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Trends in cardiac dynamics : towards coupled models of intracavity fluid dynamics and deformable wall
mechanics
G. Pelle, J. Ohayon, C. Oddou
To cite this version:
G. Pelle, J. Ohayon, C. Oddou. Trends in cardiac dynamics : towards coupled models of intracavity fluid dynamics and deformable wall mechanics. Journal de Physique III, EDP Sciences, 1994, 4 (6), pp.1121-1127. �10.1051/jp3:1994103�. �jpa-00249170�
Classification Physics Absn.acts
87.45
Trends in cardiac dynamics : towards coupled models of
intracavity fluid dynamics and deformable wall mechanics
G. Pelle, J. Ohayon and C. Oddou
Laboratoire de Mdcanique Physique and INSERMU2, Universitd Paris Val de Mame,
94010 Crdteil, France
(Received 25 August1992, iei>ised ?? December J993, accepted 2 March 1994)
Abstract. We report here preliminary results in the development of a computational model in
cardiac mechanics which takes into account the coupled effects of ventricular mechanics and intracardiac hemodynamics. In this first work, complex geometrical, architectural and rheological properties of the organ have been strongly simplified in order to propose a « quasi-analytical »
model. We assume axisymmetrical geometry of the ventricle and myocardium material to be made of a sheath of a composite, collagenic, fibrous and active muscle medium inside which the blood
dynamics is dominated by unsteady inertial effects. Moreover, we have made grossly simplifying assumptions concerning rather stringent and unusual functioning conditions about the mechanical
behavior of the input and output valvular and vascular impedances as well as the biochemical
action of the fiber. By imposing the tjme variation of the input and output flow rate and activation
function. it is possible, assuming uniformity of the pressure stresses applied to the internal wall surface at every instant of the cardiac cycle, to calculate the overall distribution of fluid pressure and velocity inside the cavity as well as the distributions of stresses and strains inside the wall. It
was shown that under the action of a given biochemical activation function, both kinematics of the wall and induced motion of the fluid are such that the boundary conditions concerning normal
pressure stresses conservation was constantly satisfied. Moreover, the results concerning the dynamics of the blood flow, as viewed through the human clinical investigations using velocimetric technology based upon color doppler ultrasound, are in accordance with those obtained from such a model, at least during the ejection phase. In particular, contrarily to the filling phase processes, the ejection dynamics is such that the time evolution of the blood velocity
measured along the cavity axis does not display any phase shift characterizing an effect similar to a
velocity propagation phenomenon. This model reveals to be interesting by its dual point of view perrnitting to characterize the cardiac performance from both the fluid and envelope kinematics data, given a few number of parameters related to the geometrical and rheological properties of the heart.
1. Introduction.
A modeling of the performance of the heart pump requires detailed studies of the interaction between the dynamics of the intracardiac and intravalvular blood flow and the mechanics of the
l122 JOURNAL DE PHYSIQUE III N° 6
cardiac walls. The anisotropic elastic behavior of the left ventricle (LV), merely due to fiber orientation, is now well-established. Several models currently exist that attempt to calculate the transmural distribution of stresses, strains and tissue pressure as well as energetic
quantities. The myocardium treated as a fluid-fiber continuum has been successfully applied to the mechanics of the LV [1-6]. Using such a description of the medium, the cardiac mechanics
have been described as the motion of a viscous, incompressible fluid containing a immersed system of elastic or contractile fibers [7]. However, the microstructure of the collagen sheath
or woven fabric surrounding a myocyte and the collagen struts interconnecting neighboring
myocytes [8, 9] can be considered and has been incorporated into a fluid-fiber-collagen
continuum description of the myocardium [10]. The intracardiac blood flow, dominated by
inertial convective effects, during the filling phase presents a vortical pattern which is flushed
out during the ejection period. These viscous effects require the resolution of Navier-Stokes
equations [7, 11]. But, neglecting the thin boundary viscous layers and wakes behind the
valves, as a rough and mean approximation, the flow can be treated as potential with
unsteadiness which plays a major role [12-14]. From these considerations, we have developed
in this paper an analysis of fluid-structure interactions in the heart which couple the mechanics of a fluid-fiber-collagen material and the dynamics of an inviscid fluid, assumed to be
unvortical.
2. Mathematical formulation and solution.
2, I EQUATIONS oF THE STRUCTURE. The unstressed reference geometry of the left ventricle is held as a cylinder of length L, inner radius R,, and outer radius R~. On the upper surface, or
base, at Z
=
0 radial displacements are allowed, but axial displacements are assumed to be
zero. The bottom surface, or apex, at Z
=
L is supposed to be a free plane which allows radial and axial displacements. The cavity inside the cylinder is filled with blood, assumed to
exert a uniform pressure P, on the endocardial surface 3 (see Fig, I).
The fluid-fiber-collagen constitutive law u~ed in thi~ study is
«=-P)+2me+KT(~le,ge,+11 -K)T)°leogeo (II
where « is the stress tensor, e is the strain tensor, T)°~ is the tension in the axial fibers, T)°' is the tension in the circumferential fibers, m is the effective shear modulus of the isotropic collagen matrix connecting the muscle fibers, is the unit matrix, and K represents the density
of the axial fibers (0 w K w I ). The collagen matrix is assumed to be incompressible, and thus
gives rise to an isotropic component of its stress field, which, merging with the hydrostatic
pressure in the liquid phase combine to give the myocardial tissue pressure P~. The fiber direction is taken to be the superimposition of two fiber sets : one axial (e=) and one
circumferential (eo). Such fiber configuration is used to eliminating the twisting degree of freedom, but the anisotropy of elastic properties is maintained.
The rheological law of the fiber is deemed to be linear in the passive state (elastic modulus E) and in the active state (elastic modulus E* and active tension at zero strain To). During the
cardiac cycle, the fiber rheological behavior is assumed to be a linear combination of these two states [3] :
T)°'
= Ei(t e,, + P (t To and T(°~
= E~(t coo + P (t To
with E~(ti
= (i p (t )i E + p (t ) E* (2)
where e~= and coo represent respectively the linearized axial and circumferential fiber strains, and p iii is the time-dependent activation function (0 w p (t ) w I). The end-diastolic state is
represented by p = 0 and the end-systolic state, by p
=
1.
cm/s ~ 12g.
?i
12.
-.
-45.
-l03. -4 s
21 41 61,81 Q 21 41 61,81
124.
S
1,21 41 61,81 1,21 41 61 al
Fig. I. Idealized model of the left ventricle. On the left, input data valvular velocity and activation function. On the right, wall velocity and pressure deduced from the structure model and used as input to described the intracardiac fluid dynamics.
Since the objective of this study was to keep the mathematical analysis as simple as and tractable as possible, the transmural delay in the mechanical activation function as well as the effects of gravity and inertial have been neglected and a small deformation analysis was used.
The basic problem was then to find a single-valued displacement (u) and tissue pressure
(P~) fields that need satisfy the equations of equilibrium,
V.«=0 (3)
the linearized condition of incompressibility,
v.u=o (4)
and the boundary conditions which postulate the absence of shear stresses and the continuity of normal stresses on lateral surfaces. Such a system together with axial force equilibrium ofapex plane determines the instantaneous mechanichal state of the thick-walled cylinder with uniform endocardial and epicardial pressures, whose solution has been given previously (10).
2.2 EQUATIONS oF THE FLUID. The velocity field
v of the incompressible intracardiac fluid is govemed by the mass conservation equation V v
=
0. Neglecting viscous effects and
assuming irrotational flow during the cardiac cycle, we can express such a velocity field from a
velocity potential ~Pby the relation : v
=
V~P. Both equations require that this velocity satisfies the Laplace equation
V~~P
=
0 15)
in the domain i2, inside ventricular cavity (see Fig, I), with Neumann conditions on the
boundary 1:
~ ~°
= v n =
~~
n (6)
dn dt
l124 JOURNAL DE PHYSIQUE III N° 6
where n refers to the internal unit vector normal to the boundary 3, d~P/dn is the normal derivative of the potential ~P and du/dt is the normal velocity of the wall determined by the
dynamics of the structure.
We have also estimated the pressure field inside the cavity from the Bernoulli equation involving three terms: the convective effects, the unsteadiness effects and a constant
depending on time. Such a constant has been postulated to be equal to the uniform endocardial pressure P~ as precedingly defined.
P (r, z, t)
= pV (r, z, t)~ p
~~ ~~' ~' ~~
+ P~(t). (7)
We recall that if our model is mechanically coherent both from structural and fluid dynamics viewpoints, it would imply that, at every instant of time, the pressure distribution calculated from such an equation should be constant on the endocardial wall. To meet such a condition,
the sum of convective and unsteadiness effects must be negligible as compared to
P~.
For an isotropic and homogeneous three-dimensional medium, we obtain the solution of this
problem by using the Boundary Element Method which allows to transform the integration of
equation (5) over all the domain i2 into an integration over the boundary 3 alone. While
following such a method, the potential ~P is generated by a distribution, localized on 3, of
sources with intensity q by unit surface
~P (M
= qIF G (M, P d3p with G (M, P
= (8)
z 4 " (MP
where G(M, P is the Green function.
The boundary condition (Eq. (6)) is then expressed by :
V(P') n =
~~ (P') n =
d (P')
=
~ ~~'~
+ lq(P ) ~~ ~~" ~ d3p. (9)
dt ~n 2
z ~nP
The integral equation (Eq. (9)) allows us to find the distribution q(P ). In order to solve such
an equation we use a numerical approximation proposed by Hess and Smith [15] where 3 is divided into N elements 3~ on which q(P is approximated by a constant q~. Thus, we obtain from equation (9) the following linear algebraic system of unknowns q~
v(P,) n
= (P, ) n =
~ (P,)
= +
(
q~
~~~" ~/~
d3p~ j10j
,-~,~, j np,
from which we derive the velocity potential ~P as defined by equation (8) and approximated in
the same manner.
Owing to the axisymmetry of our model, the problem can be solved by scanning a half-plane
with a mesh grid of fifteen by thirty elements. This number of elements has been held as
constant during the overall cycle but their sizes were geared to the variable geometry of the
model.
3. Results.
3.I STRUCTURE MoTloN. An example of the kinematics of the cardiac structure, as
obtained from such a model, is shown in figure I with emphasis on the wall internal boundary velocity. This calculation tends to simulate human heart physiology with given characteristic
rheological parameters (E
= 2 x 10~N/m~, E *
=
30 x 10~ N/m~, m
= 0.6 x 10~ N/m~, T~
=
7 x 10~N/m~ and K
= 1/3) and geometrical parameters (Rj = 2, I cm, R~/Rj = 1.5, L
=
5.9 cm). Moreover, the time variation of both normalized activation of fibers p (t) and valvular blood velocity Vo(t) are given as inputs (Fig. I). From this velocity, the temporal
distribution of cavity volume is derived, whilst supposing a blunt profile inside the
input/output section assumed to be of constant area (3.14 cm~). The cardiac cycle starts with
an isovolumic contraction followed by the ejection phase. Then the isovolumic relaxation
happens with an ensuing rapid filling phase. Finally, an auricular contraction phase occurs at the end of the cardiac cycle. The time interval has been chosen as lo ms and the heart rate as 75 bpm.
The output data of the structure model relate to the kinematics of the wall as expressed by
radial and axial velocities (V~(t) and V~(t)), provided that a cylinder remains cylindrical. This
assumption requires the action of a constant pressure along the boundary, such a pressure
being another output features of the model.
3.2 VELOCITY FIELD. Previous data, related to the structure kinematics (V~(t) and V~it ))
are used for fluid dynamics computation. Illustrative examples of the different patterns of flow field as obtained from the spatio-temporal distribution of velocity are given in figure 2. All along the isovolumetric contraction, we have observed a radial fluid displacement with small
velocity mainly due to the increase of the diameter and, during the ejection, a potential flow driven by the kinematics of the walls. Throughout diastole, we do observe during isovolumetric relaxation, an axial fluid displacement due to the increase of the cylinder length and, during the
filling phase, the radial displacement is a determining factor for the flow.
The global evolution pattern is very similar to what is measured by Doppler velocimetry.
Nevertheless, there is a discrepancy during the end of filling phase, essentially due to the fact
A B
Fig. 2. Maps of steamlines illustrating the dynamics of the flow at different phases of the cardiac cycle. A) Ejection t = lso ms, V~ II 8 cm/s, V~ =
3.4 cm/s, V~ = 4.3 cm/s and P (t
=
120 mmHg. B Rapid filling t soo ms, V~ 94 cm/s, V, = 3.7 cm/s, V~
= 2.3 cm/s and
P (t 3 mmHg.
126 JOURNAL DE PHYSIQUE III N° 6
that several effects are not taken into account (I) viscous effects and associated vortical pattern, and (it) effects due to presence of the valve.
3.3 PRESSURE FIELD. In figure 3, the distributions of pressure field are presented. It can be
noticed that the obtained pressure distribution is quasi-uniform inside the cavity and
particularly at the vicinity of the wall, in the presence of a large pressure gradient localized at the input/output a pressure gap of the order of 8 mmHg during the ejection phase and 3 mmHg during the filling phase was observed.
2.0
o-o
122 4
jf ~f
) )
e e
) I
I 1
Fig. 3. Pressure fields in half the cavity at the same instants as in the figure 2 on the left during the
ejection and on the right during the filling phase.
To verify that the uniformity of such a pressure field is maintained at every instant during the cardiac cycle, we have computed the maximal pressure difference AP (t along the endocardial wall. The results are presented in figure 4 and show that this pressure difference does not
exceed lmmHg during the cardiac cycle and that the relative pressure difference
(AP/P ) is always less than lo ilb, even during the filling phase, when the wall pressure drops
to 3 mmHg. This fact shows that, in the present model, there is a consistency between the linearization of structure mechanics which exerts a uniform pressure at the endocardium and
our assumption of unsteady potential flow in cylindrical geometry.
mmHg AP<t> AP<t>/P< t>
21 41 61 81 21 41 61 81
Fig. 4. -Absolute and relative maximal pressure difference along the endocardial wall during the cardiac cycle.
Conclusion.
Within our conjectural framework and the limits of the assumptions we made, I-e- the
cylindrical structure of the wall (made of a fibrous active material) which is uniformly pressurized under the dynamical action of an inviscid fluid, we have shown that a
« mechanically coherent » model of the ventricular hemodynamics was obtained under the action of a given biochemical activation function, the kinematics of the wall and the induced
motion of the fluid are such that the boundary conditions concerning normal stresses (pressure) conservation was constantly met.
Moreover, the results concerning the dynamics of the blood flow as viewed through
ultrasound velocimetric technology are buttressing those obtained with such a model, at least
during the ejection phase the ejection dynamics are such that the time evolution of the blood
velocity measured along the cavity axis does not display any phase shift characterizing a
propagation effect. Conversely, during the second part of the filling phase, the experimental
data are definitely showing such an effect which was not described by such a model. So far, it is then concluded that, owing to viscous and valve input effects, the model is not well adapted during the latter part of the filling phase. It is worth reminding here that such a model involves few parameters and can be easily processed using modest mathematical instruments, which is
paramount if we want to use such a model in clinical investigations.
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[3] Chadwick R. S., Mechanics of the left ventricle, Biophys J. 39 (1982) 279-288.
[4] Tozeren A., Static analysis of the left ventricle, J. Biomech. Eng. 105 (1982) 35-46.
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[6] Ohayon J. and Chadwick R. S., Theoretical analysis of the effects of a radia activation wave and
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