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Ann Thorac Surg 2003;76:1171-1180
© 2003 The Society of Thoracic Surgeons
a Department of Surgery, San Francisco, CA, USA
b Department of Bioengineering, San Francisco, CA, USA
c Department of Anesthesia, Division of Cardiothoracic Surgery, School of Medicine of the University of California, San Francisco, California, USA
d San Francisco Veterans Affairs Medical Center, San Francisco, California, USA
e Department of Biophysics, Maastricht University, Maastricht, the the Netherlands
Accepted for publication April 18, 2003.
* Address reprint requests to Dr Guccione, Division of Surgical Services (112D), San Francisco Veterans Affairs Medical Center, 4150 Clement St, San Francisco, CA 94121, USA
e-mail: julius.guccione{at}med.va.gov
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Abstract |
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METHODS: A finite element model was used to simulate the effects of Myosplint on the LV stroke volume/end-diastolic pressure (Starling) relationship and regional distributions of stress in the local muscle fiber direction (fiber stress) for a wide range of diastolic and end-systolic material properties. 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 globally dilated LV wall.
RESULTS: Both diastolic compliance and end-systolic elastance shifted to the left on the pressure-volume diagram. LV end-diastolic volume and end-systolic volumes were reduced by 7.6% and 8.6%, respectively. Mean end-diastolic and end-systolic fiber stress was decreased by 24% and 16%, respectively. Although the effect of Myosplint on the Starling relationship was not significant, there were trends toward an improvement in this relationship at low diastolic stiffness, C, high peak intracellular calcium concentration, Ca0, and high arterial elastance, EA. Of note, the effect of C was twice that of Ca0 and EA. Diastolic function would, therefore, be expected to be the prime determinant of success with Myosplint.
CONCLUSIONS: Myosplint reduces fiber stress without a decrement in the Starling relationship. Myosplint should be much more effective than partial ventriculectomy as a surgical therapy for patients with dilated cardiomyopathy and end-stage congestive heart failure.
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Introduction |
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| Drs Ratcliffe and Salahieh disclose that they have a financial relationship with Myocor, Inc (Maple Grove, MN).
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Reduction of left ventricular (LV) wall stress is a cornerstone in the treatment of heart failure. First, the time integral of systolic stress is proportional to myocardial oxygen consumption in the normal [1], and stunned LV [2], and in the failing LV with hypertrophy [3]. Second, increased ventricular wall stress causes structural changes (remodeling) in the LV [4, 5] that adversely affect ventricular performance and patient survival. Mechanisms include stress mediated myocyte hypertrophy, increased myocyte apoptosis [6], altered matrix metalloproteinase (MMP), collagen turnover [7], and myocyte Ca2+ handling [8]. Conversely, LV unloading with chronic assist device therapy has been reported to decrease beta receptor density, improve Ca2+ cycling [9], and downregulate MMPs [10].
The Myocor Myosplint (Myocor Inc., Maple Grove, MN) is a transcavitary tensioning device designed to change ventricular shape and reduce wall stress [11]. Typically, three Myosplints, each consisting of two epicardial pads connected by a tension member, are placed along the long axis of the LV. This draws the opposing anterior and posterior walls of the LV together and creates a bilobular cross-section with decreased chamber radius [11]. Using LaPlace’s law to calculate wall stress, application of the Myosplint device in dogs with pacing tachycardia induced heart failure [11] caused an acute reduction in end-diastolic and end-systolic wall stress of 40% and 18%, respectively.
Ventricular wall stress, which cannot be measured directly, has traditionally been estimated with LaPlace’s law. However, the use of LaPlace’s law in this case is not appropriate. First, the calculation of transmural variations in stress through the LV wall is impossible using an analysis based on a global force balance rather than on knowledge of material properties because LaPlace’s law can only predict the average stress across the wall thickness. Second, application of LaPlace’s law to the complex bilobar LV shape that occurs with Myosplint devices requires knowledge of the tension in these devices, which has never been measured. Finally, Myosplint devices probably are associated with regional stress variations and stress concentrations at the Myosplint anchor sites, for which LaPlace’s law cannot account.
Surgical procedures (such as partial ventriculectomy, aneurysm repair, and Myosplint) can achieve substantial LV volume and stress reduction [12, 13] but the cost of surgical volume reduction may include a reduction in LV function. Specifically, operations that alter ventricular size, shape, or regional stiffness effect both diastolic compliance and end-systolic elastance and those changes have a net effect on ventricular function [15]. For example, although partial ventriculectomy improves end-systolic elastance, diastolic compliance is reduced more and the net effect is a decrement in the Starling relationship [14, 15]. Previous studies using Myosplint devices in dogs with pacing tachycardia induced heart failure [11], and in sheep with anteroapical myocardial infarction [16] did not demonstrate a significant change in cardiac output and suggest a balanced effect of Myosplint on end-systolic elastance and diastolic compliance.
The goal of the present study was to use a realistic finite element model of a globally dilated left ventricle to simulate the effect of Myosplint on regional fiber stress and the Starling relationship. To determine the extent to which myocardial material properties influence the effectiveness of Myosplint, we used nine different combinations of diastolic and systolic material properties. We tested the hypothesis that application of the Myosplint devices decreases regional wall stress and improves LV function.
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Material and methods |
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Finite element mesh and boundary conditions
The finite element model was meshed with three-dimensional solid (continuum) elements (eight nodes, trilinear nodal displacement interpolation in prolate spheroidal coordinates, 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. These constrained degrees of freedom prevent a hole from opening up at the apex and provide longitudinal and circumferential frames of reference for displacements, respectively. Because stress and strain are independent of rigid body motion, our solutions would be the same if we instead allowed the LV base to move up and down (during diastole and systole, respectively) and used the epicardial node at the apex as a frame of reference for longitudinal and circumferential displacement. It should be noted that the apical and basal nodes were allowed to move in the radial or transmural direction. Converged solutions were obtained when the mesh was refined into 4 elements transmurally, 8 elements longitudinally, and 12 elements circumferentially (for a total of 384). The effect of element number on end-systolic and end-diastolic pressure-volume relationships differed by only 3% when the number of elements was increased from 180 to 384 elements. Finally, these results did not depend on the "path" the model took in the pressure-volume plane (e.g., it did not matter whether end-systole was simulated directly or after diastolic filling). This finite element mesh facilitated comparison between our finite element model analyses of ventricular volume reduction surgery and Myosplint. Having a more realistic geometrical shape would have necessitated an increase in the number of elements beyond our present computational capabilities.
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, W, developed by Guccione and associates [18] to describe myocardium as a nonlinear material that is anisotropic (transversely isotropic) with respect to the local muscle fiber direction:
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Myosplint simulations
The effects of a typical (25%) reduction in end-diastolic diameter at the points of application of three Myosplints devices were simulated with the use of the globally dilated heart model. Typically about 3 cm separate the Myosplints devices from the LV base and each other. To determine the epicardial diameters at these three longitudinal positions for each type of diastolic material property (normal, reduced compliance, and failing) the model was loaded passively up to an LV pressure of 20 mm Hg (2.67 kPa). Then, the radial and longitudinal displacement of the epicardial node to which each Myosplint device was attached (six nodes) were specified so that the epicardial diameter was reduced by 25% (Fig 1B). Longitudinal displacements of the nodes at the apex and base were constrained, and pressures on the endocardial and epicardial surfaces were set to zero. Residual stresses in the new unloaded configurations were taken into account by treating the initial unloaded configurations before splinting as the stress-free reference configuration.
Calculation of diastolic and End-Systolic Pressure-Volume relationships
Diastolic and end-systolic solutions were obtained at ranges of end-diastolic (i.e., 0 to 20 mm Hg) and end-systolic (i.e., 0 to 120 mm Hg) chamber pressures. The end-systolic pressure (PES) and end-systolic volume (VES), and end-diastolic pressure (PED) and end-diastolic volume (VED) relationships were fit to the following quadratic equations using least squares regression analysis [25]. The end-systolic pressure-volume relationship is:
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Calculation of stroke Volume/PED (Starling) relationship
For each combination of end-systolic and diastolic material properties, before and after simulated Myosplint procedure, a stroke volume/PED (Starling) relationship was calculated by constructing a series of pressure-volume loops that were bounded by the end-systolic and diastolic pressure-volume relationships of the simulation. VED initially corresponded to a PED of 20 mm Hg and was incrementally decreased for subsequent loops. In order to calculate the end-systolic pressure-volume point on each loop, the arterial elastance, EA, was either calculated by fixing LV pressure at 100 mm Hg to correspond to our previous methodology [19] or fixed at 2.5 ± 0.5 to correspond to EA values measured in patients with dilated cardiomyopathy [26]. However, because the end-systolic pressure-volume relationship was significantly nonlinear (Fig 2B) the following equation was used to find the intersection between the arterial elastance line and the end-systolic pressure-volume relationship:
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Calculation of diastolic and systolic fiber stress
For each combination of diastolic and systolic material properties, stress in the local muscle fiber direction was computed (using the finite element equations 8 and 10 of Costa and associates [17]) throughout the LV wall at end-diastole and end-systole of the initial pressure-volume loop (PED = 20 mm Hg; PES = 100 mm Hg). Transmural fiber stress distributions "near" the points of application of the three Myosplints were obtained from the centers of the 96 elements (where the hydrostatic pressure component of stress is most accurate) that come in contact with the splints (Fig 4A).
The 96 elements with interelement boundaries that are perpendicular to (or rotated 90 degrees in the short-axis plane away from) the Myosplints were used to compute transmural fiber stress distributions "far" from their points of application (Fig 4A). To obtain "overall" end-diastolic and end-systolic fiber stresses we calculated the mean values from the centers of all 384 finite elements. Each fiber stress value was "weighted" in these calculations by the relative volume of its corresponding element. It should be noted that the mean fiber stress 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 (nor myocardial material properties).
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Average end-systolic volume was reduced from 299.7 to 273.9 mL at 100 mm Hg (8.6%) and average end-diastolic volume was reduced from 382.1 to 353.0 mL at 20 mm Hg (7.6%). Average ejection fraction did not change (preoperative 21.47% versus Myosplint 21.48%).
The overall effect of Myosplint on the Starling relationship was not significant. However, Figure 3 documents an effect of arterial elastance. For instance, Figure 3A illustrates the effect when preoperative EA is calculated by fixing LV pressure at 100 mm Hg. Figure 3B demonstrates the effect when preoperative EA is fixed at 2, 2.5, or 3. In all cases preoperative EA = Myosplint EA. There was a trend toward an improvement in the Starling relationship at low diastolic stiffness (C) (ß1,Starling term = -0.156, p = 0.29), high peak intracellular calcium concentration (Ca0) (ß1,Starling term = 0.048, p = 0.595), and high arterial elastance (EA) (ß1,Starling term = 0.049, p = 0.510).
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The effect of Myosplint on mean fiber stress throughout the LV wall for all combinations of diastolic and systolic material properties is indicated in Table 2. End-diastolic and end-systolic stresses were obtained at 20 and 100 mm Hg, respectively. Note that mean fiber stress decreases with the 25% end-diastolic diameter reduction by 23.0% to 23.7% at end-diastole and by 13.0% to 15.8% at end-systole.
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Comment |
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In a closed circular chamber, if two opposing walls are brought together, thereby reducing the radius of curvature of the adjacent walls, and, presumably, the wall stress in those walls, one might ask why does the region of the chamber where the walls are flattened out by Myosplint not have an offsetting increase in wall stress? Qualitative static analysis indicates that the internal pressure in the ventricle has to be in perfect balance with the stresses in the wall of the ventricle and the tensile forces acting on Myosplint. In other words, Myosplint and wall of the heart act structurally in parallel to contain internal ventricular pressure. Therefore, the overall mean stresses in the ventricle are reduced with the application of Myosplint. Regional stress distributions are more difficult to qualitatively ascertain. In the vicinity of the Myosplint, big changes in radius of curvature take place and a subsequent rise in stresses is observed in those areas. But this happens only locally and the overall mean stresses are reduced as evidenced in the "far" and "near" stress figures.
The change in the curvature of the end-systolic pressure-volume relationships illustrated in Figures 3A and 3B is due to the rigid Myosplint. Specifically, at small LV end-systolic volumes, Myosplint would be expected to have little effect. However, as one moves to the right on the end-systolic pressure-volume curve the length of Myosplint relative to the diameter of the unconstrained chamber becomes proportionally smaller. This forces the end-systolic pressure-volume curve to deviate progressively to the left. Myosplint has a similar effect on diastolic compliance, but because compliance is normally nonlinear, the effect is masked.
Comparison with animal experiments
When compared with application of the Myosplint device in dogs with pacing tachycardia-induced heart failure [11], our simulations underestimate the effect of Myosplint application on both ventricular volume and fiber stress. For instance, simulations predicted that LV end-diastolic and end-systolic volumes were reduced by 7.6% and 8.6%, respectively, whereas application in dogs with heart failure [11] caused an acute reduction in end-diastolic and end-systolic volume of 38 and 51%, respectively. The reason is graphically illustrated in Figure 5. Specifically, because each Myosplint is attached to a single point on the myocardium, indentation of the epicardial surface at the point of simulated attachment is greater than that of the endocardium. The addition of an anchoring pad to the simulation would have probably produced a greater "volume" effect at the endocardium. However, we estimate (with current computer hardware [Octane 2]) refinement of the finite element mesh necessary to simulate the anchoring pads would cause simulations to run for 3 weeks before convergence. Revised simulations are not practical until software is adapted to allow the use of multiple parallel processors.
Comparison with partial ventriculectomy
In previous finite element model studies [15, 19] our simulations suggested that there is something fundamentally wrong with partial ventriculectomy from a global cardiac mechanics point of view because its net effect was a depression of ventricular function (as measured by the Starling relationship) regardless of the choice of myocardial material properties and baseline ventricular geometry. Comparison of Figure 4 in that previous study [19] with Figure 3 in the present study clearly demonstrates that partial ventriculectomy depresses ventricular function a great deal more than Myosplints do. Both surgical therapies successfully reduced regional wall stress in our model. Comparison of Table III in that previous study [19] with Table 2 in the present study reveals that a 25% end-diastolic diameter reduction decreases end-diastolic mean fiber stresses by an amount (23.0% to 23.7% depending on material properties) comparable to that of a 20% ventricular volume reduction (22.5% to 24.1% depending on whether or not residual stress is taken into account). Similarly, a 25% end-diastolic diameter reduction decreases end-systolic mean fiber stresses by an amount (13.0% to 15.8%) comparable to that of a 10% ventricular volume reduction (12.0% to 12.3%).
Fiber stress
Direct measurement of ventricular wall stress in the intact heart is impossible because of tethering from surrounding myocardium [28]. An alternative approach to quantifying ventricular wall stress is mathematical modeling based on the conservation laws of continuum mechanics. To solve the governing equations of equilibrium for a pressure vessel with such a complex geometry, boundary conditions, and material properties, computational techniques are required. The most versatile technique for analyzing cardiac mechanics is the finite element method [29].
Many finite element models of the left ventricle have been proposed. However, most have assumed that the myocardium is linearly elastic (Hooke’s law) [5, 30]. Of note, finite element analysis with a linearly elastic material is computationally efficient because iterative solutions are not required. However, description of myocardium with a nonlinear constitutive relationship is not only more realistic but may allow significantly more accurate calculations [31]. For instance, Janz and Grimm [32] found that a linear elastic model was able to predict pressure-volume relationships but unable to predict fiber elongation.
Furthermore, most large deformation nonlinear [32] and most linear [5, 30] finite element models of the left ventricle have treated the myocardium as an isotropic material. We have previously modeled the effect of the ventricular muscle-fiber distribution with finite elements that possess material anisotropy with respect to a continuously varying fiber axis [24]. In addition, the incompressibility of the heart muscle, which is composed mostly (80%) of water, has very often been incorrectly accounted for by assuming that the myocardium has a Poisson ratio close to 0.5 [5, 30]. Our previous study more correctly introduced the hydrostatic pressure as a Lagrange multiplier in the strain energy function [24]. Solutions indicated that the stiffness of passive myocardium would be 2.4 to 6.6 times greater in the fiber direction, which agrees with the results of biaxial tissue testing. Of note, this study and our previous analysis of partial ventriculectomy [19], represent the first analyses of a surgical remodeling procedure on fiber stress using a nonlinear stress-strain relationship for the diastolic myocardium that is anisotropic with respect to the local muscle fiber direction.
Effect of arterial elastance
In our prior simulations of partial ventriculectomy [19] and cellular transplantation [33], EA was held constant during the calculation of each stroke volume/PED curve. However, because the calculation of each curve began at PES of 100 mm Hg and PED of 20 mm Hg, by necessity each simulation had a different value of EA. This may have significantly biased our previous calculations [19] because EA typically rose after simulated partial ventriculectomy, and higher afterload (EA) would be expected to depress the Starling relationship. In the future, we plan to investigate this in more detail. However, in this study all Starling curves, including preoperative and Myosplint simulations, were calculated with a constant EA (2.0, 2.5, and 3.0 mm Hg/ml). This is a significant departure from our previous report [19] in which we used the relationship between stroke volume and EA described by Sagawa [34] to calculate the stroke volume/PED relationship.
Conclusion and future directions
The present model study is a significant improvement over previous attempts to predict the effect of Myosplint on ventricular function and mechanics [11]. The thickness of Myosplint is very small in comparison to other dimensions in the problem (i.e., thickness of the heart). Producing a meshing scheme that produces elements at least as small as Myosplint thickness (e.g., with four instead of one constrained node) will necessitate an increase in the degrees of freedom of the problem beyond our present computational capabilities. Because the main focus of the study is on mean fiber stresses and pressure-volume relationships that are not affected by the exact distribution of stresses right at the Myosplint interfaces (by virtue of Saint Venant’s principle) [35], we believe our Myosplint modeling simplification is acceptable. Again, increasing the number of elements from 180 to 384 changed the LV volume results by less than 2%. However, in the future, the effect of different Myosplint number, configurations, and tension needs to be performed. In addition, model predictions need to be confirmed by comparison with three-dimensional myocardial deformation and strain, measured using magnetic resonance imaging with noninvasive tags. Nevertheless, current results suggest that "shape change" therapy is a mechanically viable therapy for heart failure.
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= preoperative; 
= Myosplint) and diastolic compliance (
= preoperative; • = Myosplint). (B) Nonlinearity of end-systolic elastance before and after Myosplint. In A and B, values are average (nine end-systolic models and three diastolic models). *p < 0.05 by multivariate regression. (LV = left ventricle.)
= comparison between preoperative and "far;" p less than 0.05 by two-way ANOVA with Bonferroni correction.)
