Download left ventricle

PUBLISHING HOUSE
OF THE ROMANIAN ACADEMY
PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A,
Volume 3, Number 1-2/2002.
A GENETIC ALGORITHM FOR MODELING THE CONSTITUTIVE LAWS
FOR THE LEFT VENTRICLE
1
1
Ligia MUNTEANU , Veturia CHIROIU , Marco SCALERANDI
1
2
2
Institute of Solid Mechanics, Romanian Academy
INFM-Dipartimento di Fisica del Politecnico di Torino, Torino
Corresponding author: [email protected]
The central problem in modeling the dynamics of the heart is in identifying functional forms and
parameters of the constitutive equations, which describe the material properties of the myocardium.
The constitutive properties of myocardium are three-dimensional, anisotropic, nonlinear and timedependent. In this paper the active and passive constitutive laws for the left ventricle are determined
from experimental data by using a genetic algorithm.
Key words: myocardium, constitutive model, genetic algorithm
1. INTRODUCTION
The dynamics of the left ventricle is the result of
the contractile motion of the muscle cells in the
left ventricular wall. Heart muscle is a mixture of
muscle and collagen fibers, coronary vessels,
coronary blood and the interstitial fluid. The fibers
wind around the ventricle, and their orientation,
relative to the circumferential direction, changes
continuously from about 60 0 at the endocardium
to –60 0 at the epicardium. This anisotropy
influences the transmural distribution of wall
stress. Recent models capture some important
properties including: the nonlinear interactions
between the responses to different loading
patterns; the influence of the laminar myofiber
sheet architecture; the effects of transverse stresses
developed by the myocytes; and the relationship
between collagen fiber architecture and
mechanical properties in healing scar tissue after
myocardial infarction (Bardinet, Cohen and
Ayache [1,2], Huyghe., Van Campen, Arts and
Heethaar [4,5], Van Campen, Huyghe, Bovendeerd
and Arts [6]).
A dificult problem in modeling the dynamics of
the heart is in identifying functional forms and
__________________________
Recommended by Radu P.VOINEA
Member of the Romanian Academy
parameters of the constitutive equations, which
describe the material properties of the
myocardium.
We consider in this paper the cardiac tissue as a
mixture of an incompressible solid and an
incompressible fluid (Munteanu, Chiroiu C. and
Chiroiu V. [3]). The active and passive
constitutive laws for the left ventricle are
determined from experimental data (Bardinet,
Cohen and Ayache [1,2]) by using a genetic
algorithm.
2. MATHEMATICAL MODEL
The underformed heart, in a stress-free
reference state, is modeled as a super ellipsoid
surface S (Bardinet, Cohen and Ayache [1])
c2
2
2
2 c

1
c1
c2
c2






x
y
z
          1
a  
a 
 a 
 2 
 3
 1 

(2.1)
where the constants ai , i  1,2,3 and ci , i  1,2 are
c1  c 2  0.773 , a1  a 2  0.892 R , a3  R , with
R  0.0619 m for a particular heart. For a sphere
we have c1  c2  1 and a1  a 2  a3  R . The z axis corresponds to the z -axis of inertia of the
Constitutive laws for left ventricle
super ellipsoid model. The muscle fibres in the
ventricular wall are assumed to be parallel to the
endocardial and epicardial surfaces.
The cardiac muscle is considered to be a
mixture of two phases, a solid phase and a fluid
phase. The equations of the beating left ventricle
are composed from (Van Campen, Huyghe,
Bovendeerd and Arts [6]):
1. The equilibrium equation of the deformed
myocardicum (by neglecting the inertia forces)
 s  p  0
(2.2)
where  s is the effective Cauchy stress in the
solid representing the stress induced by the
deformation in the absence of fluid and measured
per unit bulk surface and p the intra-myocardial
pressure representing the stress in the liquid
component of the bi-phasic mixture. The total
Cauchy stress tensor in the mixture is
   s  pI ,   T .
2. Darcy’s law in Eulerian form:
q   Kp
(2.3)
J 1
 1) 2 K 0 , the parameters K 0
Nb
(permeability tensor of the underformed tissue)
and N b ( averaged porosity of the underformed
tissue) being specified. Here q is the Eulerian
with
K (
spatial fluid flow vector, K 0 the permeability
tensor of the underformed tissue, F  1  u the
deformation gradient tensor, J  det F  0 the
Jacobean of the deformation, and u the
 u1   u r 
   
displacement vector,  u 2    u  
u  u 
 3  z
3. Continuity equation (conservation of mass):
u  q  0
(2.4)
C
E
(2.5)
with:
S  Sa  S p , S p  Sc  Ss
C
cc
( J  1) 2 , J  det F  0
2
S ( E, t )  JF 1 s ( F 1 ) T , S  S T
(2.8)
Here C is the isotropic energy function, E the
Green-Lagrange strain tensor, and S ( E , t ) the
effective second Piola-Kirchhoff stress tensor,
split into an active stress S a ( E, t ) and a passive
stress S p ( E, t ) . The passive stress tensor is split
into a component resulting from elastic volume
change of the myocardial tissue S c (E) , and a
component resulting from viscoelastic shape
S s ( E, t ) described in the form of quasi-linear
viscoelasticity as:
t
S
s

d
 G(t  ) dτ S
e
d
(2.10)

Se 
W
E
(2.11)
with S e the anisotropic elastic response of the
material, G (t ) a scalar function (reduced
relaxation function) derived from a continuous
relaxation spectrum (Huyghe., Van Campen, Arts
and Heethaar [5]) and W the potential energy of
deformation per unit volume.
The form of the strain energy C (2.3) and
W are chosen so that C and W are zero in the
unstrained state and positive elsewhere, and
S c (E) and S e are zero in the underformed state.
The expression (2.3) satisfies those conditions. In
this paper we consider for the strain energy
function W the expression of the ion-core (BornMayer) repulsive energy (Delsanto, Provenzano
and Uberall [7]:
W 
0.5
() exp[ () R]dV
V V

(2.12)
where (( x)) is the repulsive energy function,
(( x)) the repulsive range function and V is the
heart volume, and :
where dot means the material time derivative.
4. Pasive constitutive laws:
Sc 
3
(2.6)
(2.7)
R  ( x  x0 ) 2  ( y  y0 ) 2  ( z  z0 ) 2
with ( x0 , y 0 , z 0 ) an arbitrary point. We suppose
that () and () depend on the angles angle
 helix  trans in the form:
()  1 ( helix ) 2 ( trans )
(2.13)
4
Ligia MUNTEANU, Veturia CHIROIU, Marco SCALERANDI
()  1 ( helix ) 2 ( trans )
We mention that the values of the angles depend
on the position x . Though the energy function
(2.12) is referring to metallic bilayers and noble
metals, we choose this form to be used in the
description of the ventricle upon a closer analysis
of the compatibility between the experimental data
and analytical form for W .
5. Active constitutive laws (Van Campen,
Huyghe, Bovendeerd and Arts [6] and Arts,
Veenstra and Reneman [8]):
T a  T a0 A(t , l , v)
(2.14)
where T a is the first order Piola-Kirchhoff nonsymmetric active stress tensor, related to the
second Piola-Kirchhoff active stress by
S a  F 1T a , and T a0 a constant associated with
the load of maximum isometric stress.
The stress tensor T a is convenient for some
purposes; it is measured relative to the initial
underformed configuration and can be determined
experimentally. The cardiac muscle is striated
across the fibre direction. The sarcomere length l
(the distance between the striations) is used as a
measure of fibre length. The experiments show
that the active stress generated by cardiac muscle
depends on time t , sarcomere length l and
dl
velocity of shortening of the sarcomeres v 
dt
(Van Campen, Huyghe, Bovendeerd and Arts [6]).
The active stress generated by the sarcomeres is
directed parallel to the fibre orientation. The
function A(t , l , v) represent the dependency on
dl
. We suppose that A(t , l , v) has the
t, l and v 
dt
form:
A(t , l , v)  f (t ) g (l )h(v)
the ventricle) and a diastole (relaxation of the
ventricle) phases.
We have supposed that at the endocardial
surface 1 a uniform intraventricular pressure p 0
is applied as an external load. The loads exerted by
the papillary muscles and by the pericardium are
neglected. The surface  3 represents the upper
end of the annulus fibrosis and is a non-contracting
surface with a circumferential fibre orientation. At
 3  1 only radial displacement u1 is allowed
(fig. 1). The set of equations (2.2)-(2.15) are
completely determined if A(t , l , v) , () , () are
known.
The principal aim of this paper is to determine
these functions from experimental data [1,2] by
using a genetic algorithm.
The set of controlling functions are noted by
P:
P  { 1 ( helix ),  2 ( trans ), 1 ( helix ),
 2 ( trans ), f (t ), g (l ), h(v)}
(2.16)
(2.15)
The set of equations (2.2)-(2.15) represent four
nonlinear coupled equations for the displacements
u k ( x, t ) , k  1,2,3 and the intramyocardial
pressure p( x, t ) . The boundary conditions are:
p( x,0)  p0 ( x) , x  1 , k  1,2,3 , t  [0, T ]
uk ( x,0)  uk0 , x  1 , k  1,2,3 , t  [0, T ]
u k ( x,0)  0 , x   3  1 , k  2,3 , t  [0, T ]
where 1 is the epicardial surface, and [0, T ] the
time interval during a cardiac cycle. The cardiac
cycle is composed from a systole (contraction of
Fig.1
Representations of surfaces
 1 - endocardium,  2 -
 3   2 - the portion where only radial
displacement is allowed. A uniform pressure p 0 is applied on
epicardium and
1 .
Constitutive laws for left ventricle
3. A GENETIC ALGORITHM FOR
EVALUATION THE CONTROLLING
FUNCTIONS
In the following we consider that the functions
Pi i  1,2,...7 are approximated by polynomials of
five degree, characterized by unknowns b j ,
j  1,2...42 .
We extract the coefficients b j
from the
experimental data concerning the strain energy W
(Demer and Yin [9]) and first Piola-Kirchhoff
stress T a (Arts, Veenstra and Reneman [8]). For
experimental measuring of W , a biaxial tissue
testing is a valuable method. Biaxial tests
generally involve excision of a thin rectangular
slab of tissue parallel to the epicardial surface so
that the muscle fibers lie within the plane of the
tissue sample and the predominant fiber direction
is aligned with one edge of the sample. The tissue
is then placed in a biaxial testing apparatus that
measures force and displacement (stress and
strain) along the orthogonal fiber and cross-fiber
axes (Demer and Yin [9]).
The active stress T a is obtainable in
experiments as a function of time, sarcomere
length and velocity of shortening of the sarcomere
(Arts, Veenstra and Reneman [8]) An objective
function  must be chosen that measures the
agreement between theoretical and experimental
data:
( P) 
K
 [W
k
k 1
 Wkexp ]2 
M
 [T
a
m
 Tma exp ]2
m 1
where Wk , k  1,2...K are the predicted
representation of W calculated as functions of b
at K points belonging to the volume between the
inner and outer wall of the left ventricle. The
quantities Wkexp , k  1,2...K are the experimental
values of W measured at the same points.
The functions Tma , m  1,2...M are M predicted
representation of T a calculated as functions of b
at different moments of time, sarcomere length and
velocity of shortening of the sarcomere, and Tma exp
the corresponding experimental values. The
controlling parameters b j , j  1,2,...42 are
determined by using a genetic algorithm (GA). GA
assures an iteration scheme that guarantees a
closer correspondence of predicted and
5
experimental values of W and T a at each
iteration.
We use a binary vector with 42 genes
representing the real values of the parameters b j ,
j  1,2...42 . The length of the vector depends on
the required precision, which in this case is six
places after the decimal point. The domain of
parameters b j  [a j , a j ] , with length 2a j is
divided into a least 15000 equal size ranges. That
means that each parameter b j , j  1,2...42 is
represented by a gene (string) of 22 bits
( 2 21  3000000  2 22 ). One individual consists of
the row of 42 genes, that is, a binary vector with
22  42 components:
b
(1) (1)
(1) ( 2) ( 2)
( 2)
( 42) ( 42)
( 42)
21 b20 ...b0 b21 b20 ...b0 ...b21 b20 ...b0

GA is linked to the problem that is to be solved
through the fitness function, which measures how
well an individual satisfies the real data. From one
generation to the next GA usually decreases the
objective function of the best model and the
average fitness of the population. The starting
population is usually randomly generated. Then,
new descendant populations are iteratively created,
with the goal of an overall objective function
decrease from generation to generation. Each new
generation is created from the current one by the
main operations:
selection, crossover and
reproduction, mutation and fluctuation. By
selection two individuals of the current population
are randomly selected (parent 1 and parent 2) with
a probability that is proportional to their fitness.
This ensures that individuals with a good
fitness have better chance to advance to the next
generation. In the crossover and reproduction
operation some crossover sites are chosen
randomly and two individuals are reproduced by
exchanging some genes between parents. In the
new produced individuals, a randomly selected
gene is changed with a random generated integer
number by the mutation operation. In the
fluctuation operation we exchange a discretized
value of an unknown parameter in a random
direction, by extending the search in the
neighbourhood of a current solution.
The fitness function is evaluated for each
individual that corresponds to the gene
representation. The alternation of generations
stops when the convergence is detected.
Otherwise, the process stops when a maximum
6
Ligia MUNTEANU, Veturia CHIROIU, Marco SCALERANDI
number of generations are reached. The alternation
of generations is stopped when convergence is
detected. If no convergence the iteration process
continues until the specified maximum number of
generations is reached
We can see clearly areas on the ventricle where the
displacements are high (for example the area A).
In conclusions, the stress T a is correctly
predicted by the genetic algorithm, the results
being qualitatively and quantitatively consistent
with the results obtained by Bovendeerd, Arts,
Huyghe, Van Campen and Reneman [10], and Van
Campen, Huyghe, Bovendeerd and Arts [6]. The
expression of the ion-core (Born-Mayer) repulsive
energy (Delsanto, Provenzano and Uberall [7])
shows good agreement with the experimental data
of biaxial experiments (Demer and Yin [9]).
Further experiments are desirable to fully
assess the applicability of this theory. Triaxial
tissue testing is ideal, but it remains challenging in
practice due to technical limitations in
simultaneously loading myocardium in three
orthogonal directions, ensuring the resulting
strains and interpreting their significance.
By separating the volume change from fiber
extension modeled by the repulsive energy
function, and shearing distortions modeled by the
repulsive range function, the resulting set of
response terms should provide an improved
foundation for myocardial constitutive modeling.
4.RESULTS OF THE GENETIC
ALGORITHM
We report in this section the results of the
genetic algorithm. Fig.2 shows the active material
behavior as obtained by GA after 317 iterations.
There are shown the time dependence of active
stress for sarcomere lengths of 1.7, 1.9, 2.1 and
2.3 m , the length dependence of active stress, and
the velocity dependence of active stress.
Figs 3 and 4 show the dependence of () and
() on angles  helix  trans given by GA after 222
iterations. The distribution of  helix and  trans from
the endocardium to the epicardium it is also
shown.
Fig. 5 shows the initial pressure p 0 applied at
the endocardial surface  1 . The visualization of
the displacement field by different values
according to the range 0-10 mm is shown in Fig. 6.
Fig.2
Active material behavior as obtained by the genetic algorithm
(time dependence of active stress for sarcomere lengths of 1.7, 1.9, 2.1
and 2.3 m , length dependence of active stress,
and velocity dependence of active stress)
Constitutive laws for left ventricle
7
Fig.4
Fig.3
 1 ( helix ) and 1 ( helix )
as given by the genetic algorithm.
Angle dependence of
.
Angle dependence of
 2 ( trans ) and  2 ( trans ) as given
by the genetic algorithm.
8
Ligia MUNTEANU, Veturia CHIROIU, Marco SCALERANDI
REFERENCES
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Fig. 5
Initial intraventricular pressure
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p0
applied at the endocardial
1
Fig. 6
Visualization of displacement field by different values
according to the range 0-10 mm
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Received March 13, 2002