Vol. 89, Issue 1, 314-322, July 2000
Ventricular motion during the ejection phase: a computational
analysis
A.
Redaelli1,
F.
Maisano2,
J. J.
Schreuder2, and
F. M.
Montevecchi1
1 Department of Bioengineering and Centro di Bioingegneria e
Innovazioni Tecnologiche in Cardiochirurgia, Politecnico di Milano, and
Instituti di Ricovero e Cura a Carattere Scientifico San Raffaele,
and 2 Division of Cardiac Surgery Instituti di Ricovero e Cura a
Carattere Scientifico San Raffaele, 20133 Milan, Italy
 |
ABSTRACT |
In the present paper, the study of the ventricular motion
during systole was addressed by means of a computational model of
ventricular ejection. In particular, the implications of ventricular motion on blood acceleration and velocity measurements at the valvular
plane (VP) were evaluated. An algorithm was developed to assess the
force exchange between the ventricle and the surrounding tissue, i.e.,
the inflow and outflow vessels of the heart. The algorithm, based on
the momentum equation for a transitory flowing system, was used in a
fluid-structure model of the ventricle that includes the contractile
behavior of the fibers and the viscous and inertial forces of the
intraventricular fluid. The model calculates the ventricular center of
mass motion, the VP motion, and intraventricular pressure gradients.
Results indicate that the motion of the ventricle affects the
noninvasive estimation of the transvalvular pressure gradient using
Doppler ultrasound. The VP motion can lead to an underestimation equal
to 12.4 ± 6.6%.
fluid-structure interaction; Doppler ultrasound; pressure
gradients; blood acceleration
| |
INTRODUCTION |
DURING CARDIAC EJECTION, biochemical
energy is converted into mechanical energy by muscle fibers and
transferred to blood. The blood is then delivered to the systemic
circulation. The mass and momentum released with the outflowing blood
(stroke volume) are responsible for ventricular motion in the opposite
direction, similar to the recoil of a gun. This motion is restrained by
the inflow and outflow vessels, which constrain the ventricle. To the
best of our knowledge, few studies have been performed concerning this
phenomenon. The center of mass (CoM) motion has been investigated with
reference to the total heart (9) and the single-ventricle heart (7). These studies demonstrated that, during
systole, CoM motion is less than 3 mm and is directed toward the
ventricle base. Cao and colleagues (3a) have calculated a
maximal excursion of 10-20 mm for the valvular plane (VP) during
the cardiac cycle.
To determine the influence of ventricular motion on ventricular
mechanics, an algorithm based on the momentum equation for a transitory
flowing system was developed. The algorithm was integrated into an
axisymmetric fluid-structure model of the ventricle (4, 19) to calculate the CoM motion during ejection under
normal conditions.
Different elastic and viscous constraints were applied to the VP. The
elastic constraint was varied to modify the extent of the maximum
displacement. The viscous constraint was introduced to damp the
oscillations due to a purely elastic constraint. The model was used to
evaluate the effect of ventricular motion on the intraventricular
pressure gradients, which were demonstrated in a previous work to be
more sensitive to changes in ventricular contraction than the maximum
ventricular pressure and the first time derivative of ventricular
pressure (dP/dt) (20); furthermore, it was used
to assess the noninvasive Doppler estimation of ventricular performance
indexes, such as the transvalvular pressure drop and dP/dt.
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MATERIALS AND METHODS |
CoM motion algorithm.
The momentum equation for the ventricular CoM is written in scalar
terms according to the vector assignment of Fig.
1. The relative velocity of blood
wVP at the outlet is
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(1)
|
where vVP is the absolute velocity of
blood at the VP and uVP is the velocity of VP.
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Fig. 1.
Scheme adopted to describe the constraints acting on the
ventricle. The viscoelastic constraint is assumed to be linear and acts
on valvular plane (VP). z, Longitudinal coordinate;
r, radial coordinate; vCoM, velocity
of the ventricular center of mass (CoM); vVP,
absolute velocity of blood at VP; uVP,
velocity of VP; wVP, relative velocity of blood
at the outlet with respect to VP; sVP and
sCoM, displacements of VP and CoM, respectively;
F, force.
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The ventricle is defined by a control volume limited by the outer
surface of the ventricular wall and by the outflow area at VP, and its
mass varies with time. During systole, mass outflows through the VP;
M is the mass of the ventricle, including blood and wall,
and its time derivative divided by the density 
is the volumetric
flow entering the ventricle (
), which is the opposite
of the volumetric flow leaving the ventricle
|
(2)
|
where A is the orifice area.
The overall momentum variation for a constant mass system equals the
external forces. In this case, where a varying mass system is taken
into account, the momentum inside the control volume changes also
according to the momentum acquired (lost) throughout the mass entering
(leaving) at the VP; the additional term to be included in the momentum
equation is the unit volume momentum of the blood crossing the VP,
vVP, multiplied for the volumetric transfer
rate across the VP taken positive for entering flows (
AwVP). The term to be added is hence
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(3)
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Therefore the momentum equation is
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(4)
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Concerning the external forces, those due to the pressure acting
on the orifice area and to the stretching of the aorta in the
longitudinal direction can be assumed to be equal in value and
opposite. This assumption is rigorously true if a straight tube of
infinite length with zero pressure at the outlet is attached to the
ventricle; only two forces act on the blood flowing in the tube: the
force at the inlet due to the ventricular pressure and the shear
stresses at the blood-tube interface due to the viscous flow. The tube
is also subjected to two forces: the forces at the tube-ventricle
interface where the tube is constrained to the ventricular wall and the
shear stresses at the blood-tube interface. By combining these two
equilibrium relationships, the force at the tube-ventricle interface
actually equalizes the force due to the ventricular pressure. For this
reason, the only external forces to be considered acting on the
mechanical system are the forces due to the elastic constraint
(k) and to the viscous drag (
).
The first right term is responsible for the backward motion due to
blood ejection.
Equation 4 can be modified by applying the chain rule to the
time derivative of the left-hand term. When the mass derivative terms
in the right-hand side are grouped and the elastic and viscous forces
are explicitly stated, it becomes
|
(5)
|
where the instantaneous and reference (end-diastolic) locations
of the VP are sVP and
sVP0, respectively, and the elastic [k(sVPsVP)]
and viscous (uVP) constraints are assumed to
be linear.
Finally, by substituting Eq. 1 in the first
right term, the following equation is obtained
![M <FR><NU>∂v<SUB>CoM</SUB></NU><DE><IT>∂t</IT></DE></FR><IT>=<A><AC>M</AC><AC>˙</AC></A></IT>[(<IT>u</IT><SUB>VP</SUB><IT>−v</IT><SUB>CoM</SUB>)<IT>+w</IT><SUB>VP</SUB>]<IT>−k</IT>(<IT>s</IT><SUB>VP</SUB><IT>−s</IT><SUB>VP0</SUB>)<IT>−&eegr;u</IT><SUB>VP</SUB>](/content/vol89/issue1/fulltext/314/img006.gif)
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(6)
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stating that the ventricular CoM acceleration depends on the
relative velocity of CoM with respect to VP (vVP
vCoM), the outflow velocity
wVP, and the mechanical constraints.
By combining Eqs. 2 and 6, the following equation
is obtained
|
(7)
|
which is the algorithm used in this study.
Fluid-structure interaction model.
Equation 6 was integrated in an axisymmetric model of the
left ventricle described in detail in previous publications
(19, 20) and depicted in Fig.
2. The model accounts for fluid-structure interaction between the intraventricular fluid and the myocardial wall.
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Fig. 2.
A: axisymmetric geometry of the left
ventricular model bounded by the myofibers; ef,
unit vector in the fiber direction. B: mesh of the fluid
domain. C: boundary conditions of the fluid domain;
s and v are the wall displacement and velocity,
respectively, calculated as the sum of the contraction of the wall and
VP motion. vr, Fluid velocity radial component;
paor, aortic pressure at the VP; t, time.
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In the reference state, the geometry of the left ventricle is assumed
to be a prolate ellipsoid with a minor semiaxis equal to 25 mm, a major
semiaxis equal to 53 mm, and an inner volume of 113 cm3
according to normal human values (5, 23). The
ellipsoid is truncated at a distance of 79.9 mm from the ventricular
apex. The ventricle wall mass is 300 g with a uniform wall
thickness. The heart rate is assumed to be 72 beats/min.
The intraventricular blood behavior is simulated by means of the
FIDAP general-purpose fluid dynamics code (Fluent, Lebanon, NH). The main assumptions are that the fluid is incompressible, homogeneous, and Newtonian ( = 1.06 × 103
kg/m3, µ = 3 × 10
3 kg · m1 · s), the walls are impermeable, and no slip
occurs at the wall. Gravitational effects are ignored because a supine
position is assumed. The fluid-dynamics mesh consists of 1,251 nodes;
nine-node (quadrilateral) isoparametric annular elements are used to
discretize the fluid domain. The full Navier Stokes equation (including
the transient term) and the continuity equation in their axisymmetric formulation are solved by using Galerkin's weighted residual approach in conjunction with finite element approximation. To account for myocardial inner surface movement, an additional free surface degree of
freedom is taken into account in accordance with Saito and Scriven's
method (21). This method requires the assignment of
displacement and velocity for each moving boundary node through time
functions. A quasi Newton solver was used for the solution method. The
time-integration technique adopted was the implicit backward Euler with
a fixed time step of 1 ms.
The wall stress and motion were calculated by means of a structural
model, constructed on the basis of the modified version of Wong's
sarcomere model (27) proposed by Montevecchi and
Pietrabissa (12) and Pietrabissa et al. (16).
A detailed description of the sarcomere model is given in the
APPENDIX. Sarcomeres are arranged to constitute a
thin-shell model composed of two sets of counterrotating fibers.
When gravitational effects and local wall inertia and viscosity are
ignored, the equation of equilibrium is
|
(8)
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where 
is the Cauchy stress tensor and is defined with
respect to the global reference frame. If f is the
stress in the fiber direction ef (Fig. 2), the
Cauchy stress tensor is obtained by rotation of the local reference
frame using the following equation
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(9)
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where the term
efefT is introduced
to refer to the global reference frame.
The fiber thin shell is discretized into five three-node axisymmetric
elements, with a total number of nodes equal to 11. Nodal forces and
displacements are calculated by integrating Eq. 7 throughout
each element, approximating the strain and stress fields quadratically.
The boundary pressure field is also approximated quadratically. A
solver based on a multiplane quasi-Newton method proposed by Buzzi
Ferraris and Tronconi (3) was used for the solution of the
wall motion.
To calculate the aortic impedance, a lumped parameter model of the
systemic circulation was used and integrated with the structural model.
It supplies the aortic pressure at the outlet section (aortic valve
plane) on the basis of the ventricular outflow. The lumped parameter
values adopted are reported in a paper by Avanzolini et al.
(1). A fourth-order Runge-Kutta method was used to solve the model with a fixed time step equal to 1 ms; the stability of the
Runge-Kutta algorithm was evaluated by comparing the results obtained
with different time steps (range 0.1-10 ms). Four cycles were
performed to allow the lumped parameter model to reach the proper
initial conditions.
Interface and solution procedures.
The calculation scheme is based on an uncoupled approach
(4). On the basis of a tentative intraventricular pressure
distribution on the moving boundary, the fiber shortening is calculated
at the 11 nodes of the thin shell model to satisfy equilibrium: the fiber shortens so that, at these nodes, the force generated by the wall
balances the force generated by the intraventricular fluid forces. The
fluid dynamic stress tensor (
ij), calculated
by FIDAP at each node of the boundary, has the following form
|
(10)
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where 
ij is the Kroeneker delta, P is
the pressure, v is the velocity, and the indicial notation
i,j denotes the adopted coordinate system (r and
x, Fig. 2). For convenience, only the pressure term was
taken into account, because the viscous forces are two orders of
magnitude lower than the isotropic pressure (8,
22). The local pressure value is given by the summation of
the imposed local pressures plus the pressure calculated at the VP by
means of the lumped parameter model. The thin-walled model provides the
estimation of the wall motion as well as the relative velocity and
displacement between CoM and VP and wVP, which
are the inputs of Eq. 6. The last unknown in Eq. 6, uVP, is calculated from Eq. 1.
The wall motion and uVP are the boundary
conditions (Fig. 2) for the fluid dynamic model, which provides a
new intraventricular pressure distribution along the boundary
(PFIDAPi; i = 1, ... , number of nodes of the
structural model). The latter is compared with the tentative one
(Pi). If the difference is within the tolerance
limit (0.1 mmHg), the time is incremented and a new time step is
performed. Otherwise the tentative values for stress distribution are
recomputed as
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(11)
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and the time step calculations are performed again.
Computations were performed with an IBM RISC/6000 model 230 workstation. Computational time for each iteration was 105 s, for a total number of iterations per time step of ~10 iterations (800 
1,200 s per step). The adopted time step was 1 ms with a
total number of ~140 time steps to simulate the ejection phase.
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RESULTS |
Two simulation sets were performed to investigate the role of the
elastic and viscous constraints, respectively. In the first set, two
simulations were performed with two different values for the elastic
constraint k (200 and 800 kg/s2) and without the
viscous constraint (equal to 0 kg/s). As control cases, two
simulations have also been performed by leaving the VP free and
maintaining VP fixed, respectively. In the second set, the influence of
the viscous constraint was studied by comparing the results of two
simulations performed with two different values of (5 and 10 kg/s)
and leaving k fixed equal to 200 kg/s2. The
simulation of the previous set with k equal to 200 kg/s2 and equal to 0 kg/s was adopted as the control
case for this set.
The values for k were chosen to obtain maximum VP
displacements of ~8 mm in the case of k = 200 kg/s2 and 2 mm in the case of k = 800 kg/s2 (3a). The values for were chosen
iteratively to halve the displacements of VP and CoM.
The overall features of the ventricular performance did not vary
significantly throughout the simulations: the ventricular stroke was
~58.4 ± 0.3 cm3 (mean ± SD) with an ejection
fraction equal to 51.3 ± 0.2%. The pressure at the VP
(P0) varied in the range of 77 to 133.6 ± 0.6 mmHg.
In turn, the intraventricular pressure gradients and VP and CoM motion
varied appreciably as discussed below.
In Fig. 3, intraventricular pressure
difference (IPD) time courses are given. They are calculated in four
points (16, 32, and 48 mm from the VP and in the ventricular apex) as
the difference between the local pressure (Pi)
and P0. IPD is shown because it was demonstrated in a
previous paper that it is the most sensitive index of ventricular
contraction (20). Results are reported with reference to
the two elastic constraints (200 and 800 kg/s2) and with
free and fixed VP. IPDs show an oscillating pattern that fades during
ejection, with the highest peak occurring in the earlier phase when
blood acceleration is maximum. As previously reported (19,
20), the computed IPD peak values at the ventricular apex
are similar to IPD values observed in vivo (6,
15, 18). The oscillating pattern is more
pronounced than in in vivo observations (14,
15) because of the viscous characteristic of the wall, which has not been accounted for in the present study
(19). The IPDs in the four panels are similar and scaled
moving from the ventricular apex toward the base. The IPD patterns are
driven by ventricular contraction and VP and CoM motion. Their
interpretation is not trivial; nevertheless, some considerations may be
argued about the effects of k on IPD when the acceleration
phase (the first 50 ms of the ejection phase) is considered. In this
phase, two peaks occur. The effect of k is evident on the
second peak with reference to the IPD time courses of the ventricles
with free VP, k equal to 200 kg/s2, and
k equal to 800 kg/s2; a lower k
promotes the motion of the ventricle in the direction opposite to the
flow, thus facilitating the ventricular ejection and decreasing the
value of the IPD peak. During the first peak, this occurrence is not
detectable because the VP motion is still small and the forces due to
the constraints are negligible. In turn, when the VP motion is impeded,
a higher IPD (first) peak occurs along with a delay of the IPD
oscillation. The difficulties encountered by the ventricle with fixed
VP in the initial phase are detectable also by another occurrence: the
comparison of an IPD peak at 30 ms close to the VP (Fig. 3,
C and D). This occurrence is due to the fact that
the wall contraction occurs mainly in the region close to the
ventricular base where blood inertia is lower (19).
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Fig. 3.
Time courses of the pressure (P) drop
evaluated between VP and four locations lying on the ventricle symmetry
axis at the apex (A), and at 48 (B), 32 (C), and 16 mm (D) from VP. Curves refer to
simulations with elastic constraint (k) = 200 and 800 kg/s2. Two control cases were also considered for which VP
was fixed and left free, respectively. In A-D,
the intraventricular pressure difference (IPD) patterns are similar,
and the IPD values are scaled moving toward the ventricular base. The
IPD pattern in the case of fixed VP differs significantly from the
other cases. This is due to the difficulties encountered by the
ventricle in the initial phase of the ejection which cause a delayed
and higher peak of the IPD in this phase (see the text for
details).
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Figure 4 shows the displacements
and the velocities of CoM and VP, respectively. At free VP, both CoM
and VP displace increasingly throughout the ejection phase. In the two
cases with elastic constraint, the motion of VP is initially negative.
The motion of CoM is positive and goes toward VP because of the
decrease of the ventricular volume. In middle/late systole, the
behavior is different, depending on the value of k. The
higher k is, the earlier the reversion of motion.
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Fig. 4.
Time courses of s and v at CoM
(A and C, respectively) and VP (B and
D, respectively). Curves refer to simulations with
k = 200 and 800 kg/s2. Two control cases
were also considered for which VP was fixed and left free,
respectively.
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In the second simulation set, the effects of a viscous constraint were
investigated. Results obtained with two values of 5 and 10 kg/s,
respectively, were compared with those obtained without viscous
constraint. The value of k was fixed equal to 200 kg/s2. Also in this simulation set, the overall features of
the ventricular performance did not vary significantly throughout the
simulations and from the previous set. The effect of the viscous
constraint on IPDs is reported in Fig. 5.
As expected, a value of different from 0 decreases the oscillations
of IPD according to a damping behavior. This occurrence is discernible
with reference to the two negative peaks in the accelerative phase and
the late systole. Also CoM and VP displacements and velocities in the
direction opposite to blood outflow decrease when the value is
increased according to a damping behavior as reported in Fig.
6.
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Fig. 5.
Time courses of the pressure drop evaluated between VP
and four locations on the ventricle symmetry axis at the apex
(A), and at 48 (B), 32 (C), and 16 mm
(D) from VP. Curves refer to simulations with viscous
constraint () = 0.5 and 10 kg/s. k = 200 kg/s2. Damps IPDs oscillations in particular with
reference to the two initial negative peaks.
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Fig. 6.
Time courses of s and v at CoM
(A and C, respectively) and VP (B and
D, respectively). Curves refer to simulations with = 0.5 and 10 kg/s. k = 200 kg/s2. Maximum
displacements and velocities in the direction opposite to blood outflow
decrease when the value increases.
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DISCUSSION |
In the present study, a model, described in detail previously
(4, 19), was used to test the motion of the
ventricular CoM. For this purpose, an algorithm was implemented to take
into account the balance of momentum with external k and constraints. Different values of k and were used to
simulate the VP motion within the physiological range.
IPD and VP and CoM motions in early systole.
During the first 20 ms of ejection, no changes can be noted when
the mechanical features of the constraints are varied. This can be
attributed to the fact that the elastic and viscoelastic constraints
are not significantly loaded. A different behavior occurs when VP is
fixed; in this case higher values of the first peak were calculated at
the apex. In the other simulations, VP shows a little movement toward
the apex during early ejection; this movement facilitates the ejection
and reduces the initial pressure peak significantly. A fixed VP
requires the generation of a higher fiber force and evokes a different
dynamic behavior throughout ejection by increasing both the first IPD
peak and the period of IPD oscillation.
IPD and VP and CoM motions in late systole.
In the subsequent part of the ejection phase, the time courses of VP
and CoM motions and of IPD differ significantly; this allows evaluation
of the effects of the constraints. For instance, a high value for
k decreases VP and CoM motion in the direction opposite to
blood outflow and increases the oscillating frequency of
sVP (Fig. 4). Furthermore, the second positive
peak of IPD increases markedly (Fig. 3). This phenomenon, however, has
not been observed in in vivo recordings (14,
15), thus suggesting an inhibition either by a
viscoelastic constraint or by a nonlinear behavior of the elastic
constraint. Indeed, the values of used in the present study
decrease the magnitude of IPD oscillations (Fig. 5). In general, the
IPD course and CoM and VP motions are damped by increasing the value of
. Furthermore they become positive sooner (Figs. 4 and 5). However,
this is not the case for the first IPD peak in the early ejection phase
where the velocities are still too low to generate a substantial
viscous force.
Doppler velocimetry implications.
There are at least two clinical implications of the present results.
They concern the Doppler estimation of the acceleration and velocity of
the outflowing blood used for the noninvasive estimation of ventricular
performance. The first implication is the estimation of the maximum
dP/dt (dP/dtmax) (2,
24) through the measurement of the maximum aortic
acceleration by using the formula dP/dtmax = c dw/dtmax, where is the density and c is the velocity of the pulse wave;
second is the estimation of the transvalvular pressure gradients by
means of the simplified Bernoulli formula (10,
13, 28) IPDtrans = 4wmax2 where IPDtrans (mmHg) is
the estimated transvalvular gradient and wmax
(cm/s) is the maximum outflow blood velocity. Doppler velocimetry
measures the absolute velocity and acceleration in a selected frame,
usually located close to VP, thus accounting for both blood and VP
velocities and accelerations. When the two velocities (accelerations)
have opposite directions, Doppler ultrasound will underestimate the
blood outflow velocity (acceleration); it will overestimate when both
are in the same direction. This inaccuracy is not known a priori
because VP motion depends on the elastic features of the fibers and the
constraints. In the case of blood acceleration, the peak occurs at the
very beginning of the ejection phase concomitant with the VP
acceleration peak that has an opposite direction; however, simulation
results show that VP acceleration is ~1.2 ± 0.4% (mean ± SD)
of the blood outflow acceleration; hence it does not significantly
affect dP/dt estimation. In turn, as far as the
transvalvular pressure gradient estimation is concerned,
wmax occurs ~40 ms after the valve
opening. Also in this case, the outflow and VP velocity have opposite
directions because the ventricular CoM moves backward (as does the VP);
this would imply that Doppler ultrasound estimates a value for the velocity peak that is lower than the blood outflow value. Simulations show a difference between the blood outflow velocity
wmax and the absolute velocity
vmax equal to +6.2 ± 3.3% with respect to blood outflow velocity, depending on the constraints. This leads to an
underestimation of transvalvular pressure gradients equal to 12.4 ± 6.6%, compared with recordings in which solid-state pressure sensors
are used.
Conclusions.
This paper describes the physics of a ventricle-like model. The model
accounts for blood-ventricular wall interaction by assuming axisymmetry
and the absence of fiber-damping behavior. As discussed previously (19, 20), the model is not able to
illustrate local fiber impairment and flow field asymmetry and
overestimates the oscillatory pattern of IPD in late ejection. In the
present study, the model was modified to simulate the
ventricle-external structure interaction, with the assumption that the
vessels connected to the ventricle have a linear viscoelastic behavior.
Results show that 1) ventricular motion aids early ejection
blood outflow; 2) the first intraventricular pressure peak
is not affected by VP motion except when VP motion is neglected;
3) the intraventricular pressure gradient time course is, in
turn, sensitive to ventricular motion; 4) the ventricular
motion does not affect ejection in terms of overall performance; and
5) VP velocity is comparable to blood outflow velocity and
can give inaccurate Doppler estimation of transvalvular pressure
gradients, whereas VP acceleration is negligible with respect to blood acceleration.
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APPENDIX |
According to Wong's model (27), the behavior of
the sarcomere is defined by the sum of two terms representing the
behavior of two nonlinear elastic springs, the former in series (SE)
with the contractile unit (CE) and the latter in parallel (PE) with the
complex SE-CE. The first term accounts for the passive elastic characteristic of the myofiber (PE); the second term differs from zero
during contraction and accounts for the active elastic response (SE)
![F<IT>=</IT>F<SUB><IT>0</IT></SUB><IT>·</IT>[e<SUP><IT>k</IT><SUB>p</SUB>(<IT>l<SUB>s</SUB>−l<SUB>s0</SUB></IT>)</SUP><IT>−1</IT>]<IT>+</IT>F<SUB>L</SUB><IT>·</IT>(e<SUP><IT>k<SUB>s</SUB>&Dgr;l</IT><SUB>SE</SUB></SUP><IT>−1</IT>)](/content/vol89/issue1/fulltext/314/img013.gif)
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(A1)
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where FL is the preload, i.e., the elastic response
of the parallel spring, PE, just before the contraction phase;
ls0 is the resting length of the
sarcomere; and F0, kp, and
ks are constants whose values are 0.0317, 0.9155/mm, and 0.4254, respectively (for a sarcomere resting length of
6.8 mm). The forces are normalized with respect to the maximum
isometric force, as reported in Wong's paper (27).
The 
lSE is defined as the sum of the
sarcomere shortening
(lsls0) and
the contractile unit shortening lCE and is
normalized with respect to the difference between the sarcomere length
at which maximum tension is generated and its resting length
(12).
CE contracts with time according to a modified version of Julian's
(11) model of skeletal muscle activation. This curve is
parametric with respect to the frequency and the maximum shortening of
CE (lCE max).
To calculate the value of lCE max, the
following two steps are necessary. First, the value of
FCE max is computed: the relationship between the
normalized maximum tension FCE max and the preload
(lCE)t=0 is calculated according
to the method of Pollack and Krueger (17). Then the
relationship between FCE max and
lCE max is computed from Eq. A1
in the hypothesis of isometric contraction. According to this
hypothesis, the preload is
![F<SUB>L</SUB><IT>=</IT>F<SUB><IT>0</IT></SUB><IT>·</IT>[e<SUP><IT>k</IT><SUB>p</SUB>(<IT>l<SUB>s</SUB>−l<SUB>s0</SUB></IT>)</SUP><IT>−1</IT>]](/content/vol89/issue1/fulltext/314/img014.gif)
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(A2)
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and lSE is equal to the shortening of
CE (lCE). When lCE
is isolated from Eq. A1, the following relationship is
obtained
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(A3)
|
In the hypothesis of incompressible material, the relationship
between the force (F) exerted by the sarcomere, normalized with respect
to the corresponding maximum isometric force (FIso max), and the normalized fiber stress (/Iso max) is
|
(A4)
|
where lsIso max is the
preload length corresponding to the maximum isometric force.
In the calculations, the effective normalized stress defined as
eff = 
· is used instead of . Is the fiber number that varies with respect to the section considered.
In the reference condition, ef and are
calculated for the 11 mesh nodes under the assumption of an
intraventricular pressure of 7 mmHg and a fiber preload length equal to
the length that gives the maximum isometric force (27,
29).
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FOOTNOTES |
Address for reprint requests and other correspondence: A. Redaelli, Dipartimento di Bioingegneria, Politecnico di Milano, P.za L. da Vinci, 32, 20133 Milano, Italy (E-mail:
redaelli{at}biomed.polimi.it).
The costs of publication of this
article were defrayed in part by the
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REFERENCES |
1.
Avanzolini, G,
Barbini P,
Cappello A,
and
Cevenini G.
CADCS simulation of the closed-loop cardiovascular system.
Int J Biomed Comput
22:
39-49,
1988[Medline].
2.
Bennett, ED,
Barclay SA,
Davis AL,
Mannering D,
and
Metha N.
Ascending aortic blood velocity and acceleration using Doppler ultrasound in the assessment of left ventricular function.
Cardiovasc Res
18:
623-638,
1984.
3.
Buzzi Ferraris, G,
and
Tronconi E.
Bunsli: a FORTRAN program for solution of system of nonlinear algebraic equations.
Comput Chem Engng
2:
129-141,
1986.
3a.
Cao, T,
Shapiro SM,
Bersohn MM,
Liu SC,
and
Ginzton LE.
Influence of cardiac motion on Doppler measurements using in vitro and in vivo models.
J Am Coll Cardiol
22:
271-276,
1993[Abstract].
4.
Dubini, G,
and
Redaelli A.
Different finite element approaches to fluid-structure interaction problems in biofluid mechanics.
Ann Biomed Eng
25:
218-231,
1997[ISI][Medline].
5.
Erbel, R,
Henkel B,
Ostlander C,
Clas W,
Brennecke R,
and
Meyer J.
Normalwerte fur die zweidimensionale Echokardiographie.
Dtsch Med Wochenschr
110:
123-128,
1985[Medline].
6.
Falsetti, HL,
Verani MS,
Chen CJ,
and
Cramer JA.
Regional pressure differences in the left ventricle.
Cathet Cardiovasc Diagn
6:
123-134,
1980[ISI][Medline].
7.
Foegel, MA,
Weinberg PM,
Fellows KE,
and
Hoffman EA.
Magnetic resonance imaging of constant total heart volume and center of mass in patients with functional single ventricle before and after staged fontan procedure.
Am J Cardiol
72:
1435-1443,
1993[ISI][Medline].
8.
Georgiadis, JG,
Mingyu W,
and
Pasipoulardies A.
Computational fluid dynamics of left ventricular ejection.
Ann Biomed Eng
20:
81-97,
1992[ISI][Medline].
9.
Hoffman, EH,
Rumberg J,
Dougherty L,
Reichek N,
and
Axel A.
A geometric view of cardiac efficiency (Abstract).
J Am Coll Cardiol
13:
86A,
1989.
10.
Holen, J,
Aaslid R,
Landmark K,
and
Simonsen S.
Determination of pressure gradient in mitral stenosis with a noninvasive ultrasound Doppler technique.
Acta Med Scand
199:
455-460,
1976[ISI][Medline].
11.
Julian, EF.
Activation in skeletal muscle contraction model with a modification for insect fibrillar muscle.
Biophys J
9:
547-570,
1969.
12.
Montevecchi, FM,
and
Pietrabissa R.
A model of multicomponent cardiac fibre.
J Biomech
20:
365-370,
1987[ISI][Medline].
13.
Panza, JA,
Petrone RK,
Fanananpazir L,
and
Maron BJ.
Utility of continuous wave Doppler echocardiography in the noninvasive assessment of left ventricular outflow tract pressure gradient in patients with hypertrophic cardiomyopathy.
Am J Cardiol
19:
91-99,
1992.
14.
Pasipoularides, A.
Clinical assessment of ventricular ejection dynamics with and without outflow obstruction.
J Am Coll Cardiol
15:
859-882,
1990[Abstract].
15.
Pasipoularides, A,
Murgo JP,
Miller JW,
and
Craig WE.
Nonobstructive left ventricular ejection pressure gradients in man.
Circ Res
61:
220-227,
1987[Abstract/Free Full Text].
16.
Pietrabissa, R,
Montevecchi FM,
and
Fumero R.
Mechanical behaviour of a model of a multicomponent cardiac fibre.
J Biomed Eng
13:
407-413,
1991[ISI][Medline].
17.
Pollack, GH,
and
Krueger JW.
Myocardial sarcomere mechanics: some parallel with skeletal muscle.
In: Cardiovascular System Dynamics, edited by Baan Y,
Noordergraff A,
and Raines J.. Cambridge, MA: MIT Press, 1978, p. 3-10.
18.
Ray, G,
Ghista DN,
and
Sandler H.
Left ventricular analysis for characterizing normal and diseased myocardium.
Adv Cardiovasc Phys
4:
161-178,
1979.
19.
Redaelli, A,
and
Montevecchi FM.
Computational evaluation of intraventricular pressure gradients based on a fluid-structure approach.
J Biomech Eng
118:
529-537,
1996[ISI][Medline].
20.
Redaelli, A,
and
Montevecchi FM.
Intraventricular pressure gradients dp/dx and aortic blood acceleration df/dt as indexes of cardiac inotropy: a comparison with dp/dt based on computer fluid dynamics.
Med Eng Phys
20:
231-241,
1998[ISI][Medline].
21.
Saito, H,
and
Scriven LE.
Study of coating flow by the finite element method.
J Comp Phys [A]
42:
53-76,
1981.
22.
Schoephoerster, RT,
Silva LC,
and
Ray G.
Evaluation of left ventricular function based on simulated systolic flow dynamics computed from regional wall motion.
J Biomech
27:
125-136,
1994[ISI][Medline].
23.
Semelka, RC,
Tomei E,
Wagner S,
Mayo J,
Kondo C,
Suzuki J,
Caputo GR,
and
Higgins CB.
Normal left ventricular dimensions and function: interstudy reproducibility of measurements with cine MR imaging.
Radiology
174:
763-768,
1990[Abstract/Free Full Text].
24.
Senda, S,
Sugawara M,
Matsumoto Y,
Kan T,
and
Matsuo H.
A noninvasive method of measuring max(dp/dt) of the left ventricle by Doppler echocardiography.
J Biomech Eng
114:
15-19,
1992[ISI][Medline].
25.
Sponitz, HM,
Sonnenblick EH,
and
Spiro D.
Relation of ultra-structure to function in intact heart. Sarcomere structure relative to pressure-volume curves of intact left ventricle of dog and cat.
Circ Res
18:
49-66,
1966[Abstract/Free Full Text].
27.
Wong, AYK
Application of Huxley's sliding filament theory to the mechanics of normal and hypertrophical cardiac muscle.
In: Cardiac Mechanics, edited by Mirsky I.. New York: Wiley, 1974, p. 411-437.
28.
Yoghanathan, AP,
Valdes-Cruz LM,
Schmidt-Dohna J,
Jimoh A,
Berry C,
Tamura T,
and
Sahn DJ.
Continuous-wave Doppler velocities and gradients across fixed tunnel obstructions: studies in vitro and in vivo.
Circulation
76:
657-666,
1987[Abstract/Free Full Text].
29.
Yoran, C,
Covell JW,
and
Ross J.
Structural basis for the ascending limb of left ventricular function.
Circ Res
32:
297-303,
1973[Abstract/Free Full Text].
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ABSTRACT
INTRODUCTION
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