Download Minimal haemodynamic modelling of the circulation

Minimal haemodynamic
modelling of the circulation
P.C.I. Spelde
Master Thesis in Applied Mathematics
April 2008
Minimal haemodynamic
modelling of the circulation
P.C.I. Spelde
First supervisors: A.E.P. Veldman and G. Rozema
Second supervisor: A.J. van der Schaft
External supervisor: N.M. Maurits (UMCG)
Institute of Mathematics and Computing Science
P.O. Box 407
9700 AK Groningen
The Netherlands
Abstract
The knowledge of the flowstructures in the human arteries is limited. The medical staff have
the wish to have a better side to this phenomenon. In a specific mathematical research of the
flow through the carotid bifurcation there is attention for this problem. To make it possible to
do 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 overly
complex, difficult to solve, and/or unstable. This thesis develops a minimal model with the
primary goal of having the possibility to reflect accurately a small part of the cardiovascular
system. The focus is just on the simplicity of the overall structure, with a reasonable reflection
of the heartfunction. A novel mixed-formulation approach to simulating blood flow in lumped
parameters CVS models is outlined that adds minimal complexity, but significantly improves
physiological accuracy.
The minimal model is shown to match a Wiggers’ diagram and was also verified to simulate
different heartdiseases. The model offers a tool that can be used in conjunction with experimental
research to improve understanding of the blood flow.
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Contents
1 Introduction
1.1 Physiology of the blood circulation system
1.1.1 The blood circulation system . . .
1.1.2 The bloodvessels . . . . . . . . . .
1.1.3 Heart . . . . . . . . . . . . . . . .
1.1.4 Cardiac function . . . . . . . . . .
1.2 Cardiovascular System Modelling . . . . .
1.2.1 Finite Elements Approach . . . . .
1.2.2 The Pressure-Volume Approach . .
1.2.3 Windkessel circuit . . . . . . . . .
1.2.4 Wiggers’ diagram . . . . . . . . . .
1.3 Summary . . . . . . . . . . . . . . . . . .
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2 Literature study
2.1 Minimal Heamodynamic Modelling of the Heart & Circulation for Clinical Application [SMITH] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Reduced and multiscale models for the human cardiovascular system; one dimensional model [FORVEN] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Reduced and multiscale models for the human cardiovascular system;lumped parameters for a cylindrical compliant vessel [FORVEN] . . . . . . . . . . . . . . .
2.3.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Computational modeling of cardiovascular response to orthostatic stress [HSKM]
2.4.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An identifiable model for dynamic simulation of the human cardiovascular system
[KRWIWAKR] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interaction between carotid baroregulation and the pulsating heart: a mathematical model [URS] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Overview Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Final Choises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Model description
3.1 The derivation of a mathematical model . . . . . . . . . . . . .
3.2 The 0D model for the circulation system, bottum-up approach
3.2.1 Conservation of mass . . . . . . . . . . . . . . . . . . . .
3.2.2 Conservation of momentum . . . . . . . . . . . . . . . .
3.3 The 0D model for the circulation system, Top-Down approach .
3.4 Hydraulical analog . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Simplification of the model . . . . . . . . . . . . . . . . . . . .
3.6 Simulating the heart with an active compartment . . . . . . . .
3.7 Valve simulation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Compartment coupling . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 The 6 compartment model . . . . . . . . . . . . . . . .
3.8.2 The 3 compartment model . . . . . . . . . . . . . . . .
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Numerical Model
4.1 A passive compartment without inertia . . . . . . . . . . . .
4.1.1 Discretisation, Jacobi like method . . . . . . . . . .
4.1.2 Discretisation, Gauss-Seidel like method . . . . . . .
4.1.3 Stability analysis of the Jacobi and Gauss-Seidel like
4.2 A passive compartment with inertia . . . . . . . . . . . . .
4.3 An active compartment . . . . . . . . . . . . . . . . . . . .
4.4 Testing the single compartment model . . . . . . . . . . . .
4.4.1 The initial conditions . . . . . . . . . . . . . . . . .
4.4.2 Including the inertial term? . . . . . . . . . . . . . .
4.5 A 3 compartment model . . . . . . . . . . . . . . . . . . . .
4.6 A 6 compartment model . . . . . . . . . . . . . . . . . . . .
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4.6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Testing the models
5.1 Verification of the models . . . . . . . . . .
5.1.1 A six compartment model . . . . . .
5.1.2 A three compartmentmodel . . . . .
5.2 Testing the model with some extreme cases
5.2.1 Heart Failure . . . . . . . . . . . . .
5.2.2 Shock . . . . . . . . . . . . . . . . .
5.3 Summary . . . . . . . . . . . . . . . . . . .
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6 Conclusions
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7 Future Work
7.1 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Investigation of a small part of the human CVS . . . . . . . . . . . . . . . . . . .
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A Dictionary
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List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
A schematic picture of the organs in the circulation system . . . . . . . . . . . .
The blood circulation through the heart . . . . . . . . . . . . . . . . . . . . . . .
An example of a pressure volume diagram together with the ESPVR and the
EDPVR lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The aorta and its hydraulic and electrical representation . . . . . . . . . . . . . .
a: A modelling lab which consider only the simplest Windkessel method. b: A
three elements Windkessel circuit . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-element Windkessel circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Wiggers’ diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A simple CVS of a human . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A closed loop model of a simple CVS of a human, see figure 2.1 . . . . . . . . . .
A possible choice of a cardiac driver function, a = 1, b = 80, c = 0.27, N = 1 . . .
A small part of an artery free of bifurcations . . . . . . . . . . . . . . . . . . . .
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. . . .
Single compartment circuit representation, P pressure, R resistance, C capacitor,
Q flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The entire model with in total 12 coupled single compartments . . . . . . . . . .
Hydraulic analog of the cardiovascular system. A bifurcation in the systemic
circulation is made into a splanchnic and an extrasplanchnic circulation. . . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
A simple model of the CVS . . . . . . . . . . . . . . . . . . . . . .
A tube free of bifurcations . . . . . . . . . . . . . . . . . . . . . . .
A cross section of a small artery free of bifurcations . . . . . . . .
A small part of the cross section . . . . . . . . . . . . . . . . . . .
Force balance in the axial direction . . . . . . . . . . . . . . . . . .
An electrical circuit, including a resistor, inductor and capacitor .
A simple cardiac driver function, with parameter values: A = 1,
C = 0.27s and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 sin2 cardiac driver function . . . . . . . . . . . . . . . . . . . . . .
3.9 sin cardiac driver function . . . . . . . . . . . . . . . . . . . . . . .
3.10 A 6 compartment model . . . . . . . . . . . . . . . . . . . . . . . .
3.11 A 3 compartment model . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
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B = 80s−1 ,
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The convergence by different time steps with the Jacobi like method . . . . . . .
The convergence for different time steps with the Gauss Seidel like method. In
this figure only the 25t h heartbeat is depicted. . . . . . . . . . . . . . . . . . . .
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4.4
4.5
The difference in result by using different driverfunctions . . . . . . . . . . . . . .
Starting with different initial conditions has no influence on the final results . . .
The difference in the results by using inertia . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
Simulation results from the closed loop model without inertia with our own program
Simulation results from the closed loop model with inertia and ventricular interaction, Results from [SMITH] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 A Wiggers’ diagram from a 6 compartment model . . . . . . . . . . . . . . . . .
5.4 A Wiggers’ diagram from a 3 compartment model . . . . . . . . . . . . . . . . .
5.5 Simulating a dystolic disfunction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Effect of diastolic dysfunction causing an increase in ventricle elastance on a PV
diagram of the left ventricle [BRWD] . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Simulating a systolic disfunction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Effect of systolic dysfunction causing a drop in cintractility on a PV diagram of
the left ventricle [BRWD] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Simulating aortic stenosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 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 ventricle
pressure diagram caused by valvular insufficiency . . . . . . . . . . . . . . . . . .
5.11 Simulating valvular insufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Simulating a heart block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
p̂ mean pressure over the whole compartment
P Pressure
Q
1. intantaneous charge on the capacitor
2. blood flow
Q̂ mean flow rate over the whole district
r internal radius
r0 reference radius
r̂ radial direction
R Resistance
s velocity profile
t time
T time interval
THF constant
u velocity
ū mean velocity
x axial direction
y variable
V Voltage
Volume
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 C 1 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, there
is a growing interest in the mathematical and numerical modelling of the human CVS (cardiovascular system). Cited to this interest, much research is devoted to complex three dimensional
simulations able to provide sufficient details of the flow field to extract local data such as wall
shear stresses. However, these computations are still quite expensive in terms of human resources needed to extract the geometry and prepare the computational model and computing
time. Since bioengineers and medical researchers do not need the flow in such detail everywhere and less detailed models have demonstrated their ability to provide useful information at
a reasonable computational cost, further research is done in the description of the CVS in less
detailed models. In the less detailed models there must be the possibility to include a small
piece 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 knowledge of others will be used to create a minimal model. Finally, this minimal model will be tested
by 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 physiology
and 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 around
by the heart. In this section in short there will be an introduction in the physiology of the blood
circulation system.
1
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- or
bodycirculation. The lungcirculation starts in the heart and provides the lungs of oxygenpoor
blood and returns back, with oxygenrich blood, to the heart. Next, the heart pumps the blood
into the bodycirculation and provides all the organs of blood before the blood returns to the
heart 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. The
bloodvessels 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 all
the organs (see fig 1.1.1). A small part of the SV will be stored in the aorta. Hereby the elastic
wall of the aorta will be stretched. When the heart is at rest the aorta contracts and pumps
the rest of the SV away. The heart pumps the blood into the arterial system with a pressure of
about 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 the
legs is about 10m/sec. This difference can be explained by the decreasing elasticity of the
arteries. Further, the pressure can be influenced by the lumen through narrowing and enlarging
(resistance regulation).
2
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 and
a small velocity in the capillaries. The velocity in the cappilaries is about 0.3mm/sec. The
advantage of this low pressure and small velocity is that the walls can be thin and the transfer
of 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 mechanisms
to pump the blood back to the heart.
• A musclepump For the veins which receive blood from the muscles, there is the muscle
pump. By the contraction of the muscle, the vein will be suppressed. By means of a valve
the 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 happens
as 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 relaxation
of the muscle, the heart can function as a pump. The four chambers can be split up in two
3
contraction time
atrium-ventricle valve
closed
aorta valve
closed
ejection time
closed
open
relaxation time
closed
closed
filling time
open
closed
systolic phase
cardiaccycle
diastolic phase
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 pumps
blood through the bodycirculation and the right atrium-ventricle pair pumps blood through the
lungcirculation. 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 time
periods:
• Contraction time The contraction of the heartmuscle causes a strong increase of the pressure 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 the
ventricle is still increasing. The aorta valve is open. The muscle of the ventricle is contracting, the volume will decrease. The valve closes as soon as the blood flows in the wrong
direction.
• 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 of
the atrium, then the atrium-ventricle valve opens.
The contraction time and the ejection time together are called the systolic phase. The relaxation
time 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 common
indicators are the pressure-volume diagram, the cardiac output and preload and afterload. These
indicators 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.
4
The two main characteristics of the PV diagram are the lines plotting the End Systolic PressureVolume 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 EDPVR
lines
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 relationship
of an elastic chamber. The ESPVR gives a measure of cardiac contractility, or the strength
of 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, when
there 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 is
defined as the amount of blood pumped from the ventricle during one heart beat. For a more
general measure, the cardiac output (CO), is defined as the amount of blood pumped into the
aorta, from the left ventricle, in litres per minute. Therefore, the CO is equal to the product of
the 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 the
peripheral tissues. The equation for the CO highlights the important dependence on the SV
and the HR. While the HR is driven by the sympathetic nervous system, the stroke volume is
dependent on the function of the heart muscle as well as on the ventricle preloads and afterloads.
5
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 a
measure of the muscle fibre length, immediately prior to contraction, while afterload is a measure
of 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 the
CVS in great detail and utilizing FE calculations to simulate these parts. The PV approach is
a simpler method, by grouping parameters and making assumptions to simplify the model as
much 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 accurate both in trend and magnitude. This kind of approach needs a micro-scale measurement of the
mechanical 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 models
using 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, particularly
around 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, rapid
diagnostic feedback. Further, it is not feasible to obtain the detailed specific measurements from
a living patient, so a model of a specific patient is difficult. Finally, these micro-scale calculations
require 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 of
elements simulating elastic chambers and blood flow, separately. The elastic chamber elements
model the PV relationship in a section of the CVS, such as a ventricle, an atrium, or a peripheral
section of the circulation system such as the arteries or veins. All these separate elastic chamber
elements are connected by the fluid flow elements which represent blood flow through different
parts of the circulation system.
For the modelling of the CVS there exist hydraulic and electrical analogs, see figure 1.4. In
the next sections we tell something more about the connection between the electric and hydraulic
analog. For the explanation of the hydraulic analogs Windkessel circuits are used. The usage of
Windkessel 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 the
hydraulic formulas and the electrical formulas we use Windkessel circuits. Windkessel circuits
6
Figure 1.4: The aorta and its hydraulic and electrical representation
circulation
element
blood flow
blood pressure
pumping function
vessels
large arteries
hydraulic
analog
flow rate
pressure
compliance
viscosity
inertia
electrical
analog
current
voltage
capacitance
resistance
inductance
Table 1.2: The hydraulic and electric analogs
are circuits which describe the load faced by the heart in pumping blood through the systemic
arterial 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 physiologist
Otto Frank in the article ”Die Grundform des Arteriellen Pulses”, published in 1899. In this
article Frank compared the heart and systemic arterial system with a closed hydraulic circuit
comprised of a waterpump connected to a chamber. The hydraulic circuit is completely filled
with water, except for a pocket of air in the chamber. When water is pumped into the system, the
water compresses the air in the pocket and pushes water back in the pump. The compressibility
of the air in the pocket simulates the elasticity and extensibility of the major artery. This is
known as the arterial compliance. The resistance which the water encounters by flowing through
the Windkessel and returning back to the pump, simulates the resistance which the blood flow
encounters by the blood flowing through the arteries. This process is known as the peripheral
resistance.
A Windkessel circuit can consist of a varying number of elements. The simplest model
consists of two elements (see figure 1.2.3), namely a compliance and a peripheral resistance. By
using 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 three
elements Windkessel circuit
model can be described by a mathematical model.
According to Ohm’s law, the drop in electrical potential across the resistor is IR R and the
drop in electrical potential across the capacitor is Q/C, where Q is the instantaneous charge on
the 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) = IR R and V (t) = Q/C. From Kirchhof’s
current law, the sum of currents into a junction must equal the sum of currents out of the same
junction: I(t) = IC + IR . By now, the current in the capacitor is given by IC = C(dV /dt). If we
now substitute IC and IR from above into Kirchhof’s current law then we finally get an electric
mathematical model which describes the 2-element Windkessel model:
I(t) = C
dV (t) V (t)
+
dt
R
(1.2)
In terms of the physiological system, I(t) is the blood flow from the heart to the aorta, V (t) is
the blood pressure in the aorta, C is the arterial compliance and R is the peripheral resistance
in the arterial system. In physiological terms the hydraulic mathematical model reads:
Q(t) = C
dP (t) P (t)
+
dt
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
8
(1.4)
Figure 1.6: 2-element Windkessel circuit
A modification of the 2-element Windkessel circuit is obtained by including an inductor in the
main branch of the circuit, as can be seen in figure (1.5b). This inductor simulates inertia of the
fluid in the hydrodynamic model. The mathematical model of this 3-element Windkessel circuit
can be found by using that the drop in electrical potential across an inductor with inductance
equals VL = L(dIL (t)/dt) and Kirchhof’s law, IL = IR + IC :
L
L)
+ C d(V (t)−V
=
IL (t) = IR (t) + IC (t) = VRR + C dVdtC = V (t)−V
R
dt
2
dV (t)
d IL (t)
V (t)
L dI(t)
= R − R dt + C dt − CL dt2
V (t)
R
−
VL
R
L
+ C dVdt(t) − C dV
dt
⇔
IL (t) +
L dIL (t)
R dt
2I
+ CL d
L (t)
dt2
=
V (t)
R
+ C dVdt(t)
In this case, the hydraulic equivalent reads:
Q(t) +
1.2.4
d2 Q
dP (t)
P (t)
L dQ
+ LCp 2 =
+ Cp
R dt
dt
R
dt
(1.5)
Wiggers’ diagram
The Wiggers’ diagram depict the pressure and volume in the heart and the ejecting activity of
the 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 the
aorta.
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. The
result of this pulling is a pressure drop in the atrium, showed at the x-top. Right before
the x-top, a c-top in the pressure of the left atrium is denoted. This c-top is caused by
the 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 valve
is opened for a fast filling of the ventricle during the isometric relaxation. This opening
causes 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 diagram
depicts the values for the right atrium and ventricle. As can be seen, the pressure in the
right 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 of
the mitral valve, the second (II) noise reflects the closure of the aortic valve. The first and
second noise give exactly the duration of the relaxation and contraction. The systolic phase
starts at the beginning of the first noise till the beginning of the second noise. The diastolic 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’ diagram
as reference material for a specific person. The phonocardiogram can be used to measure the
time needed for the different heart periods. The other three graphs can be used as reference for
our own model.
1.3
Summary
In this section the main goals of this report have been outlined, namely creating a human CVS
model 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 the
physiology of the circulation system. Finally, we gave an introduction about the possibilities of
CVS modelling. We introduced two approaches, a FE approach and a PV approach, together
with their advantages and disadvantages.
11
12
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 final
section.
2.1
2.1.1
Minimal Heamodynamic Modelling of the Heart & Circulation for Clinical Application [SMITH]
General description
This is the title of the thesis that is presented by Bram W. Smith for the degree of Doctor
of Philosophy in Mechanical Engineering at the University of Canterburry, Christchurch, New
Zealand. In this thesis Smith has the intention to make a model which contains:
• A closed-loop, stable model with minimal complexity and physiologically realistic inertia
and valve effects.
• A model parameters that can be relatively easily determined or approximated for a specific
patient using standard, commonly used techniques.
• A model that can be run on a standard desktop computer in reasonable time (eg. in the
order of 1-5 minutes)
• Accurate prediction of trends
The model presented a hydraulic, 0D, 6 compartment model, which intends to simulate the
essential haemodynamics of the CVS including the heart, and the pulmonary and systemic
circulation systems. Figure 2.1 shows a simplified diagram of the human circulation system
with in the middle the human heart. Figure 2.2 presents a closed-loop model of the same human
CVS. As can be seen in figure 2.2, the closed-loop model contains compartments which are
connected 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 driver
function 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 between
minimum and maximum values during a single heart beat:
e(t) =
N
X
2
ai e−bi (t−ci ) ,
(2.1)
i=1
where the ai , bi , ci and N are parameters that determine the shape of the driver profile. For his
simple 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 approximated as flow through a tube. The flow rate equations have directly been derived from the
14
profile of the cardiac driver function
1
0.9
0.8
0.7
e(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
Time t (sec)
0.5
0.6
0.7
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:
1 ∂
∂ux uθ ∂ux
∂ux
1 ∂P
∂ux
1 ∂ 2 ux ∂ 2 ux
∂ux
+ ur
+
+ ux
=−
+ν
+
r
+ 2
∂t
∂r
r ∂θ
∂x
ρ ∂x
r ∂r
∂r
r ∂θ 2
∂x2
(2.2)
where ux , ur and uθ are the longitudinal, radial and angular velocities, respectively, P is the
modified 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 equations
governing 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 with
constant viscosity (µ is constant).
∂r
= 0). This
• The arteries are assumed to be rigid with a constant cross sectional area ( ∂x
assumption fits with standard Windkessel circuit design involving a rigid pipe and an
elastic compartment in series. The rigid tube simulates the fluid dynamics, while the
elastic compartment simulates the compliance of the artery walls.
x
• Laminar uni-directional axi-symmetric flow is assumed (ur = 0, uθ = 0 and ∂u
∂θ = 0).
Although turbulence can occur around the valves, it takes time to develop, and is assumed
not to affect the flow profile significantly.
• The flow is assumed to be fully developed along the length of the tube meaning the velocity
x
profile is constant with respect to x ( ∂u
∂x = 0).
• Pressure is assumed constant across the cross-sectional area and the pressure gradient is
constant along the length of each section so that the pressure gradient is a function of time
only ( ∂P
∂x (t)).
With these assumptions equation (2.2) reduces to the following equation
∂u(r, t)
∂P
µ ∂
∂u(r, t)
ρ
=−
(t) +
r
,
∂t
∂x
r ∂r
∂r
15
(2.3)
where µ is the viscosity (µ = νρ) and u(r, t) is the velocity in the x-direction (ux (r, t)) as a
function 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 equation
for flow rate, assuming constant resistance and no inertial effects:
Q(t) =
where R =
resistance
8µl
πr04
P1 (t) − P2 (t)
,
R
(2.4)
is the resistance. The second equation includes inertial effects and constant
L
dQ
= P1 − P2 − QR,
dt
(2.5)
ρl
8µl
where L = πr
2 is inertia and R = πr 4 constant resistance. Around the heart where are big flow
0
0
differences the second equation will be used. Far from the heart flows a nearly constant flow, so
the first equation will be used.
2.1.4
Parameters
Smith uses the following parameters for his tests.
16
Description
Symbol
Blood properties
Blood Density
ρ
Blood Viscosity
µ
Blood Kinematic Viscosity
ν
Stressed Volume of blood in CVS
Vtot
Unstressed Volume of blood in CVS
Artery properties
Internal Artery Radius
r0
Artery length
l
Compartment properties
Chamber Elastance
Ees
EDPVR Volume Cross-over
V0
ESPVR Volume Cross-over
Vd
Constant
λ
Heart Rate
ω
Constant
a
Value
1050kg/m3
0.004N S/m2
3.8 10−6 m2 /s
1500ml
4000ml
0.0125m
0.2m
1N/m5
0m3
0m3
23000m−3
1.33beats/sec
15N/m2
Table 2.1: Constants used in a single compartment simulation
PARAMETERS
Units
Left Ventricle free wall (lvf)
Right ventricle free wall (rvf)
Septum free wall (spt)
Pericardium (pcd)
Vena-cava (vc)
Pulmonary Artery (pa)
Pulmonary Vein (pu)
Aorta (ao)
Ees
6
10 N/m5
Vd
−6
10 m3
V0
−6
10 m3
100
54
6500
1.3
72
1.9
98
0
0
2
0
0
0
0
0
0
2
200
-
λ
m−3
33000
23000
435000
30000
-
P0
N/m2
10
10
148
66.7
-
Table 2.2: Mechanical properties of the heart and circulation system
17
Parameter
Mitral Valve (mt)
Aortic Valve (av)
Tricuspid Valve (tc)
Pulmonary Valve (pv)
Pulmonary Circulation System (pul)
Systemic Circulation System (sys)
Resistance
N s/m5
6.1 106
2.75 106
1 106
1 106
9.4 106
170 106
Inertance
N s2 /m5
1.3 104
5 104
1.3 104
2 104
N/A
N/A
Table 2.3: Hydraulic properties for flow between compartments
Description
Elastance of Vena-cava
Elastance of Left Ventricle
Elastance of Pulmonary Artery
Elastance of Pulmonary Vein
Elastance of Right Ventricle
Elastance of Aorta
Resistance of Tricuspid Valve
Resistance of Pulmonary Valve
Resistance of Pulmonary Circulation
Resistance of Mitral Valve
Resistance of Aortic Valve
Resistance of Systemic Circulation
Contractility of Left Ventricle
Contractility of Right Ventricle
Symbol
Evc = 1.29 106 N/m5
P0,lvf = 9.07N/m2
Epa = 44.5 106 N/m5
Epu = 0.85 106 N/m5
P0,rvf = 20.7N/m2
Eao = 98 106 N/m5
Rtc = 3.3 106 N s2 /m5
Rpv = 1 106 N s2 /m5
Rpul = 19.3 106 N s2 /m5
Rmt = 2.33 106 N s2 /m5
Rav = 5.33 106 N s2 /m5
Rsys = 139.6 106 N s2 /m5
Ees,lvf = 377 106 N/m5
Ees,rvf = 87.8 106 N/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 the
tests is the comparison with a Wiggers’ diagram (for explanation of a Wiggers’ diagram see
paragraph 1.2.4). In table 2.5, Smith compares his results with the Wiggers’ diagram. After
Model
Target
Variable
Value
Value
Pressure in Aorta
Amp Pao
40mmHg
41.407
Avg Pao
100mmHg 119.168
Pressure in Pulmonary Artery
Amp Ppa
17mmHg
20.414
Avg Ppa 16.5mmHg
20.314
Volume in Left Ventricle
Amp Vlv
70ml
69.508
Avg Vlv
80ml
84.042
Volume in Right Ventricle
Amp Vrv
70ml
69.569
Avg Rrv
80ml 121.185
Pressure in Pulmonary Vein
Avg Ppu
2mmHg
10.112
Pressure in Vena-cava
Avg Pvc
2mmHg
1.050
% Error
3.5%
19.2%
20.1%
23.1%
−0.7%
5.1%
−0.6%
51.5%
405.6%
−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 be
calculated using Poiseuilles equation (2.4). Poiseuilles equation assumes incompressible,
Newtonian, laminar, axi-symmetric, fully developed flow through a rigid tube of constant
cross-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 in
left 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
2.2.1
Reduced and multiscale models for the human cardiovascular system; one dimensional model [FORVEN]
General description
Formaggia and Veneziani wrote this report as collection of the notes of the two lectures given
by Formaggia at the 7th VKI Lecture Series on ”Biological fluid dynamics” held on the Von
Karman Institute, Belgium, on May 2003. They give a summary of some aspects of the research
aimed at providing mathematical models and numerical techniques for the simulation of the
human 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 part
of the artery can be described by a straight cylinder with a circular cross section. For the
description of the flow through this straight cylinder the Navier-Stokes equations are used and
integrated over a generic cross section. Starting parameters are the time interval T = (0, t1 ) and
the 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 is
too expensive, so some simplifying assumptions are made:
1. All quantities are independent of the angular coordinate θ. As a consequence, every axial
section x = const remains circular during the wall motion. The tube radius r is a function
of x and t.
2. The wall displaces along the radial direction solely, thus at each point at the tube surface
they may write η= ηer , where η = r − r0 is the displacement with respect to the reference
radius r0 .
3. The vessel will expand and contract around its axis, which is fixed in time. This hypothesis 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 the
coronaries 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 component along x. The latter is indicated by ux and its expression in cylindrical coordinates is
supposed to be of the form
ux (t, r̂, x) = ū(t, x)s(r̂r −1 (x))
(2.6)
where ū is the mean velocity on Reach axial section and s : R → R is a velocity profile,
1
which must be chosen such that 0 s(y)ydy = 12 . The fact that the velocity profile does
not vary in time and space is in contrast with experimental observations and numerical
results carried out with full scale models. However, it is a necessary assumption for the
derivation of the reduced model. One may then think of s as being a profile representative
of an average flow configuration.
Finally, a momentum-flux correction coefficient is defined by:
R 2
R 2
S s dσ
S ux dσ
=
.
ψ=
Aū2
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=
Z
ux dσ = Aū;
S
• A the surface area of an axial section;
• P the pressure.
When ψ is taken constant, the reduced model looks like
(
∂Q
∂A
∂t + ∂x = 0
Q
∂Q
∂ Q2
A ∂P
∂t + ψ ∂x ( A ) + ρ ∂x + Kr ( A ) = 0
(2.7)
for x ∈ (0, l), t ∈ T , where Kr = −2πνs′ is a friction parameter and s′ the derivative of the
velocity profile.
21
FLUID
STRUCTURE
parameter
input pressure amplitude
viscosity, ν
Density, ρ
Wall Thickness, h
Reference Radius, r0
20 × 103 dyne/cm2
0.035poise
1.021kg/m3
0.05cm
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−y 2 ). This profile corresponds
with the Poisseuille solution characteristic of steady flow in circular tubes. The parabolic profile
is a variant of the profile most used: s(y) = γ −1 (γ + 2)(1 − y γ ). This power law profile is most
used, 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 first
series of tests are focussed on the single artery, the second series of tests are done with a coupling
of 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 the
results see [FORVEN].
Comparing the results with literature, Formaggia and Veneziani conclude that there is little
agreement with reality. This can be explained by the chosen model. The model is namely formed
by a closed network with a high-level of inter-dependency. In this model the flow dynamics
of 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 carotid
bifurcation 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-dependence
model must be found. The next section describes how Formaggia and Veneziani make use of a
Windkesselcircuit to accomplish this.
2.3
2.3.1
Reduced and multiscale models for the human cardiovascular system;lumped parameters for a cylindrical compliant
vessel [FORVEN]
General description
In the previous section Formaggia and Veneziani developed a 1D model. After doing different
tests they conclude that the model has a high level of interdependency and that there is little
agreement reflecting reality. So, they continu their research aimed at providing mathematical
models and numerical techniques for the simulation of the human CVS, by focussing on coupling
techniques. In this section Formaggia and Veneziani are describing a mathematical model of
the 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 of
the vessel wall, while the systemic model is based on a one-dimensional system or on a lumped
parameters model. The lumped parameters model is based on the solution of a system of ordinary differential equations for the average mass flow and pressure.
For the systemic model, a choice can be made between a one dimensional model and a
lumped parameters model. As one dimensional model you can think of a model like the one
described 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 to
the circulation at a low computational cost. An effective description of this model is by dealing
with 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 the
wall 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, see
figure 2.5. In this circular cylindrical domain, the axial section equals A(t, x) = πr 2 (t, x) where
r(t, x) is the radius of the section at x. With this consideration they want to form a simplified
model, 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 NavierStokes equations and make the same standard assumptions. After these standard assumptions
a 1D model is left.
( ∂A ∂Q
∂t + ∂x = 0 (2.8)
Q2
Q
∂Q
∂
+ Aρ ∂P
∂t + ψ ∂x
A
∂x + Kr A = 0
In order to close the system, a further equation, which is provided by the constitutive law for
the 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. By
following [FOVE], they take:
√
√
A − A0
(P − Pext ) = d(r − r0 ) = β0
,
(2.9)
A0
where Pext is a constant reference pressure, A0 = πr02 a constant reference area, d is a
√
constant related to the physical properties of the vascular tissues and β0 = A0 d/ π.
23
Now, the one dimensional model (2.8) will be integrated along x ∈ (0, l):

 k1 l dp̂
dt + Q2 − Q1 = 0
i
h 2
i R h
2
 l dQ̂ + ψ Q2 − Q1 + l A ∂P + KR Q dx = 0
dt
A2
A1
A
0 ρ ∂x
(2.10)
In this system, p̂ is the mean pressure over the whole compartment and k1 a constant. Since
this is not a satisfactory model, because it is not linear, some more assumptions are needed.
2
Q
Q2
• The quantity A22 − A11 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 integral
in 2.10 will be approximated
Z l
Z l
A ∂P
Q
Q
A0 ∂P
+ KR
+ KR
dx
dx ≈
ρ ∂x
A
ρ ∂x
A0
0
0
2.3.3
Mathematical model
With all these assumptions Formaggia and Veneziani end op with a system for the lumped
parameters description of the blood flow in the compliant cylindrical vessel. It involves the
mean values of the flow rate and the pressure over the domain, as well as the upstream and
downstream flow rate and pressure values:

 k1 l dp̂
dt + Q2 − Q1 = 0
(2.11)
 ρl dQ̂ + ρK2R l Q̂ + P2 − P1 = 0
A0 dt
A
0
The final system (2.11) will be represented by a hydraulic analog. In this analog three parameters
are used, namely:
• R the resistance induced by the blood viscosity is represented by R =
parabolic velocity profile gives
8µl
R= 4
πr0
ρKR l
.
A20
Assuming a
(2.12)
• L the inductance of the flow represents the inertial term in the momentum conservation
law and is given by
ρl
L= 2
(2.13)
πr0
• C the capacitance of the vessel represents the coefficient of the mass storage term in the
mass conservation law and is given by
C=
3πr03 l
2Eh
With this notation equation (2.11) becomes
( dp̂
C dt + Q2 − Q1 = 0
L ddtQ̂ + RQ̂ + P2 − P1 = 0
24
(2.14)
(2.15)
2.3.4
Conclusions
In an analytical test case a completely lumped parameters description of the circulation, providing a reference solution to the systemic level, is given. The aim of this test is to model blood
flow behaviour in Ω by the Navier-Stokes equations coupled with the lumped description of the
remaining network. Formaggia and Veneziani expect from this test that the presence of a local
accurate submodel does not have to modify significantly the results at the systemic level. This
is exactly what they obtain numerically. The heterogeneous model is able to compute accurately
the 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 between a 3D fluid structure model and a 1D reduced model is an effective way to greatly reduce
numerical reflections of the pressure waves.
In a last test of clinical interest, the methodology was in particular applied to a reconstructive procedure, the systemic-to-pulmonary shunt, used in cardiovascular paediatric surgery
to treat a group of complex congential malformations. The 3D-model includes the shunt, the
innominate artery (through which blood flows in) and the pulmonary, carotid and subclavian
arteries (through which blood flows out). In this case the lumped model is composed by different
blocks 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, gives
a clear idea of what can be obtained using the multiscale methodology.
2.4
2.4.1
Computational modeling of cardiovascular response to orthostatic stress [HSKM]
General description
The objective of this study is to develop a model of the cardiovascular system capable of simulating the short term (< 5min) transient and steady state heamodynamic responses to head-up tilt
and lower body negative pressure. A subobjective of this study is to develop and test a general
0D, 12 compartment model of a CVS that contains the essential features associated with the
effects of gravity. The development of the model is not completely their own, but they use the
knowledge and formulas of other investigators.
2.4.2
Mathematical Model
The model of [WHFICR] and [DAMA] is based on a closed-loop lumped parameters heamodynamic model with local blood flow to major peripheral circulatory branches. This heamodynamical model is mathematically formulated in terms of a hydraulic analog model in which inertial
effects are neglected. A single compartment circuit representation which has been used is given
in figure 2.6. The equations read
Pn−1 − Pn
Rn
Pn − Pn+1
Q2 =
Rn+1
Q1 =
Q3 =
d
[Cn × (Pn − Pbias )]
dt
25
(2.16)
(2.17)
(2.18)
Figure 2.6: Single compartment circuit representation, P pressure, R resistance, C capacitor, Q
flow rates
The flow Q1 at node Pn splits up like Q1 = Q2 + Q3 . Combining these expressions for the flow
rates leads to
dCn
Pn+1 − Pn Pn−1 − Pn Pbias − Pn
d
d
Pn =
(2.19)
+
+
·
+ Pbias
dt
Cn Rn+1
Cn Rn
Cn
dt
dt
In total 12 such first order differential equations are used to describe the entire model, see figure
2.7. To solve this model, the authors used a fourth order Runge-Kutta integration routine. The
pumping action of the heart is realized by varying the right and the left ventricular elastances
according to a predefined function of time (Er and El )


Esys −Edias
t

√
Edias +
1 − cos π
0 ≤ t ≤ Ts

2

(n−1)

0.3 T √
t−0.3 T (n−1)
(2.20)
e(t) =
E −E
√
Ts < t ≤ 32 Ts
 Edias + sys 2 dias 1 + cos 2π

0.3 T (n−1)



3
Edias
2 Ts < t ≤ T (n)
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 the
time measured with resect to the onset of ventricular
p 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 observations
of [LUD]. In accordance to these observations they model the functional form of the pressurevolume relationships of the venous compartments of the legs, the splanchnic circulation and the
abdominal venous compartment with
πC0
2∆V
arctan
∆Ptrans ,
(2.21)
∆V =
π
2∆Vmax
where ∆V represents the change in compartment volume due to a change in transmural pressure ∆Ptrans , ∆Vmax is the maximal change in compartment volume and C0 represents the
compartment 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 during
orthastatic stress.
2.4.3
Parameters
The parameters given in this section are the resistance, volume and capacitance values for the
12 different compartments.
In this tables the writers make use of the P RU = mmHg.s/ml (peripheral resistance unit) and
ZP F V (zero pressure filling volume). Further all values compound with a 71 − 75kg normal
male subject and a body surface area of 1.7 − 2.1m2 with a total blood volume of 5700ml.
27
Rlo
0.006
Rll2
0.3
Rup1
3.9
Rsup
0.06
Resistance values [P RU ]
Rkid1 Rsp1 Rll1
Rup2
4.1
3.0
3.6
0.23
Rab
Rinf
Rro
Rp
0.01
0.015 0.003 0.08
Rkid2
0.3
Rpv
0.01
Rsp2
0.18
Table 2.7: Resistance values
Compartment
Right ventricle
Pulmonary arteries
Pulmonary veins
Left ventricle
Systemic arteries
Systemic veins
Upper body
Kidney
Splanchnic
Lower limbs
Abdominal veins
Inferior vena cava
Superior vena cava
ZPFV
ml
50
90
490
50
715
Capacitance
ml/mmHg
1.2-20
4.3
8.4
0.4-10
2.0
650
150
1300
350
250
75
10
8
15
55
19
25
2
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 generated
by the model are within the range of what is considered as physiologically normal in the general
population. Representative simulated pressure waveforms are made, too. The conclusions the
authors draw after these tests are
1. They assume that the dynamics of the system can be simulated by restricting their analysis
to relatively few representative points within the CVS. Although this approach is incapable
of simulating pulse wave propagation, it does reproduce realistic values of beat-to-beat
heamodynamic parameters.
2. One potential limitation of the heamodynamic system in its present form might be the
lack of atria, which are thought to contribute significantly to ventricular filling at high
heart rates.
3. The model generates steady state and transient heamodynamic responses that compare
well to population-averaged and individual subject data.
28
2.5
2.5.1
An identifiable model for dynamic simulation of the human
cardiovascular system [KRWIWAKR]
General description
The authors describe the mathematical model in their paper as a hapy medium one. The idea
was to keep the model as exact as needed and to make it as easy as possible. For this model
they combine a compartment model with a dynamic (time and space dependent) model of the
arterial vascular system to simulate mainly the arterial part of the CVS. Both parts of the model
are calculated separately and connected afterwards by a feedback control mechanism.
2.5.2
Mathematical model
The compartment model
models the whole CVS including the heart, the systemic and the pulmonary part. This model is
divided in six compartments with a feedback control mechanism. Further it takes into account
outer 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 systemic
blood pressure are computed. These four variables are the main characteristics of the CVS and
are directly controlled through the model.
The feedback control mechanism for modelling the baro-receptor mechanism is based on
measured data. The examples show that the mechanism which increases the heart rate does not
depend linear anymore on the stress when a critical level is reached. Nevertheless, to a certain
level the authors model the behavior linearly and define a threshold for the nonlinear part of
control. With this approach there is still a partially linear model:
−KHG(ω−HF C)
BH =
( KHC − KHCe
ω
B
− T HF
+ KHF EWT+HF
HF
ω̇ =
+HF B)
ω
+ KHF (1−BH )(EW
− T HF
T HF
for
ω < HF C
(2.22)
else
Here ω (heartrate), BH, HF B (compensation terms) and EW (stress) are time dependent, the
rest are constants. HF C 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 had
been developed by [WIB].
(
At + Qx = 0
2
(2.23)
∂
φ QA + E = Kr Q
Qt + ∂x
A
For the connection between the dynamic model and the control mechanism two aspects have to
be 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 the
peripheral complex resistances (impedances) have to be calculated before. Doing this, the
29
Womersley solution of the one dimensional Navier-Stokes equation to compute impedances
in any arterial segment has to be used. With this method it is possible to compute the
Windkessel data at any end node of the modelled arterial tree. A static model is used to
compute impedances at bifurcations of the vascular system.
In a simplified model, this is done by using the solution of the axisymmetric Navier-Stokes equations with equations that describe the motion of the vessel wall and solving a Bessel equation.
This leads to
Z a
ωr 2πrdr,
(2.24)
Q=
0
where ωr is the interaction between fluid and wall and given through a term which depends on
blood density, the Womersley number, the kinematic viscosity and the complex wave-propagation
velocity. Integration over the cross sectional area yields
Q=
A0 Ec wp
(1 − Fj ),
c0 ρ
(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 for
the impedances at any arterial segment.
2.5.3
Conclusions
Comparison of the simulation results with measured data shows that with this model the behavior of the human CVS can be described very well. The measurements are done with ultrasound
techniques whereby each point is measured twice: once with a standing person and once with
a lying person. After the test they can conclude that the nonlinear reaction of the heartrate
caused by stress can be modelled very well with an easy compartmental approach for modelling
shorttime control of the human CVS.
Further they have considered the reaction of the CVS caused by changes of hydrostatic pressure (tilting table test). Again the results were very satisfying. The most satisfying of these two
tests is that under the circumstances given above all the parameters needed can be identified
with an ergometer and a tilting table. In addition to the compartment results there are some
results of the complex dynamic model which can be compared with measured data, too. The
ultrasound measured flow velocity and the computed flow velocities confirm the validity of the
model in a qualitative manner. Some tests have been carried out, but they were not finished
when this paper was published.
Finally, after the tests allready done, they conclude that the model is a good approximation for the human CVS and that combining a compartment model with a model for pulsatile
blood flow in arteries provides a distributed model which is dynamic in time and space, feedback
controlled, identifiable and verifiable through measured data.
2.6
2.6.1
Interaction between carotid baroregulation and the pulsating heart: a mathematical model [URS]
General description
Ursino presents in his paper a mathematical model of short-term arterial pressure control by
the carotid baroreceptors in pulsatile conditions. The model includes an elastance variable de30
scription of the left and the right heart, the systemic and the pulmonary circulation, the afferent
carotid baroreceptor pathway, the sympethatic and vagal efferent activities and the action of
several effector mechanisms. The model is used to simulate the interaction among the carotid
baroreflex, the pulsating heart and the effector responses in different experiments.
2.6.2
Mathematical model
Figure 2.8: Hydraulic analog of the cardiovascular system. A bifurcation in the systemic circulation is made into a splanchnic and an extrasplanchnic circulation.
Ursino uses a 0D, 12 compartment model. This model is a generalization of the model
presented 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 written
by enforcing conservation of mass at the capacities in figure 2.8 and equilibrium of forces at the
inertances.
dP
dt
dQ
dt
=
=
1
C (∆Q)
1
L (∆P −
31
RQ)
(2.26)
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 a
ventricle. The models for the left and right part are the same except for parametervalues. The
atrium is modelled by a linear capacity characterized by constant values of capacitance and
unstressed volume. The blood flow between the atrium and the ventricle is modelled by an
atrioventricular valve, mimicked as the series arrangement of an ideal unidirectional valve with
a constant resistance:
0
if Pla ≤ Plv
(2.27)
Qi,l =
Pla −Plv
if Pla > Plv
Rla
The contractile activity of the ventricle is described by means of a Voigt viscoelastic model.
Plv
dVlv
dt = Qi,l
Pmax,lv (t)
e(t)
u(t)
− Q0,l
= Pmax,lv − Rlv Q0,l
λlv Vlv − 1)
= e(t)E
max,lv
i Vu,lv ) + [1 − e(t)]P0,lv (e
h (Vlv −
(
(t)
u 0 ≤ u ≤ Tsys /T
sin2 TπT
sys (t)
=
0h
Tsysi/T ≤ u ≤ 1
Rt 1
= frac t0 T (τ ) dτ + u(to )
0 ≤ e(t) ≤ 1
(2.28)
In these equations Emax,lv is the ventricle elastance at the instant of maximum contraction, Vu,lv
is the corresponding ventricle unstressed volume, Q0,l is the cardiac output from the left ventricle
and P0,lv and kE,lv are constant parameters that characterize the exponential pressure-volume
function at diastole.
In the Voigt viscoelastic model a linear pressure-volume function at end-systole is adopted
and at diastole an exponential pressure-volume function. This is done to reflect the varying elastance during the cardiac cycle. The shifting between the end-systole and diastole is governed
by a pulsating activation function e(t), with period T equal to the heart period. In this work a
sin-square function for e(t) has been used. In this equation T equals the heart period and Tsys
stands for the duration of systole.
Since e(t) only must take values between 0 (complete relaxation) and 1 (maximum contraction), an expression for u(t) has been obtained by means of an ”integrate and fire” model. In
this expression the function f rac() 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 a
70kg body weight. The total blood volume is taken as 5300ml.
32
Capacitance
[ml/mmHg]
Csa = 0.28
Csp = 2.05
Cep = 1.67
Csv = 61.11
Cev = 50.0
Cpa = 0.76
Cpp = 5.80
Cpv = 25.37
Unstressed Volume
[ml]
Vu,sa = 0
Vu,sp = 274.4
Vu,ep = 336.6
Vu,sv = 1.121
Vu,ev = 1.375
Vu,pa = 0
Vu,pp = 123
Vu,pv = 120
Hydraulic Resistance
[mmHg.s.ml−1 ]
Rsa = 0.06
Rsp = 3.307
Rep = 1.407
Rsv = 0.038
Rev = 0.016
Rpa = 0.023
Rpp = 0.0894
Rpv = 0.0056
Inertance
[mmHg.ml.ml−2 ]
Lsa = 0.22 10−3
Lpa = 0.18 10−3
Table 2.9: Parameters characterizing the vascular system in basal condition
Left Heart
Cla = 19.23ml/mmHg
Vu,la = 25ml
Rla = 2.5 10−3 mmHg.s.ml−1
P0,lv = 1.5mmHg
λlv = 0.014ml−1
Vu,lv = 16.77ml
Emax,lv = 2.95mmHg/ml
kR,lv = 3.75 10−4 s/ml
Right Heart
Cra = 31.25ml/mmHg
Vu,ra = 25ml
Rra = 2.5 10−3 mmHg.s.ml−1
P0,rv = 1.5mmHg
λrv = 0.011ml−1
Vu,rv = 40.8ml
Emax,rv = 1.75mmHg/ml
kR,rv = 1.4 10−3 s/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/s
ka = 11.758mmHg
τz = 6.37s
τp = 2.076s
Sympathetic efferent pathway
fes,inf = 2.10spikes/s fes,0 = 16.11spiks/s
kes = 0.0675s
fes,min = 2.66spikes/s
Vagal efferent pathway
fev,0 = 3.2spikes/s
fev,inf = 6.3spikes/s
kev = 7.06spikes/s
fcs,0 = 25spikes/s
Effectors
−1
−1
GEmax,lv = 0.475mmHg.ml .v
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/ml
GR,sp = 0.695mmHg.s.ml−1 .v −1
τR,sp = 6s
DR,sp = 2s
Rsp,0 = 2.49mmHg.s/ml
GR,ep = 0.53mmHg.s.ml−1 .v −1
τR,ep = 6s
DR,ep = 2s
Rep,0 = 0.78mmHg.s/ml
GVu,sv = −265.4ml/v
τVu,sv = 20s
DVu,sv = 5s
Vusv,0 = 1435.4ml
GVu,ev = −132.5ml/v
τVu,ev = 20s
DVu,ev = 5s
Vuev,0 = 1537ml
GT,s = −0.13s/v
τT,s = 2s
DT,s = 2s
T0 = 0.58s
GT,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-KuttaFehlberg method with adjustable step length. During the simulations the integration and memorization 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. Ursino
compare his results with [KCYSTN] which has results of a dog CVS. There are similar patterns
observed.
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 systemic
resistance.
2. The dependence of heart contractility on the carotid baroreflex and on other heamodynamic influences.
3. The present model neglects the effect of changes in coronary perfusion on the end-systolic
pressure-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 by
means of monotonic exponential functions linking activity in the afferent and efferent
neural pathways.
35
2.7
Summary
There are a lot of papers written about the computational model for the human cardiovascular
system. The papers summarized were selected because we expected that they could satisfy the
requirements.
• 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 simulate the human CVS and the way of simulating the heart. What can be said is that every
author 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 will
conclude which model or combination of models we will use for further investigation. In this
table the comparence of the parametervalues are missing. This because it is very difficult to
compare the parameters, since all the models use compartments which reflects different parts of
the CVS.
36
2.1
0D/1D
0D
2.2
1D
2.3
0D
2.4
2.5
0D
1D
2.6
0D
Table 2.12: Overview table from all summarized articles
equation
including inertia
=
P
−
P
−
QR
L dQ
in
the large arteries
1
2
dt
1 dV
dP
dt = C dt
∂Q
∂A
not relevant
∂t + ∂x = 0
2
Q
∂Q
∂ Q
A ∂P
∂t + α ∂x ( A ) + ρ ∂x + Kr ( A ) = 0
C dp̂
yes
dt + Q2 − Q1 = 0
L ddtQ̂ + RQ̂ + P2 − P1 = 0
d P = Pn+1 −Pn + Pn−1 −Pn + Pbias −Pn ·
dt n
Cn Rn+1
Cn Rn
Cn
dC n
dt
d P
+ dt
bias
At + Qx = 0
2
∂
Qt + ∂x α QA + p = K Q
A
1
dP
=
(∆Q)
dt
C
dQ
1
=
(∆P
− RQ)
dt
L
2.1
2.2
2.3
2.4
2.5
2.6
# compartments
6
no compartments
5
12
6
12
circuit type
Windkessel like circuit
one closed system
Windkessel circuit
Windkessel circuit
Windkessel circuit
Windkessel circuit
2.1
2.2
2.3
2.4
2.5
2.6
driverfunction
P
−Bi (t−Ci )2
e(t) = N
i=1 Ai e
different sine waves
none
cos
none
sin2
in the large arteries
heart simulation
veins+ra,rv,lungs+la,lv
1 ventricle
rv,lv
rv,lv
heartrate input
ra,rv,la,lv
parameter values
given, from literature
given, from literature
not given
given, from literature
not given
given, from literature
37
no
not relevant
solution method
ODE15s (Matlab)
Taylor Galerkin scheme
finite difference scheme
RK4 integration routine
not applicable
5th order Runge-Kutta-Fehlberg method
most important conclusion
2.1
2.2
2.3
2.4
The results show the capability of the presented approach to create patient specific models
All these tests gave interesting outputs, but there is no comparence with the reality
the heterogeneous model is able to compute accurately the velocity and pressure fields in the domain of interest
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
2.7.2
The model is able to reproduce several aspects of carotid baroreflex control rather well
Discussion
In this discusion we will make a decision which model we are going to use to create a satisfying
model. 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 advantage
of being a detailed model. The 1D model has as disadvantage that it has a high level of
interdependency between the used equations in the summarized models. Further, the 1D model
has no realistic results.
The 0D model has the advantage of being a model which can be solved easily and the
solutions do have connections with reality. The disadvantage of the 0D model is that it is only
detailed in a small part of the circulation system.
With these advantages and disadvantages in mind plus our own demands, we choose for a 0D
model. 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 part
of 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 of
modelling:
• 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 12
compartments. Section 2.3 (with five compartments) concluded that the heterogeneous model
is able to compute accurately the velocity and pressure fields in the domain of interest, but has
the 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 the
domain of interest. So to get feeling with modelling a circulation system, we start with a model
like 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 not
enough. Although this remark, we think it is possible to create a model with only 3 compartments which satisfy our demands. We think this is possible, because the circulation system is a
single closed loop and a compartment reflects a part of the circulation what can be considered
as a whole. In our 3 compartment model, the heart is reflected by a compartment, the aorta by
a compartment and the rest of the circulation system reflected as a compartments. The results
of our 3 compartment model will be compared with the results of the 6 compartment model and
the Wiggers’ diagram.
The next question is about a good driver function. However, each model analized uses another driver function so we cannot say anything about a good one for our model. Therefor, we
will test different driver functions. After choosing a good driver function, this function will be
used in the other tests.
The third and fourth question can be seen as what part of the circulation system can be
considered as a whole. In the 6 and 12 compartment models of section 2.1 and section 2.6 a right
ventricle plus a lungcirculation is included as compartment. In the model with 3 compartments
we will include the lungcirculation, because by following one blood element through the circulation, then you will see that the whole circulation is one closed circle. Simply said, when we
take out the lung circulation we will miss a piece of the whole circulation. We will only include
it as a part of the systemic resistance.
The last question is about including inertia. As shown in section 2.1 inertia is only needed
in 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 the
CVS with 6 compartments. This model will be used to test the system of equations. These
tests will give us the reference figures for the results of the 3 compartment model. Before we
test the 6 and 3 compartment model, we will use a not coupled single compartment to find the
best cardiac driver function and if we include inertia.
39
40
Chapter 3
Model description
Several numerical models for the human CVS are summarized in section 2. Every model has his
own advantage and disadvantage. These advantages are dependent of the specific goal of the
model. The model we want to create must be able to calculate the pressure and flow on every
point 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 the
heart, the aorta, the cappilairies and the veins. It will not be easy to describe all these parts in
one 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 system
an 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 a
tube. 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 the
rest of the system, while Γw is the artery wall.
We will give two different approaches to derive our model of the human CVS. The first
approach is a bottum-up consideration, the second approach is a top-down approach which
make use of the Navier-Stokes equations.
3.2
The 0D model for the circulation system, bottum-up approach
The system which will be derived consists of two equations. One equation for the conservation
of 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 time
equals the difference of inflow and outflow.
dV
= Q = Qin − Qout
dt
(3.1)
We want to use this statement to find a relation between the pressure and the flow in a small
artery. Therefor, we have to find a relation between the pressure and the volume in a small
artery.
Since the internal radius of a small artery is influenced by the internal and external pressure
on the wall and the wall is not very flexible, we can use the linear elastic law.
σ=
E
ǫ
1 − ν2
(3.2)
0
In this formula σ is the surface stress, ǫ = r−r
r0 the rate of change caused by the stress to the
original state of the object, E the Young modulus and ν the Poisson ratio of the artery wall
(we take ν = 21 , as for incompressible tissue). So we have the following linear elastic law for a
cylindrical vessel:
4E r − r0
σ=
.
(3.3)
3 r0
However, we do not know anything about the surface stress, but only know the pressure P on
the 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 an
internal radius r, a wall thickness h, an internal pressure Pint and an external pressure Pext . We
define the transmural pressure P as the difference between the internal and external pressure
P = Pint − Pext . Further, we take as reference values the radius r0 and a surface stress σ = 0
when there is no pressure on the wall (P = 0). We take a part of the cross section and define on
this cross section a surface element by an infinitesimal angle dφ and longitudinal displacement
dx. The area of this element equals rdφdx and the force on this element equals
P rdφdx.
(3.4)
The force in the radial direction can be calculated by measuring the angle difference between
two points before and after stretching times the stress times the surface area to which the stress
is applied.
dφ
(3.5)
2hσ sin( )dx
2
After equating both forces we come to the following equation.
P rdφdx = 2hσ sin(
dφ
)dx.
2
(3.6)
dφ
By letting dφ → 0, sin( dφ
2 ) ≈ 2 and divide everything by dφdx. We end-up with a stresspressure 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 r − r0
3
rr0
(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
d 4Eh(r − r0 )
4Eh
dP
=
(3.9)
=
dr
dr
3rr0
3r 2
and
P (r0 ) = 0
4Eh
dP
(r0 ) =
dr
3r02
(3.10)
(3.11)
The linearized equation equals
P =
4Eh(r − r0 )
3r02
(3.12)
With equation (3.12) we have a linear pressure-radius relation. This relation can be rewritten
into a pressure-volume relation by substituting
V − V0 = πl(r 2 − r02 ) ≈ 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 =
3πlr03
P + V0 .
2hE
(3.13)
Taking the derivative of V finally gives the equation for the conservation of mass
3πlr03 dP
dV
=
= Q = Qin − Qout
dt
2hE dt
3.2.2
(3.14)
Conservation of momentum
The second equation for a small artery free of bifurcations is the equation for the conservation
of 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 difference
Pin − Pout between x = −l/2 and x = l/2. Further, we assume stationairy flow without radial
velocity. 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:
du
du
(3.15)
(P (x) − P (x + dx)) 2πrdr + 2πµ (r )r+dr − (r )r dx = 0
dr
dr
In this equation the velocity u(r) satisfies a normal differential equation of second order with
the standard solution:
1 dP 2
r + C1 ln(r) + C2
(3.16)
u(r) =
4µ dx
From the fact that u must be limited, it follows that C1 = 0 and from the no-slip condition on
1 dP 2
the wall r = r0 it follows that C2 = − 4µ
dx r0 . This yields a parabolic velocity profile
u(r) = −
1 dP 2
(r − r 2 )
4µ dx 0
45
(3.17)
Figure 3.5: Force balance in the axial direction
With this velocity profile, the mass flow through the artery, by use of the Poiseuille-Hagenformula, equals
Z r0
πρr04
−πρr04 dP
=
(Pin − Pout )
(3.18)
rudr =
Q = 2πρ
8µ dx
8µl
0
This equation is the equation for the conservation of momentum for a stationairy flow.
Around
the heart there is an instationary flow. So we have to look for the force needed to move a column
of 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 in
the artery equals:
F = (Pin − Pout )πr02
(3.19)
The mass of the blood in the artery equals:
m = lπr02 ρ
(3.20)
and the velocity of the blood through the artery equals:
ū =
Q
πr02
(3.21)
Using equations (3.19), (3.20) and (3.21) in Newtons second law results in
(Pin − Pout )πr02 = lπr02 ρ
Pin − Pout
d(Q/πr02 )
dt
lρ dQ
= 2
πr0 dt
(3.22)
(3.23)
Accounting for this term in the momentum equation results in the momentum equation for
instationary blood flow
lρ dQ
8µl
(3.24)
Pin − Pout = 4 Q + 2
πr0
πr0 dt
46
Final model
Combining the mass equation (3.14) and the momentum equation (3.24) results in the final
system for a small artery free of bifurcations:

3
 3πlr0 dP = Qin − Qout
2hE dt
(3.25)
 Pin − Pout = 8µl4 Q + lρ2 dQ
dt
πr
πr
0
3.3
0
The 0D model for the circulation system, Top-Down approach
Besides the mathematical bottum-up aproach for the derivation of the model of an arterie free
of 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 the
mathematical model of the description of the blood flow in the artery. To come to a reduced
model, we are integrating the Navier-Stokes equations on a generic section A. Therefor, at first
we introduce some simplifying assumptions:
1. The blood flows only in the axial direction, so we assume independence of all quantities
involved from the circumferential coordinate θ. As a consequence r is a function of x and
t.
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 )er
is 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 assume
that 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 to
the axial one. The axial component of the velocity will be denoted by ux . The expression
of ux in cylindrical coordinates is supposed to be of the form
ux (t, r̂, x) = ū(t, x)s(r̂r −1 (x)),
(3.26)
where ū is the mean velocity on each axial section and s : R → R is a velocity profile. One
may 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:
Z
dσ = πr 2 (t, x) = π(r0 (x) + η(t, x))2 ;
A(t, x) =
A
47
(3.27)
• The mean velocity can now be defined by:
ū = A−1
• It follows from equation (3.26) that
ū = A−1
⇒
⇒
R
A ux dσ
=
1
πr 2
Rr
0
Rr
y= r̂
r2
−1 )dr̂ =r
=
r̂s(r̂r
2
0
R1
1
0 s(y)ydy = 2
R1
0
0
ux dσ;
(3.28)
A
s(y)ydy = 12 .
1
πr 2
2πr̂ux dr̂ =
R1
Z
Rr
0
2πr̂ūs(r̂r −1 )dr̂ =
rys(ryr −1 )rdy
Rr
2
ū 0
r2
r̂s(r̂r −1 )dr̂
For the sake of simplicity, we will choose s(y) = 2(1 − y 2 ), which corresponds to the
Poiseuille solution characteristic of steady flows in circular tubes.
• Next we indicate by ψ the momentum flux correction coefficient, defined as
R 2
R 2
s dσ
A ux dσ
ψ=
= A
2
Aū
A
(3.29)
In general ψ will vary in time, but as consequence of equation (3.26) in our model it will
be constant.
• The mean flux is defined as
Q=
Z
ux dσ = Aū
(3.30)
A
Under the previous assumptions, the momentum and continuity equations, in the hypothesis of
constant viscosity, are
div(u) = 0
(3.31)
∂ux
1 ∂P
∂t + div(ux u) + ρ ∂x − ν∆ux = 0
with on the tube wall the kinematic condition
u = η̇
, where η̇ =
∂η
∂t
=
∂η
∂t er
on Γw
t
(3.32)
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)
dx
l l
Consider the portion P of the tube between x = x̄ − dx
2 and x = x̄ + 2 , with x̄ ∈ (− 2 , 2 ) and
w
dx > 0 small enough. The part of δP laying on the tube wall is indicated by ΓP . The reduced
model is derived by integrating system (3.31) on P and passing to the limit as dx → 0, assuming
that all quantities are smooth enough.
Before we start with integrating, we will introduce a useful theorem, which has been proven
in [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
Z
−1
¯
f =A
f dσ.
A
We have the following relation
that
∂
¯
∂t (Af )
=
¯
A ∂f
∂t
∂A
= 2πr η̇
∂t
We start with the continuity equation.
Using the divergence theorem, we get
Z
Z
Z
Z
ux +
ux +
0 = div(u) = −
Γw
P
A+
A−
P
+ 2πr η̇fw . In particular taking f = 1 we recover
(3.33)
u·n =−
Z
ux +
A−
Z
A+
ux +
Z
Γw
P
η̇ · n
(3.34)
where n is the outwardly oriented normal. Since η̇ = η̇er , we deduce
Z
∂
η̇ · n = [2η̇πr(x̄)dx + o(dx)] = A(x̄)dx + o(dx).
w
∂t
ΓP
Substituting into equation (3.34), using the expression of Q and passing to the limit as dx → 0,
we finally obtain
∂A ∂Q
+
=0
(3.35)
∂t
∂x
On the same way we are going to integrate every term of the momentum equation over P and
consider the limit as dx tends to zero.
•
Z
Z
Z
d
d
∂Q
∂ux
ux g · n =
=
ux −
ux =
(x̄)dx + o(dx)
(3.36)
dt P
dt P
∂t
δP
P ∂t
In order to eliminate the boundary integral we have exploited the fact that ux = 0 on Γw
P
and g = 0 on A− and A+
Z
•
R
= ψ[A(x̄
By using again
•
R
∂P
P ∂xR
=−
2
2
ux g · n
δP ux u · n = − A− ux + A+ ux + Γw
P
∂ψAū2
dx
dx 2
dx
2
+ dx
2 )ū (x̄ + 2 ) − A(x̄ − 2 )ū (x̄ − 2 )] =
∂x (x̄)dx
ux = 0 on Γw
P.
P div(ux u)
A−
= A(x̄ +
⇒
R
P+
R
A+
dx
2 )P (x̄
=
R
P+
+
R
dx
2 )
Γw
P
R
R
R
+ o(dx)
(3.37)
P nx
− A(x̄ −
dx
2 )P (x̄
−
∂P
P ∂x
dx
dx
= A(x̄ + dx
2 )P (x̄ + 2 ) − A(x̄ − 2 )P (x̄ −
∂A
∂AP
= ∂x (x̄)dx − P (x̄) ∂x (x̄)dx + o(dx)
= A ∂P
∂x (x̄)dx + o(dx)
49
dx
2 ) + Γw
P
R
dx
2 )−
P nx
P (x̄)[A(x̄ +
(3.38)
dx
2 )−
A(x̄ −
dx
2 )]
+ o(dx)
Since
R
δP nx
= 0 and so
R
Γw
P
P nx = P (x̄)
−P (x̄)(A(x̄ +
•
Z
∆ux =
P
R
Γw
P
dx
2 )−
Z
δP
nx + o(dx) = −P (x̄)
A(x̄ −
dx
2 ))
∇ux · n = −
Z
+ o(dx)
A−
∂ux
+
∂x
R
Z
δP\Γw
P
A+
nx + o(dx) =
∂ux
+
∂x
Z
Γw
P
∇ux · n
(3.39)
By assuming that the variation of the change of velocity along the x-axis is small compared
x
to the other terms, we neglect the term ∂u
∂x . Using this assumption and splitting n into
two vector components, nr = nr er and nx = n − nr , we may write
Z
Z
(∇ux · nx + ∇ux · er nr )dσ
(3.40)
∆ux =
Γw
P
P
Again, we neglect ∇ux · nx which is proportional to
and the fact that nr dσ = 2πrdx to get
Z
∆ux =
P
Z
Γw
P
nr ∇ux · er dσ =
Z
Γw
P
ūr
−1 ′
∂ux
∂x .
s (1)n · er dσ = 2π
Finally, we use equation (3.26)
Z
x̄+ dx
2
x̄− dx
2
ūs′ (1)dx ≈ 2π ū(x̄)s′ (1)dx
(3.41)
Substituting all these calculated integrals in the momentum equation, dividing all terms by dx
and passing to the limit as dx → 0, we can write as the reduced momentum equation:
∂Q ∂ψAū2 A ∂P
+
+
+ Kr ū = 0
∂t
∂x
ρ ∂x
(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 for
x ∈ (0, l) and t ∈ (0, T ]
(
∂Q
∂A
∂t + ∂x = 0
(3.43)
∂Q
Q
∂ Q2
A ∂P
∂t + ψ ∂x ( A ) + ρ ∂x + Kr ( A ) = 0
For closing the system, we must find a relation between the pressure and the vessel wall displacement. Therefor, we adopt a commonly used hypothesis for the wall mechanics, namely
that the inertial terms are neglegible and that the elastic stresses in the circumferential direction are dominant. With this assumption, the only normal stress acting on the wall is that due
to the pressure. This is possible because we neglected the viscous contribution. In the most
general setting, the pressure relation looks like P (t, x) = Φ(A(t, x); A0 (x), β(x)). In this expression, we have outlined that the pressure also depends on A0 = πr02 and on a set of coefficients
β = (β0 , β1 , · · · , βq ) related to the mechanical and physical properties. In this relation we require
that Φ is a C 1 function of all its arguments and is defined for all A > 0 and A0 > 0. Furthermore
∂Φ
> 0 and Φ(A0 ; A0 , β) = 0.
we require that ∂A
For this relation in literature different possibilities are given. We choose to use a linearized
pressure-radius relation 3.12
4Eh r − r0
( 2 )
(3.44)
P (t, x) =
3
r0
50
By using (r − r0 ) =
√
√
A−
√ A0
π
we have the following relation for Φ:
Φ(A; A0 , β0 ) = β0
√
√
A − A0
A0
√
4 πhE
where β0 =
3
(3.45)
It is simply to verify that indeed all requirements are satisfied.
By observing that
∂r
∂r
3r 2 ∂P (t, x)
3πr03 ∂P (t, x)
∂A
= 2πr
≈ 2πr0
= 2πr0 ( 0
)=
∂t
∂t
∂t
4Eh
∂t
2Eh
∂t
we will assume
∂P
∂A
= k1
,
∂t
∂t
where k1 =
3πr03
2Eh
(3.46)
Still we have to deal with an one dimensional model. To find a 0D model, we must perform a
further averaging of the system (3.43) in combination with equation (3.46). At first we introduce
the following notation:
• The mean flow rate over the whole district V
Z
Z Z
Z
1 l
1 l
1
ux dV =
ux dσdx =
Qdx
Q̂ =
l V
l 0 A(x)
l 0
• The mean pressure over the whole compartment
Z
1 l
P dx
p̂ =
l 0
(3.47)
(3.48)
To average further we integrate system (3.43) over x ∈ (0, l). Integration of the first equation
gives:
Z l
Z l
Z l
Z l
∂A
∂Q
∂Q
∂P
dp̂
k1
dx +
dx =
dx +
dx = k1 l + Qout − Qin = 0
∂t
∂x
∂t
∂x
dt
0
0
0
0
Integration of the second equation gives:
2
Z l
Z l
Qout Q2in
∂ Q2
A ∂P
Q
Q
A ∂P
dQ̂
∂Q
+ψ ( )+
+ Kr
+ψ
+ Kr
−
+
dx = l
dx = 0
∂t
∂x A
ρ ∂x
A
dt
A2
A1
ρ ∂x
A
0
0
Since the integrated system is not linear we introduce the following two assumptions:
Q2
Q2
1. Since we deal with short pipes, the quantity ( Aout
− Ain
) is small compared with the other
2
1
terms. So in the rest of the calculations we neglect the contribution of the convective
terms.
2. The variation of A with respect to x is small compared with that of P and Q, so in the
second equation we can rewrite the integral by saying
Z l
Z l
Q
Q
lKr
A0 ∂P
A0
A ∂P
+ Kr
+ Kr
(Pout − Pin ) +
Q̂
dx =
dx ≈
ρ ∂x
A
ρ ∂x
A0
ρ
A0
0
0
51
With the last two assumptions we finally have the following zero dimensional model
(
k1 l dp̂
dt + Qout − Qin = 0
ρl dQ̂
A0 dt
+
ρlKr
Q̂
A20
+ Pout − Pin = 0
(3.49)
We are still not finished, since in this equation we have 6 unknowns and 2 equations, so we need
some 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 p̂ and Q̂, i.e. by the unknowns
that are under the time derivative, so it is reasonable to approximate the unknowns on
the upstream and the downstream sections with the state variables, that is
p̂ = Pin ,
Q̂ = Qout
With these last assumptions we have the following system
(
k1 l dPdtin + Qout = Qin
ρl dQout
ρlKr
A0 dt + A2 Qout − Pin = −P0ut
(3.50)
0
Similarly, we can assume that the Qout and the Pin are given. Then we have
(
k1 l dPdtout − Qin = −Qout
ρl dQin
ρlKr
A0 dt + A2 Qin + Pout = Pin
(3.51)
0
This system represents a lumped parameters description of the blood flow in the compliant
cylindrical vessel and involves the mean values of the flow rate and the pressure over the domain, as well as the upstream and downstream flow rate and pressure values. This model can
be 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 analog
of the system. Therefor we start with an electrical circuit depicted in figure 3.4. By using the
standard formulas for an electrical circuit, we can calculate the voltage differences around the
inductor and the resistor.
• The voltage drop over de inductor equals:
dI
;
dt
(3.52)
VR = IR;
(3.53)
VL = L
• The voltage drop over de resistor equals:
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 = V L + V R = L
dI
+ IR.
dt
(3.54)
On the same way we can calculate the charge that the capacitor will store as a function of the
voltage difference between the two plates
•
Iin − Iout = C
dVC
.
dt
By placing these two equations (3.54) and (3.55) in an electrical system
dVC
C dt = Iin − Iout
VT = L dI
dt + iR
(3.55)
(3.56)
and comparing this system with our two final systems (3.25) and (3.50), then it can be seen
that 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)
rl
representing the resistance induced to the flow by the blood
• Further, we set R = ρK
A20
viscosity. When we use a parabolic velocity, we have
R=
53
8µl
;
πr04
(3.59)
• We set
ρl
(3.60)
A0
L represents the inertial term in the momentum conservation law and will be called the
inductance of the flow;
L=
• Finally, we set
3πr03 l
(3.61)
2Eh
C represents the coefficient of the mass storage term in the mass conservation law, due to
the capacitance of the vessel.
C=
Using the resistance, inductance and capacitance in the systems (3.50) and (3.25) we have the
following hydraulic system:
dPin
C dt + Qout = Qin
(3.62)
L dQdtout + RQout + Pout = Pin
3.5
Simplification of the model
In the creation of the final model, nothing is said about the small variation in velocity. Assuming
x
that the steady flow is fully developed, we can say that the term du
dt = 0. When applying this
assumption to the momentum equation the first term can be neglected. This has as result that
in the final model there is no inertial term
dPin
C dt + Qout = Qin
(3.63)
out
Qout = Pin −P
R
However, significant changes in velocity will occur around the heart valves and the flow probably
cannot be assumed steady during the course of a heart beat. In the next chapter we will figure
out 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 a
pump, so we need an active compartment. For the active compartment, we still make use of
the momentum equation of system (3.62) or system (3.63), but the continuity equation will be
replaced by an active pressure relation. We will use a Voigt viscoelasticity model to find an
active pressure or pump relation.
The elastance varies during the cardiac cycle as a consequence of the contractile activity of
the ventricle. At diastole when the muscle fibers are relaxed, the ventricle fills according to an
exponential PV function (the EDPVR graph), which reflects the elasticity both of the relaxed
muscle 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 measure
this point with the linearized function of Pedpvr .
Ped = P0 λ(V − V0 ) = Eed (V − V0 )
54
(3.65)
The end-systolic point will be measured by a linear PV relation (ESPVR graph), where the
slope (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 endsystolic relationship is governed by a pulsatating activation function (e(t)), called a cardiac
driver function.
P (V, t) = e(t)Pes (V ) + (1 − e(t))Ped (V ) 0 ≤ e(t) ≤ 1
P (V, t) = e(t)Ees (V − V0 ) + (1 − e(t))Eed (V − V0 )
(3.67)
(3.68)
The profile of the cardiac driver function represents the variance of elastance between minimum
and maximum values over a single heart beat. A cardiac driver function value one means
elastance is defined by the ESPVR and a value of zero uses the EDPVR to define elastance. In
literature there are different cardiac driver functions proposed. In Section 2.1 an exponential
cardiac driver function is used:
N
X
2
(3.69)
Ai e−B1 (t−Ci )
e(t) =
i=1
, which has the following shape:
profile of the cardiac driver function
1
0.9
0.8
0.7
e(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
Time t (sec)
0.5
0.6
0.7
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 following
shape:
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
55
(3.72)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 3.8: sin2 cardiac driver function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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 to
C(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)
= Qin − Qout
dt
(3.74)
By eliminating the volume the system which has to be solved for an active compartment equals
dCP
dt
= Qin − Qout
out
Qout = Pin −P
R
(3.75)
Comparing this system with (3.63) the system for a passive compartment without inertia, it
can be seen that the only difference is the time dependance of the capacitance in the active
compartment.
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 the
pressure, 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 the
valves we have the following relation:
0
if Pin ≤ Pout
(3.76)
Qout =
(Pin − Pout )/R if Pin > Pout
3.8
Compartment coupling
In section 3 we create a hydraulic model for a passive compartment and for an active compartment. 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 in
section 1.2.3. To solve the Windkessel circuit we use a PV method (section 1.2.2). In chapter 2
we 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 compartments. The two active compartments describe the right and left ventricle (subscript rv and
lv), two passive compartments decribe the lung circulation (subscript pa and pu) and two passive
compartments decribe the body circulation (subscript ao and vc). Further, all the compartments
are coupled by a resistor (subscripts pv, pul, mt, av, sys, tc). The figure below reflects such a
6 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:
(CP˙ rv ) = Qtc − Qpv
Ṗpa =
1
Cpa (Qpv − Qpul )
1
Cpu (Qpul − Qmt )
Ṗpu =
˙ lv ) = Qmt − Qav
(CP
1
Cao (Qav − Qsys )
Ṗvc = C1vc (Qsys − Qtc )
rv
Qtc = PvcR−P
if Pvc > Prv otherwise Qtc = 0
tc
Prv −Ppa
Qpv = Rpv if Prv > Ppa otherwise Qpv = 0
P −P
Qpul = paRpulpu
P −P
Qmt = puRmt lv if Ppu > Plv otherwise Qmt = 0
ao
if Plv > Pao otherwise Qav = 0
Qav = PlvR−P
av
Ṗao =
Qsys =
(3.77)
Pao −Pvc
Rsys
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 active
compartment for the left ventricle (subscript lv), one passive compartment for the aorta (subscript ao) and one passive compartment for the rest of the body circulation (subscript bc). The
compartments are coupled by a resistor (subscript av, sys, mt). See further the reflection below.
The 3 compartment model has the following system of equations:
˙ lv ) = Qmt − Qav
(CP
1
Cao (Qav − Qsys )
Ṗbc = C1bc (Qsys − Qmt )
ao
if Plv > Pao
Qav = PlvR−P
av
−Pbc
Qsys = Pao
Rsys
lv
if Pbc > Plv
Qmt = PbcR−P
mt
Ṗao =
otherwise Qav = 0
(3.78)
otherwise Qmt = 0
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 for
one compartment of the circulation system,
• One without inertia
dCPin
dt
+ Qout = Qin
out
Qout = Pin −P
R
58
(3.79)
Figure 3.11: A 3 compartment model
• and one with inertia
dCPin
+ Qout = Qin
dt
dQout
L dt + RQout − Pin
= −Pout
(3.80)
The difference between a passive and an active compartment lies in the time dependance of the
capacity. For the active pressure relation we propose three different cardiac driver functions.
•
e(t) =
N
X
2
Ai e−B1 (t−Ci )
(3.81)
i=1
•
•
e(t) = sin2 (
πT (t)
)
Tsys (t)
(3.82)
e(t) = sin(
πT (t)
)
Tsys
(3.83)
Finally, the valve regulation is described by
0
if Pin ≤ Pout
Qout =
(Pin − Pout )/R if Pin > Pout
With the system of equations we have built a 3 and a 6 compartment model.
59
(3.84)
60
Chapter 4
Numerical Model
In the previous chapter we derived a mathematical model for the human CVS. This model is
built with a number of compartments that are connected. In this section we are going to describe
how to solve the mathematical model for different single compartment models. Next, we look
to the influence of different initial values and we are going to do some tests to give answer on
the questions we stated in section 2.7.2.
• What is a good driver function?
• 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 compartment
model and describe how to solve these models.
Therefor, we start with a numerical method for solving a single compartment model. We
have three types of the single compartments, the passive compartment without inertia, the
passive compartment with inertia and the active compartment. For each single compartment
we will give a discretisation.
For the tests we use the following parameters. These are copied from [SMITH].
passive compartment
C
1/(98e6 )
R1
2.75e6
R2
170e6
L1
5e4
L2
3e5
P1
13
P3
5
active compartment
Ees
100e6
Eed
0.33e6
R1
6.1e6
R2
2.75e6
P1
80
P3
100
a
1
b
80
c
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:
dP2
dt
=
Q1 =
Q2 =
1
C (Q1
P1 −P2
R1
P2 −P3
R2
− Q2 )
(4.1)
We want to solve this system with a PV-method. Therefor we have to discretize the model and
hereafter solve at first the pressure and use next the pressure to calculate the flow. There are
different possibilities for the discretisation. It is possible to use the old and the new pressure
to calculate the new flow. By using the old pressure you discretisize with a Jacobi-like method
and by using the new pressure the discretisation is a Gauss-Seidel like method. We will do both
discretisations 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.
(n)
(n+1)
= P2
Q1
(n+1)
=
(n+1)
Q2
=
P2
+
(n)
(n)
δt
C (Q1
P1 −P2
R1
(n)
P2 −P3
R2
(n)
− Q2 )
(4.2)
This system can be placed in a matrix form:
y (n+1)



1
P2
= Ay (n) + B; y =  Q1  , A =  − R11
1
Q2
R2
δt
C
0
0



0
− δt
C
P1 
0 , B =  R
1
P3
0
R2
(4.3)
And solve the system by the following iterative process:
(n)
(n)
(n)
1. Pass the statevector [P2 , Q1 , Q2 ]T to the solver.
(n+1)
2. Calculate at the same time the pressure P2
(n+1)
and outflow Q2
.
4.1.2
(n+1)
in the compartment and the inflow Q1
Discretisation, Gauss-Seidel like method
In this case the discretized system reads:
= P2
(n+1)
=
P1 −P2
R1
=
(n+1)
P2
−P3
R2
Q1
(n+1)
Q2
(n)
(n)
δt
C (Q1
(n+1)
P2
+
(n+1)
(n)
− Q2 )
(4.4)
62
Rewritten in matrix form:
y (n+1)


1
P2
(n)



− R11
Q1
= Ay + B; y =
, A=
1
Q2
R2

− δt
C
δt
C
− Rδt
1C
δt
R2 C
δt
R1 C
− Rδt
2C


, B = 
0
P1
R1
P3
R2

 (4.5)
and solve the system with the following iterative process:
(n)
(n)
(n)
1. Pass the statevector [P2 , Q1 , Q2 ]T to the solver.
(n+1)
2. Calculate the pressure P2
(n+1)
3. Use the pressure P2
in the compartment.
(n+1)
to calculate the inflow Q1
(n+1)
and outflow Q2
.
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 section
we 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, we
substitute in system (4.2) for y (n) the Fourier component
cnk eiθk , with θk = 2πk/m where k = 0, . . . , m − 1.
We find
cn+1
eiθk = cnk eiθk A + B.
k
Since the vector B is a constant vector, it has no influence on the stability of the system and so
A is the amplification matrix. The eigenvalues of matrix A are
r
δt R1 + R2
1 1
1−4
λ1 = 0, λ2,3 = ∓
2 2
C R1 R2
For absolute stability it is required that |max(λ1 , λ2 , λ3 )| < 1. The eigenvalues λ2,3 can be
imaginary.
Real eigenvalues
1 1
| +
2 2
r
r
1−4
δt R1 + R2
|<1
C R1 R2
δt R1 + R2
<1
C R1 R2
δt R1 + R2
1<1−4
<9
C R1 R2
CR1 R2
< δt < 0
−2
R1 + R2
−3 <
1−4
63
Imaginairy eigenvalues
1 1
| +
2 2
0<
r
1−4
δt R1 + R2
|<1
C R1 R2
δt R1 + R2
1 1
− (1 − 4
)<1
4 4
C R1 R2
0<
δt R1 + R2
<1
C R1 R2
and so it can be seen that for
0 < δt <
CR1 R2
R1 + R 2
(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 table
4.1 and let the program run 90 iterationsteps. In figure 4.1 can be seen that the maximum time
step for absolute stability lies between δt = 0.0275 and δt = 0.028. Using the same parameters to
calculate the maximum time step for theoretical absolute stability then there can be a maximum
time step of δt = 0.0276. So practice conforms the theory.
14
dt=0.027
13.5
13
12.5
12
0
10
20
30
40
50
60
70
80
90
100
pressure
dt=0.0275
14
13.5
13
12.5
12
0
10
20
30
0
10
20
30
40
50
60
70
80
90
60
70
80
90
dt=0.028
15
14
13
12
11
40
50
number of iteration steps
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 substituting the Fourier component cnk eiθk in the system, the matrix A from equation (4.5) is the the
amplification matrix in the Von Neumann analysis. It can be seen that the eigenvalues λ are
+
λ1,2 = 0 λ3 = 1 − ( Rδt
1C
and that
0 ≤ δt < 2
δt
R2 C )
(4.7)
R1 R2 C
R1 + R 2
(4.8)
When we implement this passive compartment with a Gauss Seidel like discretisation into matlab, use the parameters given in table (4.1) and let it run for 90 iteration steps. than it can seen
in figure 4.2 that the practical maximum time step for absolute stability lies between δt = 0.055
and δt = 0.056. The theoretical maximum time step for absolute stability equals δt = 0.0552.
So practice conforms theory.
14
dt=0.054
13.5
13
12.5
12
0
10
20
30
40
50
60
70
80
90
100
pressure
dt=0.055
14
13.5
13
12.5
12
0
10
20
30
0
10
20
30
40
50
60
70
80
90
60
70
80
90
dt=0.056
30
20
10
0
40
50
number of iteration steps
Figure 4.2: The convergence for different time steps with the Gauss Seidel like method. In this
figure only the 25t h heartbeat is depicted.
With the Von Neumann analysis we can theoretically as well as practically conclude that passing
the P (n+1) to the solver for calculating the flow is a good choice, since there will be absolute
stability with a time step which can be up to 2 times higher. For the rest of the calculations we
will 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 on
both sections. By now we assume that the inflow is been influenced by inertia. So the following
system has to be solved:
dP2
1
dt = C (Q1 − Q2 )
1
L1 dQ
dt = P1 − P2 −
3
Q2 = P2R−P
2
R1 Q 1
(4.9)
To solve this system, at first we discretize to the time,
(n+1)
= P2
(n)
(n+1)
=
P2
Q1
(n+1)
Q2
=
(n)
(n)
δt
C (Q1 − Q2 )
(n+1)
(n)
δt
1
) + (1 − δtR
L1 (P1 − P2
L1 )Q1
+
(4.10)
(n+1)
P2
−P3
R2
and next rewrite the numerical system in matrix notation:
y (t+1) = Ay (t) + B;



1
P2
 δt


y=
Q1
,
A =  − L1
1
Q2
R
2
δt
C
− (δt)
L1 C + 1 −
δt
R2 C
2
δtR1
L1
− δt
C
(δt)2
L1 C
− Rδt
2C
and solve the system by the following iterative process:


0

δt
 , B =  L1 P1 
P3
−R
2

(4.11)
1. Pass the statevector [P2 , Q1 , Q2 ]T to the solver.
(n+1)
2. Calculate the pressure P2
(n+1)
3. Use the pressure P2
4.3
in the compartment.
(n+1)
(n+1)
to calculate the inflow Q1
and outflow Q2
.
An active compartment
The system we have to solve equals
dC(t)P2
= Q1
dt
P1 −P2
Q1 = R1
3
Q2 = P2R−P
2
− Q2
if P1 > P2 else 0
(4.12)
if P2 > P3 else 0
Again, discretize with respect to the time,
(n+1)
=
(CP2 )(n) +
C (n+1)
(n+1)
=
P1 −P2
R1
if P1 > P2
=
(n+1)
P2
−P3
R2
if P2
P2
Q1
(n+1)
Q2
+
(n)
δt
(Q1
C (n+1)
(n+1)
(n+1)
(n+1)
66
(n)
− Q2 )
else 0
> P3 else 0
(4.13)
and rewrite this in matrix notation:
y (n+1) = (I − D)−1 Ay (n) + B;


P2
y =  Q1  ,
Q2
C (n)
C (n+1)

δt
C (n+1)


C (n)
δt
A =  − R1 C (n+1) if P1 > P (n+1) else 0 − R1 C (n+1)

δt
C (n)
if P (n+1) > P3 else 0
R2 C (n+1)
R2 C (n+1)


0
 P1

if P1 > P2 else 0 
B=
 R1

P3
if
P
>
P
else
0
2
3
R2

δt
− C (n+1)


,

δt
R1 C (n+1)
− R Cδt(n+1)
2
(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. Calculate the pressure P2
(n+1)
3. Use the pressure P2
in the compartment.
(n+1)
to calculate the inflow Q1
(n+1)
and outflow Q2
.
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
X
2
Ai e−Bi (t−Ci )
(4.15)
i=1
with N = 1, A = 1, B = 0.8 and C = 0.27.
2.
πT (t)
)
Tsys
(4.16)
πT (t)
)
Tsys
(4.17)
e(t) = sin2 (
with Tsys = 0.5.
3.
e(t) = sin(
with Tsys = 0.5.
Out of these three models we choose the best model. The results plotted in figure 4.3 do we
compare with the Wiggers diagram in section 1.2.4. We conclude that the exponentional driver
function gives the most realistic results. The form of the graph of the left ventricle pressure has
the best comparison and the outflow has the most reasonable strength.
67
2
2
driver
function
exp(−b(t−c) )
1
1
0.5
0.5
0.5
pressure
0
0.5
1
0
0
0.5
0
1
150
150
150
100
100
100
50
50
50
0
1.5
inflow
sin(pi t/0.5)
1
0
−5
0
x 10
0.5
1
0
1.5
−5
0
x 10
0.5
1
0
1.5
1
1
1
0.5
0.5
0.5
0
2
outflow
sin (pi t/0.5)
0
−5
x 10
0.5
1
0
2
0
−5
x 10
0.5
1
0
2
0
0
0
−2
−2
−2
−4
0
0.5
1
−4
0
0.5
time
1
−4
0
0.5
1
−5
0
x 10
0.5
1
0
−5
x 10
0.5
1
0
0.5
1
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 proposed
in 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 the
pressure. As can be seen, for all initial pressures it converge to the same pressure, namely
12.8726mmHg. Theoretically, this value can be determined by solving the system of equations
analytically.

dP
1

 dt2 = C (Qin − Qout )
P1 −P2
Qin = R1

 Q = P2 −P3
out
R2
dP2
1 P1 − P2 P2 − P3
= (
−
)
dt
C
R1
R2
R1 + R 2
P1 R2 + P3 R1
dP2
=−
P2 +
dt
CR1 R2
CR1 R2
68
different initial conditions
60
40
pressure
20
0
5
0
0.02
−6
x 10
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
−7
x 10
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.04
0.06
0.08
0.1
time
0.12
0.14
0.16
0.18
0.2
inflow
0
−5
−10
outflow
−15
3
2
1
0
−1
0.02
Figure 4.4: Starting with different initial conditions has no influence on the final results
R +R1
t
1 R2
2
− CR
P2 (t) = P0 e
+
R2 P1 + R1 P3
R2 + R 1
In the passive compartment the initial pressure P0 = 0 and the analytical solution equals:
P (t) =
R2 P1 + R1 P3
R2 + R 1
Substituting the simulation parameters yields limit value 12.8726mmHg. The same value as to
which 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. After
a 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. In
this model, we use passive compartments without inertia and use the exponential cardiac driver
69
13
inertia on
inertia off
pressure
12.5
12
4
0
0.02
−7
x 10
0.04
inflow
3
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.08
0.1
0.12
0.14
0.16
0.18
0.2
inertia on
2
1
0
outflow
0.06
4.8
inertia off
0
0.02
−8
x 10
0.04
inertia on
4.6
4.4
4.2
4
inertia off
0
0.02
0.04
0.06
time
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 a
coupled 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:
(n)
(CPlv )(n) +
δt
+ C (n+1)
(Qmt
C (n+1)
(n)
(n)
(n)
Pao + δt
C (Qav − Qsys )
(n)
(n)
(n)
Pvc + δt
C (Qsys − Qmt )
(n+1)
=
(n+1)
=
(n+1)
=
(n+1)
=
Pvc
(n+1)
=
Plv
Plv
Pao
Pvc
Qmt
Qav
(n+1)
Qsys
(n+1)
(n+1)
−Plv
Rmt
(n+1)
=
(n+1)
> Plv
(n+1)
> Pao
if Pvc
(n+1)
−Pao
Rav
(n)
− Qav )
if Plv
(n+1)
=0
(n+1)
=0
(n+1)
otherwise Qmt
(n+1)
otherwise Qav
(n+1)
(n+1)
Pao
−Pvc
Rsys
and solve this numerical model with the following iterative process:
1. Pass the statevector [Plv , Pao , Pvc , Qmt , Qav , Qsys ]T to the solver.
(n+1)
2. Calculate the pressure Plv
(n+1)
, Pao
(n+1)
, Pvc
70
in the compartments.
(4.18)
(n+1)
(n+1)
, Pao
3. Use the pressure Plv
tween the compartments.
4.6
(n+1)
, Pvc
(n+1)
to calculate the flow Qmt
(n+1)
, Qav
(n+1)
, Qsys
be-
A 6 compartment model
In this section the steps necessary for solving the model proposed in section 3.8.1 numerically
are outlined. Start with the time discretisation:
(n)
(CPrv )(n) +
δt
+ C (n+1)
(Qtc
C (n+1)
(n)
(n)
(n)
Ppa + δt
C (Qpv − Qpul )
(n)
(n)
(n)
Ppu + δt
C (Qpul − Qmt )
(n+1)
=
(n+1)
=
(n+1)
=
(n+1)
=
Pao
(n+1)
=
(n+1)
Pvc
=
(n+1)
=
Pvc
(n+1)
=
Prv
(n+1)
=
Ppa
(n+1)
=
Ppu
Prv
Ppa
Ppu
Plv
Qtc
Qpv
Qpul
Qmt
(n+1)
=
(n+1)
=
Qav
Qsys
(n)
(CPlv )(n) +
δt
+ C (n+1)
(Qmt
C (n+1)
(n)
(n)
(n)
Pao + δt
C (Qav − Qsys )
(n)
(n)
(n)
Pvc + δt
C (Qsys − Qtc )
(n+1)
(n+1)
(n+1)
(n+1)
−Prv
Rtc
−Ppa
Rpv
(n+1)
(n+1)
(n+1)
(n)
− Qav )
(n+1)
> Prv
(n+1)
> Ppa
(n+1)
> Plv
(n+1)
> Pao
if Pvc
if Prv
(n+1)
(n)
− Qpv )
(n+1)
otherwise Qtc
(n+1)
=0
(n+1)
otherwise Qpv
(n+1)
=0
(n+1)
otherwise Qmt
(n+1)
=0
(n+1)
otherwise Qav
(n+1)
=0
(4.19)
−Ppu
Rpul
−Plv
Rmt
if Ppu
(n+1)
(n+1)
Plv
−Pao
Rav
(n+1)
(n+1)
Pao
−Pvc
Rsys
if Plv
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.
(n+1)
2. Calculate the pressure Prv
(n+1)
(n+1)
, Ppa
(n+1)
(n+1)
, Ppu
(n+1)
(n+1)
, Plv
(n+1)
(n+1)
, Pao
(n+1)
(n+1)
, Pvc
in the compartments.
(n+1)
, Pao , Pvc
to calculate the flows
3. Use the pressures Prv
, Ppa , Ppu , Plv
(n+1)
(n+1)
(n+1)
(n+1)
(n+1)
(n+1)
, Qpv , Qpul , Qmt , Qav , Qsys between the compartments.
Qtc
4.6.1
Summary
We started this section with different numerical models for a 1 compartment model. With these
models we did different tests. We showed that the system is stable and that it is better to use
a Gauss-Seidel like method for the time discretisation since it permits a timestep twice as large
as a Jacobi like method.
Further we chose
N
X
2
(4.20)
Ai e−Bi (t−Ci )
e(t) =
i=1
with N = 1, A = 1, B = 0.8 and C = 0.27 as best driver function, by using the active 1
compartment model as test model.
71
Finally, we used the passive compartment model to investigate the effect of including an
inertial term. We saw that inertia caused a retardation in the flow, but has no further influence
on 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 6
compartment 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 how
to solve the system for a specific number of compartments. In this chapter we will solve the
model for the human CVS. Since we did not find parameters for the three compartment model
in literature we have to guess these parameters. We are going to do this by solving the model
for a 6 compartment model using of the parameters postponed by [SMITH]. The results will be
compared with the results of [SMITH] and with the Wigger’s diagram. After satisfying results
we will use this results to guess reasonable parameters for the three compartment model. Besides
testing the results with others, we independently programmed the numerical model, in Matlab
and Simulink, with the same results.
5.1
5.1.1
Verification of the models
A six compartment model
To solve the system of equations for the 6 compartment model 4.6, we use the parameters given
in table (5.1). Solving the system gives the results depicted in figure (5.1): At first, we will
compare our results with that of [SMITH]. Looking at the graphs we see the same form in the
diagrams 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 and
ventricular interaction in his model, this probably explains the difference in the results. This
far, we must be satisfied with our results.
Heart
Eeslv 100e6
Eedlv 0.33e6
Eesrv
54e6
Eedrv 0.23e6
a
1
b
0.80
c
0.27
Resistance
Rav 2.75e6
Rsys 170e6
Rvc
1e6
Rpv
1e6
Rpul 9.4e6
Rmt 6.1e6
Capacity
Cao 1/98e6
Cvc 1/1.3e6
Cpa 1/72e6
Cpu 1/1.9e6
Table 5.1: Parameters in the six compartment model
73
Right Ventricle [rv]
80
130
70
Volume [Vrv] (ml)
Volume [Vlv] (ml)
Left Ventricle [lv]
140
120
SV=38
110
100
90
60
SV=38
50
40
0
0.25
0.5
0.75
1
30
1.25
0
0.25
0.5
0.75
1
1.25
100
ao
80
60
lv
40
pu
20
0
0
0.25
0.5 0.75
1
Time [t] (secs)
1.25
Pressure [Prv] (kPa)
Pressure [Plv] (kPa)
120
20
pa
10
vc
0
0
0.25
0.5 0.75
1
Time [t] (secs)
rv
1.25
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 for
the pressure with a scale of kP a 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 the
parameters.
5.1.2
A three compartmentmodel
We are satisfied with the form of the graphs, so we proceed with the validation of a three comparment model. As initial guess, we start with the parameters we use in the 6 compartment
model and by reasoning we want to find reasonable parameters. We have reasonable parameters
when 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 are
satisfied 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 interaction, 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. The
pressure in the left ventricle is shifting between the Pes and the Ped . So the stroke volume can
be calculated by SV = Ves,max − Ved,min where Ves,max is the volume of blood in the end-systolic
phase and Ved,min the volume of blood in the end-diastolic phase. Now, since we have chosen
that the Eeslv must be smaller (for a lower Pes and so a lower Ves,max ) and we are satisfied with
the 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. The
volume in the compartment is calculated by the capacity times the pressure. Since the total
volume will not be much smaller with the choice of the parameters for the left ventricle and
the pressure must be much lower, the aorta and the pulmonary vein must have less capacity, so
Cao < 1/98e6 and Cpu < 1/1.9e6.
For the resistance parameters we have to look at the flow between the compartments. In
the Wiggers’ diagram of the 6 compartment model, the outflow is somewhat too strong, so
Rav > 2.75e6. For the inflow the same can be said, so Rmt > 6.1e6. The last resistance
reflects the whole systemic part. In comparison with the 6 compartment model, it contains the
75
Pressure
[Plv] (kPa)
Wiggers’ diagram
Flow
[Q] (*10−3m3/s)
ao
filling
80
mt
40
0
6
−6
0
x 10
0.2
4
0.4
0.6
outflow
lv
0.8
1
1.2
1.4
inflow
2
0
Volume
[Vlv] (*10−3m3)
ejection
120
1.4
−6
0
x 10
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.2
1
0.8
Pressure
[Prv] (kPa)
30
20
pa
vc
10
rv
0
0
0.2
contraction
0.4
0.6
0.8
time [t] (secs)
1
1.2
1.4
relaxation
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 to
choose this resistance Rsys > 170e6.
Now that we know how to search for the correct parameters we find after comparing with
a 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 the
inflow and outflow are correct. Further, we see that the ejection time is too small in comparison
with the real Wiggers’ diagram, but this has no influence on the pressure in the ventricle. We
conclude 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 happen
with the solutions when the parameters will reflect an ill person? That is what we are looking for
in 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
Eeslv
85e6
Eedlv 0.15e6
a
1
b
0.80
c
0.27
Resistance
Rav 6.75e6
Rsys 225e6
Rmt 15.2e6
Capacity
Cao 1/175e6
Cpu 1/30e6
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 ejecting
problems, 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 myocardial
disorder. This kind of disorder limits the ability of the heart to relax so blood can enter the
ventricle during diastole. It can be characterized by an increase in ventricle filling pressure, a
decrease 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 by
increasing the ventricle elastance at end-diastole (Eedlv).
We simulate a diastolic dysfunction by increasing Eedlv with a factor of 10. The results are
shown in figure 5.5. It reflects exactly the characteristics of a diastolic dysfunction. Comparing
the result with a graph from literature, figure (5.6), we see that there is a good comparison. The
only difference is that the figure from literature is a schematic illustration specificially focussed
on 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 to
lack of oxygen. The main impact of myocardial infarction is a drop in ventricle contractility
because the weakened heart is no longer able to eject an adequate amount of blood. Decreased
contractility is simulated in the minimal model by decreasing the Eeslv . Further, systolic dysfunction can be characterized by increasing ventricle preload, a rise in ventricle volume, a drop
in stroke volume and decreased systemic pressure, [KUPA].
We simulate a systolic dysfunction by halving the ventricle contractility, see figure 5.7. This
simulation has as result that there is more volume and less pressure in the left ventricle. Figure
5.8 from literature shows the same.
Valvular disorder caused by valvular stenosis
Valvular stenosis occurs when a heart valve doesnot open properly, causing a much higher
resistance to blood flow passing through the valve decreasing the flow rate. Characteristic consequences are an increased left ventricle systolic pressure and decreased average aortic pressure.
77
Wiggers’ diagram
pressure (kPa)
20
ejection
filling
16
left ventricle pressure
12
aortic pressure
8
4
0
0
left atrium pressure
200
400
600
800
1000
1200
1400
Flow (ml/s)
outflow
400
inflow
200
0
0
200
400
600
800
1000
1200
1400
volume (ml)
250
225
200
left ventricle volume
175
150
0
200
400
600
800
1000
1200
1400
time (ms)
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 the
maximum left ventricle pressure and the maximum aortic pressure (max Plv -max Pao ). This
increased difference is caused by the larger pressure drop across the aortic valve as a result of
the 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 maximum
left 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 volume
and reduced aortic diastolic pressure, [BRWD]. Although ventricle stroke volume increases,
cardiac output decreases, as much of the blood pumped into the aorta during diastole can
return to the ventricle during systole. In literature, figure 5.10, schematically illustrated the
effect of valvular insufficiency on ventricle and aortic pressure.
In our model we simulate the valvular insuffuciency by increasing the aortic valve resistance
by a factor of 20 when the valve would normally close. The results are plotted in the figure
(5.11)
78
PV diagram of the left ventricle
16
diastolic dysfunction
normal
14
12
pressure (kPa)
10
8
6
4
2
0
160
170
180
190
200
210
volume (ml)
160
170
180
190
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 this
section we will simulate one kind of a shock. Another kind of shock which can occur we have
already seen with the systolic dysfunction. In this section we will simulate a shock from which
the patient will not die. This is not possible to simulate, because this kind of death is caused
by 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 by
taking a cardiac driver function e(t) = 0. This means that the blood will flow for a small amount
of 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 case
blood can enter the heart, but can never leave the heart, since the valves are closed. This all
can be seen in figure 5.12.
79
Figure 5.6: Effect of diastolic dysfunction causing an increase in ventricle elastance on a PV
diagram of the left ventricle [BRWD]
5.3
Summary
In this section we solved the numerical model of the human CVS. We started with solving the
6 compartment model using the parameters proposed in section 2.1. The graphs have a good
trend, but have a bad magnitude.
We decided to continue with a three compartment model and guess the parameters. As can
be 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 recognize
diastolic dysfunction, systolic dysfunction, valvular stenosis, valvular insufficiency and a heart
block. Probably the 3 compartment model can recognize more heart failures or other failures in
the CVS, but we did not test these.
80
PV diagram of the left ventricle
16
systolic dysfunction
normal
14
12
pressure (kPa)
10
8
6
4
2
0
170
180
190
200
210
220
volume (ml)
230
240
250
260
Figure 5.7: Simulating a systolic disfunction
Figure 5.8: Effect of systolic dysfunction causing a drop in cintractility on a PV diagram of the
left ventricle [BRWD]
81
normal
aortic stenosis
16
18
14
16
max Plv − max Pao
Pao
14
12
12
left ventricle
pressure (kPa)
10
10
Pao
8
8
Plv
6
6
Plv
4
4
2
0
Pmt
0
0.2
0.4
time (sec)
0.6
2
0
0.8
Pmt
0
0.2
0.4
time (sec)
0.6
0.8
Figure 5.9: Simulating aortic stenosis
Figure 5.10: 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 ventricle pressure diagram
caused by valvular insufficiency
82
normal
valvular insufficiency
16
16
14
14
amp Pao
left ventricle
pressure (kPa)
Pao
12
12
10
10
Pao
8
8
Plv
6
6
4
4
Plv
2
0
2
Pmt
0
0.2
0.4
time (sec)
0.6
Pmt
0
0.8
0
0.2
0.4
time (sec)
Figure 5.11: Simulating valvular insufficiency
83
0.6
0.8
Heart block
left ventricle
pressure (kPa)
left ventricle
volume (ml)
300
200
aortic
pressure (kPa)
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
10
0
left atrium
pressure (kPa)
0
20
20
10
0
2
1
0
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. After
we found some articles which describe a human CVS model, we had some important questions
which have been answered in this thesis:
• Are we going to use a 0D or 1D model?
• What number of compartments do we need?
• What is a good driver function?
• 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?
• Do we need inertia in our model, if so, do we always need it?
By hand of the literature study, we concluded that we want to use a 0D model, since the 0D
model has the advantage of being a model which can be solved easily and the results are in good
agreement 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 are
going to build a 6 compartment model like that of [SMITH] and after this build a 3 compartment
model according to our own ideas.
In the 0D model a system of equations describe the pressure (P ) in a compartment and
the flow (Q) at the interfaces of the compartment with the rest of the system. All compartments are coupled like a Windkessel circuit. The system for one compartment equals:
dCPin
dt + Qout = Qin
(6.1)
dQout
L dt + RQout − Pin = −Pout
85
where R is the resistance and L the inductance of the flow and C the capacity of a compartment. This system is including inertia. The system without inertia is the same, but now with
the 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:
• What is a good driver function?
• Do we need inertia in our model, if so, do we always need it?
From literature we have three different cardiac driver functions. We concluded that all three
cardiac driver functions have good results. However, the sin2 and the sin driver function have
to be tuned, before they can be used. The exponential driver function
e(t) =
N
X
2
Ai e−B1 (t−Ci )
(6.2)
i=1
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 compartments. After different tests, we can answer the last questions
• What number of compartments do we need?
• 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 be
included?
In the tests it has been shown that the 3 compartment model can reflect a human cardiovascular
system. Further, the lungcirculation is a part of the CVS so must be included. However, we
describe it with a resistance. The last question is not explicitly answered in the tests, but since
a compartment describes a part of the system, an extra compartment describing only a small
part 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 future
improvements can be made to the model and our model can be used to investigate a detailed
small part of the human CVS.
7.1
Possible improvements
The most obvious improvement is the usage of real parameters for the resistances R and capacities C. How to measure these parameters we do not know yet, this is a subject for the medical
researchers.
We can remark that if the measurement of the parameters in the passive compartments is
difficult, the equation of mass can be rewritten by using the volume in a compartment:
P2 (V (t), t) =
1
(V (t))
C
(7.1)
There are some improvements which can be made to have more detailed results, with the
drawback that the model becomes more complicated. Using passive compartments with inertia
is such an improvement. The momentum equation for the passive compartments changes in:
L
dQout
+ RQout − Pin = −Pout
dt
(7.2)
Another improvement is the choice of another driver function. As can be seen in the results
of the left ventricle pressure in comparison with the Wiggers’ diagram, the relaxation and contraction time are too large. Further, the ejection time is too small. In the search for another
driver 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 atria
and the left and right ventricle. A possible option how to include this in the model is given in
section 2.1.
A third improvement of the heart function is that the heart will stop pumping as result of a
physiological 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 compartments, 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 investigate a small part of the human CVS. To investigate a small part of the human CVS, an extra
compartment has to be included. A specific mathematical model of the small part of the human
CVS can be introduced.
An example of an application is the research to a detailed carotid artery. In this application
at first a fourth compartment for the carotid artery will be included. In the iteration first the
same system of equations will be used. After each iteration step, the calculated input and output variables of the compartment will be used as input variables for a 3D Comflo model of the
carotid artery. After one iteration step in Comflo, Comflo passes its output variables as input
variables back to the compartment model. The compartment model uses these variables in the
next 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 the
central 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 stimulated 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 exchange
with the organs.
capillaries A system of small arteries in which it is possible to exchange nutrients with the
organs.
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 circulated 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 internal carotid arteries.
central neural processing system Coordinates the activity of the muscle, monitors the organs, 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 pressure
in 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 from
the lungs.
lymph The almost colourless fluid that bathes body tissues and os found in the lymphatic
vessles 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 and
the 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 the
evironment.
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 lower
body.
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 force
since 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 in
pressure, 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 of
the heart.
91
92
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