm
Hgs
/ml]
Time [s]130 140 150 160 170 180 190 200
0.9
1
1.1
1.2
1.3
1.4
Figure 5.17: Both figures represent the case of left heart failure. Cardiac output for theleft and right ventricle (left) during the Valsalva maneuver. Baroreflex changes to systemicarterial resistance Ras during the maneuver (right).
output during the maneuver. In comparison to the physiological case, where the cardiacoutput drops to about 60 % of normal value, the cardiac output has an initial drop but thenreturns to the normal pathological value. The reason for this is not clear but is probablyconnected to the left ventricle dynamics and how they differ from the pathological case andthe physiological. This result together with the square wave form of the arterial pressure
5.5. FVC-Maneuver 68
response during the maneuver would indicate that the modelled pathological heart is lesssensitive or less adaptable to changes in intrathoracic pressure. This could be connectedwith the already heightened reflex activity, Ras ≈ 1.1 mmHgs/ml, which is accompaniedwith left heart failure.
5.5 FVC-Maneuver
This section details the hemodynamic responses of the cardiovascular system when per-forming the FVC maneuver. As previously mentioned extra attention will be given to theresponse of the pulmonary capillary system.
pith
palv
e∗ie
Pre
ssure
[mm
Hg]
Time [s]0 2 4 6 8 10 12 14 16 18 20
−40
−20
0
20
40
60
80
Figure 5.18: Changes in alveoli pressure palv during FVC maneuver plotted together withthe driving intrathoracic pressure pith.
Figure 5.18 shows the changes in the alveoli palv and intrathoracic pressure pith duringthe maneuver. Notice that they only differ when pith is negative. This is connected to thedynamics of the transmural pressure pel which is zero in the lower lung volumes that occurwhen intrathoracic pressure rises and air is pressed out of the lungs. During inspiration thelungs inflate due to a negative pith and pel increases causing a pressure difference betweenpith and palv as can be seen in the figure.
Figure 5.19 shows, on the left, the changes in pulmonary capillary volume Vpc. Theresults concur well with the ones presented in Liu [25], which can be seen on the right inthe figure. At the beginning of the maneuver, Vpc drops due to the inflation of the lungs, adrop in pre-/post capillary resistances Rcap,a/Rcap,v and the difference between inflow andoutflow of the capillaries.
The changes in pulmonary resistance can be seen in the top of figure 5.20 together withthe expected results from Liu [25] on the bottom. The drop in Vpc is followed by an increasein pulmonary capillary resistance Rpc. As expiration starts the pulmonary outflow is largerthan the inflow, as can be seen on the top in figure 5.21, which causes Vpc to recover andRpc to drop once again. Rcap,a and Rcap,v increases due to increasing pith. Rcap,a andRcap,v remains elevated until the end of the maneuver when pith and lung volume return tonormal. The two sudden drops at 8 and 12 seconds indicate a numerical error somewherein the solution. The continuity of the equations modeling the pulmonary capillaries needsto be reexamined.
The shape of the curves in figure 5.20 corresponds well to each other, with the excep-tion that the simulated results from this Master Thesis has a slight numerical error 8 and
5.5. FVC-Maneuver 69
e∗ie
Vpc
[ml]
Time [s]0 2 4 6 8 10 12 14 16 18 20
10
20
30
40
50
60
70
80
Time [s]
Vpc
[ml]
Figure 5.19: Changes in pulmonary capillary volume Vpc during FVC maneuver (left).Expected changes in Vpc according to Liu [25] (right).
Rpa
Rpv
Rpc
e∗ie
Rcap,a
Pulm
onar
yre
sist
ance
s[m
mH
gs/l
]
Time [s]
Rcap,v
Rpc
0 2 4 6 8 10 12 14 16 18 20
0
200
400
600
800
1000
1200
Rpc
Rcap,a
Rcap,v
Time [s]
R [mm
Hgs/l
]
Figure 5.20: Changes in pulmonary capillary resistances, Rpc, Rcap,a and Rcap,v during theFVC-maneuver Vpc (top). Expected changes to pulmonary resistances according to Liu [25](bottom).
12 seconds into the simulation. The reason why the simulated results for the pulmonaryresistances are much higher than that reported in Liu [25] is probably due to the slight dif-ferences in the shape of the pith curve used to simulate the maneuver. Even though Rpc isthree times larger in figure 5.20 than reported in Liu [25] this should not be seen as an error
5.5. FVC-Maneuver 70
in the simulation. The reason for this relatively large difference is most likely because of thestrong dependence of Rpc on Vpc which has the result that, in the low Vpc range, even a verysmall change in Vpc causes very large changes to Rpc. Another reason the model does notexactly replicate the results presented in Liu [25] is because the model in Liu [25] is for anisolated model of the pulmonary capillaries and in this Master Thesis the local pulmonarycapillary model has been implemented in a larger model of the entire cardiovascular systemwhich could have an effect on the absolute values of the resistances.
qout,pul
qin,pul
e∗ie
Pulm
onar
yflow
[l/m
in]
Time [s]0 2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
qin,pul
qout,pul
Time [s]
q [l/
min
]
Figure 5.21: Changes in pulmonary capillary flow qin,pul and qout,pul (top). Expectedchanges to pulmonary capillary flow according to Liu [25] (bottom).
The changes in pulmonary capillary flow can be seen on the top of figure 5.21 togetherwith the simulated results from Liu [25] on the bottom of the figure. These results aresomewhat harder to compare because in Liu [25] the transmural pulmonary arterial andvenous pressures were assumed to be constant, which is not the case in the model presentedin this Master Thesis. Because of the pulsatile nature of the heart, the pulmonary flowsfluctuate between heart beats which makes an exact comparison between the figures incon-clusive. What can be extrapolated from the figures is that, during the fast inhalation, thereis indeed a difference between inflow and outflow of the pulmonary capillaries which in turncauses the drop in Vpc as was seen in figure 5.19. This pulmonary flow difference shiftsduring the forced expiration increasing Vpc. The difference in pulmonary capillary inflowand outflow disappears as Vpc → Vpc,max and is kept zero until the end of the maneuver.That is why the curves for qin,pul and qout,pul overlap the last ten seconds of the maneuver.The two square peaks 8 and 12 seconds into the simulation are due to the same numericalerror affecting Rcap,a and Rcap,v. This is clear since a sudden change in resistance directlyaffect the flow through a system.
Figure 5.22 shows the change in lung volume Valv during the maneuver. Comparing the
5.5. FVC-Maneuver 71
e∗ie
Lung
vol
um
eValv
[l]
Time [s]0 2 4 6 8 10 12 14 16 18 20
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Va
lv -
VR
V [
l]
Time [s]
Figure 5.22: Changes in lung volume Valv during the FVC-maneuver (left). Expectedchanges to lung volume according to Liu [25] (right)
simulated results with the ones from Liu [25] in figures 5.21 and 5.22 it can be seen that theresults agree well with literature.
Rpc
[mm
Hgs
/l]
Active lung volume Valv − VRV [l]
Expiration
Inspiration
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
200
400
600
800
1000
1200
Valv - VRV [l]
R
�� [mmHgs/l]
Figure 5.23: Changes in pulmonary capillary resistance Rpc plotted against changes in alveolivolume Valv (left). Expected results according to Liu [25] (right).
The left side of the figures 5.23 and 5.24 shows the changes to Rpc and Rcap,a/Rcap,v
plotted against Valv − VRV . It can be seen that the overall behavior and shape of theresistance loops are as describe in Liu [25] on the right side of the figures but, as mentionedearlier, the shape of the loops are more square shaped and the values of the resistances arehigher than described in Liu [25].
The flow rate qair,2 changes, due to changes in Valv − VRV as can be seen in figure 5.25.The small peaks that can be observed in the figure comes from numerical approximations inthe equations describing the transmural pressure pel. The peaks are only one time step longand happen just as pith = 0 but seem to have no effect on the overall solution. This couldbe due to some error in the continuity of the equations describing the airways at pith = 0.
5.5. FVC-Maneuver 72
Rcap,a/R
cap,v
[mm
Hgs
/l]
Active lung volume Valv − VRV [l]
Expiration
Inspiration
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−100
0
100
200
300
400
500
R cap,a
, Rca
p,v [m
mHgs/
l]
Valv-VRV [l]
Figure 5.24: Changes in pulmonary pre-/post-capillary resistances Rcap,a/Rcap,v plottedagainst changes in active lung volume Valv − VRV (left). Expected results according to Liu[25] (right).
q air,2
[l/s
]
Active lung volume Valv − VRV [l]
Expiration
Inspiration
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10
−8
−6
−4
−2
0
2
4
6
8
10
VA - VRV [l]
qair
,2 [
l/s]
Figure 5.25: Airway flow response qair,2 plotted against changes in alveoli volume Valv (left).Expected results according to Liu [25] (right).
Chapter 6
Review
This chapter contains a review over the results presented, over any limitations of the systemand future work that could or needs to be done.
6.1 Achieved Results
The largest problem encountered when trying to compare simulated results to that of litera-ture and experiments is that every person is an individual with small differences in their ownhemodynamics, i.e. lung size, cardiac output, normal heart rate can all vary from personto person. This makes it hard to say that a result is false or correct. Therefore a morequalitative approach is needed with a quantitative comparison done by always consideringinter-subject variability.
When looking at the overall simulation results of the system one can observe a very goodcomparison with ones reported in literature. Both during normal conditions and pathologicalconditions with and without LVAD support the results are very similar to those reported inBaumann [1], which was the goal.
There are three points in which a deeper discussion is needed because they are part ofthe new model and therefore important to understand:
– Cardiovascular response due to orthostatic stress.
– Cardiovascular response during the Valsalva maneuver.
– The response of the intra-alveolar pulmonary capillaries during the FVC maneuver.
The first results that will be discussed are the hemodynamic responses of the model toorthostatic stress. The most important thing that the human body controls is the meantransmural aortic pressure [17]. When looking in literature [16, 18], the aortic pressure is,when standing up or performing the HUT test, mostly constant. What is different in thesimulated results from literature is the initial drop in aortic pressure at the beginning of theHUT test. By comparing the simulated results to [16, 18] one can observe a deviation fromwhat has been reported. A possible reason for this would be accountable to the systemicarterial resistance controller, the heart rate controller and stress relaxation have been im-plemented as regulatory mechanisms. The truth is that there are several more mechanismsthat help regulate the cardiovascular system, e.g. a changeable heart contractility due tobaroreflex control. It could be that the existing regulatory mechanisms are not enough
73
6.2. Limitations 74
to immediately compensate for the large hemodynamic changes that take place during theHUT test, which makes this an area of improvement.
The next results to be discussed into more detail are about the hemodynamic responses ofthe Valsalva maneuver. The results of the Valsalva maneuver both in the physiological andpathological cases are very promising. In the physiological case all four phases describedin theory could be detected and coincided with earlier results [7, 11, 24, 26, 37]. In thepathological case it was also very positive to see the change in aortic pressure response tothe square wave form indicative of left heart failure as according to literature [43].
The last results to be discussed relate to the internal hemodynamic responses of the intra-alveolar pulmonary capillaries. Even though the responses for the pulmonary capillarieswere overall very good and in accordance to literature [25, 26] one small problem lay inthe lack of comparable literature. Since the model was based on the same source as thecomparable results, even though the results were similar it is hard to draw any conclusions.The parameter values describing the airway mechanics have been taken and verified from[25] for just one simulated person. This is something that should probably be varied andimplemented for a number of people in order to more solidly verify the model. Due to thelack of time this was not possible in this Master’s Thesis.
Nevertheless, even though the internal responses of the pulmonary capillary model wereonly verified against a single source this was not the only test performed. All the globaltests done, during normal conditions, pathological conditions, Valsalva maneuver show veryrealistic hemodynamic responses. This can only be true when the equations describing theoverall dynamics of the pulmonary capillaries are correct.
6.2 Limitations
When discussing the limitations of the model it is important to know that the model is onlysupposed to be a close approximation of the human cardiovascular system. The model issupposed to display as much as possible of the complex dynamics that exist in the cardio-vascular system keeping the model as simple as possible.
There are however some important limitations to the current model. One of the largerlimitations of the model could be seen in the hemodynamic responses to orthostatic stress.
There is also, to date, no way, to simulate physical stress with the present model, i.e.exercise and an increasing cardiac output. The only way to increase cardiac output isthrough an infusion of blood but there is a limit to how much blood can be infused beforethe model crashes. With infusion the model response of an increase in cardiac output ofabout 100 % (10 l/min) is measurable before it crashes. This is probably related to theequations governing the heart mechanics because with an increase in total blood volume theventricles will fill with more blood. When too much blood is forced into the ventricles theequations describing the ESPVR and the EDPVR are no longer valid and the simulationcrashes. It also a realistic response since there is a limit to how much blood can be infusedinto a person.
6.3 Future work
There is still much that could be done in regards to future work. As mentioned earlier, amodel of the human cardiovascular system is almost never finished. Even though modelverifications have been done against literature a future Master’s Thesis could still be theverification of this model. This should be done by someone with a medical background since
6.3. Future work 75
there is much more to verification than just comparing the results to literary references. Itrequires a lot of foreknowledge and sound judgement.
Something that could be done relatively quickly, but was left out due to a shortage oftime, is the reimplementation of the unstressed volume controller in the Matlab-code. Thisonly requires an implementation of the differential equation describing the control loop inthe Matlab-code. Due to the complexity of the inner mechanics of the axial flow pump thePI-controller could be left as implemented in Simulink.
Another field of further study could be the reduction of the model and the future develop-ment of automatic control strategies for LVADs. As the results in this Master’s Thesis haveshown, the LVAD, when in use, almost completely takes over the workings of the left ven-tricle. Instead of supporting the heart, the LVAD takes over and in some cases reduces thechances of a normal recovery. With new control strategies, the LVAD could be adjusted towork with the heart and further recovery making them suitable as therapy oriented devicesand not only for medium to long term support until transplantation.
Chapter 7
Acknowledgements
This Master Thesis Project was done in cooperation with the Automation and ControlInstitute (ACIN) at Vienna University of Technology, the Research Group in CardivascularDynamics and Artificial organs at the Medical University of Vienna (MUV) situated at theGeneral Hospital of Vienna and Umeå University.
First and foremost I would like to thank Univ.-Prof. Dr. techn. A. Kugi (ACIN) andUniv.-Prof. Dr. techn. H. Schima (MUV) for their support and for the provition of awork area. A great thanks go out to my supervisors Dipl.-Ing. Dr. techn. FrancescoMoscato (MUV), Dr.-Ing. Wolfgang Kemmetmüller (ACIN) and Katrin Speicher (ACIN)for supporting me, coming with suggestions and correcting my work. A special thanks alsogoes out to DI Michael Baumann.
76
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Appendix A
Parameters and Variables
A list of all variables with explanatory comments can be found in the source code cvmodel.c.A list of all parameters and their values, also including comments, can be found in theparameter file data_CV_model_BAUMANN_Michael.m.
80
Appendix B
Full Model Schematic
81
82
pla
Cla
pith
0 mmHg
Rd,mv
Lmv
D1
Ri,mv D2
Rd,av
Lav
D3
Ri,av D4
Rlv
Clv
plv
pith
0 mmHg
Rao
Cao
pao
pith
0 mmHg
Lao
Cas
0 mmHg
Ras Rleg,1
Cleg
portho,leg
Cv
Rleg,2
0 mmHg0 mmHg
pv
Rkid,1
Ckid
pabd
Rkid,2
portho,kid
0 mmHgRsp,1
Csp
pabd
Rsp,2
portho,sp
0 mmHg
Rhead,1
Chead
Rhead,2
portho,head
0 mmHg
Rsvc
Csvcpith
0 mmHg
Rivc
Civc
pith
0 mmHg
Cra
pith
0 mmHg
praRd,pv
Lpv
D7
Ri,pv D8
Rd,tv
Ltv
D5
Ri,tv D6
Rrv
Crv
pith
0 mmHg
prvRpul
Cpul
pith
0 mmHg
pap
Ccap
pith
0 mmHg
pcap,aRcap,aRpc/2Rpc/2Rcap,v
palv
ptmb
0 mmHg
ppcRvp
Cvp
pith
0 mmHg
pvp
Cannula and Pumppao
Control
pao pao,ref
pao Control
pao pao,ref
fhr
Airway
Mechanics
0 mmHg
pith
pel
ptm
RLti
palv
RS
qair,2
Rcoll
qair,1
pC
pref
Figure B.1: Schematic of full cardivascular model.
83
pla
Cla
pith
0mmHg
Rd,mv Lmv D1
Ri,mv Lmv D2
Rd,av Lav D3
Ri,av Lav D4
Rlv
Clv
plv
pith
0mmHg
Rao
Cao
pao
pith
0mmHg
Lao
Cas
pith
0mmHg
Ras Rleg,1
Cleg
portho,leg
Cv
Rleg,2
0mmHg0mmHg
pv
Rkid,1
Ckid
Rkid,2
portho,kid
0mmHg
Rsp,1
Csp
Rsp,2
portho,sp
0mmHg
Cra
pith
0mmHg
praRd,pv Lpv D7
Ri,pv Lpv D8
Rd,tv Ltv D5
Ri,tv Ltv D6
Rrv
Crv
pith
0mmHg
prvRpulLpul
Cpul
pith
0mmHg
papRcap
Ccap
pith
0mmHg
Rvp
Cvp
pith
0mmHg
pvp
Cannula and Pumppao
Control
pao pao,ref
pap
Control
pappla
pao Control
pao pao,ref
fhr
Figure B.2: Schematic representing the human cardiovascular system according to Baumann[1].
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