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J Thorac Cardiovasc Surg 2001;122:592-599
© 2001 The American Association for Thoracic Surgery
Surgery for Acquired Cardiovascular Disease (ACD) |
From the Departments of Surgery,a Bioengineering,b and Anesthesia,c University of California, San Francisco, Calif.
This study was supported by National Institutes of Health grants R01-HL-58759 (Dr Guccione) and R01-HL-63348 (Dr Ratcliffe).
Received for publication Sept 18, 2000. Revisions requested Jan 5, 2001; revisions received Jan 30, 2001. Accepted for publication Feb 5, 2001. Address for reprints: Julius M. Guccione, PhD, Division of Surgical Services (112D), Department of Veterans Affairs Medical Center, 4150 Clement St, San Francisco, CA 94121. (E-mail: Julius.Guccione@ med.va.gov).
Abstract
Objectives: Residual stress is the stress (force per unit area) that remains when all external loads (eg, left ventricular chamber and pericardial pressures) are removed. It has been suggested that ventricular volume reduction surgery can reconstitute the residual stress-strain state of the left ventricle. To determine the extent to which residual stress is involved, we used a mathematical (finite element) model to simulate the effect of volume reduction operations on left ventricular stroke volume/end-diastolic pressure (Starling) relationships, as well as on regional distributions of stress in the local muscle fiber direction (fiber stress).
Methods: The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction. An elastance model for active fiber stress was incorporated in an axisymmetric geometric model of the dilated, poorly contractile left ventricular wall.
Results: When residual stress is implemented in the model simulation of volume reduction operations, the additional decrease in stroke volume at fixed left ventricular end-diastolic pressure is small (10% volume reduction: 2.0% at 1 mm Hg and 2.0% at 20 mm Hg; 20% volume reduction: 2.2% at 1 mm Hg and 3.1% at 20 mm Hg). Furthermore, there is little change in the mean fiber stress throughout the left ventricular wall (10% volume reduction: +1.0% at end-diastole and –0.3% at end-systole; 20% volume reduction: +2.1% at end-diastole and –1.0% at end-systole).
Conclusions: These results suggest that residual stress produced by volume reduction operations has little effect on left ventricular function and the mean fiber stresses at end-diastole and end-systole.
Ventricular volume reduction surgery (VVRS) is a surgical therapy first performed by a Brazilian surgeon, Randas Batista, which includes the resection of a viable slice of the lateral left ventricular (LV) wall in patients with dilated cardiomyopathy (DCM) and end-stage congestive heart failure.
1 The primary objective of this therapy is to reduce the size of the left ventricle to restore a more optimal physiologic volume/mass relationship. Although early results with this procedure in uncontrolled trials were promising, subsequent clinical results have been mixed. 2 The development of a sound physiologic rationale for VVRS is necessary before the results of clinical trials can be understood. However, experimental analysis has been hampered by the lack of a large animal model of DCM.
In general, an unloaded deformable body may not be in a state of zero stress. Residual stresses are the internal forces in a body that remain after all of the external loads are removed. If the residual stresses are relieved by one or more cuts, the body adopts a new stress-free shape, and this shape change is described by measuring the residual strains. 3 In the case of VVRS, a stress-free shape is obtained after the slice is resected, and residual stresses are created when the LV chamber is reconstructed. Kresh and Wechsler 4 suggested that these stresses are responsible for the apparent increase in ventricular performance that results from VVRS.
In the studies of Dickstein, 5 Ratcliffe, 6 and their colleagues, mathematical (finite element) models were used to simulate the global effects of VVRS on cardiac pump mechanics. Those studies were uniform in predicting observed clinical benefits of VVRS, such as improved ejection fraction and enhanced elastance. Cardiac muscle is known to be significantly stiffer in the direction of the muscle fibers than in directions perpendicular to the muscle fibers (anisotropy). However, neither of the above models incorporated anisotropic material properties nor residual stresses in the unpressurized ventricle, and in fact, predicted results may have been quite different if anisotropy and residual stress were taken into consideration. For instance, Guccione and associates 7 used a thick-walled cylinder model of the left ventricle and showed that transmural fiber angle distribution, material anisotropy, torsion, and residual stress all combine to minimize the gradients in fiber stress and strain in the LV wall.
Cylindrical models probably are confined, at best, to describing the mechanics of a narrow equatorial cross-section of the LV wall. Moreover, the success of an operation that surgically remodels ventricular size, shape, or regional stiffness (ie, at the suture line) depends on how the procedure affects both end-diastolic and end-systolic pressure-volume relationships and how those changes affect ventricular function. Therefore, the purpose of this article was to extend these models to an axisymmetric geometry that allowed us to compute stroke volume/end-diastolic pressure (Starling) relationships, 8 as well as transmural distributions of end-diastolic and end-systolic fiber stress from the apex to the base of the dilated, poorly contractile LV wall. To determine the extent to which residual stress is involved, we used a 3-dimensional finite element model to simulate the effect of VVRS under 2 different conditions: (1) when the unloaded postoperative configuration is stress free and (2) when the unloaded postoperative configuration is subjected to residual stress.
Methods
Overview of finite element model
A finite element model of the dilated poorly contractile left ventricle was developed(Figure 1, A). Finite element meshes were created and loaded with a range of physiologic intraventricular pressures with the use of the 3-dimensional method of Costa and associates 9 for large elastic deformations of ventricular myocardium, together with the mathematical descriptions for diastolic and systolic myocardial material properties (stress-strain relations) of Guccione and associates. 10 Finite element modeling and analysis were performed on a Unix-based workstation (Octane; SGI, Mountain View, Calif).
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6 The diameter/length ratio is 0.836. The muscle fiber orientation throughout the left ventricle was assumed to vary (linearly) in the transmural direction from 60° clockwise from the circumferential direction at the endocardium to 60° counterclockwise from the circumferential direction at the epicardium on the basis of experimental measurements made by Streeter and associates 11 in canine hearts, owing to the absence of these data from patients with DCM. The sarcomere length throughout the left ventricle was assumed to be 1.845 µm, which is in accordance with the suggestion of Kresh and Wechsler 4 that a dilated, volume-overloaded heart is devoid of residual stress and with the stress-free sarcomere length measured in rats by Rodriguez and associates. 12
Finite element mesh and boundary conditions
The finite element model was meshed with 3-dimensional solid (continuum) elements (8 nodes, trilinear nodal displacement interpolation in prolate spheroidal coordinates, and constant hydrostatic pressure within each element). Longitudinal displacement of all nodes at the apex and base and circumferential displacement of the epicardial node at the base were constrained. Converged solutions were obtained when the mesh was refined into only 5 elements transmurally and 14 elements longitudinally.
Material properties
Diastolic material properties
Both diastolic and systolic material properties of the LV wall were assumed to be homogeneous and anisotropic. Diastolic material properties were described by the strain energy potential developed by Guccione and associates 10 to describe myocardium as a nonlinear material that is anisotropic (transversely isotropic) with respect to the local muscle fiber direction:
W = c/2 {exp[bfE211 + bt(E222 + E233 + E223 + E232) + bfs (E212 + E221 + E213 + E231)] - 1} (1)
where W is the strain energy potential, E11 is the fiber strain, E22 is the crossfiber in-plane strain, E33 is the radial strain, E23 is the shear in the transverse plane, and E12 and E13 are the shear strain in the fiber-cross-fiber and fiber-radial coordinate planes, respectively. Guccione and colleagues 7 found that the material constants (C = 0.876 kPa, bf = 18.48, bt = 3.58, and bfs = 1.627) allowed a cylindrical model of the left ventricle to match epicardial strains measured in an intact canine heart preparation during passive LV filling.
Systolic material properties
The end-systolic material properties of the LV wall were obtained by defining the stress components referred to fiber coordinates as the sum of the end-diastolic 3-dimensional stress components derived from the strain energy potential (by partial differentiation with respect to the corresponding strain component) and an active fiber-directed component, which was a function of time, peak intracellular calcium concentration, and sarcomere length. 10 To simulate poorly contracting myocardium, we reduced the peak intracellular calcium concentration from 4.35 to 1.8 µmol/L.
VVRS simulations
The effects of VVRS were simulated with the use of the globally dilated heart model. Ten percent and 20% of LV mass was removed from the initial unloaded configuration as slices or wedges subtending angles of 36° and 72°, respectively(Figure 1
, B). Then new unloaded configurations were obtained by closing up the openings(Figure 1, C). Specifically, circumferential displacements of the nodes on the resected surfaces were prescribed, longitudinal displacements of the nodes at the apex and base were constrained (to preserve the apex-to-base distance or height of the left ventricle), and pressures on the endocardial and epicardial surfaces were set to zero. In one set of simulations, the new unloaded configurations were considered to be stress free. In another set, residual stresses in the new unloaded configurations were taken into account by treating the initial unloaded configurations with the wedges missing as the stress-free reference configurations.
Calculation of diastolic and systolic pressure-volume relationships
As in the study by Ratcliffe and associates, 6 diastolic and systolic solutions were obtained at ranges of diastolic (ie, 0-40 mm Hg) and systolic (ie, 0-120 mm Hg) chamber pressures. The end-systolic pressure (PES) and end-systolic volume (VES) from the finite element model were fit to the following linear equation by means of least-squares regression analysis 13 (Microsoft EXCEL, Redmond, Wash):
PES = EES(VES - V0) (2)
where V0 is the volume intercept and EES is the slope of the LV elastance. The end-diastolic pressure (PED) and end-diastolic volume (VED) were fit to the following quadratic equation with the use of least-squares regression analysis 13 (Microsoft EXCEL, Redmond, Wash):
PED = ß0 + ß1 VED + ß2V2ED (3)
where ß0, ß1, and ß2 are the stiffness parameters of the LV diastolic compliance.
Calculation of stroke volume/PED(Starling) relationship
For each simulation (DCM and 10% and 20% VVRS with and without residual stress), a stroke volume/PED relationship was calculated from the diastolic and systolic pressure-volume relationships, assuming that arterial elastance (EA) 14 was constant. As in the study by Ratcliffe and associates, 4 EA was calculated according to the following equation:
SV = VED - V01 + EA/EES (4)
where SV is the stroke volume. The initial LV pressure-volume loop was constructed so that the PED was 20 mm Hg and the PES was 100 mm Hg. Initial EA was calculated, PED was reduced incrementally, successive values of stroke volume were calculated with equation 4, and corresponding values of VES and PES were used to construct successive pressure-volume loops.
Calculation of diastolic and systolic fiber stress
For each simulation (DCM and 10% and 20% VVRS with and without residual stress), stress in the local muscle fiber direction was computed (with the finite element equations 8 and 10 of Costa and associates 9) throughout the LV wall at end-diastole and end-systole of the initial pressure-volume loop (end-diastolic pressure = 20 mm Hg; end-systolic pressure = 100 mm Hg). To obtain overall end-diastolic and end-systolic fiber stresses, we calculated the mean values from the centers of the 70 finite elements (where the hydrostatic pressure component of stress is most accurate). It should be noted that these values do not correspond to those calculated by a global force balance (the law of Laplace), which does not take into account the transmural variation in muscle fiber orientation (or myocardial material properties).
Results
End-diastolic and end-systolic finite element meshes of 20% VVRS when residual stress is taken into account are shown inFigure 2. In each panel, the ventricular wall is represented by a wire frame, and the endocardial surface has been rendered. The diastolic model is loaded with 20 mm Hg of intracavitary LV pressure, and the systolic model is loaded with 100 mm Hg of pressure.
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6 the compliance in each case is shifted a greater amount to the left than the elastance. The results when residual stress is not taken into account are essentially the same as those inFigure 3, with very slight differences in numerical values for end-diastolic and end-systolic volume and at corresponding end-diastolic and end-systolic pressures, respectively. The largest difference was only 0.8% between end-diastolic volume with (137.7 mL) and without (138.8 mL) residual stress taken into account for 20% VVRS at an end-diastolic pressure of 18.75 mm Hg.
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6 stroke volume was reduced in all cases because the decrease in diastolic compliance was not sufficiently compensated by the improvements in end-systolic elastance. Again, these results suggest that residual stress produced by VVRS has little effect on LV function.
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The principal result of this finite element simulation of VVRS is that the residual stress produced by this surgical procedure has little effect on ventricular function and mechanics. This is a very important result because it suggests that it may not be necessary to take residual stress into account when simulating the effects of other surgical procedures (eg, LV aneurysm repair). Moreover, this study confirms the results of the study by Ratcliffe and colleagues 6 (ie, VVRS increases ejection fraction but, more importantly and unfortunately, decreases stroke volume), and it provides the first realistic estimates of fiber stress before and after VVRS.
Kresh and Wechsler 4 have previously suggested that an argument for the observed improvement in contractile function and cardiac energetic efficiency can be formulated on the basis of the residual wall stress-strain state of the myocardium. In contrast, we do not believe that residual stress produced by VVRS is the principal model by which this procedure restores myocardial function because it had such a small effect on our model simulations of postoperative LV function and mechanics. Kresh and Wechsler did state, however, that their analysis was confined to circumferential strain-stress consideration and that a 3-dimensional mapping of the residual (end-systolic) regional strains would provide important additional information that may help optimize the planned surgical remodeling (resection size and shape) of the heart beyond that of performing a simple excision. We computed transmural distributions of end-diastolic and end-systolic stresses referred to the local muscle fiber direction through several LV regions from apex to base. At nearly every location, VVRS had a much greater affect on these stresses than on residual stress.
Finite element modeling and assumptions
We have made several refinements to the previous Ratcliffe model, 6 including a more realistic LV geometry, diastolic myocardial material properties that are anisotropic with respect to the local muscle fiber orientation, systolic contraction based on experimental measurements of active tension-sarcomere length relationships, and residual stress in the unpressurized ventricle. Our model of VVRS is therefore the most realistic to date. One remaining limitation of the present model is that there are no regional differences in material properties (ie, at the suture line). However, we do not believe that this limitation affects our main conclusions.
Another limitation of the present model is that standard assumptions are made concerning the ventricular muscle. Short-term and midterm follow-up from The Cleveland Clinic has been previously reported, and event-free survival (freedom from death, LV assist device, or return of New York Heart Association class IV failure) was only 50% and 37% at 1 and 2 years, respectively. 15 Therefore, it seems that the current model applies to most patients who undergo VVRS. However, it is clear that some of the patients who undergo VVRS do well. For instance, event-free survivors in The Cleveland Clinic study experienced improvement in New York Heart Association class (3.7 to 2.2) and increased oxygen consumption (11.7 to 16.0 mL · kg–1 · min–1). 15 Therefore, it seems reasonable that heterogeneity in clinical outcomes is caused by patient-to-patient variation in systolic and diastolic material properties. However, the model can account for differences in myocardial properties. For instance, if extracellular matrix is replaced by fibrosis in some patients, the material constant C in equation 1 should be increased. This was done in a previous finite element model 16 to simulate a stiff LV aneurysm.
Along those lines, we have previously tested the hypothesis that preoperative diastolic function determines the stroke volume/end-diastolic pressure (Starling) relationship after partial ventriculectomy. 17 That study predicted that the left ventricle with the stiffest diastolic pressure-volume relationship (as might occur when the ventricle has a significant amount of replacement fibrosis) had the best result after VVRS. However, all degrees of diastolic stiffness had a negative effect on the Starling relationship, suggesting that variation in diastolic material properties is not responsible for improvement in function after VVRS.
A second hypothesis is that acute volume and fiber stress reduction after VVRS leads to an improvement in systolic elastance in some patients. It has been previously demonstrated that vasodilators 18 and LV assist devices 19 improve systolic function (preload recruitable stroke work and elastance, respectively) in the postischemic myocardium. If myocardium in some patients with DCM (eg, those with recent cardiac dilation) 20 behaves in a similar fashion, those patients would probably experience an improvement in the Starling relationship after VVRS.
Although finite element techniques exist that can model interaction between fluid motion in a chamber and the structure and stiffness of the chamber wall, our model does not have that capability and does not incorporate the effect of mitral regurgitation. This may be of particular importance in the interpretation of the stroke volume/end-diastolic pressure curves. The increase in end-diastolic pressure and the corresponding increase in left atrial pressure needed to attain comparable stroke work may be effectively offset by the removal of mitral regurgitation.
Conclusion and future directions
It is interesting and perhaps surprising that both the finite element model of Ratcliffe and colleagues 6 and the one used in the present study simulated end-diastolic pressure-volume relationships that shifted more to the left (to lower LV volumes) than the end-systolic pressure-volume relations, especially because the myocardial material properties of these models were quite different (ie, isotropic vs anisotropic). Moreover, this was still the case when we repeated our simulations of VVRS with a more ellipsoidal unloaded DCM model (cavity volume, 127.4 mL; wall volume, 230.2 mL; diameter/length ratio, 0.682). Because the net effect of VVRS was a depression of ventricular function (as measured by the Starling relationship) regardless of the choice of myocardial material properties and baseline ventricular geometry, this suggests that there is something fundamentally wrong with this surgical therapy from a global cardiac mechanics point of view.
An alternative surgical therapy in the not too distant future might be to replace the resected viable slice of the lateral LV wall with a contractile patch made from a myocardial tissue culture that is of comparable dimensions and has greater contractility. To get an idea of how much this new procedure would improve LV function, we ran another simulation with the peak intracellular calcium concentration increased from 1.8 to 2.8 µmol/L in 20% of our DCM model. The resulting end-diastolic and end-systolic LV volumes, stroke volume, and ejection fraction were 246.4 mL, 205.5 mL, 40.9 mL, and 16.6%, respectively. Most important, the stroke volume increased 11% from the DCM value inTable 2. Thus it appears that replacing poorly contracting myocardium with a contractile patch would be superior to VVRS, at least as far as LV function is concerned. At present, however, such myocardial tissue cultures are only 1 mm thick, and they contract rather weakly. 21
In summary, VVRS successfully reduces regional stress. Twenty percent VVRS, for example, reduced both end-diastolic and end-systolic mean fiber stress by at least 25%, irrespective of whether residual stress was taken into account. The stress distributions in the myocardium are important to regional ventricular function because regional coronary blood flow, 22 myocardial oxygen consumption, 23 hypertrophy, 24 and remodeling 25 are all influenced by ventricular wall stress. However, VVRS causes LV pump function to deteriorate, even when myocardial anisotropy and residual stress are taken into account. These results confirm the predictions of our previous simple finite element simulation 6 and show that the Batista operation is an inappropriate therapy for DCM. On the other hand, surgical therapies that both reduce ventricular fiber stress and maintain or increase stroke volume are ideal. Currently, we are using finite element simulations to develop such a therapy.
Acknowledgments
We thank Kevin D. Costa, PhD, for modifying our finite element software so that it could incorporate residual stress and Evan A. Zamir, BS, for assistance with preliminary model simulations.
References
This article has been cited by other articles:
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