A simulation of three-dimensional systolic flow dynamics in a spherical ventricle: Effects of...

Annals of Biomedical Engineering. Vol. 24, pp. 48-57, 1996 0090-6964/96 $10.50 + .00 Printed in the USA. All rights reserved. Copyright �9 1995 Biomedical Engineering Society

A Simulation of Three-Dimensional Systolic Flow Dynamics in a Spherical Ventricle: Effects of Abnormal Wall Motion

ERICK GONZALEZ and RICHARD T. SCHOEPHOERSTER

Mechanical Engineering Department, Florida International University, Miami, FL 33199

Abstract--Alterations in left ventricle (LV) wall motion induced by ischemia will affect flow dynamics, and these altered flow fields can be used to evaluate LV pumping efficiency. LV chamber flow fields were obtained in this study by solving the discretized three-dimensional Navier-Stokes equations for viscous, incompressible unsteady flow by using the finite analytic method. Several cases of abnormal wall motion (AWM) were simulated by a manipulation of the boundary conditions to produce regions of hypokinesis, akinesis, and dyskinesis. These solutions were used to determine the central ejection region (CER), defined as the region of flow domain in which the obtained velocity field vectors are aligned -+ 3 ~ from the LV long axis. A CER coefficient was computed from information on the location and orientation of the CER within the LV cavity. Con- traction of the spherical ventricle produced a vector field that was symmetric with respect to the long axis. For the simulations of AWM, an asymmetrical flow pattern developed, became more pronounced with increasing severity of AWM, and resulted in a shorter CER that shifted toward the ischemic region. The CER coefficients decreased monotonically with increased severity in AWM from 0.948 in the normal case to a low of 0.164 for the most severe case of AWM. The CER coefficient quantitatively displayed the sensitivity of the flow patterns to even moderate degrees of hypokinesis. In addition, visualization of the three- dimensional flow field reinforced the necessity of three-dimensional simulations to capture aspects of the flow that existing methods of two-dimensional flow imaging that use ultrasound may miss.

KeywordsmLeft ventricular function, Intraventricular flow patterns, Central ejection region, Intraventricular pressure gradient, Abnormal wall motion.

INTRODUCTION

Ischemic coronary artery disease often manifests itself in the form of abnormal wall motion (AWM) of the left ventricle (LV) during systolic ejection as a result of a lack of blood supply to the myocardium in the affected region. This lack may result in localized regions of hypokinetic, akinetic, and dyskinetic LV wall motion. The degree to

Acknowledgment--This work was supported by a research grant from The Whitaker Foundation.

Address correspondence to Richard T. Schoephoerster, Mechanical Engineering Department, Florida International University, Miami, FL 33199, U.S.A.

(Received 12Jun95, Revised 23Aug95, Revised, Accepted 8Sep95)

which this dysfunctional contraction affects the efficiency of the LV to eject its contents and thereby maintain ade- quate blood flow traditionally has been evaluated through global parameters of ejection fraction and temporal pressure gradients (dp/dt) (7). In addition, magnitudes of spatial pressure gradients along the LV long axis have been correlated with the strength of contractions (8,14). How- ever, global parameters of LV function often are not ad- equate in characterizing the state of ischemic coronary artery disease because of the local nature of the disease in its early stages and compensatory actions in the unaffected regions of the myocardium (18,20). Attempts to localize ischemic regions of the myocardium and quantify the degree of ischemia rely on various techniques of imaging LV wall motion, including echocardiography, angiography, computed tomography, and magnetic resonance. Much of the work in this area has been directed toward detailing the actual movement of the LV wall (6,12,13,19) and toward basing evaluative parameters on regional shortening. These parameters convey information on the localized strength of contractions, but do not provide a quantitative index of global LV function on the basis of regional wall motion.

The LV chamber flow certainly is influenced by the regional motion of the LV wall. Alterations in LV wall motion will affect chamber flow dynamics, and these altered flow fields can be used to evaluate LV pumping efficiency. Recently, we have developed a two-dimensional numerical model of LV systolic flow dynamics (16,17), and introduced a new parameter to assess LV function; this parameter is the central ejection region (CER), which is determined from the chamber flow patterns. The CER coefficient seems to provide a qualitative and quantitative global measure of LV function that regional wall motion analysis alone cannot provide, and is a parameter that is sensitive to regional and temporal AWM and the resultant compensatory actions, which cannot be detected by global parameters. However, for proper diag- nosis, evaluation, and localization of the ischemic region along the three-dimensional outline of the myocardium, a three-dimensional approach to the problem is required. Therefore, we have extended our analysis to three dimen-

48

Three-Dimensional Simulation of Systolic Flow Dynamics 49

sions and present herein our initial results for a spherical model.

METHODS

Model Description

The geometry for the problem at hand is the three- dimensional surface of the LV cavity. For initial model development and validation, the LV was assumed to be spherical in shape and to have a single outlet port (Fig. 1). Because we are interested only in the contraction charac- teristics of the LV, only the systolic portion of the heart cycle was modeled. The blood was assumed to behave as a Newtonian fluid, and the flow was assumed to be laminar. Under these assumptions, the governing equations for the flow within the LV cavity were the equation of continuity (Eq. 1) and the Navier-Stokes equations (Eqs. 2 through 4) for incompressible, unsteady flow with constant properties:

au Ov aw - - + + = 0 ( l ) ox gy-gz

Ou Ou Ou Ou at + U -~x + V ~y + w oz -

_ ---- ( eq2u 02U 02U~ 1 Op + v + - - + -~z2] (2) pox \-~X 2 Oy 2

Ov 3v av Ov ~ y - - ot + U-~x + v + W oz

{ a2v 027 02v~ - 1 - 3 P + v ! k ' ~ x 2 + +

P Oy ~y2 -~z 2} (3)

aw Ow Ow Ow - - + u + v + w - Ot -~x ~ y Oz

o2w -LOP+ \ox 2 + +

ooz v T V ) (4)

The wall boundary conditions were no-slip and imperme- able and had an imposed velocity computed from the LV wall motion. The boundary condition for the outlet was a uniform velocity distribution determined from conserva- tion of mass within the LV chamber, The flow initially was (end-diastole) assumed to be stationary. Motion of the LV wall was obtained from successive three-dimensional outlines of the model LV shown in Fig. 1 in which a small reduction in radius R was made with each time step. This represents the form of the input data for this computational model, which is the three-dimensional surface of the LV endocardium at discrete time steps within the systolic phase of the cardiac cycle.

The heart, in its pumping action, is not rigidly con- strained within the pericardium and, therefore, undergoes some amount of rigid body motion combined with the contraction of the heart walls. In computing wall veloci-

U outlet

lllllll,lllll 0.436R

I Rq

I Uwall

FIGURE 1. Three-dimensional spherical model of the LV, including wall and inlet velocity boundary conditions.

ties for the determination of boundary conditions, this rigid body motion of the heart was accounted for in our numerical model by using the center of the outlet as the inertial reference point, as is standard practice (3). There- fore, the LV cavity was assumed to be tethered at this point, and all wall and fluid movement was computed relative to this point.

The wall velocity boundary conditions were obtained as follows. The flow domain was divided into a 24 x 24 x 24 uniform grid mesh. On the basis of the chamber surface shape at end-diastole, each node was designated as either a wall node or an active node (interior flow domain) depending on its proximity to the LV chamber outline via a "stair-stepping" approximation. This approach was vi- able because the flow in general was perpendicular to the wall at any given point, and a detailed flow analysis near the wall was not necessary. The velocity boundary condition for each interior wall node that describes the LV cavity boundary was obtained by using the next successive chamber three-dimensional surface shape. The wall node was assumed to move to the closest point on the next chamber surface, and, therefore, the average velocity of the wall at that particular nodal location over the time period between successive LV outlines was determined by dividing this distance by the time increment between successive chamber outlines. The assumption that any point on the wall moves to the next closest point on the outline obtained from the succeeding time step was used for the following reasons. It was very difficult to determine the

50 E. GONZALEZ and R. T. SCHOEPHOERSTER

actual corresponding points from one outline to another because of the irregularity of LV wall movement. If the time step from one frame to the next, At, was small (and, therefore, &x), the error introduced by using the vector to the next closest point instead of the actual corresponding point on the ensuing outline was negligible compared with errors introduced by the imaging process itself in deter- mining the LV cavity outline.

Numerical Procedures

The numerical solution was obtained by discretizing the governing differential equations by using the finite analytic (FA) method (2). In the FA formulation, the discretized algebraic equation for each small element was obtained from the local analytic solution of the partial differential equations goveming the physical systems. The principle and procedures used in obtaining the FA solutions are illustrated in brief below.

The FA method was used to generate an algebraic representation of the momentum equations (Eqs. 2, 3, and 4) used to solve for U, V, and W, and a staggered grid was used to allow for generation of a pressure equation from the continuity equation (Eq. 1). We first considered Eq. 2 (or Eq. 3 or Eq. 4), which can be expressed in the following dimensionless form:

OU OU OU OU - - + U + V + W -

ot -@ oz

1 { O2U 02U 02U~ \ (5)

wherefis the pressure gradient, an inhomogeneous source term. Equation 5 was solved numerically with proper boundary and initial conditions. When using the FA method, the problem is first subdivided into small three- dimensional elements. An analytic solution for the partial differential equation (PDE) may then be obtained in each such element.

Because Eq. 5 was nonlinear, it was linearized locally in the element by approximating the convective velocities U, V, and W and the source term f with Up, Vp, Wp, and fp, which were constant over the element and were known from the previous time step. In this fashion, the overall nonlinear effect could still be approximately preserved by assembly of the local analytic solutions. Equation 5 can be simplified further by replacing the unsteady term with a simple backward finite difference formula. In this way, the unsteady momentum equation can be reduced to a quasisteady PDE, and the unsteady term is absorbed into a constant source term g. The results of these simplifica- tions is a PDE of the form that follows:

OU 8U OU O2U c?2U 02U 2A + 28 + 2 c 0z - 0x 2 § + g

(6)

where A, B, and C are constants. The analytic solution of this PDE resulted in an algebraic equation in which the coefficients were determined from nodal boundary conditions. A three-dimensional 19-point element was used (Fig. 2), and the discretized algebraic equations were obtained from the superposition of two-dimensional FA solutions in the x-y, y-z, and z-x domains. This simplifica- tion greatly reduced computation time and storage space requirements. To solve for pressure, the velocity variables in the continuity equation were replaced by the pressure variable, and a staggered grid was used to avoid possible unrealistic pressure and velocity fields resulting from the finite difference representation of the pressure gradient term in the equation of continuity. The system of FA algebraic equations was obtained for all elements. The solution of these algebraic equations with the proper boundary and initial conditions provided the numerical solution to the problem. The sequence of numerical oper- ation followed the SIMPLER algorithm (15). Details of the FA method, including the algebraic coefficients, are found in Bravo (2).

Solutions were obtained at time steps midway between the time of acquisition of each of the LV chamber outlines, beginning with the onset of systole. Once all boundary conditions had been set, and assuming the flow initially was stationary, a convergent solution was obtained

FIGURE 2. Three-dimensional-19-point FA element.


for this first time step. Convergence for each time step was defined as a normalized difference of 0.001 in computed velocities from the previous iteration. For the next time step, nodes were again divided into wall and active nodes based on the second chamber outline on the same stationary grid mesh, and the reference point (center of outlet) of the second outline was aligned with the reference point of the first outline. All interior nodes retained the values of the flow parameters computed from the previous time step, and therefore, a quasiunsteadiness is achieved. Boundary conditions were calculated, and a convergent solution was obtained. This process was repeated for all remaining systolic LV outlines. Grid convergence was determined by comparing solutions at 24 • 24 x 24 nodes with those at 39 • 39 • 39 nodes with insignificant differences (<1%) in computed flow variables.

CER

The CER is determined on the basis of the assumption that a normally functioning ventricle will contract uniformly and thereby produce a uniform flow pattern within the LV cavity, with velocity vectors generally being aligned with the LV long axis. Although "normal" LV wall movement is very hard to define, clinical evidence indicates that, on average, healthy human LVs contract uniformly with respect to the long axis (4). The flow patterns within the LV cavity produced by the motion of the LV walls may be characterized by what we term the CER. The CER is arbitrarily defined as the region of flow domain in which the obtained velocity field vectors are aligned +--3 ~ from the LV long axis. Therefore, the CER is the region of flow domain that is aligned for ejection. When the LV is contracting uniformly, the CER should follow the centroidal axis of the LV cavity. To quantify the correlation between the CER and the LV centroidal axis, we defined the CER coefficient for three-dimensional geometries as

1 N (dBL_dcFR, I N Z dl~L ] CCER (7)

where dBL is the average distance at any particular short axis cross-section of the LV from the LV wall to the centroid of the cross-section, and dcER is the distance from the cross-section centroid to the center of the CER; both are measured along a direction perpendicular to the long axis (see Fig. 3). Summation is over N computational planes along the long axis. If the CER does not appear in a particular short axis plane, dCER is set to dBL. There- fore, the range of CCE R values is 0 ~< CCE R ~< l, where CCE R = 1 indicates a perfectly uniform contraction of the LV, and CCE R = 0 indicates a LV producing no CER over the given time period.

FIGURE 3. Graphical representation of the CER.

Computer Simulations

Normal LV contraction was simulated assuming a si- nusoidal systolic flow waveform with a maximum Reyn- olds number of 1,000, which was determined on the basis of outlet diameter and velocity. Although typical peak Reynolds numbers may be in the range of 4,000, this lower value was arbitrarily chosen to ensure laminar flow within the numerical model and convergence within a rea- sonable length of time. The radius of the spherical cavity, R, was then obtained as a function of time and normalized to produce a systolic ejection fraction (change in volume


divided by initial volume) of 50%. The systolic phase was divided into 10 equal time steps, and the spherical outlines of these time steps were used as inputs to the numerical code.

In addition to the simulation of normal LV contraction, several cases of AWM were simulated by the manipulation of the boundary conditions to produce a region of hypokinesis, akinesis, or dyskinesis (Fig. 4). This was accomplished by multiplying the computed velocity boundary condition for several regions along the LV model by the factor S, as follows:

UwaLl (ischemic region) = S �9 Uwalt (normal case) (8)

The magnitude of this S factor corresponds to the degree of AWM for the particular region affected. In other words, multiplication of the wall velocity in a certain region by a factor S = 0.3 resulted in a regional wall motion of 30% of the normal wall motion for the affected region. For 0.0 < S < 1.0, a region of hypokinesis was generated. S = 0.0 produced an akinetic region, and S < 0.0 produced a region of dyskinetic motion. To gauge the sensitivity of the CER coefficient to the relative size of the ischemic region, several sizes were used in the simulations, as shown in Fig. 5. The ischemic regions extend from the apex to mid-ventricle in the long axis plane (x-y) either 90 ~ or 122 ~ and extend circumferentially at mid- ventricle (y-z plane) either 90 ~ or 180 ~ resulting in four sizes, designated SM, L, XL, and XXL.

AWM simulations were performed for one time step only; the Reynolds number for that time step was 500 for each simulation. As indicated by Eq. 5, all governing equations were nondimensionalized before the solution procedures were applied. Accordingly, all velocity variables were normalized by the outlet velocity, Uout le t , and all lengths were normalized by the outlet diameter, Doutlet. For the AWM simulations, the outlet velocity was adjusted to compensate for the lower velocities along the ischemic wall region and to still conserve mass within the LV chamber. All velocities were then renormalized by this adjusted outlet velocity. With a constant outlet diameter, an identical Reynolds number for each simulation required a constant outlet velocity, and this, in turn, required that

�9 s , . �9 - ? , Normal Hypokmesls Akmests Dyskmests

S= 1.0 O.O<S< 1.0 S=0.0 S<0.0

FIGURE 4. Types of regional A W M simulated in this study and the corresponding ranges of S. The two outlines shown for each case represent successive outlines of the LV endocardium.

( io~ (a) b)

(c) (d)

FIGURE 5. The four regions of ischemia (cross-hatched area on sphere surface) simulated in this study: (a) SM, (b) L, (c) XL, and (d) XXL.

the ejection fraction, defined as the change in LV chamber volume divided by the original volume, be identical for each AWM simulation. Therefore, this model of AWM represented an ischemic LV with compensatory contraction of the unaffected myocardium.

RESULTS

Contraction of the spherical ventricle produced a vector field that was directed toward the outlet and was symmetric with respect to the long axis, and this symmetry is depicted in all three orthogonal views (Fig. 6). Because the LV contraction was uniform throughout systole, flow fields at all time steps were similar. The CER is symmetric with respect to the LV centroidal axis (which is identical to the long axis for a spherical ventricle); it has a wide base at the apex and narrows toward the outlet. There is a


FIGURE 6. Velocity vector field and CER for the normally contracting LV, as viewed from the two long axis planes (x-y and z-x) and the short axis plane (y-z).

large acceleration of the flow field near the outlet in the long axis lanes (x-y and z-x), so that the outlet velocities are nearly 3.5 times the wall velocities at the apex. In the transverse plane (y-z) at mid-ventricle, the flow is directed entirely toward the long axis, which runs through the centroid of that plane.

Simulations of AWM were run with the four sizes of ischemic regions depicted in Fig. 5 and with simulation factors ranging from 0.9 to - 0 . 2 in increments of 0.1. Representative vector flow patterns and CERs produced with the small (SM) ischemic region are depicted in Figs. 7-9. In the plane that bisects the region of ischemia (x-y), an asymmetrical flow pattern that became more asymmetrical with increasing severity of AWM developed, culmi- nating in small pockets of reverse or "paradoxical" flow near the ischemic region for the akinetic (S = 0.0) and dyskinetic (S = - 0 . 2 ) cases. This resulted in a CER that shifted toward the ischemic region and shortened as AWM became more severe. In the long axis plane that lies par- allel to the ischemic region (z-x), the vector flow patterns remained symmetrical with respect to the long axis for all degrees of AWM. Note that the CER in this plane, however, was very short as a result of the large component of velocity perpendicular to this plane. This was best depicted by the velocity vector field in the transverse plane (y-z) at mid-ventricle. Hence the flow became more generally directed toward the ischemic region as AWM increased, as did the portion of the CER in this plane.

Similar flow patterns and CERs were produced with all other simulations of AWM. The computed CER coeffi-

FIGURE 7. Velocity vector field and CER (shaded region) in the x-y plane for the SM region of ischemia.

cients (Fig. 10) decreased monotonically with increased severity in AWM to the point of akinesis, and then the CER coefficients remained nearly constant or displayed a slight increase with dyskinesis. The CER coefficient was

FIGURE 8. Velocity vector field end CER (shaded region) in the z-x plane for the SM region of ischemia.


coefficient of 0.09, regardless of the size of the ischemic region. Under more severe hypokinesis, this sensitivity decreased to 0.1:0.06 for the XL and XXL cases and to 0.1:0.03 for the SM and L simulations. Therefore, the CER coefficient appeared to be more sensitive to changes in size of the ischemic region along the long axis plane than along the short axis plane. The CER coefficient was insensitive to the degree of akinetic AWM.

For a comparison, values of intraventricular pressure obtained from the solution of the governing equations are shown in Fig. 11. Pressure values along a line coinciding with the long axis of the LV, as well as along a line directed from the outlet center to the center of the ischemic region and a similar line to the opposite side of the chain-

~ 0.8

FIGURE 9. Velocity vector field and CER (shaded region) in the y-z plane for the SM region of ischemia.

not affected by the size of the iscl'emic region for moderate degrees of hypokinesis (0.6 ~ S ~< 1.0), but it did decrease with increasing size of ischemia for more severe states of hypokinesis (0.0 ~< S ~< 0.4). A high degree of sensitivity was demonstrated in the CER coefficient, although this sensitivity varied somewhat with the severity of AWM. In general, the CER coefficient appeared to be most sensitive for AWM of moderate hypokinesis, for which a change in S of 0.1 (which corresponds to 10% of normal motion) resulted in an average change in the CER

~ 0.4 t 0.2~

0.0 1.0

= 0.8

~- 0.6

�9 -- o.4

0,2 z

0.0

0.8

(b)

(c)

0 .6"

0.4-

0.2-

0.0 ' ~ 0.0 0 .2 0 .4 0 .6 0 .8 ~.0

Apex Normalized Base Distance

FIGURE 10. CER coefficients for all simulations.

FIGURE 11. Pressure values along a line extending from the outlet center to the middle of the ischemic region (a), along the long axis (b), and along a line to the opposite wall (within the x-y plane) (c) for the SM region of ischemia. Pressure values are drawn as curves for S = -0 .2 , 0.0, 0.3, 0.6, and 1.0, with S increasing in the direction shown.


ber, were plotted versus normalized distance from apex to base (outlet center). Representative results are presented for the SM case, in which the pressures were normalized by the pressure at the apex to better illustrate differences in pressure gradients from the wall. Although the overall pressure drop from apex to aortic valve outlet did not change appreciably, the gradient near the ischemic wall decreased progressively from the normal to the dyskinetic case. However, very little change was observed in the pressure gradient along the long axis, and no change in pressure gradient occurred opposite the ischemic region on the normal, unaffected wall region.

DISCUSSION

These results are similar to those obtained with a previous two-dimensional solution of flow in a circular model (16), in that the flow generally was directed perpendicular to the walls toward the long axis and then out through the orifice for a normally contracting ventricle. This was compared with a closed-form solution of inviscid flow in a spherical chamber (22), in which flow was more generally directed from the apex of the cavity toward the outflow, resulting in a much flatter profile of the velocity component in the direction of the long axis. The three-dimensional results presented herein confirm that differences exist between viscous and inviscid flow solutions, and these differences are unaltered by three-dimensional effects.

The CER in three dimensions shows promise of pro- viding a qualitative and quantitative global measure of LV function that is sensitive to regional AWM and to the resultant compensatory actions, which cannot be detected by global parameters. In previous two-dimensional simulations (17), very little change was observed in the CER coefficient as the factor S was decreased from the normal condition (S = 1.0) to as low as 0.3. The sensitivities achieved in this ideal three-dimensional model, with a newly defined CER coefficient, were highest under conditions of moderate hypokinesis. If this also is the case for real LV geometries, the CER coefficient may prove to be valuable in the early detection of LV dysfunction. In addition, the three-dimensional simulations provide a com- plete picture of the nature of the flow induced by ventricular wall motion, and, as a result, more informed conclusions may be drawn about the measure of contractility along the three-dimensional surface of the LV myocardium. For example, the flow patterns (depicted in Fig. 8) of a plane that does not intersect the ischemic region are very similar to those of the normal case shown in Fig. 6. If this plane had been chosen for two-dimensional evaluation, the effects of AWM on the LV chamber flow would not have been detectable. Finally, because the ejection fraction was held constant for the AWM simulations, this

global parameter (ejection fraction) would not have detected the abnormal LV function because of the compensatory actions of the unaffected regions of the myocardium intrinsic to the model simulations, as described earlier. However, we also must note that the simulations with normal, uniform contraction of the myocardium will produce a similar CER coefficient regardless of the input ejection fraction and therefore, is not sensitive to global malfunctions of the LV. Thus, the CER coefficient and the ejection fraction both are diagnostic parameters of global function; one is sensitive to regional abnormalities, and one is sensitive to global abnormalities. When used in combination, these diagnostic parameters may provide a clearer understanding of the current function of the LV in question.

There have been other recent attempts to model LV flow dynamics with the use of numerical simulation techniques (10,21,23). All of these models produced similar velocity vector fields and pressure gradients for a normal or uniformly contracting LV, as displayed in Figs. 6 and 11. However, only in this study were any simulations performed of LV AWM and its effect on systolic flow dynamics. For example, a previous work (10) noted changes in ejection pressure gradients in a global sense with changes in chamber geometry (eccentricity of ellip- tical shape) and outlet valve stenosis. Our results show that ejection pressure gradients may change locally in the region of ischemia, but that AWM does not seem to change the global (long axis) ejection pressure gradient appreciably (Fig. 11).

This model is limited to the systolic phase of the cardiac cycle. We recognize that diastolic filling flow patterns may influence systolic flow, and that diastolic flow patterns also may correlate with LV dysfunction (11). In addition, the spherical model of the LV cavity was chosen for its simplicity, whereas a surface depicting the three- dimensional geometry of the LV certainly will pose a sig- nificantly increased level of computational complexity. Finally, the method used to determine wall motion between time steps assumed pure translation of the myocardium and did not allow for rotation, or "wringing," of the LV, which is known to occur in vivo because of the nature of the myocardial fibers. We are currently extending this model to include flow chambers with actual LV geometry, including the diastolic phase, which will be reconstructed from two-dimensional, multiplanar images obtained from either echocardiography or magnetic resonance imaging.

After additional validation and extensions of the model to more adequately replicate real conditions, we believe that these type of simulations can be used to gain better understanding of the underlying mechanisms of the intraventricular flow observed in vivo through Doppler or magnetic resonance imaging techniques. Beppu et al. (1) used contrast and Doppler echocardiography to correlate altered


LV flow dynamics in systole and diastole with apical akinesis and dyskinesis and found that the blood near the apex was forced toward the dyskinetic apical area. This finding agrees with vector flow fields depicted in Fig. 7 for the dyskinetic case. Garrahy et al. (9) used Doppler flow mapping and found flow patterns similar to those in Fig. 6 for subjects with normal functioning hearts, with the entire flow field being directed toward the outflow tract. For subjects who displayed hypokinesis, little difference was observed from the normal case. As noted previously, however, the Cce R in this model was most sensitive for moderate degrees of hypokinesis. This result emphasizes the utility of computer simulations using models as described herein to provide details of the expected LV chamber flow field that future higher resolution Dopp- ler flow mapping may soon provide. In addition to corre- lating hemodynamics effects with LV wall dysfunction, altered apical flow patterns also have been used as a pre- dictor of apical thrombus formation, a major complication often associated with acute myocardial infarction (5). This information can be very important when long-term anti- coagulant therapy is being considered.

CONCLUSIONS

We have shown the development of a three-dimensional model of LV systolic flow dynamics, including an index of pumping efficiency, the CER. The CER coefficient quantitatively displayed the sensitivity of the flow patterns to even moderate degrees of hypokinesis. In addition, visualization of the three-dimensional flow field reinforced the necessity of three-dimensional simulations to capture aspects of the flow that existing methods of two-dimensional flow imaging that use ultrasound may miss. Future computer simulations using actual LV wall motion as input to the model have the potential to provide valuable details of LV flow patterns, which can be used by clinicians to assess the effects of and/or the necessity of certain cardiac surgical procedures (such as coronary by- pass grafting) or antithrombolytic therapy. In addition, the concept of a CER may be beneficial when methods are developed to quantify measurements of the three-dimensional LV flow field using such modalities as magnetic resonance imaging. Care must be taken, however, in ex- trapolating any of the results of this idealized model to a clinical situation without further in vitro and in vivo validation.

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NOMENCLATURE

A, B, C = constant coefficients of the general PDE to be solved

AWM = abnormal wall movement CER = central ejection region CCE R = central ejection region coefficient DBL = average distance from LV wall to

LV centroid in short axis plane DCE R = distance from LV centroid to center

of CER in short axis plane Doutlet = LV outlet (aortic valve) diameter f = source term in dimensionless mo-

mentum equation (pressure gradient)

FA =

g =

L V

N =

p = p =

PDE = Re =

S =

SM, L, XL, XXL =

t =

U, V, W

U , V , W =

Up, Vp, Wp, fp =

Uwall ~-~

Uoutlet

x, y, z = X , Y , Z =

p =

finite analytic

source term in linearlized momentum equation including pressure gradient and unsteady term

left ventricle number of computat ional planes along the LV long axis fluid pressure nondimensional form of fluid pressure partial differential equation

UoutletDoutlet/V, Reynolds number

factor defining degree of A W M along an ischemic segment notations for four sizes of ischemic regions modeled time

velocity components in the x, y, and z directions nondimensional forms of velocity components values of U, V, W, and f applied at center node of element velocity boundary condition at the LV wall velocity boundary condition at the LV outlet (aortic valve)

spatial coordinates nondimensional forms of the spatial coordinates

fluid density fluid kinematic viscosity

A simulation of three-dimensional systolic flow dynamics in a spherical ventricle: Effects of...

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