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Minimal haemodynamic

modelling of the circulation

P.C.I. Spelde

Master Thesis in Applied MathematicsApril 2008

Minimal haemodynamic

modelling of the circulation

P.C.I. Spelde

First supervisors: A.E.P. Veldman and G. RozemaSecond supervisor: A.J. van der SchaftExternal supervisor: N.M. Maurits (UMCG)

Institute of Mathematics and Computing ScienceP.O. Box 4079700 AK GroningenThe Netherlands

Abstract

The knowledge of the flowstructures in the human arteries is limited. The medical staff havethe wish to have a better side to this phenomenon. In a specific mathematical research of theflow through the carotid bifurcation there is attention for this problem. To make it possible todo this research a mathematical model of the whole cardiovascular system (CVS) is needed.

Models found in literature simulate specific areas of the CVS while others are either overlycomplex, difficult to solve, and/or unstable. This thesis develops a minimal model with theprimary goal of having the possibility to reflect accurately a small part of the cardiovascularsystem. The focus is just on the simplicity of the overall structure, with a reasonable reflectionof the heartfunction. A novel mixed-formulation approach to simulating blood flow in lumpedparameters CVS models is outlined that adds minimal complexity, but significantly improvesphysiological accuracy.

The minimal model is shown to match a Wiggers’ diagram and was also verified to simulatedifferent heartdiseases. The model offers a tool that can be used in conjunction with experimentalresearch to improve understanding of the blood flow.

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Contents

1 Introduction 11.1 Physiology of the blood circulation system . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The blood circulation system . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 The bloodvessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Cardiac function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Cardiovascular System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Finite Elements Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 The Pressure-Volume Approach . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Windkessel circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.4 Wiggers’ diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Literature study 132.1 Minimal Heamodynamic Modelling of the Heart & Circulation for Clinical Ap-

plication [SMITH] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Reduced and multiscale models for the human cardiovascular system; one dimen-sional model [FORVEN] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Reduced and multiscale models for the human cardiovascular system;lumped pa-rameters for a cylindrical compliant vessel [FORVEN] . . . . . . . . . . . . . . . 222.3.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Computational modeling of cardiovascular response to orthostatic stress [HSKM] 252.4.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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2.4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 An identifiable model for dynamic simulation of the human cardiovascular system[KRWIWAKR] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Interaction between carotid baroregulation and the pulsating heart: a mathemat-ical model [URS] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7.1 Overview Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7.3 Final Choises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Model description 413.1 The derivation of a mathematical model . . . . . . . . . . . . . . . . . . . . . . . 413.2 The 0D model for the circulation system, bottum-up approach . . . . . . . . . . 42

3.2.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 The 0D model for the circulation system, Top-Down approach . . . . . . . . . . . 473.4 Hydraulical analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Simplification of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.6 Simulating the heart with an active compartment . . . . . . . . . . . . . . . . . . 543.7 Valve simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 Compartment coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8.1 The 6 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . 573.8.2 The 3 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Numerical Model 614.1 A passive compartment without inertia . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Discretisation, Jacobi like method . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Discretisation, Gauss-Seidel like method . . . . . . . . . . . . . . . . . . . 624.1.3 Stability analysis of the Jacobi and Gauss-Seidel like method . . . . . . . 63

4.2 A passive compartment with inertia . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 An active compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Testing the single compartment model . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.1 The initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.2 Including the inertial term? . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 A 3 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 A 6 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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4.6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Testing the models 735.1 Verification of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 A six compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2 A three compartmentmodel . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Testing the model with some extreme cases . . . . . . . . . . . . . . . . . . . . . 765.2.1 Heart Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Conclusions 85

7 Future Work 877.1 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Investigation of a small part of the human CVS . . . . . . . . . . . . . . . . . . . 88

A Dictionary 89

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List of Figures

1.1 A schematic picture of the organs in the circulation system . . . . . . . . . . . . 21.2 The blood circulation through the heart . . . . . . . . . . . . . . . . . . . . . . . 31.3 An example of a pressure volume diagram together with the ESPVR and the

EDPVR lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 The aorta and its hydraulic and electrical representation . . . . . . . . . . . . . . 71.5 a: A modelling lab which consider only the simplest Windkessel method. b: A

three elements Windkessel circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 2-element Windkessel circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 A Wiggers’ diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 A simple CVS of a human . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 A closed loop model of a simple CVS of a human, see figure 2.1 . . . . . . . . . . 142.3 A possible choice of a cardiac driver function, a = 1, b = 80, c = 0.27,N = 1 . . . 152.4 A small part of an artery free of bifurcations . . . . . . . . . . . . . . . . . . . . 202.5 A simple cylindrical artery as a part of the vascular system, where the Γw is the

wall of the artery, Γ1 and Γ2 are the interfaces with the rest of the system. . . . 232.6 Single compartment circuit representation, P pressure, R resistance, C capacitor,

Q flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 The entire model with in total 12 coupled single compartments . . . . . . . . . . 272.8 Hydraulic analog of the cardiovascular system. A bifurcation in the systemic

circulation is made into a splanchnic and an extrasplanchnic circulation. . . . . . 31

3.1 A simple model of the CVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 A tube free of bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 A cross section of a small artery free of bifurcations . . . . . . . . . . . . . . . . 443.4 A small part of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5 Force balance in the axial direction . . . . . . . . . . . . . . . . . . . . . . . . . . 463.6 An electrical circuit, including a resistor, inductor and capacitor . . . . . . . . . 533.7 A simple cardiac driver function, with parameter values: A = 1, B = 80s−1,

C = 0.27s and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8 sin2 cardiac driver function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.9 sin cardiac driver function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.10 A 6 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.11 A 3 compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 The convergence by different time steps with the Jacobi like method . . . . . . . 644.2 The convergence for different time steps with the Gauss Seidel like method. In

this figure only the 25th heartbeat is depicted. . . . . . . . . . . . . . . . . . . . 65

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4.3 The difference in result by using different driverfunctions . . . . . . . . . . . . . . 684.4 Starting with different initial conditions has no influence on the final results . . . 694.5 The difference in the results by using inertia . . . . . . . . . . . . . . . . . . . . . 70

5.1 Simulation results from the closed loop model without inertia with our own program 745.2 Simulation results from the closed loop model with inertia and ventricular inter-

action, Results from [SMITH] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 A Wiggers’ diagram from a 6 compartment model . . . . . . . . . . . . . . . . . 765.4 A Wiggers’ diagram from a 3 compartment model . . . . . . . . . . . . . . . . . 785.5 Simulating a dystolic disfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Effect of diastolic dysfunction causing an increase in ventricle elastance on a PV

diagram of the left ventricle [BRWD] . . . . . . . . . . . . . . . . . . . . . . . . . 805.7 Simulating a systolic disfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.8 Effect of systolic dysfunction causing a drop in cintractility on a PV diagram of

the left ventricle [BRWD] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.9 Simulating aortic stenosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.10 A theoretical figure. On the left a normal left ventricle pressure, in the middle a

left ventricle pressure caused by aortic stenosis and on the right a left ventriclepressure diagram caused by valvular insufficiency . . . . . . . . . . . . . . . . . . 82

5.11 Simulating valvular insufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.12 Simulating a heart block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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Nomenclature

a 1. constant to define the exponential cardiac driver function

2. acceleration

A cross sectional area

b constant to define the exponential cardiac driver function

BH compensation term

c constant to define the exponential cardiac driver function

C Capacitance

d constant related to the physical properties of the vascular tissues

dx longitudinal displacement

dφ infinitesimal angle

e(t) cardiac driver function

er unit vector in radial direction

E Elastance

EW stress

f axisymmetric function

F force

h wall thickness

HFB compensation term

HFC described threshold

I current

l artery length

k1 constant

Kr friction parameter

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KHC constant

KHG constant

L inductance

m mass

p mean pressure over the whole compartment

P Pressure

Q 1. intantaneous charge on the capacitor

2. blood flow

Q mean flow rate over the whole district

r internal radius

r0 reference radius

r radial direction

R Resistance

s velocity profile

t time

T time interval

THF constant

u velocity

u mean velocity

x axial direction

y variable

V VoltageVolume

wp wave propagation

β0 constant

β set of coefficients related to the mechanical and physical properties

γ constant

Γw wall of the artery

Γ1,2 interface with the rest of the system

x

ǫ rate of change

η vessel wall displacement

θ circumferential coordinate

λ constant

µ viscosity

ν 1. kinematic viscosity

2. Poisson ratio of the artery wall

ρ density

σ surface stress

Φ C1 function

ψ momentum flux correction coefficient

ω heart rate

ωr interaction between fluid and wall

A general axial section

P portion of the tube

S general axial section

V the whole district

CO Cardiac Output

CVS CardioVascular System

EDPVR End Diastolic Pressure Volume relationship

ESPVR End Systolic Pressure Volume relationship

FE Finite Elements

HR Heart Rate

PRU Peripheral Resistance Unit

PV Pressure-Volume

SV Stroke Volume

ZPFV Zero Pressure Filling Volume

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Chapter 1

Introduction

Death cause number one in the Western world is cardiovascular disease [WE]. Therefor, thereis a growing interest in the mathematical and numerical modelling of the human CVS (cardio-vascular system). Cited to this interest, much research is devoted to complex three dimensionalsimulations able to provide sufficient details of the flow field to extract local data such as wallshear stresses. However, these computations are still quite expensive in terms of human re-sources needed to extract the geometry and prepare the computational model and computingtime. Since bioengineers and medical researchers do not need the flow in such detail every-where and less detailed models have demonstrated their ability to provide useful information ata reasonable computational cost, further research is done in the description of the CVS in lessdetailed models. In the less detailed models there must be the possibility to include a smallpiece of the CVS as a three dimensional model.

In this thesis we do research to a model which

1. Is simple,

2. Needs little computational time and

3. Can accurately reflect a small part of the human CVS.

To create such a model, we start with a literature study to other CVS models. Next the knowl-edge of others will be used to create a minimal model. Finally, this minimal model will be testedby using the outcomes of models of others and a standard Wiggers’ diagram.

Before starting with the research to different models, we give an introduction in the physiologyand in the modelling techniques of blood circulation systems.

1.1 Physiology of the blood circulation system

In the blood circulation system the blood flows through the bloodvessels and is pumped aroundby the heart. In this section in short there will be an introduction in the physiology of the bloodcirculation system.

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Figure 1.1: A schematic picture of the organs in the circulation system

1.1.1 The blood circulation system

The blood circulation system consists of a pulmonary- or lungcirculation and a systemic- orbodycirculation. The lungcirculation starts in the heart and provides the lungs of oxygenpoorblood and returns back, with oxygenrich blood, to the heart. Next, the heart pumps the bloodinto the bodycirculation and provides all the organs of blood before the blood returns to theheart as oxygenpoor blood. One circulation takes about 0.8sec.

1.1.2 The bloodvessels

In the circulation system blood flows through a system of bloodvessels, the vascular system. Thebloodvessels are divided into three different groups, the arteries, the capillaries and the veins.

Arteries

After the heart pumps blood away, all of the blood pumped out of the heart (SV - stroke volume)flows into the main artery, the aorta. Most of the SV flows at once into the arterial system to allthe organs (see fig 1.1.1). A small part of the SV will be stored in the aorta. Hereby the elasticwall of the aorta will be stretched. When the heart is at rest the aorta contracts and pumpsthe rest of the SV away. The heart pumps the blood into the arterial system with a pressure ofabout 120 mmHg (systole). The pressure generated by the aorta is about 80mmHg (diastole).

The blood flows with a velocity of about 4m/sec out of the heart. The velocity in thelegs is about 10m/sec. This difference can be explained by the decreasing elasticity of thearteries. Further, the pressure can be influenced by the lumen through narrowing and enlarging(resistance regulation).

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Capillaries

When the blood flows through the organs, the arteries split up in a large system of small arteries,called the capillaries. Because of the large system of small arteries, there is a low presure anda small velocity in the capillaries. The velocity in the cappilaries is about 0.3mm/sec. Theadvantage of this low pressure and small velocity is that the walls can be thin and the transferof nutrients with the organs is easy. After the split the capillaries come together in the veins.

Veins

In the veins the flow resistance and pressure drop is small. There are three kinds of mechanismsto pump the blood back to the heart.

• A musclepump For the veins which receive blood from the muscles, there is the musclepump. By the contraction of the muscle, the vein will be suppressed. By means of a valvethe blood in the vein will be pressed in the right direction.

• Arterial-Vein coupling When an arterie and a vein are close to each other the same happensas with a musclepump, but now with an artery which applies pressure on the vein.

• Breath By the underpressure in the chest, the hollow veins are working as a suction-pipe.

1.1.3 Heart

Figure 1.2: The blood circulation through the heart

The heart is a muscle which contains four chambers. By the periodic contraction and relaxationof the muscle, the heart can function as a pump. The four chambers can be split up in two

3

atrium-ventricle valve aorta valve

contraction time closed closedsystolic phase

ejection time closed opencardiaccycle

relaxation time closed closeddiastolic phase

filling time open closed

Table 1.1: Summary of the working of the valves

separate atrium-ventricle couples, a left and the right part. The left atrium-ventricle pair pumpsblood through the bodycirculation and the right atrium-ventricle pair pumps blood through thelungcirculation. The left and the right part are working synchron.

Working of the heart

The process of periodic contraction and relaxation of the muscle can be divided in four timeperiods:

• Contraction time The contraction of the heartmuscle causes a strong increase of the pres-sure in the ventricle. At this moment, the atrium-ventricle valve and aorta valve are closed.The volume will be the same (iso-volumetric contraction).

• Ejection time The bloodpressure will be the same as in the artery. The pressure in theventricle is still increasing. The aorta valve is open. The muscle of the ventricle is con-tracting, the volume will decrease. The valve closes as soon as the blood flows in the wrongdirection.

• Relaxation time Relaxation of the heartmuscle. The atrium-ventricle valve is still closed.There is no change in volume (iso-volumetric relaxation).

• Filling time When the bloodpressure of the ventricle drops beneath the bloodpressure ofthe atrium, then the atrium-ventricle valve opens.

The contraction time and the ejection time together are called the systolic phase. The relaxationtime and the filling time are called the diastolic phase.

1.1.4 Cardiac function

The performance of the heart is indicated by the cardiac function. Three of the most commonindicators are the pressure-volume diagram, the cardiac output and preload and afterload. Theseindicators are used by health professionals to study the patient condition.

The pressure-volume diagram

The pressure-volume (PV) diagram is used to explain the pumping mechanics of the ventricle.

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The two main characteristics of the PV diagram are the lines plotting the End Systolic Pressure-Volume relationship (ESPVR) and the End Diastolic Pressure-Volume relationship (EDPVR),which define the upper and lower limits of the cardiac cycle, respectively.

The cardiac cycle is divided in four parts. In the cardiac cycle, the four time periods of the

Figure 1.3: An example of a pressure volume diagram together with the ESPVR and the EDPVRlines

pumping process can be recognized.The EDPVR is a measure of the capacitance (C) of the ventricle. The capacitance, defined

as the inverse of the elastance (E), is the common term used to describe the PV relationshipof an elastic chamber. The ESPVR gives a measure of cardiac contractility, or the strengthof contraction, which is defined as the rate at which the heartmuscle reaches peak wall stress.When there is a diastolic failure, the compliance of the heart wall decreases. Even so, whenthere is a reduction in contractility the slope of the ESPVR line decreases.

Cardiac Output

The main measure of blood flow on a beat by beat basis is the stroke volume (SV). The SV isdefined as the amount of blood pumped from the ventricle during one heart beat. For a moregeneral measure, the cardiac output (CO), is defined as the amount of blood pumped into theaorta, from the left ventricle, in litres per minute. Therefore, the CO is equal to the product ofthe SV and heart rate (HR):

CO = SV ×HR (1.1)

The CO is used to define the capability of the heart to pump nutrient rich blood to theperipheral tissues. The equation for the CO highlights the important dependence on the SVand the HR. While the HR is driven by the sympathetic nervous system, the stroke volume isdependent on the function of the heart muscle as well as on the ventricle preloads and afterloads.

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Preload and Afterload

Preload and afterload are generally intended to be measures of ventricular boundary conditions,indicating the state of the ventricle before and after contraction, respectively. Preload is ameasure of the muscle fibre length, immediately prior to contraction, while afterload is a measureof the cardiac muscle stress required to eject blood from a ventricle.

1.2 Cardiovascular System Modelling

Most of the modelling systems for the human CVS can be divided into Finite Element (FE)or Pressure-Volume (PV) approaches. The FE approach involves breaking down parts of theCVS in great detail and utilizing FE calculations to simulate these parts. The PV approach isa simpler method, by grouping parameters and making assumptions to simplify the model asmuch as possible, while still attempting to simulate the essential dynamics.

1.2.1 Finite Elements Approach

With FE techniques it is possible to get micro-scaled results that can theoretically be very accu-rate both in trend and magnitude. This kind of approach needs a micro-scale measurement of themechanical properties such as the elastic properties and dimensions. With this measurements,FE equations can simulate the dynamics of the component being modelled on a micro-scale.

The FE approach on micro-scale is helping to improve understanding. Examples of modelsusing FE techniques are the micro-scale structures of the heart in [NIGRSMHU], [LEHUSM]and [STHU], or the attempts to model the complex fluid flow dynamics in the heart, particularlyaround the heart valves in [PEQU] and [GLHUMC]. Although these micro-scale results exists,FE techniques have a lack of flexibility which make them not suitable for patient-specific, rapiddiagnostic feedback. Further, it is not feasible to obtain the detailed specific measurements froma living patient, so a model of a specific patient is difficult. Finally, these micro-scale calculationsrequire significant computation time, making these models unsuitable for immediate feedback.

1.2.2 The Pressure-Volume Approach

PV methods are lumped parameter modelling methods where the CVS is divided into a series ofelements simulating elastic chambers and blood flow, separately. The elastic chamber elementsmodel the PV relationship in a section of the CVS, such as a ventricle, an atrium, or a peripheralsection of the circulation system such as the arteries or veins. All these separate elastic chamberelements are connected by the fluid flow elements which represent blood flow through differentparts of the circulation system.

For the modelling of the CVS there exist hydraulic and electrical analogs, see figure 1.4. Inthe next sections we tell something more about the connection between the electric and hydraulicanalog. For the explanation of the hydraulic analogs Windkessel circuits are used. The usage ofWindkessel circuits is because most of the PV approaches utilize these circuits.

1.2.3 Windkessel circuit

For most of the calculations the hydraulic formulas are used. To show the connection between thehydraulic formulas and the electrical formulas we use Windkessel circuits. Windkessel circuits

6

Figure 1.4: The aorta and its hydraulic and electrical representation

circulation hydraulic electricalelement analog analog

blood flow flow rate currentblood pressure pressure voltage

pumping function compliance capacitancevessels viscosity resistance

large arteries inertia inductance

Table 1.2: The hydraulic and electric analogs

are circuits which describe the load faced by the heart in pumping blood through the systemicarterial system and the relation between blood pressure and blood flow in the aorta.

One of the first descriptions of a Windkessel circuit was given by the German physiologistOtto Frank in the article ”Die Grundform des Arteriellen Pulses”, published in 1899. In thisarticle Frank compared the heart and systemic arterial system with a closed hydraulic circuitcomprised of a waterpump connected to a chamber. The hydraulic circuit is completely filledwith water, except for a pocket of air in the chamber. When water is pumped into the system, thewater compresses the air in the pocket and pushes water back in the pump. The compressibilityof the air in the pocket simulates the elasticity and extensibility of the major artery. This isknown as the arterial compliance. The resistance which the water encounters by flowing throughthe Windkessel and returning back to the pump, simulates the resistance which the blood flowencounters by the blood flowing through the arteries. This process is known as the peripheralresistance.

A Windkessel circuit can consist of a varying number of elements. The simplest modelconsists of two elements (see figure 1.2.3), namely a compliance and a peripheral resistance. Byusing the basic laws of an electrical circuit (Ohm’s law and Kirchhof’s laws), the Windkessel

7

Figure 1.5: a: A modelling lab which consider only the simplest Windkessel method. b: A threeelements Windkessel circuit

model can be described by a mathematical model.According to Ohm’s law, the drop in electrical potential across the resistor is IRR and the

drop in electrical potential across the capacitor is Q/C, where Q is the instantaneous charge onthe capacitor and dQ

dt= IC . From Kirchhof’s voltage law, the net change in electrical potential

around each loop of the circuit is zero; therefor V (t) = IRR and V (t) = Q/C. From Kirchhof’scurrent law, the sum of currents into a junction must equal the sum of currents out of the samejunction: I(t) = IC + IR. By now, the current in the capacitor is given by IC = C(dV/dt). If wenow substitute IC and IR from above into Kirchhof’s current law then we finally get an electricmathematical model which describes the 2-element Windkessel model:

I(t) = CdV (t)

dt+V (t)

R(1.2)

In terms of the physiological system, I(t) is the blood flow from the heart to the aorta, V (t) isthe blood pressure in the aorta, C is the arterial compliance and R is the peripheral resistancein the arterial system. In physiological terms the hydraulic mathematical model reads:

Q(t) = CdP (t)

dt+P (t)

R(1.3)

Now, we use the hydraulic equivalent to evaluate what happens during diastole. During diastole,there is no inflow, so Q(t) = 0 and an exact solution exists:

P (t) = P (0)e−RCt (1.4)

8

Figure 1.6: 2-element Windkessel circuit

A modification of the 2-element Windkessel circuit is obtained by including an inductor in themain branch of the circuit, as can be seen in figure (1.5b). This inductor simulates inertia of thefluid in the hydrodynamic model. The mathematical model of this 3-element Windkessel circuitcan be found by using that the drop in electrical potential across an inductor with inductanceequals VL = L(dIL(t)/dt) and Kirchhof’s law, IL = IR + IC :

IL(t) = IR(t) + IC(t) = VR

R+ C dVC

dt= V (t)−VL

R+ C d(V (t)−VL)

dt= V (t)

R− VL

R+ C dV (t)

dt− C dVL

dt

= V (t)R− L

RdI(t)dt

+ C dV (t)dt

− CLd2IL(t)dt2

⇔

IL(t) + LRdIL(t)dt

+ CLd2IL(t)dt2

= V (t)R

+ C dV (t)dt

In this case, the hydraulic equivalent reads:

Q(t) +L

R

dQ

dt+ LCp

d2Q

dt2=P (t)

R+ Cp

dP (t)

dt(1.5)

1.2.4 Wiggers’ diagram

The Wiggers’ diagram depict the pressure and volume in the heart and the ejecting activity ofthe heart:

• The first diagram shows the electrocardiogram (ecg). We will not use this ecg.

9

Figure 1.7: A Wiggers’ diagram

• The second diagram represents the pressure in the left atrium, left ventricle and in theaorta.The top at a of the left atrium and ventricle is the result of the contraction of the atrium.Next the ventricle contraction occurs. During ventricle ejection, the atrium is pulled. Theresult of this pulling is a pressure drop in the atrium, showed at the x-top. Right beforethe x-top, a c-top in the pressure of the left atrium is denoted. This c-top is caused bythe opening of the aortic valve. In the second part of the ejection of the left ventricle,the pressure in the atrium is increasing due to the filling with blood until the mitral valveis opened for a fast filling of the ventricle during the isometric relaxation. This openingcauses the y-top.

• The third diagram depicts the volume in the left ventricle and the flow velocity in the aorta.

10

• The fourth diagram is similar to the second diagram, with the difference that this diagramdepicts the values for the right atrium and ventricle. As can be seen, the pressure in theright side of the heart is lower then in the left side of the heart.

• In the last diagram a phonocardiogram is showed. The first (I) noise reflects the closing ofthe mitral valve, the second (II) noise reflects the closure of the aortic valve. The first andsecond noise give exactly the duration of the relaxation and contraction. The systolic phasestarts at the beginning of the first noise till the beginning of the second noise. The di-astolic phase starts at the beginning of the second noise till the beginning of the first noice.

What is the connection with our reseach? As said before, we want to use the Wiggers’ diagramas reference material for a specific person. The phonocardiogram can be used to measure thetime needed for the different heart periods. The other three graphs can be used as reference forour own model.

1.3 Summary

In this section the main goals of this report have been outlined, namely creating a human CVSmodel on a low level. Further it must be possible to describe a small part of the CVS detailed.To make it possible to create a good model for the human CVS, we gave an introduction in thephysiology of the circulation system. Finally, we gave an introduction about the possibilities ofCVS modelling. We introduced two approaches, a FE approach and a PV approach, togetherwith their advantages and disadvantages.

11

Chapter 2

Literature study

In this literature study we are going to search for existing models. The models found in literature,which at a first glance satisfy the requirements mentioned in the introduction, will be described.Based on the summaries we will make decisions about the model we will use in further studies.Every section contains the summary of one model. The conclusions will be presented in the finalsection.

2.1 Minimal Heamodynamic Modelling of the Heart & Circu-

lation for Clinical Application [SMITH]

2.1.1 General description

This is the title of the thesis that is presented by Bram W. Smith for the degree of Doctorof Philosophy in Mechanical Engineering at the University of Canterburry, Christchurch, NewZealand. In this thesis Smith has the intention to make a model which contains:

• A closed-loop, stable model with minimal complexity and physiologically realistic inertiaand valve effects.

• A model parameters that can be relatively easily determined or approximated for a specificpatient using standard, commonly used techniques.

• A model that can be run on a standard desktop computer in reasonable time (eg. in theorder of 1-5 minutes)

• Accurate prediction of trends

The model presented a hydraulic, 0D, 6 compartment model, which intends to simulate theessential haemodynamics of the CVS including the heart, and the pulmonary and systemiccirculation systems. Figure 2.1 shows a simplified diagram of the human circulation systemwith in the middle the human heart. Figure 2.2 presents a closed-loop model of the same humanCVS. As can be seen in figure 2.2, the closed-loop model contains compartments which areconnected by resistors and inductors in series and can be seen as a Windkessel circuit.

For the pulsation of the heart, Smith uses a cardiac driver function e(t). This cardiac driverfunction utilizes the ESPVR and EDPVR (see 1.1.4) as the upper and lower limits of cardiac

13

Figure 2.1: A simple CVS of a human

Figure 2.2: A closed loop model of a simple CVS of a human, see figure 2.1

chamber elastance. The profile of the driver function represents the variance of elastance betweenminimum and maximum values during a single heart beat:

e(t) =N

∑

i=1

aie−bi(t−ci)2 , (2.1)

where the ai, bi, ci and N are parameters that determine the shape of the driver profile. For hissimple model he takes a = 1, b = 80s−1, c = 0.27s and N = 1. See for the shape figure 2.3.

With respect to the heart Smith makes some assumptions, which will be described below.

2.1.2 Assumptions

The first assumption that Smith makes, is that blood, which flows through the CVS, is approx-imated as flow through a tube. The flow rate equations have directly been derived from the

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time t (sec)

e(t)

profile of the cardiac driver function

Figure 2.3: A possible choice of a cardiac driver function, a = 1, b = 80, c = 0.27,N = 1

Navier-Stokes equations in cylindrical coordinates:

∂ux∂t

+ ur∂ux∂r

+uθr

∂ux∂θ

+ ux∂ux∂x

= −1

ρ

∂P

∂x+ ν

[

1

r

∂

∂r

(

r∂ux∂r

)

+1

r2∂2ux∂θ2

+∂2ux∂x2

]

(2.2)

where ux, ur and uθ are the longitudinal, radial and angular velocities, respectively, P is themodified pressure relative to hydrostatic, ρ is the density and ν is the kinematic viscosity.

The next assumptions are standard assumptions, which will be applied to all equationsgoverning fluid flow.

• Blood is assumed to be incompressible, so ρ is constant.

• For the heart, the fluid is assumed to behave in a continuous, Newtonian manner withconstant viscosity (µ is constant).

• The arteries are assumed to be rigid with a constant cross sectional area ( ∂r∂x

= 0). Thisassumption fits with standard Windkessel circuit design involving a rigid pipe and anelastic compartment in series. The rigid tube simulates the fluid dynamics, while theelastic compartment simulates the compliance of the artery walls.

• Laminar uni-directional axi-symmetric flow is assumed (ur = 0, uθ = 0 and ∂ux

∂θ= 0).

Although turbulence can occur around the valves, it takes time to develop, and is assumednot to affect the flow profile significantly.

• The flow is assumed to be fully developed along the length of the tube meaning the velocityprofile is constant with respect to x (∂ux

∂x= 0).

• Pressure is assumed constant across the cross-sectional area and the pressure gradient isconstant along the length of each section so that the pressure gradient is a function of timeonly (∂P

∂x(t)).

With these assumptions equation (2.2) reduces to the following equation

ρ∂u(r, t)

∂t= −∂P

∂x(t) +

µ

r

∂

∂r

(

r∂u(r, t)

∂r

)

, (2.3)

15

where µ is the viscosity (µ = νρ) and u(r, t) is the velocity in the x-direction (ux(r, t)) as afunction of radius and time only.

2.1.3 Mathematical Model

For the description of the compartments Smith has the choice between two different equations.Both equations are derived from equation (2.3). The first equation used is Poiseuille’s equationfor flow rate, assuming constant resistance and no inertial effects:

Q(t) =P1(t)− P2(t)

R, (2.4)

where R = 8µlπr40

is the resistance. The second equation includes inertial effects and constant

resistance

LdQ

dt= P1 − P2 −QR, (2.5)

where L = ρl

πr20is inertia and R = 8µl

πr40constant resistance. Around the heart where are big flow

differences the second equation will be used. Far from the heart flows a nearly constant flow, sothe first equation will be used.

2.1.4 Parameters

Smith uses the following parameters for his tests.

16

Description Symbol Value

Blood properties

Blood Density ρ 1050kg/m3

Blood Viscosity µ 0.004NS/m2

Blood Kinematic Viscosity ν 3.8 10−6m2/sStressed Volume of blood in CVS Vtot 1500mlUnstressed Volume of blood in CVS 4000ml

Artery properties

Internal Artery Radius r0 0.0125mArtery length l 0.2m

Compartment properties

Chamber Elastance Ees 1N/m5

EDPVR Volume Cross-over V0 0m3

ESPVR Volume Cross-over Vd 0m3

Constant λ 23000m−3

Heart Rate ω 1.33beats/secConstant a 15N/m2

Table 2.1: Constants used in a single compartment simulation

PARAMETERS Ees Vd V0 λ P0

Units 106N/m5 10−6m3 10−6m3 m−3 N/m2

Left Ventricle free wall (lvf) 100 0 0 33000 10Right ventricle free wall (rvf) 54 0 0 23000 10Septum free wall (spt) 6500 2 2 435000 148Pericardium (pcd) - - 200 30000 66.7

Vena-cava (vc) 1.3 0 - - -Pulmonary Artery (pa) 72 0 - - -Pulmonary Vein (pu) 1.9 0 - - -Aorta (ao) 98 0 - - -

Table 2.2: Mechanical properties of the heart and circulation system

17

Resistance InertanceParameter Ns/m5 Ns2/m5

Mitral Valve (mt) 6.1 106 1.3 104

Aortic Valve (av) 2.75 106 5 104

Tricuspid Valve (tc) 1 106 1.3 104

Pulmonary Valve (pv) 1 106 2 104

Pulmonary Circulation System (pul) 9.4 106 N/ASystemic Circulation System (sys) 170 106 N/A

Table 2.3: Hydraulic properties for flow between compartments

Description Symbol

Elastance of Vena-cava Evc = 1.29 106N/m5

Elastance of Left Ventricle P0,lvf = 9.07N/m2

Elastance of Pulmonary Artery Epa = 44.5 106N/m5

Elastance of Pulmonary Vein Epu = 0.85 106N/m5

Elastance of Right Ventricle P0,rvf = 20.7N/m2

Elastance of Aorta Eao = 98 106N/m5

Resistance of Tricuspid Valve Rtc = 3.3 106Ns2/m5

Resistance of Pulmonary Valve Rpv = 1 106Ns2/m5

Resistance of Pulmonary Circulation Rpul = 19.3 106Ns2/m5

Resistance of Mitral Valve Rmt = 2.33 106Ns2/m5

Resistance of Aortic Valve Rav = 5.33 106Ns2/m5

Resistance of Systemic Circulation Rsys = 139.6 106Ns2/m5

Contractility of Left Ventricle Ees,lvf = 377 106N/m5

Contractility of Right Ventricle Ees,rvf = 87.8 106N/m5

Table 2.4: Parameter values for the closed loop model

18

2.1.5 Conclusions

Smith performs different tests to verify all the specific possibilities of his model. One of thetests is the comparison with a Wiggers’ diagram (for explanation of a Wiggers’ diagram seeparagraph 1.2.4). In table 2.5, Smith compares his results with the Wiggers’ diagram. After

Model TargetVariable Value Value % Error

Pressure in AortaAmp Pao 40mmHg 41.407 3.5%Avg Pao 100mmHg 119.168 19.2%Pressure in Pulmonary Artery

Amp Ppa 17mmHg 20.414 20.1%Avg Ppa 16.5mmHg 20.314 23.1%Volume in Left VentricleAmp Vlv 70ml 69.508 −0.7%Avg Vlv 80ml 84.042 5.1%Volume in Right Ventricle

Amp Vrv 70ml 69.569 −0.6%Avg Rrv 80ml 121.185 51.5%Pressure in Pulmonary Vein

Avg Ppu 2mmHg 10.112 405.6%Pressure in Vena-cavaAvg Pvc 2mmHg 1.050 −47.5%

Table 2.5: Comparison of the results with the Wiggers’ diagram

doing different tests, Smith comes with the following conclusions:

• The blood flow rate is primarily dependent on the pressure gradient across the resistor.If the effects of inertia are either ignored or negligible, the equation for flow rate can becalculated using Poiseuilles equation (2.4). Poiseuilles equation assumes incompressible,Newtonian, laminar, axi-symmetric, fully developed flow through a rigid tube of constantcross-section.

• Tests prove the stability of the closed loop CVS model.

• The model is seen to capture the major dynamics of the CVS including the variations inleft ventricle pressure, aortic pressure and ventricle volume.

• The decrease in cardiac output is in good agreement with readily available clinical data.

• The results show the capability of the presented approach to create patient specific models.

19

2.2 Reduced and multiscale models for the human cardiovascu-lar system; one dimensional model [FORVEN]


Formaggia and Veneziani wrote this report as collection of the notes of the two lectures givenby Formaggia at the 7th VKI Lecture Series on ”Biological fluid dynamics” held on the VonKarman Institute, Belgium, on May 2003. They give a summary of some aspects of the researchaimed at providing mathematical models and numerical techniques for the simulation of thehuman CVS.

At first they derive an one dimensional model. Hereto, they start with the mathematical

Figure 2.4: A small part of an artery free of bifurcations

description of a small part of an artery free of bifurcations. They assume that the small partof the artery can be described by a straight cylinder with a circular cross section. For thedescription of the flow through this straight cylinder the Navier-Stokes equations are used andintegrated over a generic cross section. Starting parameters are the time interval T = (0, t1) andthe vessel length x = (0, l).

2.2.2 Assumptions

Describing a small part of the artery as a straight cylinder with the Navier-stokes equations istoo expensive, so some simplifying assumptions are made:

1. All quantities are independent of the angular coordinate θ. As a consequence, every axialsection x = const remains circular during the wall motion. The tube radius r is a functionof x and t.

2. The wall displaces along the radial direction solely, thus at each point at the tube surfacethey may write η= ηer, where η = r− r0 is the displacement with respect to the referenceradius r0.

3. The vessel will expand and contract around its axis, which is fixed in time. This hypothe-sis is indeed consistent with that of axial symmetry. However, it precludes the possibility

20

of accounting for the effects of displacements of the artery axis such as occuring in thecoronaries because of the heart movement.

4. The pressure P is constant on each section, so it only depends on x and t.

5. The body forces are neglected.

6. The velocity components orthogonal to the x axis are negligible compared to the compo-nent along x. The latter is indicated by ux and its expression in cylindrical coordinates issupposed to be of the form

ux(t, r, x) = u(t, x)s(rr−1(x)) (2.6)

where u is the mean velocity on each axial section and s : R → R is a velocity profile,which must be chosen such that

∫ 10 s(y)ydy = 1

2 . The fact that the velocity profile doesnot vary in time and space is in contrast with experimental observations and numericalresults carried out with full scale models. However, it is a necessary assumption for thederivation of the reduced model. One may then think of s as being a profile representativeof an average flow configuration.

Finally, a momentum-flux correction coefficient is defined by:

ψ =

∫

S u2xdσ

Au2=

∫

S s2dσ

A.

where A is the cross sectional area and S the general axial section.

2.2.3 Mathematical model

With all these assumptions, the main variables are:

• Q the mean flow, defined as

Q =

∫

Suxdσ = Au;

• A the surface area of an axial section;

• P the pressure.

When ψ is taken constant, the reduced model looks like

∂A∂t

+ ∂Q∂x

= 0∂Q∂t

+ ψ ∂∂x

(Q2

A) + A

ρ∂P∂x

+Kr(QA

) = 0(2.7)

for x ∈ (0, l), t ∈ T , where Kr = −2πνs′ is a friction parameter and s′ the derivative of thevelocity profile.

21

parameter

input pressure amplitude 20× 103dyne/cm2

FLUID viscosity, ν 0.035poiseDensity, ρ 1.021kg/m3

STRUCTURE Wall Thickness, h 0.05cmReference Radius, r0 0.5cm

Table 2.6: Parameters used in the one dimensional model

2.2.4 Parameters

For the model specific parameters see [FORVEN, page 1.37].

2.2.5 Conclusions

Before a test can be done, a velocity profile has to be chosen. Hereto several options exist.Formaggia and Veneziani chose for the parabolic profile s(y) = 2(1−y2). This profile correspondswith the Poisseuille solution characteristic of steady flow in circular tubes. The parabolic profileis a variant of the profile most used: s(y) = γ−1(γ + 2)(1 − yγ). This power law profile is mostused, since it has been found experimentally that the velocity profile is, on average, rather flat.

Now that a velocity profile has been selected, three sets of tests are distinguished. The firstseries of tests are focussed on the single artery, the second series of tests are done with a couplingof 55 main arteries and the last series of tests are an improvement of the second series of tests.The improvements in the third test are made by taking inertia of the wall into account. For theresults see [FORVEN].

Comparing the results with literature, Formaggia and Veneziani conclude that there is littleagreement with reality. This can be explained by the chosen model. The model is namely formedby a closed network with a high-level of inter-dependency. In this model the flow dynamicsof the blood in a specific vascular district is stricly related to the global, systemic dynamics.However, in [ARFEL], it is shown that even a strong reduction in the vascular lumen in a carotidbifurcation does not mean a relevant reduction of the blood supply to the brain.

So, to make the results more realistic another way of coupling parts of a high inter-dependencemodel must be found. The next section describes how Formaggia and Veneziani make use of aWindkesselcircuit to accomplish this.

2.3 Reduced and multiscale models for the human cardiovas-

cular system;lumped parameters for a cylindrical compliantvessel [FORVEN]


In the previous section Formaggia and Veneziani developed a 1D model. After doing differenttests they conclude that the model has a high level of interdependency and that there is littleagreement reflecting reality. So, they continu their research aimed at providing mathematicalmodels and numerical techniques for the simulation of the human CVS, by focussing on couplingtechniques. In this section Formaggia and Veneziani are describing a mathematical model ofthe CVS which couples a local system with a systemic model. The local system is based on the

22

solutions of the incompressible Navier-Stokes equations possibly coupled with the dynamics ofthe vessel wall, while the systemic model is based on a one-dimensional system or on a lumpedparameters model. The lumped parameters model is based on the solution of a system of ordi-nary differential equations for the average mass flow and pressure.

For the systemic model, a choice can be made between a one dimensional model and alumped parameters model. As one dimensional model you can think of a model like the onedescribed in section 2.2. A lumped parameters model is described below.

A lumped parameter models

A lumped parameters model provides a systemic description of the main phenomena related tothe circulation at a low computational cost. An effective description of this model is by dealingwith separate ’compartments’ and their interaction. To develop a lumped parameters model

Figure 2.5: A simple cylindrical artery as a part of the vascular system, where the Γw is thewall of the artery, Γ1 and Γ2 are the interfaces with the rest of the system.

they start with a seperate compartment, for which they consider a simple cylindrical artery, seefigure 2.5. In this circular cylindrical domain, the axial section equals A(t, x) = πr2(t, x) wherer(t, x) is the radius of the section at x. With this consideration they want to form a simplifiedmodel, therefor some assumptions have been introduced, as described next.

2.3.2 Assumptions

As in the one dimensional model of section 2.2, Formaggia and Veneziani start with the Navier-Stokes equations and make the same standard assumptions. After these standard assumptionsa 1D model is left.

∂A∂t

+ ∂Q∂x

= 0∂Q∂t

+ ψ ∂∂x

(

Q2

A

)

+ Aρ∂P∂x

+KrQA

= 0(2.8)

In order to close the system, a further equation, which is provided by the constitutive law forthe vessel tissues, is needed. So another assumption is made:

• The vessel wall displacement η is related to the pressure P by an algebraic linear law. Byfollowing [FOVE], they take:

(P − Pext) = d(r − r0) = β0

√A−

√A0

A0, (2.9)

where Pext is a constant reference pressure, A0 = πr20 a constant reference area, d is aconstant related to the physical properties of the vascular tissues and β0 = A0d/

√π.

23

Now, the one dimensional model (2.8) will be integrated along x ∈ (0, l):

k1ldpdt

+Q2 −Q1 = 0

l dQdt

+ ψ[

Q22

A2− Q2

1A1

]

+∫ l

0

[

Aρ∂P∂x

+KRQA

]

dx = 0(2.10)

In this system, p is the mean pressure over the whole compartment and k1 a constant. Sincethis is not a satisfactory model, because it is not linear, some more assumptions are needed.

• The quantity(

Q22

A2− Q2

1A1

)

is so small compared to the other terms in short pipes that it

can be discarded.

• The variation of A with respect to x is small compared to that of P and Q, so the integralin 2.10 will be approximated

∫ l

0

[

A

ρ

∂P

∂x+KR

Q

A

]

dx ≈∫ l

0

[

A0

ρ

∂P

∂x+KR

Q

A0

]

dx


With all these assumptions Formaggia and Veneziani end op with a system for the lumpedparameters description of the blood flow in the compliant cylindrical vessel. It involves themean values of the flow rate and the pressure over the domain, as well as the upstream anddownstream flow rate and pressure values:

k1ldpdt

+Q2 −Q1 = 0

ρlA0

dQdt

+ ρKRl

A20Q+ P2 − P1 = 0

(2.11)

The final system (2.11) will be represented by a hydraulic analog. In this analog three parametersare used, namely:

• R the resistance induced by the blood viscosity is represented by R = ρKRl

A20

. Assuming a

parabolic velocity profile gives

R =8µl

πr40(2.12)

• L the inductance of the flow represents the inertial term in the momentum conservationlaw and is given by

L =ρl

πr20(2.13)

• C the capacitance of the vessel represents the coefficient of the mass storage term in themass conservation law and is given by

C =3πr30l

2Eh(2.14)

With this notation equation (2.11) becomes

C dpdt

+Q2 −Q1 = 0

LdQdt

+RQ+ P2 − P1 = 0(2.15)

24

2.3.4 Conclusions

In an analytical test case a completely lumped parameters description of the circulation, pro-viding a reference solution to the systemic level, is given. The aim of this test is to model bloodflow behaviour in Ω by the Navier-Stokes equations coupled with the lumped description of theremaining network. Formaggia and Veneziani expect from this test that the presence of a localaccurate submodel does not have to modify significantly the results at the systemic level. Thisis exactly what they obtain numerically. The heterogeneous model is able to compute accuratelythe velocity and pressure fields in the domain of interest.

In a test case where a 3D-1D coupling is made, it could be concluded that the coupling be-tween a 3D fluid structure model and a 1D reduced model is an effective way to greatly reducenumerical reflections of the pressure waves.

In a last test of clinical interest, the methodology was in particular applied to a recon-structive procedure, the systemic-to-pulmonary shunt, used in cardiovascular paediatric surgeryto treat a group of complex congential malformations. The 3D-model includes the shunt, theinnominate artery (through which blood flows in) and the pulmonary, carotid and subclavianarteries (through which blood flows out). In this case the lumped model is composed by differentblocks describing the rest of the pulmonary circulation, the upper and lower body, the aorta,the coronary system and the heart. This application to the systemic-to-pulmonary shunt, givesa clear idea of what can be obtained using the multiscale methodology.

2.4 Computational modeling of cardiovascular response to or-

thostatic stress [HSKM]


The objective of this study is to develop a model of the cardiovascular system capable of simulat-ing the short term (< 5min) transient and steady state heamodynamic responses to head-up tiltand lower body negative pressure. A subobjective of this study is to develop and test a general0D, 12 compartment model of a CVS that contains the essential features associated with theeffects of gravity. The development of the model is not completely their own, but they use theknowledge and formulas of other investigators.

2.4.2 Mathematical Model

The model of [WHFICR] and [DAMA] is based on a closed-loop lumped parameters heamody-namic model with local blood flow to major peripheral circulatory branches. This heamodynam-ical model is mathematically formulated in terms of a hydraulic analog model in which inertialeffects are neglected. A single compartment circuit representation which has been used is givenin figure 2.6. The equations read

Q1 =Pn−1 − Pn

Rn(2.16)

Q2 =Pn − Pn+1

Rn+1(2.17)

Q3 =d

dt[Cn × (Pn − Pbias)] (2.18)

25

Figure 2.6: Single compartment circuit representation, P pressure, R resistance, C capacitor, Qflow rates

The flow Q1 at node Pn splits up like Q1 = Q2 +Q3. Combining these expressions for the flowrates leads to

d

dtPn =

Pn+1 − PnCnRn+1

+Pn−1 − PnCnRn

+Pbias − Pn

Cn·(

dCndt

)

+d

dtPbias (2.19)

In total 12 such first order differential equations are used to describe the entire model, see figure2.7. To solve this model, the authors used a fourth order Runge-Kutta integration routine. Thepumping action of the heart is realized by varying the right and the left ventricular elastancesaccording to a predefined function of time (Er and El)

e(t) =

Edias +Esys−Edias

2

(

1− cos

(

π t

0.3√T (n−1)

))

0 ≤ t ≤ Ts

Edias +Esys−Edias

2

(

1 + cos

(

2πt−0.3

√T (n−1)

0.3√T (n−1)

))

Ts < t ≤ 32Ts

Edias32Ts < t ≤ T (n)

(2.20)

In this equation Edias and Esys represent the end-diastolic and end-systolic elastance values,respectively. Further T (i) denotes the cardiac cycle length of the ith beat and t denotes thetime measured with resect to the onset of ventricular contraction. The systolic time interval,Ts, is determined by the Bazett formula, Ts(n) ≈ 0.3

√

T (n− 1). The atria are not represented,because their function is partially absorbed into the function of adjacent compartments.

For the change of volume in the compartments the authors refer to experimental observationsof [LUD]. In accordance to these observations they model the functional form of the pressure-volume relationships of the venous compartments of the legs, the splanchnic circulation and theabdominal venous compartment with

∆V =2∆V

πarctan

(

πC0

2∆Vmax∆Ptrans

)

, (2.21)

where ∆V represents the change in compartment volume due to a change in transmural pres-sure ∆Ptrans, ∆Vmax is the maximal change in compartment volume and C0 represents thecompartment capacitance at the baseline transmural pressure. Finally, the total blood volume

26

Figure 2.7: The entire model with in total 12 coupled single compartments

is modified as a function of time to simulate fluid sequestration into the interstitium duringorthastatic stress.

2.4.3 Parameters

The parameters given in this section are the resistance, volume and capacitance values for the12 different compartments.In this tables the writers make use of the PRU = mmHg.s/ml (peripheral resistance unit) andZPFV (zero pressure filling volume). Further all values compound with a 71 − 75kg normalmale subject and a body surface area of 1.7− 2.1m2 with a total blood volume of 5700ml.

27

Resistance values [PRU ]

Rlo Rup1 Rkid1 Rsp1 Rll1 Rup2 Rkid2 Rsp20.006 3.9 4.1 3.0 3.6 0.23 0.3 0.18

Rll2 Rsup Rab Rinf Rro Rp Rpv0.3 0.06 0.01 0.015 0.003 0.08 0.01

Table 2.7: Resistance values

ZPFV CapacitanceCompartment ml ml/mmHg

Right ventricle 50 1.2-20Pulmonary arteries 90 4.3Pulmonary veins 490 8.4

Left ventricle 50 0.4-10Systemic arteries 715 2.0Systemic veins

Upper body 650 8Kidney 150 15

Splanchnic 1300 55Lower limbs 350 19

Abdominal veins 250 25Inferior vena cava 75 2Superior vena cava 10 15

Table 2.8: Volume and capacitance values

2.4.4 Conclusions

After several tests have been done, it shows that all major heamodynamic parameters generatedby the model are within the range of what is considered as physiologically normal in the generalpopulation. Representative simulated pressure waveforms are made, too. The conclusions theauthors draw after these tests are

1. They assume that the dynamics of the system can be simulated by restricting their analysisto relatively few representative points within the CVS. Although this approach is incapableof simulating pulse wave propagation, it does reproduce realistic values of beat-to-beatheamodynamic parameters.

2. One potential limitation of the heamodynamic system in its present form might be thelack of atria, which are thought to contribute significantly to ventricular filling at highheart rates.

3. The model generates steady state and transient heamodynamic responses that comparewell to population-averaged and individual subject data.

28

2.5 An identifiable model for dynamic simulation of the humancardiovascular system [KRWIWAKR]


The authors describe the mathematical model in their paper as a hapy medium one. The ideawas to keep the model as exact as needed and to make it as easy as possible. For this modelthey combine a compartment model with a dynamic (time and space dependent) model of thearterial vascular system to simulate mainly the arterial part of the CVS. Both parts of the modelare calculated separately and connected afterwards by a feedback control mechanism.


The compartment model

models the whole CVS including the heart, the systemic and the pulmonary part. This model isdivided in six compartments with a feedback control mechanism. Further it takes into accountouter influences on the system like changes in hydrostatic pressure and external exposures. Here,centered curves of the beat volume, the heart rate, the peripheral resistance and the systemicblood pressure are computed. These four variables are the main characteristics of the CVS andare directly controlled through the model.

The feedback control mechanism for modelling the baro-receptor mechanism is based onmeasured data. The examples show that the mechanism which increases the heart rate does notdepend linear anymore on the stress when a critical level is reached. Nevertheless, to a certainlevel the authors model the behavior linearly and define a threshold for the nonlinear part ofcontrol. With this approach there is still a partially linear model:

BH = KHC −KHCe−KHG(ω−HFC)

ω =

− ωTHF

+KHF EW+HFBTHF

for ω < HFC

− ωTHF

+KHF (1−BH)(EW+HFB)THF

else

(2.22)

Here ω (heartrate), BH,HFB (compensation terms) and EW (stress) are time dependent, therest are constants. HFC is the described threshold.

Dynamic model of the arterial tree

For the simulation of the human CVS a one-dimensional blood flow model is used, which hadbeen developed by [WIB].

At +Qx = 0

Qt + ∂∂x

(

φQ2

A+ E

)

= KrQA

(2.23)

For the connection between the dynamic model and the control mechanism two aspects have tobe considered:

• A function for the aortic flow has to be created that depends on the cardiac output.

• The dynamic blood flow simulation uses a Windkessel circuit as outflow condition, so theperipheral complex resistances (impedances) have to be calculated before. Doing this, the

29

Womersley solution of the one dimensional Navier-Stokes equation to compute impedancesin any arterial segment has to be used. With this method it is possible to compute theWindkessel data at any end node of the modelled arterial tree. A static model is used tocompute impedances at bifurcations of the vascular system.

In a simplified model, this is done by using the solution of the axisymmetric Navier-Stokes equa-tions with equations that describe the motion of the vessel wall and solving a Bessel equation.This leads to

Q =

∫ a

0ωr2πrdr, (2.24)

where ωr is the interaction between fluid and wall and given through a term which depends onblood density, the Womersley number, the kinematic viscosity and the complex wave-propagationvelocity. Integration over the cross sectional area yields

Q =A0Ecwpc0ρ

(1− Fj), (2.25)

where Fj depends on the Womersley number and wp is the complex wave-propagation velocity.Using this the momentum and continuity equations can be solved and give an exact solution forthe impedances at any arterial segment.

2.5.3 Conclusions

Comparison of the simulation results with measured data shows that with this model the behav-ior of the human CVS can be described very well. The measurements are done with ultrasoundtechniques whereby each point is measured twice: once with a standing person and once witha lying person. After the test they can conclude that the nonlinear reaction of the heartratecaused by stress can be modelled very well with an easy compartmental approach for modellingshorttime control of the human CVS.

Further they have considered the reaction of the CVS caused by changes of hydrostatic pres-sure (tilting table test). Again the results were very satisfying. The most satisfying of these twotests is that under the circumstances given above all the parameters needed can be identifiedwith an ergometer and a tilting table. In addition to the compartment results there are someresults of the complex dynamic model which can be compared with measured data, too. Theultrasound measured flow velocity and the computed flow velocities confirm the validity of themodel in a qualitative manner. Some tests have been carried out, but they were not finishedwhen this paper was published.

Finally, after the tests allready done, they conclude that the model is a good approxima-tion for the human CVS and that combining a compartment model with a model for pulsatileblood flow in arteries provides a distributed model which is dynamic in time and space, feedbackcontrolled, identifiable and verifiable through measured data.

2.6 Interaction between carotid baroregulation and the pulsat-

ing heart: a mathematical model [URS]


Ursino presents in his paper a mathematical model of short-term arterial pressure control bythe carotid baroreceptors in pulsatile conditions. The model includes an elastance variable de-

30

scription of the left and the right heart, the systemic and the pulmonary circulation, the afferentcarotid baroreceptor pathway, the sympethatic and vagal efferent activities and the action ofseveral effector mechanisms. The model is used to simulate the interaction among the carotidbaroreflex, the pulsating heart and the effector responses in different experiments.


Figure 2.8: Hydraulic analog of the cardiovascular system. A bifurcation in the systemic circu-lation is made into a splanchnic and an extrasplanchnic circulation.

Ursino uses a 0D, 12 compartment model. This model is a generalization of the modelpresented by [URANBE], see figure 2.8, based on a Windkessel circuit.

The vascular components

Equations relating pressure P and flow Q in all points of the vascular system have been writtenby enforcing conservation of mass at the capacities in figure 2.8 and equilibrium of forces at theinertances.

dPdt

= 1C

(∆Q)dQdt

= 1L(∆P −RQ)

(2.26)

31

The heart as a pump

Since the vascular system is split into a pulmonary and a systemic part, the heart is split,in a right and a left part. Further the left and right part are split into an atrium and aventricle. The models for the left and right part are the same except for parametervalues. Theatrium is modelled by a linear capacity characterized by constant values of capacitance andunstressed volume. The blood flow between the atrium and the ventricle is modelled by anatrioventricular valve, mimicked as the series arrangement of an ideal unidirectional valve witha constant resistance:

Qi,l =

0 if Pla ≤ PlvPla−Plv

Rlaif Pla > Plv

(2.27)

The contractile activity of the ventricle is described by means of a Voigt viscoelastic model.

Plv = Pmax,lv −RlvQ0,ldVlv

dt= Qi,l −Q0,l

Pmax,lv(t) = e(t)Emax,lv(Vlv − Vu,lv) + [1− e(t)]P0,lv(eλlvVlv − 1) 0 ≤ e(t) ≤ 1

e(t) =

sin2[

πT (t)Tsys(t)

u]

0 ≤ u ≤ Tsys/T

0 Tsys/T ≤ u ≤ 1

u(t) = frac[

∫ t

t01

T (τ)dτ + u(to)]

(2.28)

In these equations Emax,lv is the ventricle elastance at the instant of maximum contraction, Vu,lvis the corresponding ventricle unstressed volume, Q0,l is the cardiac output from the left ventricleand P0,lv and kE,lv are constant parameters that characterize the exponential pressure-volumefunction at diastole.

In the Voigt viscoelastic model a linear pressure-volume function at end-systole is adoptedand at diastole an exponential pressure-volume function. This is done to reflect the varying elas-tance during the cardiac cycle. The shifting between the end-systole and diastole is governedby a pulsating activation function e(t), with period T equal to the heart period. In this work asin-square function for e(t) has been used. In this equation T equals the heart period and Tsysstands for the duration of systole.

Since e(t) only must take values between 0 (complete relaxation) and 1 (maximum contrac-tion), an expression for u(t) has been obtained by means of an ”integrate and fire” model. Inthis expression the function frac() resets the variable u(t) to zero as soon as it reaches the value+1.

2.6.3 Parameters

All parameters used by Ursino are taken from literature, suitably rescaled for a subject with a70kg body weight. The total blood volume is taken as 5300ml.

32

Capacitance Unstressed Volume Hydraulic Resistance Inertance[ml/mmHg] [ml] [mmHg.s.ml−1] [mmHg.ml.ml−2]

Csa = 0.28 Vu,sa = 0 Rsa = 0.06 Lsa = 0.22 10−3

Csp = 2.05 Vu,sp = 274.4 Rsp = 3.307Cep = 1.67 Vu,ep = 336.6 Rep = 1.407Csv = 61.11 Vu,sv = 1.121 Rsv = 0.038Cev = 50.0 Vu,ev = 1.375 Rev = 0.016Cpa = 0.76 Vu,pa = 0 Rpa = 0.023 Lpa = 0.18 10−3

Cpp = 5.80 Vu,pp = 123 Rpp = 0.0894Cpv = 25.37 Vu,pv = 120 Rpv = 0.0056

Table 2.9: Parameters characterizing the vascular system in basal condition

Left Heart Right Heart

Cla = 19.23ml/mmHg Cra = 31.25ml/mmHgVu,la = 25ml Vu,ra = 25mlRla = 2.5 10−3mmHg.s.ml−1 Rra = 2.5 10−3mmHg.s.ml−1

P0,lv = 1.5mmHg P0,rv = 1.5mmHgλlv = 0.014ml−1 λrv = 0.011ml−1

Vu,lv = 16.77ml Vu,rv = 40.8mlEmax,lv = 2.95mmHg/ml Emax,rv = 1.75mmHg/mlkR,lv = 3.75 10−4s/ml kR,rv = 1.4 10−3s/ml

Table 2.10: parameters describing the right and left heart

33

Carotid sinus afferent pathways

Pn = 92mmHg fmin = 2.52spikes/s fmax = 47.78spikes/ska = 11.758mmHg τz = 6.37s τp = 2.076s

Sympathetic efferent pathway

fes,inf = 2.10spikes/s fes,0 = 16.11spiks/s kes = 0.0675sfes,min = 2.66spikes/s

Vagal efferent pathway

fev,0 = 3.2spikes/s fev,inf = 6.3spikes/s kev = 7.06spikes/sfcs,0 = 25spikes/s

Effectors

GEmax,lv= 0.475mmHg.ml−1.v−1 τEmax,lv

= 8s DEmax,lv= 2s Emaxlv,0

= 2.392mmHg/ml

GEmax,rv = 0.282mmHg.ml−1.v−1 τEmax,rv = 8s DEmax,rv = 2s Emaxrv,0 = 1.412mmHg/mlGR,sp = 0.695mmHg.s.ml−1.v−1 τR,sp = 6s DR,sp = 2s Rsp,0 = 2.49mmHg.s/mlGR,ep = 0.53mmHg.s.ml−1.v−1 τR,ep = 6s DR,ep = 2s Rep,0 = 0.78mmHg.s/mlGVu,sv = −265.4ml/v τVu,sv = 20s DVu,sv = 5s Vusv,0 = 1435.4mlGVu,ev = −132.5ml/v τVu,ev = 20s DVu,ev = 5s Vuev,0 = 1537mlGT,s = −0.13s/v τT,s = 2s DT,s = 2s T0 = 0.58sGT,v = 0.09s/v τT,v = 1.5s DT,v = 0.2s

Table 2.11: Basal values of parameters for regulatory mechanisms

2.6.4 Tests

Numerical integration of differential equations is performed using the fifth order Runge-Kutta-Fehlberg method with adjustable step length. During the simulations the integration and mem-orization steps were as low as 0.01s. Several tests have been carried out.

The series of tests done by Ursino are tests to see if there are satisfying results. Ursinocompare his results with [KCYSTN] which has results of a dog CVS. There are similar patternsobserved.

34

2.6.5 Conclusions

After all these tests, Ursino made some interesting conclusions:

1. A new feature of the present model is the characterization of the heart as a pulsatile pump.

2. Despite the unavoidable limitations involved in modeling a complex physiological system,the model is able to reproduce several aspects of carotid baroreflex control rather well.

Further he gave the main limitations and simplifications of the model:

1. The absence of local autoregulation mechanisms in the control of peripheral systemicresistance.

2. The dependence of heart contractility on the carotid baroreflex and on other heamody-namic influences.

3. The present model neglects the effect of changes in coronary perfusion on the end-systolicpressure-volume function.

4. The absence of vagal afferents in the model, especially cardiopulmonary baroreceptors.

5. The description of the central neural processing system, which was simply mimicked bymeans of monotonic exponential functions linking activity in the afferent and efferentneural pathways.

35

2.7 Summary

There are a lot of papers written about the computational model for the human cardiovascularsystem. The papers summarized were selected because we expected that they could satisfy therequirements.

• Is simple,

• Needs little computional time and

• Can accurately reflect a small part of the human CVS.

The big difference between all models is the number of compartments which are needed to sim-ulate the human CVS and the way of simulating the heart. What can be said is that everyauthor is satisfied with his own model and concludes that his model is good for simulation.

2.7.1 Overview Table

In the scheme below we will present an overview table of the models discussed. Finally, we willconclude which model or combination of models we will use for further investigation. In thistable the comparence of the parametervalues are missing. This because it is very difficult tocompare the parameters, since all the models use compartments which reflects different parts ofthe CVS.

36

Table 2.12: Overview table from all summarized articles0D/1D equation including inertia

2.1 0D LdQdt

= P1 − P2 −QR in the large arteriesdPdt

= 1CdVdt

2.2 1D ∂A∂t

+ ∂Q∂x

= 0 not relevant∂Q∂t

+ α ∂∂x

(Q2

A) + A

ρ∂P∂x

+Kr(QA

) = 0

2.3 0D C dpdt

+Q2 −Q1 = 0 yes

LdQdt

+RQ+ P2 − P1 = 0

2.4 0D d

dtPn =

Pn+1−Pn

CnRn+1+

Pn−1−Pn

CnRn+

Pbias−Pn

Cn·

dCn

dt

+ d

dtPbias no

2.5 1D At +Qx = 0 not relevant

Qt +∂∂x

(

αQ2

A+ p

)

= K QA

2.6 0D dPdt

= 1C

(∆Q) in the large arteriesdQdt

= 1L(∆P −RQ)

# compartments circuit type heart simulation

2.1 6 Windkessel like circuit veins+ra,rv,lungs+la,lv

2.2 no compartments one closed system 1 ventricle

2.3 5 Windkessel circuit rv,lv

2.4 12 Windkessel circuit rv,lv

2.5 6 Windkessel circuit heartrate input

2.6 12 Windkessel circuit ra,rv,la,lv

driverfunction parameter values solution method

2.1 e(t) =∑N

i=1Aie−Bi(t−Ci)2 given, from literature ODE15s (Matlab)

2.2 different sine waves given, from literature Taylor Galerkin scheme

2.3 none not given finite difference scheme

2.4 cos given, from literature RK4 integration routine

2.5 none not given not applicable

2.6 sin2 given, from literature 5th order Runge-Kutta-Fehlberg method

37

most important conclusion

2.1 The results show the capability of the presented approach to create patient specific models

2.2 All these tests gave interesting outputs, but there is no comparence with the reality

2.3 the heterogeneous model is able to compute accurately the velocity and pressure fields in the domain of interest

2.4 The model generates steady state and transient hemodynamic responses that compare well to population-averaged

and individual subject data

2.5 Comparison of the results of simulation with measured data show that with this model the behavior of the human

cardiovascular system can be modeled very well

2.6 The model is able to reproduce several aspects of carotid baroreflex control rather well

2.7.2 Discussion

In this discusion we will make a decision which model we are going to use to create a satisfyingmodel. The first question we have to answer is:

• Are we going to use a 0D or 1D model?

In the summarized papers, both 0D and 1D models are used. The 1D model has the advantageof being a detailed model. The 1D model has as disadvantage that it has a high level ofinterdependency between the used equations in the summarized models. Further, the 1D modelhas no realistic results.

The 0D model has the advantage of being a model which can be solved easily and thesolutions do have connections with reality. The disadvantage of the 0D model is that it is onlydetailed in a small part of the circulation system.

With these advantages and disadvantages in mind plus our own demands, we choose for a 0Dmodel. We make this choise, because the advantage of the 0D model is exactly what we wanted.Further, the test cases in the papers show good results and less computational time. Finally,the disadvantage of the 0D model is an advantage for us, because we want a small detailed partof the CVS and a less detailed part for the rest of the system.

By now we know that we use a 0D model, there are some questions about the way ofmodelling:

• What number of compartments do we need?

• What is a good driver function?

• Do we need a lung circlation in our model?

• How to model the body circulation, such that a small part of the can be included?

• Do we need inertia in our model, if so, do we always need it?

When we are looking to the number of compartments, the various models make use of 5, 6 or 12compartments. Section 2.3 (with five compartments) concluded that the heterogeneous modelis able to compute accurately the velocity and pressure fields in the domain of interest, but hasthe problem that it doesnot give the parameter values used. Following the conclusion of 2.3,6 compartments must be enough to compute accurately the velocity and pressure fields in thedomain of interest. So to get feeling with modelling a circulation system, we start with a modellike presented in section 2.1. In this paper all parameters needed are given.

However, Smith concluded in 2.1 that his model is not ready to compute the velocity and

38

pressure in the domain of interest. Following the conclusion of 2.1, 6 compartments are notenough. Although this remark, we think it is possible to create a model with only 3 compart-ments which satisfy our demands. We think this is possible, because the circulation system is asingle closed loop and a compartment reflects a part of the circulation what can be consideredas a whole. In our 3 compartment model, the heart is reflected by a compartment, the aorta bya compartment and the rest of the circulation system reflected as a compartments. The resultsof our 3 compartment model will be compared with the results of the 6 compartment model andthe Wiggers’ diagram.

The next question is about a good driver function. However, each model analized uses an-other driver function so we cannot say anything about a good one for our model. Therefor, wewill test different driver functions. After choosing a good driver function, this function will beused in the other tests.

The third and fourth question can be seen as what part of the circulation system can beconsidered as a whole. In the 6 and 12 compartment models of section 2.1 and section 2.6 a rightventricle plus a lungcirculation is included as compartment. In the model with 3 compartmentswe will include the lungcirculation, because by following one blood element through the circu-lation, then you will see that the whole circulation is one closed circle. Simply said, when wetake out the lung circulation we will miss a piece of the whole circulation. We will only includeit as a part of the systemic resistance.

The last question is about including inertia. As shown in section 2.1 inertia is only neededin the large arteries. However, we will do some tests to see if inertia is needed in our model.

2.7.3 Final Choises

For our research we are going to use the 0D equations to model a human cardiovascular system.We are going to make two different models. The first model, based on section 2.1, reflects theCVS with 6 compartments. This model will be used to test the system of equations. Thesetests will give us the reference figures for the results of the 3 compartment model. Before wetest the 6 and 3 compartment model, we will use a not coupled single compartment to find thebest cardiac driver function and if we include inertia.

39

Chapter 3

Model description

Several numerical models for the human CVS are summarized in section 2. Every model has hisown advantage and disadvantage. These advantages are dependent of the specific goal of themodel. The model we want to create must be able to calculate the pressure and flow on everypoint of the CVS. We use a 0D model, such as the models in section 2.1, 2.3 and 2.6.

3.1 The derivation of a mathematical model

We start with a very simple model of the CVS. This simple model (figure 3.1) contains one

Figure 3.1: A simple model of the CVS

41

central artery free of bifurcations. This central artery represents the pumpingfunction of theheart, the aorta, the cappilairies and the veins. It will not be easy to describe all these parts inone single equation. So we split up the central arterie in different parts, called compartments.Every compartment posesses a variable pressure and at the interfaces with the rest of the systeman inflow and an outflow. The compartments are coupled to from a Windkesselcircuit.

Since we deal with one single artery free of bifurcations, the artery can be reflected by atube. We start with a stiff tube, later we will include the elasticity of the arterie in the model.In this tube we take an oriented length-axis x, with a length l. For every x, a general axial

Figure 3.2: A tube free of bifurcations

section A(t, x) will be defined. Furthermore, the sections Γ1 and Γ2 are the interfaces with therest of the system, while Γw is the artery wall.

We will give two different approaches to derive our model of the human CVS. The firstapproach is a bottum-up consideration, the second approach is a top-down approach whichmake use of the Navier-Stokes equations.

3.2 The 0D model for the circulation system, bottum-up ap-

proach

The system which will be derived consists of two equations. One equation for the conservationof mass and one equation for the conservation of momentum.

42

3.2.1 Conservation of mass

We start the derivation of this equation with the statement that the change of volume in timeequals the difference of inflow and outflow.

dV

dt= Q = Qin −Qout (3.1)

We want to use this statement to find a relation between the pressure and the flow in a smallartery. Therefor, we have to find a relation between the pressure and the volume in a smallartery.

Since the internal radius of a small artery is influenced by the internal and external pressureon the wall and the wall is not very flexible, we can use the linear elastic law.

σ =E

1− ν2ǫ (3.2)

In this formula σ is the surface stress, ǫ = r−r0r0

the rate of change caused by the stress to theoriginal state of the object, E the Young modulus and ν the Poisson ratio of the artery wall(we take ν = 1

2 , as for incompressible tissue). So we have the following linear elastic law for acylindrical vessel:

σ =4E

3

r − r0r0

. (3.3)

However, we do not know anything about the surface stress, but only know the pressure P onthe artery wall. So, we have to look for a relation between the pressure and the surface stress.

Consider a cross section of a small artery free of bifurcations. This cross section has aninternal radius r, a wall thickness h, an internal pressure Pint and an external pressure Pext. Wedefine the transmural pressure P as the difference between the internal and external pressureP = Pint − Pext. Further, we take as reference values the radius r0 and a surface stress σ = 0when there is no pressure on the wall (P = 0). We take a part of the cross section and define onthis cross section a surface element by an infinitesimal angle dφ and longitudinal displacementdx. The area of this element equals rdφdx and the force on this element equals

Prdφdx. (3.4)

The force in the radial direction can be calculated by measuring the angle difference betweentwo points before and after stretching times the stress times the surface area to which the stressis applied.

2hσ sin(dφ

2)dx (3.5)

After equating both forces we come to the following equation.

Prdφdx = 2hσ sin(dφ

2)dx. (3.6)

By letting dφ → 0, sin(dφ2 ) ≈ dφ2 and divide everything by dφdx. We end-up with a stress-

pressure relation for a small artery.rP = hσ. (3.7)

43

Figure 3.3: A cross section of a small artery free of bifurcations

This stress-pressure relation is substituted in the linear elastic law (3.3), yielding:

P =4Eh

3

r − r0rr0

(3.8)

a pressure-radius relation.This is a non-linear equation. Because we want a linear equation, we will linearize equation

(3.8) with a Taylor series. Since

dP

dr=

d

dr

(

4Eh(r − r0)

3rr0

)

=4Eh

3r2(3.9)

and

P (r0) = 0 (3.10)

dP

dr(r0) =

4Eh

3r20(3.11)

The linearized equation equals

P =4Eh(r − r0)

3r20(3.12)

With equation (3.12) we have a linear pressure-radius relation. This relation can be rewritteninto a pressure-volume relation by substituting

V − V0 = πl(r2 − r20) ≈ 2πlr0(r − r0)

44

Figure 3.4: A small part of the cross section

into equation (3.12). What we get is a pressure-volume relation

V =3πlr302hE

P + V0. (3.13)

Taking the derivative of V finally gives the equation for the conservation of mass

dV

dt=

3πlr302hE

dP

dt= Q = Qin −Qout (3.14)

3.2.2 Conservation of momentum

The second equation for a small artery free of bifurcations is the equation for the conservationof momentum. To find this equation, we start again with the small artery free of bifurcations(figure 3.1) in which axial symmetric flow is assumed. This flow is driven by a pressure differencePin − Pout between x = −l/2 and x = l/2. Further, we assume stationairy flow without radialvelocity. For the derivation we start with a force balance in the axial direction (see figure 3.2.2).Equating all the forces results in the following equation:

(P (x)− P (x+ dx)) 2πrdr + 2πµ

(

(rdu

dr)r+dr − (r

du

dr)r

)

dx = 0 (3.15)

In this equation the velocity u(r) satisfies a normal differential equation of second order withthe standard solution:

u(r) =1

4µ

dP

dxr2 + C1 ln(r) + C2 (3.16)

From the fact that u must be limited, it follows that C1 = 0 and from the no-slip condition onthe wall r = r0 it follows that C2 = − 1

4µdPdxr20. This yields a parabolic velocity profile

u(r) = − 1

4µ

dP

dx(r20 − r2) (3.17)

45

Figure 3.5: Force balance in the axial direction

With this velocity profile, the mass flow through the artery, by use of the Poiseuille-Hagen-formula, equals

Q = 2πρ

∫ r0

0rudr =

−πρr408µ

dP

dx=πρr408µl

(Pin − Pout) (3.18)

This equation is the equation for the conservation of momentum for a stationairy flow. Aroundthe heart there is an instationary flow. So we have to look for the force needed to move a columnof blood in the artery. Again, we use figure (3.1) to derive the force needed for the movement.

To find this force, we make use of Newtons second law F = ma. The force on the blood inthe artery equals:

F = (Pin − Pout)πr20 (3.19)

The mass of the blood in the artery equals:

m = lπr20ρ (3.20)

and the velocity of the blood through the artery equals:

u =Q

πr20(3.21)

Using equations (3.19), (3.20) and (3.21) in Newtons second law results in

(Pin − Pout)πr20 = lπr20ρ

d(Q/πr20)

dt(3.22)

Pin − Pout =lρ

πr20

dQ

dt(3.23)

Accounting for this term in the momentum equation results in the momentum equation forinstationary blood flow

Pin − Pout =8µl

πr40Q+

lρ

πr20

dQ

dt(3.24)

46

Final model

Combining the mass equation (3.14) and the momentum equation (3.24) results in the finalsystem for a small artery free of bifurcations:

3πlr302hE

dPdt

= Qin −Qout

Pin − Pout = 8µlπr40

Q+ lρ

πr20

dQdt

(3.25)

3.3 The 0D model for the circulation system, Top-Down ap-proach

Besides the mathematical bottum-up aproach for the derivation of the model of an arterie freeof bifurcations, the Navier-Stokes equations can be used to find the same mathetical model, too.We will use a reduced model of the Navier-Stokes equations for an incompressible fluid for themathematical model of the description of the blood flow in the artery. To come to a reducedmodel, we are integrating the Navier-Stokes equations on a generic section A. Therefor, at firstwe introduce some simplifying assumptions:

1. The blood flows only in the axial direction, so we assume independence of all quantitiesinvolved from the circumferential coordinate θ. As a consequence r is a function of x andt.

2. The flexible artery wall can move in radial direction only. So, if η is the wall displacement,er the unit vector in the radial direction and r0 the reference radius, then η = (r − r0)eris the wall displacement with respect to the reference radius r0.

3. Since the expansion and contraction of the vessel is only in radial direction, we assumethat the x−axis is fixed is time.

4. Next we assume that the artery has an ideal wall, and there is no pressure loss at the wall.So on each section the pressure P is constant and only dependent on x and t.

5. By the ideal wall, the velocity fields orthogonal to the x−axis are negligible compared tothe axial one. The axial component of the velocity will be denoted by ux. The expressionof ux in cylindrical coordinates is supposed to be of the form

ux(t, r, x) = u(t, x)s(rr−1(x)), (3.26)

where u is the mean velocity on each axial section and s : R → R is a velocity profile. Onemay think of s as being a profile representative of an average flow configuration.

6. We neglegt the body forces, such as gravity.

and at second give some expressions:

• A general axial section A will be measured by:

A(t, x) =

∫

A

dσ = πr2(t, x) = π(r0(x) + η(t, x))2; (3.27)

47

• The mean velocity can now be defined by:

u = A−1

∫

A

uxdσ; (3.28)

• It follows from equation (3.26) that∫ 10 s(y)ydy = 1

2 .

u = A−1∫

Auxdσ = 1

πr2

∫ r

0 2πruxdr = 1πr2

∫ r

0 2πrus(rr−1)dr = 2r2u

∫ r

0 rs(rr−1)dr

⇒ r2

2 =∫ r

0 rs(rr−1)dr

y= rr=

∫ 10 rys(ryr

−1)rdy

⇒∫ 10 s(y)ydy = 1

2

For the sake of simplicity, we will choose s(y) = 2(1 − y2), which corresponds to thePoiseuille solution characteristic of steady flows in circular tubes.

• Next we indicate by ψ the momentum flux correction coefficient, defined as

ψ =

∫

Au2xdσ

Au2=

∫

As2dσ

A(3.29)

In general ψ will vary in time, but as consequence of equation (3.26) in our model it willbe constant.

• The mean flux is defined as

Q =

∫

A

uxdσ = Au (3.30)

Under the previous assumptions, the momentum and continuity equations, in the hypothesis ofconstant viscosity, are

div(u) = 0∂ux

∂t+ div(uxu) + 1

ρ∂P∂x− ν∆ux = 0

(3.31)

with on the tube wall the kinematic condition

u = η on Γwt (3.32)

, where η =∂η∂t

= ∂η∂t

er is the vessel wall velocity and with the following boundary conditions:

Qin(t) = Q(t, 0), Pin(t) = P (t, 0), Qout(t) = Q(t, l), Pout(t) = P (t, l)

Consider the portion P of the tube between x = x− dx2 and x = x+ dx

2 , with x ∈ (− l2 ,

l2) and

dx > 0 small enough. The part of δP laying on the tube wall is indicated by ΓwP. The reduced

model is derived by integrating system (3.31) on P and passing to the limit as dx→ 0, assumingthat all quantities are smooth enough.

Before we start with integrating, we will introduce a useful theorem, which has been provenin [QUFO].

48

Theorem 3.3.1 Let f : Ωt × I → R be an axisymmetric function, i.e. ∂f∂θ

= 0. Let us indicate

by fw the value of f on the wall boundary and by f its mean value on each axial section, defined

by

f = A−1

∫

A

fdσ.

We have the following relation ∂∂t

(Af) = A ∂f∂t

+ 2πrηfw. In particular taking f = 1 we recover

that∂A

∂t= 2πrη (3.33)

We start with the continuity equation.Using the divergence theorem, we get

0 =

∫

P

div(u) = −∫

A−

ux +

∫

A+

ux +

∫

ΓwP

u · n = −∫

A−

ux +

∫

A+

ux +

∫

ΓwP

η · n (3.34)

where n is the outwardly oriented normal. Since η = ηer, we deduce∫

ΓwP

η · n = [2ηπr(x)dx+ o(dx)] =∂

∂tA(x)dx+ o(dx).

Substituting into equation (3.34), using the expression of Q and passing to the limit as dx→ 0,we finally obtain

∂A

∂t+∂Q

∂x= 0 (3.35)

On the same way we are going to integrate every term of the momentum equation over P andconsider the limit as dx tends to zero.

•∫

P

∂ux∂t

=d

dt

∫

P

ux −∫

δP

uxg · n =d

dt

∫

P

ux =∂Q

∂t(x)dx+ o(dx) (3.36)

In order to eliminate the boundary integral we have exploited the fact that ux = 0 on ΓwP

and g = 0 on A− and A

+

•∫

Pdiv(uxu) =

∫

δPuxu · n = −

∫

A−u2x +

∫

A+ u2x +

∫

ΓwP

uxg · n= ψ[A(x + dx

2 )u2(x+ dx2 )−A(x− dx

2 )u2(x− dx2 )] = ∂ψAu2

∂x(x)dx+ o(dx)

(3.37)

By using again ux = 0 on ΓwP.

•∫

P

∂P∂x

= −∫

A−P +

∫

A+ P +∫

ΓwP

Pnx

= A(x+ dx2 )P (x+ dx

2 )−A(x− dx2 )P (x− dx

2 ) +∫

ΓwP

Pnx

⇒∫

P

∂P∂x

= A(x+ dx2 )P (x+ dx

2 )−A(x− dx2 )P (x− dx

2 )− P (x)[A(x+ dx2 )−A(x− dx

2 )] + o(dx)

= ∂AP∂x

(x)dx− P (x)∂A∂x

(x)dx+ o(dx)

= A∂P∂x

(x)dx+ o(dx)

(3.38)

49

Since∫

δPnx = 0 and so

∫

ΓwP

Pnx = P (x)∫

ΓwP

nx + o(dx) = −P (x)∫

δP\ΓwP

nx + o(dx) =

−P (x)(A(x + dx2 )−A(x− dx

2 )) + o(dx)

•∫

P

∆ux =

∫

δP

∇ux · n = −∫

A−

∂ux∂x

+

∫

A+

∂ux∂x

+

∫

ΓwP

∇ux · n (3.39)

By assuming that the variation of the change of velocity along the x-axis is small comparedto the other terms, we neglect the term ∂ux

∂x. Using this assumption and splitting n into

two vector components, nr = nrer and nx = n− nr, we may write

∫

P

∆ux =

∫

ΓwP

(∇ux · nx +∇ux · ernr)dσ (3.40)

Again, we neglect ∇ux · nx which is proportional to ∂ux

∂x. Finally, we use equation (3.26)

and the fact that nrdσ = 2πrdx to get

∫

P

∆ux =

∫

ΓwP

nr∇ux ·erdσ =

∫

ΓwP

ur−1s′(1)n ·erdσ = 2π

∫ x+ dx2

x− dx2

us′(1)dx ≈ 2πu(x)s′(1)dx

(3.41)

Substituting all these calculated integrals in the momentum equation, dividing all terms by dxand passing to the limit as dx→ 0, we can write as the reduced momentum equation:

∂Q

∂t+∂ψAu2

∂x+A

ρ

∂P

∂x+Kru = 0 (3.42)

where Kr = −2πνs′(1) is a friction parameter. By choosing a parabolic profile, Kr = 8πν.The reduced system we have after all assumptions is the following one dimensional model forx ∈ (0, l) and t ∈ (0, T ]

∂A∂t

+ ∂Q∂x

= 0∂Q∂t

+ ψ ∂∂x

(Q2

A) + A

ρ∂P∂x

+Kr(QA

) = 0(3.43)

For closing the system, we must find a relation between the pressure and the vessel wall dis-placement. Therefor, we adopt a commonly used hypothesis for the wall mechanics, namelythat the inertial terms are neglegible and that the elastic stresses in the circumferential direc-tion are dominant. With this assumption, the only normal stress acting on the wall is that dueto the pressure. This is possible because we neglected the viscous contribution. In the mostgeneral setting, the pressure relation looks like P (t, x) = Φ(A(t, x);A0(x), β(x)). In this expres-sion, we have outlined that the pressure also depends on A0 = πr20 and on a set of coefficientsβ = (β0, β1, · · · , βq) related to the mechanical and physical properties. In this relation we requirethat Φ is a C1 function of all its arguments and is defined for all A > 0 and A0 > 0. Furthermorewe require that ∂Φ

∂A> 0 and Φ(A0;A0, β) = 0.

For this relation in literature different possibilities are given. We choose to use a linearizedpressure-radius relation 3.12

P (t, x) =4Eh

3(r − r0r20

) (3.44)

50

By using (r − r0) =√A−

√A0√

πwe have the following relation for Φ:

Φ(A;A0, β0) = β0

√A−

√A0

A0where β0 =

4√πhE

3(3.45)

It is simply to verify that indeed all requirements are satisfied.By observing that

∂A

∂t= 2πr

∂r

∂t≈ 2πr0

∂r

∂t= 2πr0(

3r204Eh

∂P (t, x)

∂t) =

3πr302Eh

∂P (t, x)

∂t

we will assume∂A

∂t= k1

∂P

∂t, where k1 =

3πr302Eh

(3.46)

Still we have to deal with an one dimensional model. To find a 0D model, we must perform afurther averaging of the system (3.43) in combination with equation (3.46). At first we introducethe following notation:

• The mean flow rate over the whole district V

Q =1

l

∫

V

uxdV =1

l

∫ l

0

∫

A(x)uxdσdx =

1

l

∫ l

0Qdx (3.47)

• The mean pressure over the whole compartment

p =1

l

∫ l

0Pdx (3.48)

To average further we integrate system (3.43) over x ∈ (0, l). Integration of the first equationgives:

∫ l

0

∂A

∂tdx+

∫ l

0

∂Q

∂xdx =

∫ l

0k1∂P

∂tdx+

∫ l

0

∂Q

∂xdx = k1l

dp

dt+Qout −Qin = 0

Integration of the second equation gives:

∫ l

0

[

∂Q

∂t+ ψ

∂

∂x(Q2

A) +

A

ρ

∂P

∂x+Kr

Q

A

]

dx = ldQ

dt+ψ

[

Q2out

A2− Q2

in

A1

]

+

∫ l

0

[

A

ρ

∂P

∂x+Kr

Q

A

]

dx = 0

Since the integrated system is not linear we introduce the following two assumptions:

1. Since we deal with short pipes, the quantity (Q2

out

A2− Q2

in

A1) is small compared with the other

terms. So in the rest of the calculations we neglect the contribution of the convectiveterms.

2. The variation of A with respect to x is small compared with that of P and Q, so in thesecond equation we can rewrite the integral by saying

∫ l

0

[

A

ρ

∂P

∂x+Kr

Q

A

]

dx ≈∫ l

0

[

A0

ρ

∂P

∂x+Kr

Q

A0

]

dx =A0

ρ(Pout − Pin) +

lKr

A0Q

51

With the last two assumptions we finally have the following zero dimensional model

k1ldpdt

+Qout −Qin = 0ρlA0

dQdt

+ ρlKr

A20Q+ Pout − Pin = 0

(3.49)

We are still not finished, since in this equation we have 6 unknowns and 2 equations, so we needsome more assumptions.

1. At first, we will assume that two values are given, for instance Qin and Pout.

2. Secondly, the dynamics of the system is represented by p and Q, i.e. by the unknownsthat are under the time derivative, so it is reasonable to approximate the unknowns onthe upstream and the downstream sections with the state variables, that is

p = Pin, Q = Qout

With these last assumptions we have the following system

k1ldPin

dt+Qout = Qin

ρlA0

dQout

dt+ ρlKr

A20Qout − Pin = −P0ut

(3.50)

Similarly, we can assume that the Qout and the Pin are given. Then we have

k1ldPout

dt−Qin = −Qout

ρlA0

dQin

dt+ ρlKr

A20Qin + Pout = Pin

(3.51)

This system represents a lumped parameters description of the blood flow in the compliantcylindrical vessel and involves the mean values of the flow rate and the pressure over the do-main, as well as the upstream and downstream flow rate and pressure values. This model canbe considered as an elementary compartment for the description of a more complex system.

3.4 Hydraulical analog

The model above is a mathematical model. In this section we are creating a hydraulic analogof the system. Therefor we start with an electrical circuit depicted in figure 3.4. By using thestandard formulas for an electrical circuit, we can calculate the voltage differences around theinductor and the resistor.

• The voltage drop over de inductor equals:

VL = LdI

dt; (3.52)

• The voltage drop over de resistor equals:

VR = IR; (3.53)

52

Figure 3.6: An electrical circuit, including a resistor, inductor and capacitor

• And the voltage drop over the inductor and the resistor in total equals:

VT = VL + VR = LdI

dt+ IR. (3.54)

On the same way we can calculate the charge that the capacitor will store as a function of thevoltage difference between the two plates

•Iin − Iout = C

dVCdt

. (3.55)

By placing these two equations (3.54) and (3.55) in an electrical system

C dVC

dt= Iin − Iout

VT = LdIdt

+ iR(3.56)

and comparing this system with our two final systems (3.25) and (3.50), then it can be seenthat by grouping parameters, there is a comparison between both systems.

• By comparing the voltage difference with the pressure difference, we can say that

VC = Pin;VT = Pin − Pout; (3.57)

• The current can be compared with the flow rate

I = Q; (3.58)

• Further, we set R = ρKrl

A20

representing the resistance induced to the flow by the blood

viscosity. When we use a parabolic velocity, we have

R =8µl

πr40; (3.59)

53

• We set

L =ρl

A0(3.60)

L represents the inertial term in the momentum conservation law and will be called theinductance of the flow;

• Finally, we set

C =3πr30l

2Eh(3.61)

C represents the coefficient of the mass storage term in the mass conservation law, due tothe capacitance of the vessel.

Using the resistance, inductance and capacitance in the systems (3.50) and (3.25) we have thefollowing hydraulic system:

C dPin

dt+Qout = Qin

LdQout

dt+RQout + Pout = Pin

(3.62)

3.5 Simplification of the model

In the creation of the final model, nothing is said about the small variation in velocity. Assumingthat the steady flow is fully developed, we can say that the term dux

dt= 0. When applying this

assumption to the momentum equation the first term can be neglected. This has as result thatin the final model there is no inertial term

C dPin

dt+Qout = Qin

Qout = Pin−Pout

R

(3.63)

However, significant changes in velocity will occur around the heart valves and the flow probablycannot be assumed steady during the course of a heart beat. In the next chapter we will figureout if we can use the simple model everywhere.

3.6 Simulating the heart with an active compartment

In the model above we derived a passive circulation system. However, the circulation needs apump, so we need an active compartment. For the active compartment, we still make use ofthe momentum equation of system (3.62) or system (3.63), but the continuity equation will bereplaced by an active pressure relation. We will use a Voigt viscoelasticity model to find anactive pressure or pump relation.

The elastance varies during the cardiac cycle as a consequence of the contractile activity ofthe ventricle. At diastole when the muscle fibers are relaxed, the ventricle fills according to anexponential PV function (the EDPVR graph), which reflects the elasticity both of the relaxedmuscle and of its external constraints.

Pedpvr(V ) = P0(eλ(V −V0) − 1) (3.64)

In this exponential function P0, λ and V0 define gradient, curvature and volume at zero pressure,respectively. For the active pressure relation we only need the end-diastolic point. We measurethis point with the linearized function of Pedpvr.

Ped = P0λ(V − V0) = Eed(V − V0) (3.65)

54

The end-systolic point will be measured by a linear PV relation (ESPVR graph), where theslope (usually called the end-systolic elastance) is denoted by Ees.

Pes(V ) = Ees(V − V0) (3.66)

These two relations are plot in figure (1.3). The shifting from the end-diastolic to the end-systolic relationship is governed by a pulsatating activation function (e(t)), called a cardiacdriver function.

P (V, t) = e(t)Pes(V ) + (1− e(t))Ped(V ) 0 ≤ e(t) ≤ 1 (3.67)

P (V, t) = e(t)Ees(V − V0) + (1− e(t))Eed(V − V0) (3.68)

The profile of the cardiac driver function represents the variance of elastance between minimumand maximum values over a single heart beat. A cardiac driver function value one meanselastance is defined by the ESPVR and a value of zero uses the EDPVR to define elastance. Inliterature there are different cardiac driver functions proposed. In Section 2.1 an exponentialcardiac driver function is used:

e(t) =

N∑

i=1

Aie−B1(t−Ci)2 (3.69)

, which has the following shape:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time t (sec)

e(t)

profile of the cardiac driver function

Figure 3.7: A simple cardiac driver function, with parameter values: A = 1, B = 80s−1,C = 0.27s and N = 1

In Section 2.6 a sin2 cardiac driver function is used:

e(t) = sin2(πT (t)

Tsys(t)) (3.70)

where T is the heart period and Tsys the duration of the systolic phase which has the followingshape:In Section 2.2 and 2.3 a sin function is used:

e(t) = sin(πT (t)

Tsys) (3.71)

Writing down the system which has to be solved for an active compartment,

P (V, t) = e(t)Ees(V − Ves,0) + (1− e(t))Eed(V − Ved,0)

Qout = Pin−Pout

R

(3.72)

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.8: sin2 cardiac driver function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.9: sin cardiac driver function

Notice that the volume is included as an extra unknown.By choosing the Volume such that Ves,0 = Ved,0 = 0 we can rewrite the equation for the

pressure toC(t)P (V, t) = V (t) where C(t) = 1/(e(t)Ees + (1− e(t))Eed) (3.73)

Taking the derivative on both sides and using equation (3.1) gives

dC(t)P (t)

dt= Qin −Qout (3.74)

By eliminating the volume the system which has to be solved for an active compartment equals

dCPdt

= Qin −QoutQout = Pin−Pout

R

(3.75)

Comparing this system with (3.63) the system for a passive compartment without inertia, itcan be seen that the only difference is the time dependance of the capacitance in the activecompartment.

3.7 Valve simulation

The heart ejects blood in a small time of one heart beat. The rest of the time the heart relax,contracts or fills. The four periods are scheduled by the valves. When we are calculating thepressure, the flow is calculated by Qout = (Pin − Pout)/R. If we use the theory, then the valves

56

opens when Pin > Pout and the valves close when Pin ≤ Pout. So for closing and opening thevalves we have the following relation:

Qout =

0 if Pin ≤ Pout(Pin − Pout)/R if Pin > Pout

(3.76)

3.8 Compartment coupling

In section 3 we create a hydraulic model for a passive compartment and for an active compart-ment. Now, we must describe a whole human CVS. Therefor, we have to couple compartments.For the coupling of the compartments, we make use of a Windkessel circuit, as described insection 1.2.3. To solve the Windkessel circuit we use a PV method (section 1.2.2). In chapter 2we make the choice to create a 6 and a 3 compartment model.

3.8.1 The 6 compartment model

The 6 compartment model we propose contain two active compartments and four passive com-partments. The two active compartments describe the right and left ventricle (subscript rv andlv), two passive compartments decribe the lung circulation (subscript pa and pu) and two passivecompartments decribe the body circulation (subscript ao and vc). Further, all the compartmentsare coupled by a resistor (subscripts pv, pul, mt, av, sys, tc). The figure below reflects such a6 compartment model. By coupling the different elements in the system, we have the following

Figure 3.10: A 6 compartment model

57

system of equations:

˙(CPrv) = Qtc −QpvPpa = 1

Cpa(Qpv −Qpul)

Ppu = 1Cpu

(Qpul −Qmt)

˙(CPlv) = Qmt −Qav

Pao = 1Cao

(Qav −Qsys)

Pvc = 1Cvc

(Qsys −Qtc)

Qtc = Pvc−Prv

Rtcif Pvc > Prv otherwise Qtc = 0

Qpv =Prv−Ppa

Rpvif Prv > Ppa otherwise Qpv = 0

Qpul =Ppa−Ppu

Rpul

Qmt =Ppu−Plv

Rmtif Ppu > Plv otherwise Qmt = 0

Qav = Plv−Pao

Ravif Plv > Pao otherwise Qav = 0

Qsys = Pao−Pvc

Rsys

(3.77)

with the statevector [Plv, Pao, Pvc, Prv , Ppa, Ppu, Qtc, Qpv, Qpul, Qmt, Qav , Qsys].

3.8.2 The 3 compartment model

In the 3 compartment model we only include the important compartments. We use one activecompartment for the left ventricle (subscript lv), one passive compartment for the aorta (sub-script ao) and one passive compartment for the rest of the body circulation (subscript bc). Thecompartments are coupled by a resistor (subscript av, sys, mt). See further the reflection below.The 3 compartment model has the following system of equations:

˙(CPlv) = Qmt −Qav

Pao = 1Cao

(Qav −Qsys)

Pbc = 1Cbc

(Qsys −Qmt)

Qav = Plv−Pao

Ravif Plv > Pao otherwise Qav = 0

Qsys = Pao−Pbc

Rsys

Qmt = Pbc−Plv

Rmtif Pbc > Plv otherwise Qmt = 0

(3.78)

with the following statevector [Plv, Pao, Pbc, Qav , Qsys, Qmt]

3.9 Summary

In this section we derive, on a bottum-up and a top-down approach, two hydraulic systems forone compartment of the circulation system,

• One without inertia

dCPin

dt+Qout = Qin

Qout = Pin−Pout

R

(3.79)

58

Figure 3.11: A 3 compartment model

• and one with inertia

dCPin

dt+Qout = Qin

LdQout

dt+RQout − Pin = −Pout

(3.80)

The difference between a passive and an active compartment lies in the time dependance of thecapacity. For the active pressure relation we propose three different cardiac driver functions.

•

e(t) =N

∑

i=1

Aie−B1(t−Ci)

2(3.81)

•e(t) = sin2(

πT (t)

Tsys(t)) (3.82)

•e(t) = sin(

πT (t)

Tsys) (3.83)

Finally, the valve regulation is described by

Qout =

0 if Pin ≤ Pout(Pin − Pout)/R if Pin > Pout

(3.84)

With the system of equations we have built a 3 and a 6 compartment model.

59

Chapter 4

Numerical Model

In the previous chapter we derived a mathematical model for the human CVS. This model isbuilt with a number of compartments that are connected. In this section we are going to describehow to solve the mathematical model for different single compartment models. Next, we lookto the influence of different initial values and we are going to do some tests to give answer onthe questions we stated in section 2.7.2.


• Do we need inertia in our model,if so, do we always need it?

With the answers on these questions we give a numerical model for an ideal 3 and 6 compartmentmodel and describe how to solve these models.

Therefor, we start with a numerical method for solving a single compartment model. Wehave three types of the single compartments, the passive compartment without inertia, thepassive compartment with inertia and the active compartment. For each single compartmentwe will give a discretisation.

For the tests we use the following parameters. These are copied from [SMITH].

passive compartment active compartment

C 1/(98e6) Ees 100e6

R1 2.75e6 Eed 0.33e6

R2 170e6 R1 6.1e6

L1 5e4 R2 2.75e6

L2 3e5 P1 80P1 13 P3 100P3 5 a 1

b 80c 0.27

Tsys 0.5

Table 4.1: Parameters in the single compartment tests

61

4.1 A passive compartment without inertia

The system of equations for a single passive compartment equals:

dP2dt

= 1C

(Q1 −Q2)

Q1 = P1−P2R1

Q2 = P2−P3R2

(4.1)

We want to solve this system with a PV-method. Therefor we have to discretize the model andhereafter solve at first the pressure and use next the pressure to calculate the flow. There aredifferent possibilities for the discretisation. It is possible to use the old and the new pressureto calculate the new flow. By using the old pressure you discretisize with a Jacobi-like methodand by using the new pressure the discretisation is a Gauss-Seidel like method. We will do bothdiscretisations and perform a stability analysis to see the difference.

4.1.1 Discretisation, Jacobi like method

We start with the discretisation of the system with respect to the time.

P(n+1)2 = P

(n)2 + δt

C(Q

(n)1 −Q

(n)2 )

Q(n+1)1 =

P1−P (n)2

R1

Q(n+1)2 =

P(n)2 −P3

R2

(4.2)

This system can be placed in a matrix form:

y(n+1) = Ay(n) +B; y =

P2

Q1

Q2

, A =

1 δtC

− δtC

− 1R1

0 01R2

0 0

, B =

0P1R1P3R2

(4.3)

And solve the system by the following iterative process:

1. Pass the statevector [P(n)2 , Q

(n)1 , Q

(n)2 ]T to the solver.

2. Calculate at the same time the pressure P(n+1)2 in the compartment and the inflow Q

(n+1)1

and outflow Q(n+1)2 .

4.1.2 Discretisation, Gauss-Seidel like method

In this case the discretized system reads:

P(n+1)2 = P

(n)2 + δt

C(Q

(n)1 −Q

(n)2 )

Q(n+1)1 =

P1−P (n+1)2R1

Q(n+1)2 =

P(n+1)2 −P3

R2

(4.4)

62

Rewritten in matrix form:

y(n+1) = Ay(n) +B; y =

P2

Q1

Q2

, A =

1 δtC

− δtC

− 1R1

− δtR1C

δtR1C

1R2

δtR2C

− δtR2C

, B =

0P1R1P3R2

(4.5)

and solve the system with the following iterative process:

1. Pass the statevector [P(n)2 , Q

(n)1 , Q

(n)2 ]T to the solver.

2. Calculate the pressure P(n+1)2 in the compartment.

3. Use the pressure P(n+1)2 to calculate the inflow Q

(n+1)1 and outflow Q

(n+1)2 .

Now we have the two possible discretisations, we compare these two by a stability analysis.

4.1.3 Stability analysis of the Jacobi and Gauss-Seidel like method

Before we start a stability analysis we want to remark that for the calculations in this sectionwe assume that the input (P1) and output (P3) pressure are constant.

Jacobi like method

To investigate the stabilty of both discretized systems with the Von Neumann analysis, wesubstitute in system (4.2) for y(n) the Fourier component

cnkeiθk , with θk = 2πk/m where k = 0, . . . ,m− 1.

We findcn+1k eiθk = cnke

iθkA+B.

Since the vector B is a constant vector, it has no influence on the stability of the system and soA is the amplification matrix. The eigenvalues of matrix A are

λ1 = 0, λ2,3 =1

2∓ 1

2

√

1− 4δt

C

R1 +R2

R1R2

For absolute stability it is required that |max(λ1, λ2, λ3)| < 1. The eigenvalues λ2,3 can beimaginary.

Real eigenvalues

|12

+1

2

√

1− 4δt

C

R1 +R2

R1R2| < 1

−3 <

√

1− 4δt

C

R1 +R2

R1R2< 1

1 < 1− 4δt

C

R1 +R2

R1R2< 9

−2CR1R2

R1 +R2< δt < 0

63

Imaginairy eigenvalues

|12

+1

2

√

1− 4δt

C

R1 +R2

R1R2| < 1

0 <1

4− 1

4(1− 4

δt

C

R1 +R2

R1R2) < 1

0 <δt

C

R1 +R2

R1R2< 1

and so it can be seen that for

0 < δt <CR1R2

R1 +R2(4.6)

we have absolute stability.To compare the theroretical stability with the practical stability, we implement the passive

compartment with the Jacobi like discretisation in matlab. We use the parameters from table4.1 and let the program run 90 iterationsteps. In figure 4.1 can be seen that the maximum timestep for absolute stability lies between δt = 0.0275 and δt = 0.028. Using the same parameters tocalculate the maximum time step for theoretical absolute stability then there can be a maximumtime step of δt = 0.0276. So practice conforms the theory.

0 10 20 30 40 50 60 70 80 90 10012

12.5

13

13.5

14

dt=

0.02

7

0 10 20 30 40 50 60 70 80 9012

12.5

13

13.5

14

pres

sure

dt=

0.02

75

0 10 20 30 40 50 60 70 80 9011

12

13

14

15

number of iteration steps

dt=

0.02

8

Figure 4.1: The convergence by different time steps with the Jacobi like method

64

Gauss-Seidel like method

Again the stability of this system will be analyzed by the Von Neumann analysis. After substi-tuting the Fourier component cnke

iθk in the system, the matrix A from equation (4.5) is the theamplification matrix in the Von Neumann analysis. It can be seen that the eigenvalues λ are

λ1,2 = 0 λ3 = 1− ( δtR1C

+ δtR2C

) (4.7)

and that

0 ≤ δt < 2R1R2C

R1 +R2(4.8)

When we implement this passive compartment with a Gauss Seidel like discretisation into mat-lab, use the parameters given in table (4.1) and let it run for 90 iteration steps. than it can seenin figure 4.2 that the practical maximum time step for absolute stability lies between δt = 0.055and δt = 0.056. The theoretical maximum time step for absolute stability equals δt = 0.0552.So practice conforms theory.

0 10 20 30 40 50 60 70 80 90 10012

12.5

13

13.5

14

dt=

0.05

4

0 10 20 30 40 50 60 70 80 9012

12.5

13

13.5

14

pres

sure

dt=

0.05

5

0 10 20 30 40 50 60 70 80 900

10

20

30

number of iteration steps

dt=

0.05

6

Figure 4.2: The convergence for different time steps with the Gauss Seidel like method. In thisfigure only the 25th heartbeat is depicted.

With the Von Neumann analysis we can theoretically as well as practically conclude that passingthe P (n+1) to the solver for calculating the flow is a good choice, since there will be absolutestability with a time step which can be up to 2 times higher. For the rest of the calculations wewill pass P (n+1) to the solver for the calculations of the inflow and outflow.

65

4.2 A passive compartment with inertia

We will only include inertia at the inflow or at the outflow section of the compartment, never onboth sections. By now we assume that the inflow is been influenced by inertia. So the followingsystem has to be solved:

dP2dt

= 1C

(Q1 −Q2)

L1dQ1

dt= P1 − P2 −R1Q1

Q2 = P2−P3R2

(4.9)

To solve this system, at first we discretize to the time,

P(n+1)2 = P

(n)2 + δt

C(Q

(n)1 −Q

(n)2 )

Q(n+1)1 = δt

L1(P1 − P

(n+1)2 ) + (1 − δtR1

L1)Q

(n)1

Q(n+1)2 =

P(n+1)2 −P3

R2

(4.10)

and next rewrite the numerical system in matrix notation:

y(t+1) = Ay(t) +B;

y =

P2

Q1

Q2

, A =

1 δtC

− δtC

− δtL1

− (δt)2

L1C+ 1− δtR1

L1

(δt)2

L1C1R2

δtR2C

− δtR2C

, B =

0δtL1P1

− P3R2

(4.11)

and solve the system by the following iterative process:

1. Pass the statevector [P2, Q1, Q2]T to the solver.




(n+1)2 .

4.3 An active compartment

The system we have to solve equals

dC(t)P2

dt= Q1 −Q2

Q1 = P1−P2R1

if P1 > P2 else 0

Q2 = P2−P3R2

if P2 > P3 else 0

(4.12)

Again, discretize with respect to the time,

P(n+1)2 = (CP2)(n)+

C(n+1) + δtC(n+1) (Q

(n)1 −Q

(n)2 )

Q(n+1)1 =

P1−P (n+1)2R1

if P1 > P(n+1)2 else 0

Q(n+1)2 =

P(n+1)2 −P3

R2if P

(n+1)2 > P3 else 0

(4.13)

66

and rewrite this in matrix notation:

y(n+1) = (I −D)−1Ay(n) +B;

y =

P2

Q1

Q2

,

A =

C(n)

C(n+1)δt

C(n+1) − δtC(n+1)

− C(n)

R1C(n+1) if P1 > P (n+1) else 0 − δtR1C(n+1)

δtR1C(n+1)

C(n)

R2C(n+1) if P (n+1) > P3 else 0 δtR2C(n+1) − δt

R2C(n+1)

,

B =

0

P1R1

if P1 > P2 else 0

P3R2

if P2 > P3 else 0

(4.14)

For solving the system, do the following iterative process:

1. Pass the statevector [P2, Q1, Q2]T to the solver.




(n+1)2 .

Which driver function do we want to use?

For the cardiac driver function there are different models proposed in chapter 2.

1.

e(t) =

N∑

i=1

Aie−Bi(t−Ci)2 (4.15)

with N = 1, A = 1, B = 0.8 and C = 0.27.

2.

e(t) = sin2(πT (t)

Tsys) (4.16)

with Tsys = 0.5.

3.

e(t) = sin(πT (t)

Tsys) (4.17)

with Tsys = 0.5.

Out of these three models we choose the best model. The results plotted in figure 4.3 do wecompare with the Wiggers diagram in section 1.2.4. We conclude that the exponentional driverfunction gives the most realistic results. The form of the graph of the left ventricle pressure hasthe best comparison and the outflow has the most reasonable strength.

67

0 0.5 10

0.5

1

driv

erfu

nctio

n

0 0.5 10

0.5

1

exp(−b(t−c)2) sin2(pi t/0.5) sin(pi t/0.5)

0 0.5 10

0.5

1

0 0.5 10

50

100

150

pres

sure

0 0.5 10

0.5

1

1.5x 10

−5

inflo

w

0 0.5 1−4

−2

0

2x 10

−5

outfl

ow

0 0.5 10

50

100

150

0 0.5 10

0.5

1

1.5x 10

−5

0 0.5 1−4

−2

0

2x 10

−5

time

0 0.5 10

50

100

150

0 0.5 10

0.5

1

1.5x 10

−5

0 0.5 1−4

−2

0

2x 10

−5

Figure 4.3: The difference in result by using different driverfunctions

4.4 Testing the single compartment model

Now we have solved the different single compartments we can answer the questions we proposedin section 2.7.2. Before we answer the questions, we have a look at the initial conditions.

4.4.1 The initial conditions

The passive compartment without inertia is simulated with different initial conditions for thepressure. As can be seen, for all initial pressures it converge to the same pressure, namely12.8726mmHg. Theoretically, this value can be determined by solving the system of equationsanalytically.

dP2dt

= 1C

(Qin −Qout)

Qin = P1−P2R1

Qout = P2−P3R2

dP2

dt=

1

C(P1 − P2

R1− P2 − P3

R2)

dP2

dt= −R1 +R2

CR1R2P2 +

P1R2 + P3R1

CR1R2

68

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

20

40

60

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

5x 10

−6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−1

0

1

2

3x 10

−7

time

outfl

ow

inflo

w

pre

ssur

edifferent initial conditions

Figure 4.4: Starting with different initial conditions has no influence on the final results

P2(t) = P0e−R2+R1

CR1R2t+R2P1 +R1P3

R2 +R1

In the passive compartment the initial pressure P0 = 0 and the analytical solution equals:

P (t) =R2P1 +R1P3

R2 +R1

Substituting the simulation parameters yields limit value 12.8726mmHg. The same value as towhich our numerical model converge.

4.4.2 Including the inertial term?

What is the influence of inertia on the pressure and the flow? As can be seen in figure 4.5,inertia causes a retardation in the flow, and so causes a higher pressure in the beginning. Aftera small time the system with inertia converges to the same limit value.

4.5 A 3 compartment model

The creation of a 3 compartment model is simple: we couple three single compartments. Inthis model, we use passive compartments without inertia and use the exponential cardiac driver

69

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.212

12.5

13

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

1

2

3

4x 10

−7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.24

4.2

4.4

4.6

4.8x 10

−8

time

outfl

ow

inf

low

p

ress

ure

inertia off

inertia on

inertia off

inertia on

inertia off

inertia on

Figure 4.5: The difference in the results by using inertia

function. For the discretisation are we using the Gauss-Seidel like method. Since we have acoupled system there is no prescribed pressure input and output between the compartments.

For solving the mathematical model, we start with the discretisation to the time:

P(n+1)lv = (CPlv)(n)+

C(n+1) + δtC(n+1) (Q

(n)mt −Q

(n)av )

P(n+1)ao = P

(n)ao + δt

C(Q

(n)av −Q

(n)sys)

P(n+1)vc = P

(n)vc + δt

C(Q

(n)sys −Q

(n)mt )

Q(n+1)mt =

P(n+1)vc −P (n+1)

lv

Rmtif P

(n+1)vc > P

(n+1)lv otherwise Q

(n+1)mt = 0

Q(n+1)av =

P(n+1)lv

−P (n+1)ao

Ravif P

(n+1)lv > P

(n+1)ao otherwise Q

(n+1)av = 0

Q(n+1)sys = P

(n+1)ao −P (n+1)

vc

Rsys

(4.18)

and solve this numerical model with the following iterative process:

1. Pass the statevector [Plv, Pao, Pvc, Qmt, Qav , Qsys]T to the solver.

2. Calculate the pressure P(n+1)lv , P

(n+1)ao , P

(n+1)vc in the compartments.

70

3. Use the pressure P(n+1)lv , P

(n+1)ao , P

(n+1)vc to calculate the flow Q

(n+1)mt , Q

(n+1)av , Q

(n+1)sys be-

tween the compartments.

4.6 A 6 compartment model

In this section the steps necessary for solving the model proposed in section 3.8.1 numericallyare outlined. Start with the time discretisation:

P(n+1)rv = (CPrv)(n)+

C(n+1) + δtC(n+1) (Q

(n)tc −Q

(n)pv )

P(n+1)pa = P

(n)pa + δt

C(Q

(n)pv −Q

(n)pul)

P(n+1)pu = P

(n)pu + δt

C(Q

(n)pul −Q

(n)mt )

P(n+1)lv = (CPlv)(n)+

C(n+1) + δtC(n+1) (Q

(n)mt −Q

(n)av )

P(n+1)ao = P

(n)ao + δt

C(Q

(n)av −Q

(n)sys)

P(n+1)vc = P

(n)vc + δt

C(Q

(n)sys −Q

(n)tc )

Q(n+1)tc = P

(n+1)vc −P (n+1)

rv

Rtcif P

(n+1)vc > P

(n+1)rv otherwise Q

(n+1)tc = 0

Q(n+1)pv =

P(n+1)rv −P (n+1)

pa

Rpvif P

(n+1)rv > P

(n+1)pa otherwise Q

(n+1)pv = 0

Q(n+1)pul =

P(n+1)pa −P (n+1)

pu

Rpul

Q(n+1)mt =

P(n+1)pu −P (n+1)

lv

Rmtif P

(n+1)pu > P

(n+1)lv otherwise Q

(n+1)mt = 0

Q(n+1)av =

P(n+1)lv

−P (n+1)ao

Ravif P

(n+1)lv > P

(n+1)ao otherwise Q

(n+1)av = 0

Q(n+1)sys = P

(n+1)ao −P (n+1)

vc

Rsys

(4.19)

and solve this numerical model with the following iterative process:

1. Pass the statevector [Prv, Ppa, Ppu, Plv , Pao, Pvc, Qtc, Qpv, Qpul, Qmt, Qav, Qsys]T to the solver.

2. Calculate the pressure P(n+1)rv , P

(n+1)pa , P

(n+1)pu , P

(n+1)lv , P

(n+1)ao , P

(n+1)vc in the compartments.

3. Use the pressures P(n+1)rv , P

(n+1)pa , P

(n+1)pu , P

(n+1)lv , P

(n+1)ao , P

(n+1)vc to calculate the flows

Q(n+1)tc , Q

(n+1)pv , Q

(n+1)pul , Q

(n+1)mt , Q

(n+1)av , Q

(n+1)sys between the compartments.

4.6.1 Summary

We started this section with different numerical models for a 1 compartment model. With thesemodels we did different tests. We showed that the system is stable and that it is better to usea Gauss-Seidel like method for the time discretisation since it permits a timestep twice as largeas a Jacobi like method.

Further we chose

e(t) =

N∑

i=1

Aie−Bi(t−Ci)2 (4.20)

with N = 1, A = 1, B = 0.8 and C = 0.27 as best driver function, by using the active 1compartment model as test model.

71

Finally, we used the passive compartment model to investigate the effect of including aninertial term. We saw that inertia caused a retardation in the flow, but has no further influenceon the flow. So we decided to use only compartments without inertia.

After these tests had been performed we continue with the discretisation of the 3 and the 6compartment models.

72

Chapter 5

Testing the models

In the last two chapters we derived a model for the human CVS and we gave a scheme howto solve the system for a specific number of compartments. In this chapter we will solve themodel for the human CVS. Since we did not find parameters for the three compartment modelin literature we have to guess these parameters. We are going to do this by solving the modelfor a 6 compartment model using of the parameters postponed by [SMITH]. The results will becompared with the results of [SMITH] and with the Wigger’s diagram. After satisfying resultswe will use this results to guess reasonable parameters for the three compartment model. Besidestesting the results with others, we independently programmed the numerical model, in Matlaband Simulink, with the same results.

5.1 Verification of the models

5.1.1 A six compartment model

To solve the system of equations for the 6 compartment model 4.6, we use the parameters givenin table (5.1). Solving the system gives the results depicted in figure (5.1): At first, we willcompare our results with that of [SMITH]. Looking at the graphs we see the same form in thediagrams and in the volume-time diagram equal SV. The difference with the results of [SMITH]is that we have less volume and more pressure in the ventricles. Since Smith is using inertia andventricular interaction in his model, this probably explains the difference in the results. Thisfar, we must be satisfied with our results.

Heart Resistance Capacity

Eeslv 100e6 Rav 2.75e6 Cao 1/98e6Eedlv 0.33e6 Rsys 170e6 Cvc 1/1.3e6Eesrv 54e6 Rvc 1e6 Cpa 1/72e6Eedrv 0.23e6 Rpv 1e6 Cpu 1/1.9e6

a 1 Rpul 9.4e6b 0.80 Rmt 6.1e6c 0.27

Table 5.1: Parameters in the six compartment model

73

0 0.25 0.5 0.75 1 1.2590

100

110

120

130

140Left Ventricle [lv]

Vol

ume

[Vlv

] (m

l)

0 0.25 0.5 0.75 1 1.2530

40

50

60

70

80Right Ventricle [rv]

Vol

ume

[Vrv

] (m

l)

0 0.25 0.5 0.75 1 1.250

20

40

60

80

100

120

Time [t] (secs)

Pre

ssur

e [P

lv] (

kPa)

0 0.25 0.5 0.75 1 1.250

10

20P

ress

ure

[Prv

] (kP

a)

Time [t] (secs)

SV=38 SV=38

ao

pu

lv pa

rv vc

Figure 5.1: Simulation results from the closed loop model without inertia with our own program

However, comparing our results with a Wiggers’ diagram, we see a good comparison in form,but a very bad comparison in magnitude. We see a bad comparence, because we calculate forthe pressure with a scale of kPa and our results are more realistic with a scale of mmHg. finally,we conclude that we are satisfied with the form of the graph, but not with the choice of theparameters.

5.1.2 A three compartmentmodel

We are satisfied with the form of the graphs, so we proceed with the validation of a three com-parment model. As initial guess, we start with the parameters we use in the 6 compartmentmodel and by reasoning we want to find reasonable parameters. We have reasonable parameterswhen the graphs do have good comparence with a Wiggers’ diagram.

Finding reasonable parameters

In our search for good parameters we start with the left ventricle compartment (heart). We aresatisfied about the form of the graphs of the left ventricle, so we do not change the parameters(a, b and c) of the cardiac driver function. We have to change the Eeslv and the Eedlv. The

74

Figure 5.2: Simulation results from the closed loop model with inertia and ventricular interac-tion, Results from [SMITH]

Eeslv is the slope of the linear pressure volume relation which measures the end-systolic point.Since the pressure is too high, the end-systolic point has to be lower and so Eeslv < 100e6. Thepressure in the left ventricle is shifting between the Pes and the Ped. So the stroke volume canbe calculated by SV = Ves,max−Ved,min where Ves,max is the volume of blood in the end-systolicphase and Ved,min the volume of blood in the end-diastolic phase. Now, since we have chosenthat the Eeslv must be smaller (for a lower Pes and so a lower Ves,max) and we are satisfied withthe SV, we must have a lower Ved,min. This is achievd by setting Eedlv < 0.33e6.

We continue with the aortic compartment and the body circulation compartment. Thevolume in the compartment is calculated by the capacity times the pressure. Since the totalvolume will not be much smaller with the choice of the parameters for the left ventricle andthe pressure must be much lower, the aorta and the pulmonary vein must have less capacity, soCao < 1/98e6 and Cpu < 1/1.9e6.

For the resistance parameters we have to look at the flow between the compartments. Inthe Wiggers’ diagram of the 6 compartment model, the outflow is somewhat too strong, soRav > 2.75e6. For the inflow the same can be said, so Rmt > 6.1e6. The last resistancereflects the whole systemic part. In comparison with the 6 compartment model, it contains the

75

0 0.2 0.4 0.6 0.8 1 1.2 1.40

40

80

120

Wiggers’ diagramP

ress

ure

[Plv

] (kP

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6x 10

−6

Flo

w[Q

] (*1

0−3 m

3 /s)

0 0.2 0.4 0.6 0.8 1 1.2 1.40.8

1

1.2

1.4x 10

−6

Vol

ume

[Vlv

] (*1

0−3 m

3 )

0 0.2 0.4 0.6 0.8 1 1.2 1.40

10

20

30

Pre

ssur

e[P

rv] (

kPa)

time [t] (secs)

filling ejection

contraction relaxation

ao

lv mt

outflow inflow

pa

rv vc

Figure 5.3: A Wiggers’ diagram from a 6 compartment model

Rsys, Rvc, Rpv, Rpul, Cvc, Crv and Cpa. Since, this resistance describes a large area, we have tochoose this resistance Rsys > 170e6.

Now that we know how to search for the correct parameters we find after comparing witha Wiggers’ diagram the following parameters, see table 5.2: In the resulting Wiggers’ diagram(figure (5.4)) the volume in the left ventricle is a little bit too large, but the pressure and theinflow and outflow are correct. Further, we see that the ejection time is too small in comparisonwith the real Wiggers’ diagram, but this has no influence on the pressure in the ventricle. Weconclude that we have found a three compartment model with reasonable parameters.

5.2 Testing the model with some extreme cases

The model as presented above simulates the human CVS of a healthy person. What will happenwith the solutions when the parameters will reflect an ill person? That is what we are looking forin this section. Since our model is built around the heart, we are going to look for heart diseases.There are different heart diseases. There are the heart failures and there are the shocks.

76

Heart Resistance Capacity

Eeslv 85e6 Rav 6.75e6 Cao 1/175e6Eedlv 0.15e6 Rsys 225e6 Cpu 1/30e6

a 1 Rmt 15.2e6b 0.80c 0.27

Table 5.2: The parameters in a three compartment model

5.2.1 Heart Failure

A heart failure can be caused by several problems in the heart. For example the filling or ejectingproblems, called diastolic or systolic dysfunction, respectively, caused by a myocardial disorder.Further there are the valvular disorders, like the valvular stenosis or the valvular insufficiency.In the next paragraphs we investigate if our system reacts correctly on this kind of heart failures.

Diastolic dysfunction caused by a myocardial disorder

The diastolic dysfunction can be caused by several disorders. One such disorder is the myocardialdisorder. This kind of disorder limits the ability of the heart to relax so blood can enter theventricle during diastole. It can be characterized by an increase in ventricle filling pressure, adecrease in ventricle volume, diminished cardiac output and in the absence of reflex responses,a drop in end-systolic pressure, [BRWD]. The effect of this dysfunction can be simulated byincreasing the ventricle elastance at end-diastole (Eedlv).

We simulate a diastolic dysfunction by increasing Eedlv with a factor of 10. The results areshown in figure 5.5. It reflects exactly the characteristics of a diastolic dysfunction. Comparingthe result with a graph from literature, figure (5.6), we see that there is a good comparison. Theonly difference is that the figure from literature is a schematic illustration specificially focussedon end-diastolic function and does not show changes in end-diastolic pressure.

Systolic dysfunction caused by a myocardial disorder

The systolic dysfunction is caused by a myocardial infarction where myocardium died due tolack of oxygen. The main impact of myocardial infarction is a drop in ventricle contractilitybecause the weakened heart is no longer able to eject an adequate amount of blood. Decreasedcontractility is simulated in the minimal model by decreasing the Eeslv. Further, systolic dys-function can be characterized by increasing ventricle preload, a rise in ventricle volume, a dropin stroke volume and decreased systemic pressure, [KUPA].

We simulate a systolic dysfunction by halving the ventricle contractility, see figure 5.7. Thissimulation has as result that there is more volume and less pressure in the left ventricle. Figure5.8 from literature shows the same.

Valvular disorder caused by valvular stenosis

Valvular stenosis occurs when a heart valve doesnot open properly, causing a much higherresistance to blood flow passing through the valve decreasing the flow rate. Characteristic con-sequences are an increased left ventricle systolic pressure and decreased average aortic pressure.

77

0 200 400 600 800 1000 1200 14000

4

8

12

16

20Wiggers’ diagram

pres

sure

(kP

a)

0 200 400 600 800 1000 1200 1400

0

200

400

Flo

w (

ml/s

)

0 200 400 600 800 1000 1200 1400150

175

200

225

250

time (ms)

volu

me

(ml)

aortic pressure

left ventricle pressure

left atrium pressure

outflow

left ventricle volume

inflow

filling ejection

contraction relaxation

Figure 5.4: A Wiggers’ diagram from a 3 compartment model

Looking at figure 5.10 from literature, then we see an increase in the difference between themaximum left ventricle pressure and the maximum aortic pressure (max Plv-max Pao). Thisincreased difference is caused by the larger pressure drop across the aortic valve as a result ofthe higher resistance.

In our model we model the valvular stenosis by increasing the aortic valve resistance (Rav)by a factor of 5. In figure 5.9 it can be seen that there is a significant increase in the maximumleft ventricle pressure along with a drop in the average pressure in the aorta.

Valvular disorder caused by valvular insufficiency

Aortic insufficience is characterized by an increase in the left ventricle volume and stroke volumeand reduced aortic diastolic pressure, [BRWD]. Although ventricle stroke volume increases,cardiac output decreases, as much of the blood pumped into the aorta during diastole canreturn to the ventricle during systole. In literature, figure 5.10, schematically illustrated theeffect of valvular insufficiency on ventricle and aortic pressure.

In our model we simulate the valvular insuffuciency by increasing the aortic valve resistanceby a factor of 20 when the valve would normally close. The results are plotted in the figure(5.11)

78

160 170 180 190 200 210 160 170 180 1900

2

4

6

8

10

12

14

16

volume (ml)

pres

sure

(kP

a)PV diagram of the left ventricle

diastolic dysfunctionnormal

Figure 5.5: Simulating a dystolic disfunction

5.2.2 Shock

A general definition of shock is, tissue damage due to lack of oxygen and other nutrients. In thissection we will simulate one kind of a shock. Another kind of shock which can occur we havealready seen with the systolic dysfunction. In this section we will simulate a shock from whichthe patient will not die. This is not possible to simulate, because this kind of death is causedby other physical influences.

The shock we are going to model is the heart block.

Heart block

When a heart block occurs, the ventricle doesnot contract anymore. This can be modelled bytaking a cardiac driver function e(t) = 0. This means that the blood will flow for a small amountof time by the peristaltic movement of the arteries, but after some time this will stop, too. Since,there is no driver function anymore, we have to deal with an end-diastolic function. In this caseblood can enter the heart, but can never leave the heart, since the valves are closed. This allcan be seen in figure 5.12.

79

Figure 5.6: Effect of diastolic dysfunction causing an increase in ventricle elastance on a PVdiagram of the left ventricle [BRWD]

5.3 Summary

In this section we solved the numerical model of the human CVS. We started with solving the6 compartment model using the parameters proposed in section 2.1. The graphs have a goodtrend, but have a bad magnitude.

We decided to continue with a three compartment model and guess the parameters. As canbe seen in figure 5.4, this has very satisfying results.

In the second part of this section we simulate heart failures. We showed that it can recognizediastolic dysfunction, systolic dysfunction, valvular stenosis, valvular insufficiency and a heartblock. Probably the 3 compartment model can recognize more heart failures or other failures inthe CVS, but we did not test these.

80

170 180 190 200 210 220 230 240 250 2600

2

4

6

8

10

12

14

16

volume (ml)

pres

sure

(kP

a)

PV diagram of the left ventricle

systolic dysfunctionnormal

Figure 5.7: Simulating a systolic disfunction

Figure 5.8: Effect of systolic dysfunction causing a drop in cintractility on a PV diagram of theleft ventricle [BRWD]

81

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16normal

left

vent

ricle

pr

essu

re (

kPa)

time (sec)0 0.2 0.4 0.6 0.8

0

2

4

6

8

10

12

14

16

18aortic stenosis

time (sec)

Pao

Plv

Pmt

max Plv − max Pao

Pao

Plv

Pmt

Figure 5.9: Simulating aortic stenosis

Figure 5.10: A theoretical figure. On the left a normal left ventricle pressure, in the middle aleft ventricle pressure caused by aortic stenosis and on the right a left ventricle pressure diagramcaused by valvular insufficiency

82

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16normal

left

vent

ricle

pr

essu

re (

kPa)

time (sec)0 0.2 0.4 0.6 0.8

0

2

4

6

8

10

12

14

16valvular insufficiency

time (sec)

amp Pao

Pao

Plv

Pmt

Pao

Plv

Pmt

Figure 5.11: Simulating valvular insufficiency

83

0 5 10 15 20 25

200

300Heart block

left

vent

ricle

vo

lum

e (m

l)

0 5 10 15 20 250

10

20

left

vent

ricle

pr

essu

re (

kPa)

0 5 10 15 20 250

10

20

aort

icpr

essu

re (

kPa)

0 5 10 15 20 250

1

2

left

atriu

m

pres

sure

(kP

a)

time (sec)

Figure 5.12: Simulating a heart block

84

Chapter 6

Conclusions

This thesis is a research to a mathematical model of the human CVS. This model

1. Is simple,

2. Needs little computional time and

3. Can accurately reflect a small part of the human CVS.

To create such a model, we started with a literature study to other human CVS models. Afterwe found some articles which describe a human CVS model, we had some important questionswhich have been answered in this thesis:

• Are we going to use a 0D or 1D model?



• Do we need a lung circulation in our model?

• How do we model the body circulation, such that a small part of CVS can be included?


By hand of the literature study, we concluded that we want to use a 0D model, since the 0Dmodel has the advantage of being a model which can be solved easily and the results are in goodagreement with reality. Further, the 0D model can accurately reflect a small part of the CVS.This is exactly what we are looking for.

All other questions couldnot be answered with the literature study. We decide that we aregoing to build a 6 compartment model like that of [SMITH] and after this build a 3 compartmentmodel according to our own ideas.

In the 0D model a system of equations describe the pressure (P ) in a compartment andthe flow (Q) at the interfaces of the compartment with the rest of the system. All compart-ments are coupled like a Windkessel circuit. The system for one compartment equals:

dCPin

dt+Qout = Qin

LdQout

dt+RQout − Pin = −Pout

(6.1)

85

where R is the resistance and L the inductance of the flow and C the capacity of a compart-ment. This system is including inertia. The system without inertia is the same, but now withthe inertial term neglegted.

This system can be solved numerically with the following iterative process:

1. Pass the statevector to the solver.

2. Calculate the pressure in the compartment.

3. Use the new pressure to calculate the flow between the compartments.

With the usage of a single compartment model two questions can be answered:



From literature we have three different cardiac driver functions. We concluded that all threecardiac driver functions have good results. However, the sin2 and the sin driver function haveto be tuned, before they can be used. The exponential driver function

e(t) =

N∑

i=1

Aie−B1(t−Ci)2 (6.2)

doesnot have this problem, so we decided to use this one in our model.For the choice of using inertia, we implemented both models and compared the graphs. The

model with inertia shows a retardation of the flow in comparison with the model without inertia.However, both results converge to the same pressure and flow. We decided that in our model,which must be simple, we do not need the inertial term.

After we chose the right equations, we coupled at first 6 compartments and later 3 com-partments. After different tests, we can answer the last questions


• Do we need a lung circulation in our model?

• How must we model the body circulation, such that a small part of the CVS can beincluded?

In the tests it has been shown that the 3 compartment model can reflect a human cardiovascularsystem. Further, the lungcirculation is a part of the CVS so must be included. However, wedescribe it with a resistance. The last question is not explicitly answered in the tests, but sincea compartment describes a part of the system, an extra compartment describing only a smallpart of the CVS can alwaysbe included.Finally, we can conclude that we have a satisfying model for the human CVS, which

1. Is simple,

2. Can be solved numerically with a desktop computer in less than 5 minutes and

3. Can contain a specific compartment to describe a small part of the CVS.

86

Chapter 7

Future Work

In this report a minimal mathematical model for the human CVS is developed. In the futureimprovements can be made to the model and our model can be used to investigate a detailedsmall part of the human CVS.

7.1 Possible improvements

The most obvious improvement is the usage of real parameters for the resistances R and capac-ities C. How to measure these parameters we do not know yet, this is a subject for the medicalresearchers.

We can remark that if the measurement of the parameters in the passive compartments isdifficult, the equation of mass can be rewritten by using the volume in a compartment:

P2(V (t), t) =1

C(V (t)) (7.1)

There are some improvements which can be made to have more detailed results, with thedrawback that the model becomes more complicated. Using passive compartments with inertiais such an improvement. The momentum equation for the passive compartments changes in:

LdQoutdt

+RQout − Pin = −Pout (7.2)

Another improvement is the choice of another driver function. As can be seen in the resultsof the left ventricle pressure in comparison with the Wiggers’ diagram, the relaxation and con-traction time are too large. Further, the ejection time is too small. In the search for anotherdriver function one must look for a function which has a larger slope in the increasing pressure,the top must be weakened and finally the slope of the decreasing pressure must be smaller.

A second improvement to the heart is to use the interaction between the left and right atriaand the left and right ventricle. A possible option how to include this in the model is given insection 2.1.

A third improvement of the heart function is that the heart will stop pumping as result of aphysiological damage. This is not the case as can be seen in (5.12).

A last improvement of the model is the inclusion of more compartments. The more com-partments, the more detailed the information will be.

87

7.2 Investigation of a small part of the human CVS

Besides the improvement of the model described in this thesis, the model can be used to inves-tigate a small part of the human CVS. To investigate a small part of the human CVS, an extracompartment has to be included. A specific mathematical model of the small part of the humanCVS can be introduced.

An example of an application is the research to a detailed carotid artery. In this applicationat first a fourth compartment for the carotid artery will be included. In the iteration first thesame system of equations will be used. After each iteration step, the calculated input and out-put variables of the compartment will be used as input variables for a 3D Comflo model of thecarotid artery. After one iteration step in Comflo, Comflo passes its output variables as inputvariables back to the compartment model. The compartment model uses these variables in thenext iteration step. On this way every small part of the human CVS can be investigated.

88

Appendix A

Dictionary

abdomen The portion of the body which lies between the thorax and the pelvis.

abdominal venous The vein through the abdomen.

afferent pathway A chain of nerve fibers along which impulses passes from receptors to thecentral nervous system.

afterload Measure of the cardiac muscle stress required to eject blood from a ventricle.

aorta The main arterie.

arteries Part of the circulation system through which flows blood to the organs.

atrium Part of the heart which collect blood from the veins and pumps it into the ventricle.

baro receptor A cell or sense organ found in the wall of the body’s major arteries and stimu-lated by changes in blood pressure.

bloodvessels An elastic tubular channel through which the blood circulates.

bodycirculation Part of the human circulation system in which nutrients will be exchangewith the organs.

capillaries A system of small arteries in which it is possible to exchange nutrients with theorgans.

cardiac output Amount of blood pumped into the aorta in litres per minute.

cardiovascular system The heart and the bloodvessels by which blood is pumped and circu-lated through the body.

carotid artery An artery that supplies the head and neck with oxygenated blood.

carotid bifurcation (see carotid artery;) It divides in the neck to form the external and inter-nal carotid arteries.

central neural processing system Coordinates the activity of the muscle, monitors the or-gans, constructs and also stops input from the senses and initiates actions.

89

clavicle Articulates with the shoulder on one end and the breast bone on the other.

contraction time The contraction of the heartmuscle causes a strong increase of the pressurein the ventricle. The valve is closed.

coronaries The bloodvessels which supply blood to and from the heart muscle.

diastole The period of time when the heart relaxes after contraction.

diastolic dysfunction Filling problem of the heart.

diastolic phase The relaxation time and the filling time together.

effector mechanism Binding to a proteine and thereby altering the activity of that proteine.

efferent Carrying outward or away from a central part.

ejection time The heartmuscle is contracting. The valve is open.

epinephrine A drug that increases the contractile strength of the cardiac muscle.

ergometer An apparatus for measering force or power; especially, muscular effort of men.

extrasplanchnic circulation One part of the bifurcation in the systemic circulation.

filling time The bloodpressure in the ventricle is beneath the bloodpressure in the atrium.

heamodynamic The study of bloodflow.

heart block The ventricle doesnot contract anymore.

heart rate Heart beats per minute.

infarction Lack of oxygen.

interstitium Is a solution which bathes and surrounds the cells of multicellular animals.

lumen The cavity or channel within a tubular structure.

lungcirculation Part of the human circulation system in which oxygen will be absorbed fromthe lungs.

lymph The almost colourless fluid that bathes body tissues and os found in the lymphaticvessles that drain the tissues of fluid that filters across the bloodvessel walls from blood.

lymphatic system The tissues and organs that produce and store cells that fight infection andthe network of vessels that carry lymph.

myocardial disorder Limitation of the ability of heart to relax.

nutrients A substance used in an organism’s physiology which must be taken in from theevironment.

organs A group of tissues that perform a specific function or a group of functions.

orthostatic Pertaining or caused by standing upright.

90

paediatric The medical study of diagnosis and treatment of diseases and disorders.

pericardium A relatively stiff walled passive elastic chamber that encapsulates the heart.

preload Measure of the fibre length, immediately prior to contraction.

relaxation time The relaxation of the heartmuscle.

shock Tissue dammage due to lack of oxygen and other nutrients.

shunt A passage or anastomosis between two natural channels, especially between blood vessels.Such structures may be formed physiologically.

splanchnic circulation The part of the bifurcation in the systemic circulation to the lowerbody.

subclavian artery Situated under the clavicle.

sympathetic efferent activities Carrying out to the sympathetic nervous system.

systole The time at which ventricle contraction occurs.

systolic dysfunction Ejecting problems of the heart.

systolic phase The contraction time and the ejection time together.

stroke volume The amount of blood pumped from the ventricle during one heart beat.

tilt As any vehicle rounds a curve at speed, independent objects inside it exert centrifugal forcesince their inherent momentum forward no longer lies along the line of the vehicle’s.

tissue A collection of interconnected cells that perform a similar function within a organism.

transmural Through any wall.

Unstressed volume is the volume in a chamber that does not contribute to an increase inpressure, or the relaxed volume of a chamber.

vagal efferent activities Carrying outward to the vagus nerve.

vagus nerve Enervates the gut, heart and larynx.

valvular insuffiency A leaky state of one or more of the cardiac valves.

valvular stenosis It not properly opening of a heart valve.

vascular system The cardiovascular and lymphatic system’s collectively.

veins Part of the circulation system through which blood flows to the heart.

ventricle Part of the heart which pumps blood into the arterial system.

Wiggers’ diagram Depict the pressure and volume in the heart and the ejecting activity ofthe heart.

91

Bibliography

[FORVEN] L. Formaggia and A. Veneziana, Reduces and Multiscale models for the humancardiovascular system, 13th may 2003

[ARFEL] J.P. Archie and R.P Feldtman, Critical Stenosis of the internal carotid artery, Surgery,89:67-70, 1981

[VIJIMA] A. Viedma, C. Jiminez-Ortiz and V. Marco, Extended willis circle model to explainclinical observations in periorbital arterial flow, 30(3):265-272, 1997

[SMITH] B.W. Smith, Minimal Haemodynamic Modelling of the Heart & Circulation for Clini-cal Application, 14 April 2004, University of Canterburry, Christchurch, New Zealand

[HSKM] T. Heldt, E.B. Shim, R.D. Kamm and R.G. Mark, Computational modeling ofcardiovascular response to orthostatic stress, J Appl Physiol 92: 1239-1254, 2002;http://www.jap.org

[DAMA] TL Davis and RG Mark, Teaching physiology through simulation of hemodynamics.Comput Cardiol 17:649-652, 1990

[WHFICR] RJ White, DG Fitzjerrell and RC Croston, Cardiovascular modelling: simulatingthe human response to exercise, lower body negative pressure, zero gravity and clinicakconditions. In: Advances in Cardiovascular Physics. Basel: Karger, 1983, vol.5, pt.I,p 162-184

[LUD] J. Ludbrook, Aspects of venous function in the lower limbs, Spingfield, IL:Thomas,1966

[DAV] TL Davis, Teaching physiology through simulation of hemodynamics (master’s thesis).Cambridge, MA: Massachusetts Institute of technology, 1991

[DEKAST] RW De Boer, JM Karemaker and J Stracke, Hemodynamic fluctuations and barore-flex sensitivity in human: a beat to beat model. J Appl Physiol 253: 680-689, 1987

[KRWIWAKR] J. Kropf, M. Wibmer, S. Wassertheurer and J. Krocza, An Identifiable Modelfor Dynamic Simulation of the Human Cardiovascular System

[WIB] M. Wibmer, Dynamische Herzkreislaufsimulation unter Berucksichtigung verteilterparameter, Master’s thesis, Vienna University of Technology, Austria, 2003

[URS] M. Ursino, Interaction between carotid baroregulation and the pulsating heart: amathematical model, Am. J. Physiol. 275 (Heart Circ physiol. 44):H1733-H1747, 1998

93

[URANBE] M. Ursino, M. Antonucci, E. Belardinelli, The role of active changes in venous ca-pacity by the carotid baroreflex: analysis with a mathematical model, Am. J. Physiol.267 (Heart Circ Physiol. 36): H2531-H2546, 1994

[KCYSTN] T, Kubota, H. Chishaki, T. Yoshida, K. Sunagawa, A. Takeshita, Y. Nose, How toencode arterial pressure into carotid sinus nerve to invoke natural baroreflex, Am. J.Physiol. 263 (Heart Circ physiol. 32):H307-H313, 1992

[CHAB] M.W. Chapleau and F.M. Abboud, Contrasting effects of static and pulsatile pressureon carotid baroreceptor activity in dogs, Circ. Res. 61:648-658, 1987

[LEZI] M.N. Levy and H. Zieske, Autonomic ocntrol of cardiac pacemaker activity and atri-oventricular transmission, J. Appli. Physiol. 27:465-470,1969

[MEDERINO] J. Melbin, D.K. Detweiler, R.A. Riffle and A. Noordergraaf, Coherence of cardiacoutput with rate changes, AM. J. Physiol. 243 (Heart Circ. Phsiol. 12):H499-H504,1982

[WH] F.M. White, Viscous Fluid Flow, second edition. McGraw-Hill, Inc, USA, 1991

[FU] Y. Fung, Biomechanics: Motion, flow, stress and growth. Springer-Verlag, San Diego,1990

[THPSYW] J.E. Tsitlik, H.R. Halperin, A.S. Popel, A.A. Shoukas, F.C. Yin and N. Westerhof,Modeling the circulation with three-terminal elactrical networks containing specialnonlinear capacitors, Ann Biomed Eng, 20(6):595-616, 1992

[MESRCH] F.M. Melchior, R.S. Sriwasan and J,B, Charles, Mathematical modeling of hu-man cardiovascular system for simulation of orthostatic response, Am. J. Physiol,262:H1920-H1933, 1992

[BAFI] B. Bartlett and D. Fyfe, Handbook of mathematical formulae for engineers and sci-entists. Denny publications ltd., London, 1974

[KALEPL] W Kahle, H.Leonardt, W.Platzer, Sesam, Atlas van de anatomie, deel 2 inwendigeorganen, Bosch en Keuning, Baarn, 12e druk, 1996

[BUMOMOZI] W.G. Burgerhout, A.Mook, J.J. de Morree, W.G.Zijlstra, Fysiologie, leerboekvoor paramedische opleidingen, wetenschappelijke uitgeverij Bunge, Utrecht, 1e druk,1995

[ROHU] R.H.Rozendal, P.A.J.B.M.Huijing, inleiding in de kinesologie van de mens, Robijns,Houten, zesde druk, 1996

[VR] J.H. Vrijenhoek, Pathalogie en geneeskunde voor fysiotherapie, bewegingstherapie energotherapie, de tijdstroom, Utrecht, 3e druk, 1997

[KE] D.R. Kerner, Solving Windkessel Models with MLAB, http://www.civilized.com

[WE] N. Westerhof, Special issue on mechanics of the cardiovascular system, J. Biomech.,36(5):621-622, 2003

94

[PEQU] C.S. Peskin and D.M. McQueen, Cardiac Fluid Dynamics, Crit Rev Biomed Eng,20(5-6):451-9, 1992

[GLHUMC] L. Glass, P. Hunter, A. McCulloch, Theory of heart, Biomechanics, Biophysics, andNonlinear Dynamics of Cardiac Function, Springer Verlag New York, inc., (1991)

[NIGRSMHU] P.M. Nielsen, I.J. Le Grice, B.H. Smaill and P.J. Hunter, Mathematical modelof geometry and fibrous structure of the heart, Am. J Physiol, 260(4 Pt 2):H1365-78,(1991)

[LEHUSM] I. Legrice, P. Hunter, B. Smaill, Laminar structure of the heart: a mathematicalmodel, Am. J. Physiol, 272:H2466-76, (1997)

[STHU] C. Stevens and P.J. Hunter, Sarcomere length changes in a 3d methematical model ofthe pig ventricles, Prog Biophys Mol Biol, 82(1-3):229-41, 0079-6107 Journal ArticleReview Review, Tutorial, (2003)

[MO] T.A Moore, sixideas that shaped PHYSICS, unit E:Electromagnetic fields are dy-namic, WCB/McGraw-Hill, ISBN 0-07-043054-3

[FOVE] L. Formaggia and A. Veneziani, One dimensional models for blood flow in the humancardiovascular system, VKI Lecture Notes, Biologic fluid dynamics, 07-03-2003, 2003.

[QUFO] A. Quarteroni and L. Formaggia, Modelling of living systems, chapter Mathemati-cal Modelling and Numerical simulation of the Cardiovascular System, Handbook ofnumerical Analysis, Elsevier Science, 2003.

[BRWD] E. Braunwald, Heart Disease, A text book of cardiovascular medicine, W.B. SaundersCompany, Philadelphia, 5th edition, 1997

[KUPA] A. Kumar and J.E. Parrillo, Shock: Classification, pathophysiology, and approach tomanagement, In J.E. Parillo and R.C. Bone, editors, Critical Care Medicine, Principlesof diagnosis and management, Mosby, St. Louis, Missouri, 1995

95

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