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Verslagnummer BMTE02.16
Extension of a finite element
model of left ventricular mechanics
with a right ventricle.
P.M.J van den Berg
458919
Eindhoven, Mei 2002
Extension of a finite element model of left ventricular mechanics with a right ventricle
Abstract
A finite element model of the mechanics of the left ventricle (LV) and right ventricle (RV) of the
heart is made. The current model of the left ventricle of Kerckhoffs is extended with a right
ventricle. Only the gross features of the geometry of the RV are taken into account, just as the LV.
The new mesh is made in two steps: first a disc has been generated and next that disc is
transformed to both ventricles. The volumes of the cavities and walls of both ventricles are
estimated with the article of Arts [2]. In this paper, cavity and wall volumes in the heart are
estimated using the assumption that the cardiac myofibers strive for an optimum amount of work
and shortening during a cardiac cycle. In the new model the LV cavity volume is 40 ml, RV cavity
volume is 53 ml, LV wall volume is 105 ml and RV wall volume is 32 ml.
To test if the new model is formulated correctly, the filling phase of the cardiac cycle was
simulated. A uniform pressure is prescribed at the endocardial sides of the LV and RV, while the
epicardial pressure is assumed to be zero. The prescribed end-diastolic pressure of the LV is 1.0
kPa, while the prescribed end-diastolic pressure of the RV is 0.2 kPa. Myocardial tissue was
considered isotropic and virtually incompressible.
From this simulation the end-diastolic volume of the LV was found to be 78 ml, while the enddiastolic volume of the RV is 90 ml. So the volumes of both ventricles increases with increasing
prescribed pressure.
End-diastolic principal strains were calculated on the epicardial surface of the RV for pressures of
0.5 kPa, 0.7 kPa and 0.9 kPa. The calculated values are up to ten times higher than values
measured by Waldman [4].
It is concluded that the existing model of the left ventricle is successfully extended with a right
ventricle. The global hemodynamics and local mechanics obtained in a first simulation of the
filling phase are promising. To obtain more realistic results probably RV material properties or
wall thickness have to be adjusted.
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Table of Contents
ABSTRACT........................................................................................................................................................... 2
CHAPTER 1: INTRODUCTION ........................................................................................................................ 4
CHAPTER 2: MATERIAL AND METHODS ................................................................................................... 5
2.1 ANATOMY OF THE HEART .............................................................................................................................. 5
2.2 GEOMETRY OF THE EXISTING MODEL OF THE LEFT VENTRICLE ...................................................................... 5
2.3 GEOMETRY OF THE NEW MODEL OF THE LEFT VENTRICLE AND RIGHT VENTRICLE ......................................... 6
2.4 PARAMETER VALUES ..................................................................................................................................... 7
2.5 MESH GENERATION ........................................................................................................................................ 8
2.5.1 Disc generation .................................................................................................................................. 8
2.5.2 Transformation of the disc to the right- and left ventricle ................................................................... 9
2.6 SIMULATION OF THE FILLING PHASE .............................................................................................................. 9
2.6.1 Passive material behaviour ...............................................................................................................10
2.6.2 Prescribed pressures ..........................................................................................................................10
2.6.3 Boundary conditions .........................................................................................................................11
CHAPTER 3: RESULTS.................................................................................................................................... 12
3.1 CALCULATION OF THE VOLUMES OF THE CAVITIES AND WALLS ................................................................... 12
3.2 GEOMETRY OF THE MODEL OF THE RIGHT- AND LEFT VENTRICLE ................................................................ 12
3.3 FILLING PHASE ............................................................................................................................................. 13
3.3.1 Global hemodynamics....................................................................................................................... 13
3.3.2 Local mechanics................................................................................................................................ 14
3.3.3 Comparison with experimental data ................................................................................................. 14
CHAPTER 4: DISCUSSION ............................................................................................................................. 16
CHAPTER 5: CONCLUSIONS......................................................................................................................... 17
CHAPTER 6: FURTHER RESEARCH ........................................................................................................... 18
ACKNOWLEDGEMENTS................................................................................................................................ 19
REFERENCES.................................................................................................................................................... 19
APPENDIX A ...................................................................................................................................................... 20
APPENDIX B ...................................................................................................................................................... 27
Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 1: Introduction
The heart is an organ that exists of four chambers: two ventricles and two atria. The right ventricle
(RV) pumps the blood into pulmonary circulation, while the left ventricle (LV) pumps the blood
into the systemic circulation.
The global function of the heart finds its origin in the co-operative action of the contracting
muscle fibres in the cardiac wall. To study the relation between mechanics at the cardiac and
muscle fibre level, mathematical models have been developed. These models can be used for
research about the basic functions of the heart. Also normal and pathological conditions can be
simulated with these models. They can also be used as a diagnostic tool. Clinical measurements,
such as MR tagging data, can be combined with the model to get more information about the
underlying cause of an abnormal function of the heart.
An example of a mathematical model of the mechanics of the heart is the model of Kerckhoffs.
This model calculates stresses and strains in the wall of the LV when a cavity pressure is applied
and muscle fibers in the wall contract. A limitation of this model is that only the LV is modelled.
The interaction between the LV and RV cannot be taken into account in the current model. Pacing
of the heart can be done on the LV or RV. In the current model pacing is limited to the epicard of
the LV only.
The aim of this study is to extend the existing model of the LV with the RV. Only the gross
features of the geometry of the RV will take into account, just as in the model of the LV.
First the anatomy of the right and LV will be described and a description of the geometry of the
existing model of the LV will follow. Then the geometry of the new model will be discussed
including the choice of the parameter values. The mesh generation of the new model will follow
and thereafter a simulation of the mechanics of the heart during the filling phase will be
discussed. The results of the mesh generation and the simulation will be presented and a
discussion and conclusions of this study will follow.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 2: Material and Methods
2.1 Anatomy of the heart
The heart exists of two separate pumps: a right part that pumps the blood through the lungs and a
left part that pumps the blood to the rest of the body. Each of these parts is in turn a two-chamber
pump composed of an atrium and a ventricle. The atrium functions as a weak primer pump for
the ventricle, helping to move the blood into the ventricle. The ventricle supplies the main force
that propels the blood through either the pulmonary or the systemic circulation.
In figure 2.1 a longitudinal and transversal section of the anatomy of the heart are shown. It can
be seen from figure 2.1b that the wall of the left ventricle (LV) is much thicker than the wall of the
right ventricle (RV). This is a result of the fact that the LV must pump the same amount of blood
against a three time higher pressure than the RV. The wall between the RV and LV is called the
septum.
Figure 2.1: Pictures of the anatomy of the heart. (a) Longitudinal section of the heart with the right ventricle RV
and the left ventricle LV. When the heart is cut across the white line, the transversal section as shown in (b) is
obtained.
The wall of the heart exists of cardiac muscle fibers. These fibers take care of the contracting of
the heart so blood can be pumped out of the chamber. These fibers have an orientation that varies
from endocardium (inside of the heart) to epicardium (outside of the heart).
2.2 Geometry of the existing model of the left ventricle
In the existing model of the LV, the endocardial and epicardial surfaces are approximated by
truncated confocal ellipsoids. The geometry is defined by the volume Vw, of the wall, the volume
Vlv,0 of the enclosed cavity, the focal length of the ellipsoids C and the height h above the equator
at which the ellipsoids are truncated (figure 2.2). The values for these parameters as used in the
model are listed in table 2.1 [1].
Table 2.1: Geometry parameter values for the model of the left ventricle [1].
Vw (ml)
140
Vlv,0 (ml)
40
C (mm)
43
h (mm)
24.8
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Extension of a finite element model of left ventricular mechanics with a right ventricle
z
ξ =ξen
base
ξ =ξep
h
O
C
y
φ
θ = 1/2π
θ = 3/4π
x
Vw
θ=π
a
b
Figure 2.2: (a) Geometry of the existing model of the left ventricle showing the focal length C, the truncation height
above the equator h, the volume of the wall Vw, origin O, Cartesian coordinates (x,y,z) and circumferential ellipsoid
coordinate φ; (b) cross-section through the LV showing global ellipsoid coordinates ξ and θ in the plane of constant
φ; dotted lines indicate levels of constant θ at intervals of π/12 [1].
The Cartesian coordinates (x, y, z) of the points in the wall are described using ellipsoidal
coordinates (ξ, θ, φ):
x = C sinh(ξ) sin(θ) cos(φ)
y = C sinh(ξ) sin(θ) sin(φ)
z = C cosh(ξ) cos(θ)
(2.1)
The circumferential ellipsoid coordinate φ is equal to arctan (y/x). Surfaces of constant φ represent
planes passing through the longitudinal axis. Surfaces of constant radial ellipsoid coordinate ξ
represent ellipsoids with focal length C, centred about the origin. In particular, the endocardial
and epicardial surfaces are surfaces of constant ξ=ξen and ξ=ξep respectively. Surfaces of constant
longitudinal ellipsoid coordinate θ represent hyperboloids. Planes of constant ξ, θ and φ intersect
mutually perpendicular.
2.3 Geometry of the new model of the left ventricle and right ventricle
The new model will have a right ventricle in addition to the left ventricle. The endocardial and
epicardial surfaces are approximated by truncated confocal ellipsoids. The geometry in the new
model is defined by the volume Vw, free of the LV free wall, the volume Vsep of the septum, the
volume Vw, rv of the RV wall, the volume Vcav, lv of the LV cavity, the volume Vcav, rv of the RV cavity
the focal length of the ellipsoids C, the height h above the equator at which the ellipsoids are
truncated, the length Rx1 of the middle of the LV to the endocardial surface of the RV wall, and the
length Rx2 of the middle of the LV to the epicardial surface of the RV wall (figure 2.3).
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Vw, rv
Vcav, rv
Rx2
Rx1
Vsep
Vw, free
O
Vcav, lv
Figure 2.3: Geometry of the right and left ventricle showing the focal length C, the truncation height above the
equator h, the volume of the free wall of the LV Vw,free, the volume of the septum Vsep, the volume of the wall of the
RV Vw,rv, the volume of the cavity of the LV Vcav, lv, the volume of the cavity of the RV Vcav, rv, origin O, the length of
O to the endocardial surface of the RV Rx1, the length of O to the epicardial surface of the RV Rx2 , Cartesian
coordinates (x, y, z) and circumferential ellipsoid coordinate φ.
The Cartesian coordinates (x, y, z) of the points in the wall of the LV and of the septum are
described using ellipsoidal coordinates (ξ, θ, φ), just as in the model of only the LV:
x = C sinh(ξ) sin(θ) cos(φ)
y = C sinh(ξ) sin(θ) sin(φ)
z = C cosh(ξ) cos(θ)
(2.2)
The points in the wall of the RV are described using ellipsoidal coordinates (ξ, θ, φ), Rx1 and Rx2:
x = {Rx1+((Rx2-Rx1)(ξ -ξen,rv)/(ξen,rv-ξep,rv))}sin(θ) cos(φ)
y = C sinh(ξ) sin(θ) sin(φ)
z = C cosh(ξ) cos(θ)
(2.3)
ξen,rv and ξep,rv are the values of ξ at the endocardial and epicardial surface of the right ventricle,
respectively. For the description of the circumferential ellipsoid coordinate φ, the radial ellipsoid
coordinate ξ and the longitudinal ellipsoid coordinate θ see section 2.2. The only difference is that
in the right ventricle planes with constant θ don't intersect the planes with constant ξ
perpendicularly.
2.4 Parameter values
Because almost no experimental data of the volumes of a human heart were found in the
literature, the volumes of the cavity and wall of the RV are approximated with the article of Arts
[2]. In this paper, cavity and wall volumes in the heart are estimated using the assumption that the
cardiac myofibers strive for an optimum amount of work and shortening during a cardiac cycle.
According to Arts [2] the volume of a cavity of the heart can be estimated by the following
equation:
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Vcav =
Vwall
3
Vcav ,ref
1 + 3
Vwall
3( e f −e f ,ref )
e
− 1
(2.4)
with Vcav the volume of the cavity [ml], Vwall the volume of the wall [ml], Vcav,ref the volume of some
reference state, ef the strain of the myofiber [-], ef,ref the strain of the myofiber in the same
reference state [-].
The amount of work per stroke Estroke [kJ] is given by [2]:
E stroke = p ejectVstroke = Vwall wstroke
(2.5)
With peject the ejection pressure [kPa], Vstroke the stroke volume [ml] and wstroke the mechanical
stroke work per unit of tissue volume [kJm-3].
To calculate the volumes of the RV and LV, the assumption is made that half of the stroke volume
of the LV is the result of the free wall and the other half is the result of the septum. Another
assumption is that the reference volume of the cavity (Vcav,ref) is de volume after ejection. In table
2.2 the values of the parameters are listed that will be used to calculate the volumes of the RV and
LV in the unloaded state.
Table 2.2: Values of the parameters to calculate the volumes of the right and left ventricle with the article of Arts [2].
wstroke [kJm-3]
5.0
e
3( e f − e f ,ref )
1.15
peject. lv [kpa]
peject. rv [kpa]
Vstroke [ml]
Vcav,ref/Vwall
16
3.2
36
0.3
When equations (2.4) and (2.5) and the values of the parameters in table 2.2 are used, the volumes
of the cavities and walls can be calculated. With these volumes the dimensions of the geometry of
the new model are determined. The values of C and h are the same as the existing model.
2.5 Mesh generation
The mesh of the RV and LV is generated by the finite element program SEPRAN. From
experience it has turned out that it is not possible to make the mesh at once. Therefore first a
mesh of a disc is generated and with transformation rules, the disc is transformed into the RV
and LV.
2.5.1 Disc generation
The disc that is generated is shown in figure 2.4. Figure 2.4a is a view of the top of the disc and
figure 2.4b a cross-sectional view. The disc is divided into an upper part (ACDH) and a lower part
(HDEG). The upper part will be transformed into the septum and subendocardial part of the LV
free wall (figure 2.4c). The lower part is again divided into 2 parts. The part IDEF will form the
subepicardial layer of the LV free wall, while the part HIFG will form the wall of the RV. Surface
HI is defined twice so that after transformation a cavity between the upper and lower part of the
disc can be made. This will be the cavity of the RV (figure 2.4c).
The total number of elements of the disc is 108. The elements are 27-noded hexahedral elements
(quadratic brick-elements), where the displacement-field is interpolated quadratically. The
program for generating the disc can be found in appendix A.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
1
6
2
3
7 8
A
B
C
H
I
D
G
5
4
a
F
E
b
c
Figure 2.4: Different views of the disc that is generated in the finite element program. (a) top view of the disc with
the 8 volumes; (b) cross sectional view of the disc; (c) cross-section of the disc after transformation to the right and
left ventricle.
2.5.2 Transformation of the disc to the right- and left ventricle
After the disc has been generated, it is transformed to the right and left ventricle. First, the
coordinates (xdisc, ydisc, zdisc) of points in the disc are transformed into ellipsoid coordinates (ξ,θ, φ)
according to:
ξ = zdisc
φ = arctan (ydisc/xdisc)
θ=π+
r
rmax
(2.6)
(θmin - π)
with rmax the radius of the disc. θmin and r are defined as:
θmin = arccos {h/(C cosh(ξ))}
r = √(x2 + y2)
(2.7)
(2.8)
with h the height above the equator at which the ellipsoids are truncated (h=24.8 mm).
Next the points are transformed into points in the cardiac geometry according to equations (2.2)
for the left ventricle (ACDH and IDEF) and equations (2.3) for the right ventricle (HIFG). In
appendix B the program (MATLAB) to transform the disc to the both ventricles can be found.
2.6 Simulation of the filling phase
To test if the new mesh is formulated correctly the first phase of the cardiac cycle, the (diastolic)
filling phase, is simulated. In that phase the ventricles are filling with blood so cavity pressures
and volumes increase. To simulate the filling phase, pressures are prescribed at the endocardia of
the RV and LV.
During filling, the mechanics of the cardiac tissue is governed by conservation of momentum
given by:
∇·σ=0
(2.9)
where σ represents the Cauchy stress in the tissue. Conservation of the moment of momentum is
equivalent to the condition that the Cauchy stress tensor σ is symmetric.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
In the model by Kerckhoffs, the myocardial tissue is assumed to consist of a connective tissue
matrix in which muscle fibers are embedded. Each constituent contributes to the total stress σ in
the tissue in the following way:
σ = σp + σaefef
(2.10)
With σp the passive stress as a result of deformation and σaefef the stress generated by the muscle
fibers, parallel to the fiber direction ef. Since solely the filling phase is simulated only passive
material behaviour is taken into account.
2.6.1 Passive material behaviour
The passive constitutive behaviour of the tissue is described by a relation between the Cauchy
stress tensor σ and a strain energy function W(E):
σ =
1
F . ∂W(E)/∂E . Fc
det(F)
(2.11)
In the model of Kerckhoffs the passive tissue is assumed to behave transversely isotropic and
virtually incompressible. This behaviour is represented by the following strain energy function:
W(E) = Ws(E) + Wv(E)
Ws(E) = a0 [exp(a1 IE2 + a2 IIE + a3(ef . E . ef)2)- 1]
Wv(E) = a4 [det(2E + I) - 1]2
(2.12)
(2.13)
(2.14)
The term Ws describes the tissue response due to change in shape, while the term Wv is related to
the compressibility of the tissue. The invariants IE and IIE are defined as:
IE = trace(E)
IIE = 1/2 (trace(E . E) - IE2)
(2.15)
(2.16)
In this simulation the material is assumed to behave totally isotropic, so a3 is zero. The other used
values for the material parameters in equation (2.12) - (2.14) are listed in table 2.3 [3].
Table 2.3: Material parameter values for passive behaviour of the heart [3].
a0 (kPa)
0.5
a1 (-)
3.0
a2 (-)
6.0
a3 (-)
0
a4 (kPa)
55
2.6.2 Prescribed pressures
A uniform pressure is prescribed at the endocardial surfaces of the LV and RV. From t =0 ms to
t = 200 ms the pressure of the LV is prescribed from p = 0 to 1 kPa and the pressure of the RV
from 0 to 0.2 kPa in 5 timesteps. The epicardial pressure is assumed to be zero during the
simulation.
To compare the local mechanics of the simulation with the literature, after the first simulation
another 3 simulations will be done. Only the prescribed pressures of the LV and RV and the
timesteps will be different. The prescribed pressures and the number of timesteps of the first and
the next 3 simulations are listed in table 2.4.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Table 2.4: Prescribed pressures of the left and right ventricle of the different
simulations and the timesteps in which the pressure is raised from zero to
the prescribed pressure.
Simulation
1
2
3
4
Plv (kPa)
1
2.5
3.5
4.5
Prv (kPa)
0.2
0.5
0.7
0.9
Timesteps
5
20
40
50
2.6.3 Boundary conditions
The motion of the basal plane in the z-direction is suppressed. Also the circumferential motion of
three endocardial points in the basal plane lying on the coordinate axes is suppressed. In figure
2.6 the three points where the radial motion is suppressed are shown.
Figure 2.6: Points where the circumferential motion is suppressed.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 3: Results
Firstly the values of the volumes of the cavities and walls of the ventricles will be presented.
Secondly the geometry of the new mathematical model of the cardiac mechanics of the RV and LV
will be presented. Then the results of the simulation of the filling phase of the both ventricles will
be shown. The results of the simulation will be divided in two sections: the global hemodynamics
of both ventricles and the local mechanics of the RV in terms of the end-diastolic principal strains.
3.1 Calculation of the volumes of the cavities and walls
With the use of equations (2.4) and (2.5) and the values of the parameters in table 2.2 the volumes
of the cavities and walls of both ventricles are calculated. The values of these volumes are listed in
table 3.1
Table 3.1: Calculated values of the cavity and wall volume of the right and left ventricle with use of the article of Arts
[2].
Vcav. lv [ml]
35
Vcav, rv [ml]
61
Vwall, lv free [ml]
58
Vwall, sep [ml]
46
Vwall, rv [ml]
23
The volume of the LV cavity is held the same as in the existing model (40 ml) and also the values
of C and h (table 2.1). Therefore the volume of the wall of the LV is dependent on the volume of
the RV; the volume of the wall of the LV decreases with increasing volume of the wall of the RV.
The values for the volumes of the RV are chosen such that the wall of the RV is not too thin and
the values are as close as possible to the values in table 3.1. The values of the parameters for the
new model are listed in table 3.2.
Table 3.2: Values of the parameters for the new model of the left and right ventricle.
Rx1
(mm)
Rx2
(mm)
C
(mm)
h
(mm)
Vcav,lv
(ml)
Vcav, rv
(ml)
48
50
43
24.8
40
53
Vw, lv
(septum +
free wall)
(ml)
105
Vw, rv
(ml)
32
3.2 Geometry of the model of the right- and left ventricle
In figure 3.1 the 3 dimensional mesh of the left and right ventricle is shown. In figure 3.1a more
elements are shown than used for the simulation. In each direction (longitudinal, circumferential
and radial) the total number of elements used for the simulation is half of the elements in the
shown mesh. So the total elements shown in figure 3.1a is eight times the total elements in the
mesh used for the simulation. Figure 3.1b is the bottom view of the mesh. In this view the real
number of the elements are shown.
In figure 3.1 can be seen that there are 3 quadratic elements across the wall of the LV, 1 quadratic
element across the wall of the RV, 4 quadratic elements in the longitudinal direction without the
bottom, 4 quadratic elements in the bottom and 8 elements circumferential. Also can be seen that
the wall of the RV is thicker at the transition between the wall of the RV and LV than at the RV
free wall.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
a
b
Figure 3.1: (a) Mesh of the left- and right ventricle. The total elements used for the simulation is 1/8 of the total
elements shown; (b) bottom view of the mesh (quadratic elements).
3.3 Filling phase
3.3.1 Global hemodynamics
In figure 3.2a the pressure as function of the volume of the two ventricles are shown. The enddiastolic volume of the LV is 78 ml, while the end-diastolic volume of the RV is 90 ml. The
volume increase is 38 ml for both ventricles.
To get information of the consequences to the global hemodynamics of the LV when the RV is
added, a comparison is made with the results of only the LV. The difference in volume as function
of pressure between the two models is shown in figure 3.2b. The maximal absolute difference in
volume between the new and old model is 0.85 ml at 0.85 kPa. This corresponds to a maximal
difference of about 1.1 %.
Figure 3.2: (a) Pressure-volume diagram of the filling phase of the left and right ventricle; (b) differences in volume
of the left ventricle between the model of only left ventricle and the new model with the right ventricle
(Vlvnew - Vlvold)as function of the pressure.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
3.3.2 Local mechanics
In figure 3.3 the distribution of the largest end-diastolic principal strain (E2) is shown at Plv = 2.5
kPa and Prv = 0.5 kPa (simulation 2). It can be seen that the largest E2 occurs at the endocardial
side of both ventricles. The maximum value of E2 is at the base of the RV and is 0.6. The
minimum value of E2 is at the base of the epicardial side of the LV and is about 0.1. There is a
mainly circumferential gradient in E2 along the RV. E2 increases from the LV to the most left part
of the RV and decreases from that point to the LV. This gradient doesn't appear in the LV. In the
RV the maximum epicardial value of E2 is about 0.43.
Figure 3.3: Different views of the distribution of the largest end-diastolic principal strain (E2) of the left and right
ventricle when Plv= 2.5 kPa and Prv = 0.5 kPa are prescribed (simulation 2). The dots in the right figure indicate
the nodal points, which are taken into account in the calculation of the average value of E2 of simulation 2 till 4.
These values are compared with experimental data (section 3.3.3).
The direction of E2 is circumferential for almost all the nodal points. Only at the second column
from the right the direction of E2 is longitudinal (see figure 3.3). The distribution of E2 is the same
when the prescribed pressures are increased (simulations 3 and 4). Only the values of E2 increase
with increasing pressures.
3.3.3 Comparison with experimental data
The average values of the end-diastolic principal strains in the wall of the RV of simulation 2 till 4
can be compared with the results of the experiments of Waldman et al [4]. They measured
principal strains of a part of the epicardial wall of the RV.
To compare the end-diastolic principal strains of the simulation, a region in the wall of the RV
must be chosen from which the strains are calculated. In figure 3.3 the region of the wall of the
RV from which the average values of the end-diastolic principal strains are calculated is indicated
with dots. A total of 25 nodal points are taken into account in the calculation. The prescribed
pressures in the experiment of Waldman [4] of the RV are the same as the pressures of the
simulations 2 till 4 (see table 2.4). The prescribed pressures of the LV in the experiment were
different from the pressures in the simulation.
The end-diastolic principal strains in the RV calculated for the simulations 2 till 4 and as
measured by Waldman [4] are listed in table 3.3. E1 is the minimum eigenvalue of the strain
tensor, while E2 is the maximum eigenvalue of the strain tensor. In the filling phase E2 represents
the largest lengthening and E1 the smallest lengthening.
The calculated averages of the end-diastolic principal strains for the three prescribed pressures are
larger than the measured values. The standard deviation is smaller for the calculated values than
for the measured values. The end-diastolic principal strains of the simulations increase with
increasing prescribed pressure, just as the measured principal strains.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Table 3.3: End-diastolic principal strains of the right ventricle calculated for simulation 2 till 4 and measured by
Waldman [4] (mean ± std).
Component
prv = 0.5 kPa
prv = 0.7 kPa
prv = 0.9 kPa
model
exp. by
Waldman [4]
model
exp. by
Waldman [4]
model
exp. by
Waldman [4]
E1
0.22 ± 0.05
0.02 ± 0.05
0.27 ± 0.05
0.06 ± 0.05
0.30 ± 0.05
0.10 ± 0.06
E2
0.31 ± 0.06
0.13 ± 0.08
0.36 ± 0.06
0.19 ± 0.09
0.40 ± 0.06
0.28 ± 0.11
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 4: Discussion
In the geometry of the new model (figure 2.3) only the gross features were taken into account
since it was preferred to get a working model of the left ventricle (LV) and right ventricle (RV)
first. In a later stage, the anatomy of the both ventricles can be modelled in more detail and even
differences between individuals might be modelled.
The attachment of the RV to the LV at the apex is chosen as figure 2.3, because it is easy to model.
In reality, the attachment is located somewhere between the apex and about a third of the height
of the ventricle above the apex. With the chosen generation of the disc it will be easy to translate
the attachment to a higher point than is modelled since the apex is divided into 2 volumes: one for
the LV and one for the RV. The transformation rules that are used to transform the disc into a LV
and RV are chosen because of their simplicity and their similarity with the existing model of the
LV. Definition of the geometry through mapping of a disc enables a straightforward definition of
a local wall-bound coordinate system. Thus, it is easy to describe fibre orientation, which is
usually defined with respect to such a coordinate system.
Because of the scarce information on the volume of the wall and cavity of the RV in the literature,
these volumes have been approximated with equations derived by Arts [2]. These volumes were
used as guidelines to determine the volumes of RV cavity and wall in the final model, in which
the LV cavity volume was chosen identical to that in the original LV model. In the final model, LV
wall volume is smaller than in the existing model because a part of the wall of the LV is now the
wall of the RV.
The RV wall is thicker at the transition with the LV than at the RV free wall. This is the result of
the choice to limit the number of brick elements radially across the LV wall, while maintaining
equal radial dimensions of those elements. Choosing one element across the RV wall might be
too less to accurately predict local mechanics. Radial mesh refinement should be performed to
assess the accuracy of the finite element solution. Tetrahedral elements would yield much more
flexibility in meshing the geometry. Unfortunately, tetraeders have not been implemented in
SEPRAN until now.
The filling phase has been simulated to test if the new mesh is formulated correctly. The global
hemodynamics of the filling phase is what was expected: the volume of both ventricles increase
with increasing prescribed pressure. This is not trivial, because Nash [5] had already tried to test a
new model of the RV and LV and the volume of the RV decreases instead of increases. The
difference between the volume of the LV of the new model and the existing model is maximal 1.1
%. So when these pressures are prescribed the influence of the RV is not very high.
End-diastolic principal strains in the simulation are about 2-3 times higher than those measured
by Waldman [4]. The difference in the values of the maximal principal strain E1 decreases with the
applied pressures: at 0.7 kPa the calculated value of E1 is about 10 times higher, while at 0.9 kPa it
is about 3 times higher than the experimental value. Possible explanations for the discrepancy are
1) the assumption of isotropic material behaviour, and 2) the relatively thin RV wall in the model.
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Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 5: Conclusions
The existing model of the mechanics of the cardiac left ventricle was successfully extended with
the right ventricle. A first simulation of the filling phase of both ventricles showed that the new
mesh is formulated correctly. The global hemodynamics of the simulation was as was expected:
the volume of both ventricles increases with increasing pressure. The end-diastolic principal
strains of the right ventricle were maximal up to 10 times higher than the measured principal
strains but for most of the prescribed pressures still in the order of measured principal strains.
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Chapter 6: Further research
The first step to a new model of the mechanics of the cardiac left- and right ventricle is made.
Suggestions for further research are described in this section.
First of all some research should be done on the new mesh of both ventricles. The minimum
volume of the wall of the right ventricle has a limit as a result of the chosen transformation. This
will be avoided to take other transformation rules or other elements, so there are no limitations
for the volumes of the wall and cavities of the ventricles. This is an advantage when the model will
be used for clinical measurements, where a detailed model is needed for the individual patient.
Secondly a whole cardiac cycle should be simulated with the new model. Before this can be done,
the wall of both ventricles should be filled with cardiac muscle fibers. The optimal orientation of
the cardiac muscle fibers in the left ventricle is known, so only the orientation of fibers in the
right ventricle should be found in literature. Also the activation times of the fibers in the right
ventricle should be known and implemented in the model.
Thirdly research should be done to the material properties of the right ventricle. In the new model
the assumption is made that the material properties of the right ventricle are the same as the left
ventricle.
Also the volume of the cavity and wall of the right ventricle should be measured experimentally to
verify the calculations of the volumes with equations.
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Acknowledgements
I want to thank Peter Bovendeerd and Roy Kerckhoffs for their help and supervision during my
practical training. They were always willing to help with the problems I had to overcome during
this practical training.
References
[1] Bovendeerd PHM, Arts T, Huyghe JM, Campen DH, Reneman RS. Dependence of local left
ventricular wall mechanics on myocardial fiber orientation: A model study. J. Biomech 1992; 25:
1129-1140.
[2] Arts T, Bovendeerd P, Delhaas T, Prinzen F. Modeling the relation between cardiac pump
function and myofiber mechanics. Submitted to J. Biomech 2002.
[3] Rijcken J, Bovendeerd PHM, Schoofs AJG, Campen DH, Arts MGJ. Optimization cardiac fiber
orientation for homogeneous fiber strain at beginning ejection. J. Biomech 1997 ; 30: 1041-1049.
[4] Waldman LK, Allen JJ, Pavelec RS, McCulloch AD. Distributed mechanics of the canine right
ventricle: effects of varying preload. J. Biomech 1996; 29: 373-381.
[5] Nash M. Mechanics and material properties of the heart using an anatomically accurate
mathematical model. PhDtesis report 1998 on Internet.
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Appendix A
This is the program in SEPRAN to generate the disc.
# Rightnew.in
# mesh file for an ellipsoidal left and right ventricular model.
#
# Quadratic elements.
CONSTANTS
#
integers
#
nelmw1
=2
# number of elements in LV wall thickness
nelmw2
=1
# number of elements in RV wall thickness
nelmarc1
=1
# number of elements in 1/8 circumference
nelmarc2
=2
# number of elements in 1/4 circumference
nelmlen
=4
# number of elements in length (without bottom)
#
ratio = 1
#
lijn
=2
# 1 = lineair, 2 = kwadratisch
rechthoek = 6
# 5 = lineair, 6 = kwadratisch
volume = 14
# 13= lineair, 14= kwadratisch
#
reals
#
factor = 1d0
#
Rbasis = 1.06066017d0
#radius of disk is 1.5
Rb = 1.5d0
Rapex = 0.24748737d0
#radius of apex is 0.35
apexstr= 0.70710678d0
Ra = 0.27904736d0
ksi_i = -0.37053269d0
ksi_o = -0.67556696d0
ksi_m = -0.57388887d0
# coordinates of points (computed in compcons)
#
END
#
MESH3D
#
points
#
#points for top outer contour
P1=
($Rbasis,$Rbasis,$ksi_i)
P2=
(0,$Rb,$ksi_i)
P3=
(-$Rbasis,$Rbasis,$ksi_i)
P4=
(-$Rbasis,-$Rbasis,$ksi_i)
p5=
(0,-$Rb,$ksi_i)
P6= ($Rbasis,-$Rbasis,$ksi_i)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
#points for middle outer contour (with double points on the left side)
P7=
($Rbasis,$Rbasis,$ksi_m)
p8=
(0,$Rb,$ksi_m)
P9= (-$Rbasis,$Rbasis,$ksi_m)
P10= (-$Rbasis,$Rbasis,$ksi_m)
P11= (-$Rbasis,-$Rbasis,$ksi_m)
P12= (-$Rbasis,-$Rbasis,$ksi_m)
P13= (0,-$Rb,$ksi_m)
P14= ($Rbasis,-$Rbasis,$ksi_m)
#points for bottom outer contour
P15= ($Rbasis,$Rbasis,$ksi_o)
P16= (0,$Rb,$ksi_o)
P17= (-$Rbasis,$Rbasis,$ksi_o)
P18= (-$Rbasis,-$Rbasis,$ksi_o)
P19= ((0,-$Rb,$ksi_o)
P20= ($Rbasis,-$Rbasis,$ksi_o)
#points for top inner contour
P21= ($Rapex,$Rapex,$ksi_i)
P22= (0,$Ra,$ksi_i)
p23= (-$Rapex,$Rapex,$ksi_i)
P24= (-$Rapex,-$Rapex,$ksi_i)
P25= (0,-$Ra,$ksi_i)
P26= ($Rapex,-$Rapex,$ksi_i)
#points for middle inner contour (with double points on the left side)
P27= ($Rapex,$Rapex,$ksi_m)
P28= (0,$Ra,$ksi_m)
P29= (-$Rapex,$Rapex,$ksi_m)
P30= (-$Rapex,$Rapex,$ksi_m)
P31= (-$Rapex,-$Rapex,$ksi_m)
P32= (-$Rapex,-$Rapex,$ksi_m)
P33= (0,-$Ra,$ksi_m)
P34= ($Rapex,-$Rapex,$ksi_m)
#points for bottom inner contour
P35= ($Rapex,$Rapex,$ksi_o)
P36= (0,$Ra,$ksi_o)
P37= (-$Rapex,$Rapex,$ksi_o)
P38= (-$Rapex,-$Rapex,$ksi_o)
P39= (0,-$Ra,$ksi_o)
P40= ($Rapex,-$Rapex,$ksi_o)
#point for center top
P41= (0,0,$ksi_i)
#point for center middle
P42= (0,0,$ksi_m)
#point for center bottom
P43= (0,0,$ksi_o)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
#points for top inner cornerpoints
P44= (0,-$apexstr,$ksi_i)
P45= ($apexstr,0,$ksi_i)
P46= (0,$apexstr,$ksi_i)
P47= (-$apexstr,0,$ksi_i)
#points for middle inner cornerpoints
P48= (0,-$apexstr,$ksi_m)
P49= ($apexstr,0,$ksi_m)
P50= (0,$apexstr,$ksi_m)
P51= (-$apexstr,0,$ksi_m)
#points for bottom inner cornerpoints
P52= (0,-$apexstr,$ksi_o)
P53= ($apexstr,0,$ksi_o)
P54= (0,$apexstr,$ksi_o)
P55= (-$apexstr,0,$ksi_o)
#
curves
#
#top endocard
C1=
arc$lijn (P1,P2,P41,nelm = $nelmarc1)
C2= arc$lijn (P2,P3,P41,nelm = $nelmarc1)
C3=
arc$lijn (P3,P4,P41,nelm = $nelmarc2)
C4= arc$lijn (P4,P5,P41,nelm = $nelmarc1)
C5=
arc$lijn (P5,P6,P41,nelm = $nelmarc1)
C6= arc$lijn (P6,P1,P41,nelm = $nelmarc2)
#top middlecard
C7=
arc$lijn (P7,P8,P42,nelm = $nelmarc1)
C8= arc$lijn (P8,P9,P42,nelm = $nelmarc1)
C9= arc$lijn (P8,P10,P42,nelm = $nelmarc1)
C10= arc$lijn (P9,P11,P42,nelm = $nelmarc2)
C11= arc$lijn (P10,P12,P42,nelm = $nelmarc2)
C12= arc$lijn (P11,P13,P42,nelm = $nelmarc1)
C13= arc$lijn (P12,P13,P42,nelm = $nelmarc1)
C14= arc$lijn (P13,P14,P42,nelm = $nelmarc1)
C15= arc$lijn (P14,P7,P42,nelm = $nelmarc2)
#top epicard
C16= arc$lijn (P15,P16,P43,nelm = $nelmarc1)
C17= arc$lijn (P16,P17,P43,nelm = $nelmarc1)
C18= arc$lijn (P17,P18,P43,nelm = $nelmarc2)
C19= arc$lijn (P18,P19,P43,nelm = $nelmarc1)
C20= arc$lijn (P19,P20,P43,nelm = $nelmarc1)
C21= arc$lijn (P20,P15,P43,nelm = $nelmarc2)
#'spokes' of the top
C22= line$lijn (P1,P7,nelm = $nelmw1)
C23= line$lijn (P2,P8,nelm = $nelmw1)
C24= line$lijn (P3,P9,nelm = $nelmw1)
C25= line$lijn (P4,P11,nelm = $nelmw1)
C26= line$lijn (P5,P13,nelm = $nelmw1)
C27= line$lijn (P6,P14,nelm = $nelmw1)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
C28=
C29=
C30=
C31=
C32=
C33=
line$lijn (P7,P15,nelm = $nelmw2)
line$lijn (P8,P16,nelm = $nelmw2)
line$lijn (P10,P17,nelm = $nelmw2)
line$lijn (P12,P18,nelm = $nelmw2)
line$lijn (P13,P19,nelm = $nelmw2)
line$lijn (P14,P20,nelm = $nelmw2)
#apex endocard
C34= arc$lijn (P21,P22,P44,nelm = $nelmarc1)
C35= arc$lijn (P22,P23,P44,nelm = $nelmarc1)
C36= arc$lijn (P23,P24,P45,nelm = $nelmarc2)
C37= arc$lijn (P24,P25,P46,nelm = $nelmarc1)
C38= arc$lijn (P25,P26,P46,nelm = $nelmarc1)
C39= arc$lijn (P26,P21,P47,nelm = $nelmarc2)
#apex middle
C40= arc$lijn (P27,P28,P48,nelm = $nelmarc1)
C41= arc$lijn (P28,P29,P48,nelm = $nelmarc1)
C42= arc$lijn (P28,P30,P48,nelm = $nelmarc1)
C43= arc$lijn (P29,P31,P49,nelm = $nelmarc2)
C44= arc$lijn (P30,P32,P49,nelm = $nelmarc2)
C45= arc$lijn (P31,P33,P50,nelm = $nelmarc1)
C46= arc$lijn (P32,P33,P50,nelm = $nelmarc1)
C47= arc$lijn (P33,P34,P50,nelm = $nelmarc1)
C48= arc$lijn (P34,P27,P51,nelm = $nelmarc2)
#apex epicard
C49= arc$lijn (P35,P36,P52,nelm = $nelmarc1)
C50= arc$lijn (P36,P37,P52,nelm = $nelmarc1)
C51= arc$lijn (P37,P38,P53,nelm = $nelmarc2)
C52= arc$lijn (P38,P39,P54,nelm = $nelmarc1)
C53= arc$lijn (P39,P40,P54,nelm = $nelmarc1)
C54= arc$lijn (P40,P35,P55,nelm = $nelmarc2)
#'spokes' of the bottom
C55= line$lijn (P21,P27,nelm = $nelmw1)
C56= line$lijn (P22,P28,nelm = $nelmw1)
C57= line$lijn (P23,P29,nelm = $nelmw1)
C58= line$lijn (P24,P31,nelm = $nelmw1)
C59= line$lijn (P25,P33,nelm = $nelmw1)
C60= line$lijn (P26,P34,nelm = $nelmw1)
C61=
C62=
C63=
C64=
C65=
C66=
line$lijn (P27,P35,nelm = $nelmw2)
line$lijn (P28,P36,nelm = $nelmw2)
line$lijn (P30,P37,nelm = $nelmw2)
line$lijn (P32,P38,nelm = $nelmw2)
line$lijn (P33,P39,nelm = $nelmw2)
line$lijn (P34,P40,nelm = $nelmw2)
#connecting curves between top en bottom, inner ellipsoid linker ventricle
C67= line$lijn (P1,P21,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C68= line$lijn (P2,P22,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C69= line$lijn (P3,P23,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C70= line$lijn (P4,P24,nelm = $nelmlen, ratio = $ratio, factor = $factor)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
C71= line$lijn (P5,P25,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C72= line$lijn (P6,P26,nelm = $nelmlen, ratio = $ratio, factor = $factor)
#connecting curves between top en bottom, outer ellipsoid linker ventricle +
#inner ellipsoid right ventricle
C73= line$lijn (P7,P27,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C74= line$lijn (P8,P28,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C75= line$lijn (P9,P29,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C76= line$lijn (P10,P30,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C77= line$lijn (P11,P31,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C78= line$lijn (P12,P32,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C79= line$lijn (P13,P33,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C80= line$lijn (P14,P34,nelm = $nelmlen, ratio = $ratio, factor = $factor)
#connecting curves between top en bottom, outer ellipsoid total
C81= line$lijn (P15,P35,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C82= line$lijn (P16,P36,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C83= line$lijn (P17,P37,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C84= line$lijn (P18,P38,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C85= line$lijn (P19,P39,nelm = $nelmlen, ratio = $ratio, factor = $factor)
C86= line$lijn (P20,P40,nelm = $nelmlen, ratio = $ratio, factor = $factor)
#curves to divide apex
C87= line$lijn (P22,P25,nelm = $nelmarc2, ratio = $ratio, factor = $factor)
C88= line$lijn (P28,P33,nelm = $nelmarc2, ratio = $ratio, factor = $factor)
C89= line$lijn (P36,P39,nelm = $nelmarc2, ratio = $ratio, factor = $factor)
#
surfaces
#
#top plane
S1=
coons$rechthoek (-C1,C22,C7,-C23)
S2=
coons$rechthoek (-C2,C23,C8,-C24)
S3=
coons$rechthoek (-C3,C24,C10,-C25)
S4=
coons$rechthoek (-C4,C25,C12,-C26)
S5=
coons$rechthoek (-C5,C26,C14,-C27)
S6= coons$rechthoek (-C6,C27,C15,-C22)
S7=
coons$rechthoek (-C7,C28,C16,-C29)
s8=
coons$rechthoek (-C9,C29,C17,-C30)
S9= coons$rechthoek (-C11,C30,C18,-C31)
S10= coons$rechthoek (-C13,C31,C19,-C32)
S11= coons$rechthoek (-C14,C32,C20,-C33)
s12= coons$rechthoek (-C15,C33,C21,-C28)
#bottom 'ring'
S13= coons$rechthoek (-C34,C55,C40,-C56)
S14= coons$rechthoek (-C35,C56,C41,-C57)
S15= coons$rechthoek (-C36,C57,C43,-C58)
S16= coons$rechthoek (-C37,C58,C45,-C59)
S17= coons$rechthoek (-C38,C59,C47,-C60)
S18= coons$rechthoek (-C39,C60,C48,-C55)
S19= coons$rechthoek (-C40,C61,C49,-C62)
S20= coons$rechthoek (-C42,C62,C50,-C63)
S21= coons$rechthoek (-C44,C63,C51,-C64)
S22= coons$rechthoek (-C46,C64,C52,-C65)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
S23=
S24=
coons$rechthoek (-C47,C65,C53,-C66)
coons$rechthoek (-C48,C66,C54,-C61)
#inner ellipsoid LV, top part
S25= coons$rechthoek (-C68,-C1,C67,C34)
S26= coons$rechthoek (-C69,-C2,C68,C35)
S27= coons$rechthoek (-C70,-C3,C69,C36)
S28= coons$rechthoek (-C71,-C4,C70,C37)
S29= coons$rechthoek (-C72,-C5,C71,C38)
S30= coons$rechthoek (-C67,-C6,C72,C39)
#middle ellipsoid LV, top part
S31= coons$rechthoek (C73,C40,-C74,-C7)
S32= coons$rechthoek (-C74,C8,C75,-C41)
S33= coons$rechthoek (-C75,C10,C77,-C43)
S34= coons$rechthoek (-C77,C12,C79,-C45)
S35= coons$rechthoek (C79,C47,-C80,-C14)
S36= coons$rechthoek (C80,C48,-C73,-C15)
#outer ellipsoid total top part
S37= coons$rechthoek (C81,C49,-C82,-C16)
S38= coons$rechthoek (C82,C50,-C83,-C17)
S39= coons$rechthoek (C83,C51,-C84,-C18)
S40= coons$rechthoek (C84,C52,-C85,-C19)
S41= coons$rechthoek (C85,C53,-C86,-C20)
S42= coons$rechthoek (C86,C54,-C81,-C21)
#inner ellipsoid RV top part
S43= coons$rechthoek (C74,C42,-C76,-C9)
S44= coons$rechthoek (C76,C44,-C78,-C11)
S45= coons$rechthoek (C78,C46,-C79,-C13)
#bottom
S46= coons$rechthoek (-C34,-C39,-C38,-C87)
S47= coons$rechthoek (C87,-C37,-C36,-C35)
S48= coons$rechthoek (-C40,-C48,-C47,-C88)
S49= coons$rechthoek (C88,-C45,-C43,-C41)
S50= coons$rechthoek (-C49,-C54,-C53,-C89)
S51= coons$rechthoek (C89,-C52,-C51,-C50)
#planes between top parts ellipsoid LV
S52= coons$rechthoek (-C67,C22,C73,-C55)
S53= coons$rechthoek (-C68,C23,C74,-C56)
S54= coons$rechthoek (-C69,C24,C75,-C57)
S55= coons$rechthoek (-C70,C25,C77,-C58)
S56= coons$rechthoek (-C71,C26,C79,-C59)
S57= coons$rechthoek (-C72,C27,C80,-C60)
#planes between top parts ellipsoid total
S58= coons$rechthoek (-C73,C28,C81,-C61)
S59= coons$rechthoek (-C74,C29,C82,-C62)
S60= coons$rechthoek (-C76,C30,C83,-C63)
S61= coons$rechthoek (-C78,C31,C84,-C64)
S62= coons$rechthoek (-C79,C32,C85,-C65)
S63= coons$rechthoek (-C80,C33,C86,-C66)
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Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
#bottom dividing planes
S64= coons$rechthoek (C87,C59,-C88,-C56)
S65= coons$rechthoek (C88,C65,-C89,-C62)
#bottom (as S46-S51)
S66= coons$rechthoek (C88,-C46,-C44,-C42)
#
volumes
#
#top LV
V1=
brick$volume (S13,S53,S25,S52,S31,S1,orientation 225272)
V2= brick$volume (S14,S54,S26,S53,S32,S2,orientation 225222)
V3=
brick$volume (S15,S55,S27,S54,S33,S3,orientation 225222)
V4= brick$volume (S16,S56,S28,S55,S34,S4,orientation 225222)
V5=
brick$volume (S17,S57,S29,S56,S35,S5,orientation 225272)
V6= brick$volume (S18,S52,S30,S57,S36,S6,orientation 225272)
#bottom LV
V7=
brick$volume (S13,S18,S48,S64,S46,S17,orientation 456565)
V8= brick$volume (S14,S64,S49,S15,S47,S16,orientation 455455)
#bottom LV outer
V9= brick$volume (S19,S24,S50,S65,S48,S23,orientation 456565)
#top LV outer
V10= brick$volume (S19,S59,S31,S58,S37,S7,orientation 227272)
V11= brick$volume (S24,S58,S36,S63,S42,S12,orientation 227272)
V12= brick$volume (S23,S63,S35,S62,S41,S11,orientation 227272)
#top RV
V13= brick$volume (S20,S60,S43,S59,S38,S8,orientation 227272)
V14= brick$volume (S21,S61,S44,S60,S39,S9,orientation 227272)
V15= brick$volume (S22,S62,S45,S61,S40,S10,orientation 227272)
#bottom RV
V16= brick$volume (S20,S65,S51,S21,S66,S22,orientation 455455)
#
meshvolume
#
#linker ventricle
velm1 = (V1,V12)
#rechter ventricle
velm2 = (V13,V16)
#
renumber
#
plot, curve = 2, eyepoint= (0,0,50)
#
END
#
#
PROBLEM
END
#2 = curve incl. nummer
26
Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
Appendix B
The following text is the program in MATLAB to transform the disc to the left- and right ventricle.
%transrightnew.m
%transformation for left and right ventricles
clear all;
mfile = '/users/wfw3/petra/right/som3/ellipsoid6/meshoutput';
C = 43;
Rx1= 48;
Rx2=50;
ztop = 24.78;
ksi_i = 0.37053269d0;
ksi_o = 0.67556696d0;
ksi_m = ksi_i + (2*((ksi_o - ksi_i)/3))
[coor,topo,surf,elgrp]=sep_readmesh(mfile);
x=coor(:,1);
y=coor(:,2);
z=coor(:,3);
r = sqrt(x.^2+y.^2);
apex=find(abs(r)<1e-3);
y(apex)=0.001;
r = sqrt(x.^2+y.^2)/max(r)*1.5;
phi = atan2(y,x);
ksi = -z;
theta_n = r - 1;
k=0;
l=0;
for i = 1:length(x),
thetamin = acos(ztop/(C*cosh(ksi(i))));
rc = (thetamin-pi)/1.5;
b = pi + rc;
theta(i) = rc*theta_n(i) + b;
if isempty(find(topo(1,:,:)==i))
x2(i) = (Rx1+ ((ksi(i) - ksi_m)*(Rx2 - Rx1)/(ksi_o - ksi_m))).*sin(theta(i)).*cos(phi(i));
y2(i) = C*sinh(ksi(i)).*sin(theta(i)).*sin(phi(i));
z2(i) = C*cosh(ksi(i)).*cos(theta(i));
k=k+1;
else
x2(i) = C*sinh(ksi(i)).*sin(theta(i)).*cos(phi(i));
y2(i) = C*sinh(ksi(i)).*sin(theta(i)).*sin(phi(i));
z2(i) = C*cosh(ksi(i)).*cos(theta(i));
l=l+1;
end
end
27
Biomedical Engineering
Extension of a finite element model of left ventricular mechanics with a right ventricle
coor = [x2' y2' z2'];
sep_plotsol(coor, topo, elgrp, 1, 1:elgrp(1,3) , [1 2 3], []);
view(3);
axis off;
axis equal;
axis([1.1*min(coor(:,1)) 1.1*max(coor(:,1)), ...
1.1*min(coor(:,2)) 1.1*max(coor(:,2)), ...
1.1*min(coor(:,3)) 1.1*max(coor(:,3))]);
sep_plotsol(coor, topo, elgrp, 2, 1:elgrp(2,3) , [1 2 3], []);
view(3);
axis off;
axis equal;
axis([1.1*min(coor(:,1)) 1.1*max(coor(:,1)), ...
1.1*min(coor(:,2)) 1.1*max(coor(:,2)), ...
1.1*min(coor(:,3)) 1.1*max(coor(:,3))]);
figure
sep_plotsol(coor, topo, elgrp, 2, 1, [1 2 3], []);
save /users/wfw3/petra/right/som3/ellipsoid6/coordinatenright3 coor
28
Biomedical Engineering
