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J Thorac Cardiovasc Surg 1998;116:566-577
© 1998 Mosby, Inc.
SURGERY FOR ADULT CARDIOVASCULAR DISEASE |
San Francisco, Calif
Supported by California Heart Association grant-in-aid 97-241.
Received for publication Nov 24, 1997. Revisions requested Jan 13, 1998; revisions received May 14, 1998. Accepted for publication May 29, 1998. Address for reprints: Mark B. Ratcliffe, MD, 112D, San Francisco Veterans Affairs Medical Center, 4150 Clement St, San Francisco, CA 94121.
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Abstract |
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 Top Abstract IntroductionMethodsResultsDiscussionConclusion and future directionsReferences |
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionConclusion and future directionsReferences |
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1-5 The operation, first performed by a Brazilian surgeon, Randas Batista, is known as ventricular volume reduction surgery (VVRS) and 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. Although early results with this procedure in uncontrolled trials have been promising, experimental analysis of the effect of the Batista operation on LV mechanics is limited. 6-10
The success of an operation that surgically remodels ventricular size, shape, or regional stiffness depends on how the procedure affects both end-systolic and end-diastolic pressure-volume relationships and how those changes affect ventricular function (stroke-work/end-diastolic volume; preload recruitable stroke work) [PRSW], 11 and stroke-work/end-diastolic pressure relationships [Starling]). 12 End-systolic and end-diastolic pressure-volume relationships, respectively termed end-systolic elastance and diastolic compliance, are mechanically determined by respective LV regional material properties (stiffness) and unloaded ventricular shapes. 13 VVRS will change the unloaded end-systolic and end-diastolic ventricular shapes and regional stiffness in and around the surgical repair site. As a consequence, postoperative ventricular elastance and compliance are altered and may shift by different absolute amounts. As a result, the stroke work/end-diastolic volume and stroke work/end-diastolic pressure relationships may shift. At present, the effect of VVRS on systolic elastance, diastolic compliance, and the resultant net effect on ventricular function is unknown.
Finite element models relate regional material properties, structural shape, and regional stress as a system of linear equations. 14 Typically a structure, in this case the LV, is composed of "elements," material properties (regional stiffness) are assigned to each element, external loads are applied (intracavitary pressure), and regional stresses and structural deformation are calculated. In this study, we describe a finite element model of the dilated LV with reduced contractility in which the effects of partial apical and lateral wall ventricular resections are simulated.
The primary goal of this study is to model the effect of VVRS on end-systolic elastance, diastolic compliance, and ventricular function (stroke work/end-diastolic volume and stroke work/end-diastolic pressure [Starling] relationships). We also test the hypothesis that 2 different types of partial ventriculectomy involving either the lateral wall or apex had different effects on elastance, compliance, and ventricular function. Finally, we test the hypothesis that VVRS decreases myocardial energy expenditure.
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Methods |
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, steps 3 and 4). Finite element meshes were created and loaded with a range of physiologic intraventricular pressures with the use of commercially available computer-assisted design and simulation software (Ideas, release 2.0, SDRC, Inc, Milford, Ohio). Nonlinear material properties were assigned and the models solved with the use of a commercially available nonlinear finite element solver (ABAQUS, Standard module, version 5.4, Hibbet, Karlson and Sorenson, Inc, Providence, RI). Finite element modeling and analysis were performed on a Unix-based workstation (SPARCsystem 10, model 40, Sun Microsystems, Inc, Mountain View, Calif).
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15-18
Initial unloaded end-systolic shape
The unloaded end-systolic LV shape was derived from the diastolic shape by assuming that LV wall volume and aspect ratio are constant and that average circumferential and longitudinal strains are equal to the strain of an unloaded sarcomere during contraction. End-systolic dimensions were calculated by simultaneously solving the following 4 equations (Mathematica, version 2.2, Wolfram Research Inc, Champaign, Ill):
[b22,ESa2,ES - b21,ESa1,ES] = [b22,EDa2,ED - b21,EDa1,ED] (1)
19-21
Finite element mesh and boundary conditions
The finite element model was meshed with axisymmetric solid (continuum) elements (4 nodes, bilinear with reduced integration, element type CAX4RH, ABAQUS, standard module, version 5.4, Hibbet, Karlson and Sorenson, Inc, Providence, RI). A single axisymmetric hydrostatic fluid-filled cavity (element type FAX2, ABAQUS) was used to model the endocardial cavity. This element provides a convenient means by which endocardial cavitary pressure and volume can be calculated. Nodes at the apex along the ordinate axis were constrained to displace in the y direction (long axis) only. The base of the LV is composed primarily of fibrous tissue and was therefore constrained in all degrees of freedom (Fig 1
, step 2).
Material properties
Diastolic material properties
Both diastolic and systolic material properties of the LV wall were assumed to be homogeneous and isotropic. Diastolic material properties were described by the strain energy potential constitutive equation developed by Ogden 22 to describe nonlinear isotropic rubber-like materials. The Ogden constitutive equation is shown below:
i is the length, li, divided by the undeformed reference length at zero force, loi. The parameters µi and 
i are material constants, i being termed the power law exponent. Determination of µi, and i for the myocardium in different species, and by different methods, has yielded similar values. 13 Stiffness parameters from the normal canine ventricle 23 where µi = 16 and i = 16.0 were used in the diastolic model.
Systolic material properties
The end-systolic material properties of the LV wall were also described by means of the Ogden constitutive equation. End-systolic material stiffness parameters were empirically chosen so that the end-systolic pressure-volume relationship is approximately linear and passes through LV pressure of 120 mm Hg and LV volume of 200 mL. In general, µi determines the initial slope of the pressure-volume relationship and i determines the shape of the relationship. Stiffness parameters of µi = 59.722 and i = 9.5 were empirically found to produce a nearly linear pressure-volume relationship and to pass through target LV pressure and volume. These stiffness parameters are significantly less stiff than those of normal systolic myocardium. 13
Finally, both zero-pressure systole and zero-pressure diastole occur at lambda = 1. However, the rest length is different for systole and diastole (equations 3 and 4).
VVRS simulations
The effects of VVRS were simulated with the use of the globally dilated heart model. Apical and lateral VVRS were tested at 80% and 90% of original ventricular mass. More specifically, 10% and 20% of LV mass was removed from both the unloaded diastolic and systolic shapes. Apical resections were performed by first determining the position of a simulated apical ventriculectomy (Fig 1, step 3, exploded view: dashed line) associated with a 10% or 20% LV mass reduction. The cut edge of the ventriculectomy was parallel to y (long) axis and had y-coordinates of a1AR and a2AR. The cut edge of the ventricle was then moved to the midline (x = 0) (Fig 1, step 3, expanded view of apex: x) and the y-coordinates of the cut edge then became the new major semiaxes. Minor semiaxes were unchanged (Fig 1, step 3: b1 and b2).
Lateral resections were performed by first determining the position of a simulated lateral ventriculectomy (Fig 1, step 4) associated with a 10% or 20% LV mass reduction. The ventricle was incised parallel to the y (long) axis and the cut edges of the ventricle were apposed, creating new minor semiaxes (Fig l, step 4: b1LR and b2LR). The wall thickness was unchanged (b2 – b1 = b2LR – b1LR). Minor semiaxes were unchanged (Fig 1, step 4: a1 and a2). No attempt was made to alter material properties around the ventricular incision.
Calculation of systolic and diastolic pressure-volume relationships
Diastolic solutions were obtained at a range of diastolic intracavitary pressures (ie, 0-40 mm Hg). End-systolic solutions were obtained at a range of systolic intracavitary pressures (ie, 0-120 mm Hg). The pressure, PES, and volume at end-systole, VES, derived from the finite element model were fit to the following linear equation by means of least squares regression analysis 24 (Microsoft EXCEL, Redmond, Wash):
PES = EES[VES - V0] (6)
where V0 is the volume intercept and EES is the slope of the LV elastance. The pressure, PED, and volume at end-diastole, VED, were fit to the following quadratic equation with the use of least squares regression analysis 24 (Microsoft EXCEL, Redmond, Wash):
PED = ß0 + ß1VED + ß2VED2 (7)
where ß0, ß1, and ß2 are the stiffness parameters of the LV diastolic compliance.
Calculation of stroke work/PED (Starling) and stroke work/VED (PRSW) relationships
For each simulation (DCM, 10% and 20% lateral and apical VVRS), stroke work/PED and stroke work/VED relationships were calculated from the diastolic and systolic pressure-volume regressions, assuming that arterial elastance, EA, 25 was constant. EA was calculated according to the following equation:
24 (Microsoft EXCEL, Redmond, Wash).
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26 was calculated from the initial pressure-volume loop (PES = 100 and PED = 20) and the diastolic and systolic pressure-volume regressions. Pressure-volume area is equal to the stroke work (area ABCD; Fig 2) plus the potential energy (area OCD, Fig 2). Pressure-volume area/mass is our attempt to normalize energy expenditure for different amounts of remaining LV myocardium. Pressure-volume area/mass is an average term and therefore an approximation given that different myocardial regions experience different stresses and therefore have different regional energy expenditures. |
Results |
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TopAbstractIntroductionMethodsResultsDiscussionConclusion and future directionsReferences |
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) shifts compliance and elastance further to the left than apical VVRS (Fig 6).
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, B. Ten-percent apical VVRS, 20% apical VVRS, 10% lateral VVRS, and 20% lateral VVRS progressively shift the end-diastolic pressure intercept to the left. Note that all curves intersect at approximately PED = 5 mm Hg. To the right of the point of intersection, VVRS causes a progressive decrease in the Starling relationship slope (DCM 0.028 J/mm Hg; 20% lateral VVRS 0.023 J/mm Hg). However, the decrease is relatively small, causing only a 3–mm Hg difference between DCM and VVRS at stroke work of 0.5 J. Note that this effect is increased with a larger amount of resected LV mass and is greater in lateral VVRS than in apical VVRS. In summary, any amount of either apical or lateral VVRS decreases the slope of the Starling relationship but increases the slope of the stroke work/end-diastolic volume relationship. The effect of VVRS on end-diastolic and end-systolic LV volume, stroke volume, and ejection fraction is seen in Table III. End-diastolic volume was obtained at 20 mm Hg and end-systolic volume was obtained at 100 mm Hg. Note that stroke volume decreases while ejection fraction increases with VVRS. Stroke volume was reduced in all cases because the decrease in diastolic compliance was not sufficiently compensated by the improvements in end-systolic elastance.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionConclusion and future directionsReferences |
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VVRS causes an increase in ejection fraction while stroke volume decreases. However, ejection fraction is a poor index of ventricular function, 27,28 and it is therefore incorrect to conclude that cardiac function has improved if VVRS increases ejection fraction.
The effects of apical and lateral VVRS on compliance, elastance, and ventricular function are qualitatively similar. The effect of 20% lateral VVRS is quantitatively largest, whereas 10% lateral VVRS is similar to 20% apical VVRS. At present, cardiac surgeons resect either an ellipse of lateral free wall myocardium or an inverted triangle or "shield," in which the wide part of the triangle or shield is toward the atrioventricular groove. However, an apical resection would be easier to surgically reconstruct and might not interfere with the papillary muscles. These results support the potential use of apical partial ventriculectomy.
VVRS causes a decrease in energy expenditure per gram of residual myocardium. Although not calculated in this study, wall stress throughout systole is also decreased. 6,27 Both of these factors may lead to a reverse remodeling of eccentric hypertrophy and may lead to potential subacute and chronic beneficial effects from VVRS. The overall effect of VVRS must be a combination of the acute hemodynamic effects of VVRS on LV function and subacute and chronic effects of decreased energy expenditure and stress on ventricular hypertrophy.
Finite element modeling and assumptions
The accuracy of finite element analysis simulations are dependent on the accuracy of initial assumptions, including (1) the material property law, (2) the input stiffness parameters, and (3) the initial unloaded structural geometry. Given accurate initial conditions, however, finite element analysis is well suited to model the effect of LV volume reduction surgery on elastance, compliance, and ventricular function. However, the following assumptions must be recognized.
Structure
Although more realistic than previous models, the initial end-diastolic and end-systolic shapes are approximate. This is in part because there are no experimental measurements of the normal or pathologic animal or human LV at zero left and right ventricular intracavitary pressure. Furthermore, Fung 29 has demonstrated significant residual wall stress even in the absence of ventricular pressure. Residual stress has not been measured in dilated cardiomyopathy but may have a significant impact on VVRS, which in effect increases residual pre-strain and stress.
The right ventricle is not included in the model, and transseptal pressure is assumed to be the same as transmural pressure in the lateral LV wall. Right ventricular pressure has been shown to increase LV elastance and compliance, 30 and the absence of the right ventricle may lead to a shift in elastance and compliance to the right.
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 work/end-diastolic pressure curves (Fig 8, A). As above, the increase in end-diastolic pressure and corresponding increase in left atrial pressure needed to attain comparable stroke work may be effectively offset by the removal of mitral regurgitation.
The model was constructed so that average circumferential and longitudinal unloaded strains across the LV wall were equal to 0.85. This, in turn, is based on the relative change in sarcomere length between the relaxed and the contracted states. Spotnitz, Sonnenblick, and Spiro 19 measured sarcomere length in normal intact hearts and found sarcomere lengths between 1.87 and 1.95 µm in diastole and between 1.5 and 1.6 µm in systole. These values agree with values from isolated intact and skinned ventricular muscle strips. 20,21
Material properties
Scientists have used different types of material property laws in finite element analysis models. Material property laws have ranged from Hookean (linear stress-strain relation) 31 to nonlinear, 13and both isotropic 13 and anisotropic (directional) 32 descriptions have been used. In addition, although an anisotropic material property law that is linked to myocardial architecture (fiber angles) is clearly preferable,3234 these descriptions are complex and knowledge of myocardial fiber angles in the dilated cardiomyopathic LV is unknown. A nonlinear isotropic material property law, such as that developed by Ogden 22 to describe nonlinear isotropic rubber-like materials, represents a reasonable first-order approximation until such structural information becomes available.
Although the incision probably undergoes stiffness changes similar to other wounds and myocardial infarction, 35 stiffness in the region of the incision has not been experimentally measured. Therefore no attempt was made to alter material properties around the ventricular incision. Also, the fibrous base of the LV is assumed to be fixed in the diastolic position.
Finite element mesh
The finite elements used are 2-dimensional axisymmetric elements. Although these elements are computationally efficient, they can only be used to describe axisymmetric solids in rotation and cannot be used to describe variation in the third dimension.
VVRS in the apical position was performed by "gathering" the LV at the point of resection. True resection with linear closure in the operating room would lead to a more complex non-axisymmetric 3-dimensional shape. As above, no attempt was made to alter material properties around the ventricular incision.
Mathematical models of VVRS
There are analytic solutions that relate chamber compliance to stress and strain in simple shapes with linear material properties that undergo small deformations. An example is the linear elastic solution of an incompressible thin-walled sphere under internal pressure described by Lamé, 36
Despite its shortcomings, our model in conjunction with diastolic constitutive parameters derived from actual resting stress-extension tests of canine myocardium produces realistic human pressure-volume curves. Proceeding in the other direction (ie, starting with a time varying elastance model and then characterizing stress-strain behavior by way of global wall equilibrium) could very well produce the correct trend while giving quantitatively unreliable data.
Comparison with preliminary clinical results
Early reports have documented a heterogeneous effect of VVRS on ventricular function. 3,5 End-diastolic and end-systolic LV volumes have been consistently decreased 4,10 and ejection fraction has consistently increased. 4,5,7 Although most reports document a decrease in stroke volume, 4,10 stroke volume 8 and cardiac index 5 have been noted to improve. However, because postoperative ventricular end-diastolic pressure is often decreased 4,5 and heart rate increased, 4 these data are difficult to interpret.
The effects of VVRS on load-independent end-systolic and diastolic pressure-volume relationships have been measured with ventriculography, 9 conductance catheters, 10 and 2-dimensional echocardiography. 3 End-systolic elastance has been found to increase by twofold to threefold 9,10 and end-diastolic compliance significantly decreases. 3,10 Because of the heterogeneous effect of VVRS on end-systolic elastance in their study, Gorscan and associates 3 found that PRSW was not changed. The Starling relationship has not been measured. Although these preliminary results are consistent with our finite element analysis model and support the findings of this study, the effect of VVRS on ventricular function (Starling and PRSW) needs to be studied in more detail.
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Conclusion and future directions |
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TopAbstractIntroductionMethodsResultsDiscussionConclusion and future directionsReferences |
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Our finite element simulation of Batista's operation documents an acute increase in the stroke volume/end-diastolic volume relationship. However, decreased diastolic compliance causes a small decrease in the Starling relationship (3 mm Hg difference between DCM and VVRS at stroke work = 0.5 J). On the other hand, the increase in left atrial pressure needed to maintain stroke work may be effectively offset by the surgical correction of mitral regurgitation. The overall effect of VVRS is a combination of the acute hemodynamic effects of VVRS on LV function and subacute and chronic effects of decreased energy expenditure and stress on ventricular hypertrophy. This analysis suggests that VVRS may be an appropriate therapy for patients with DCM.
This model is an initial approximation. However, one of the major strengths of the finite element method is the ability to incorporate complex realistic geometry, anisotropic material properties, regional differences in material properties (ie, at the suture line), and residual stress in the unpressurized ventricle. In the future, we intend to progressively refine the model. Simulations should be repeated with varied systolic and diastolic material parameters. In addition, future directions should include the measurement of unloaded LV systolic and diastolic shapes in both the normal and pathologic heart. However, we believe that this initial approximation provides useful information provided that the assumptions incorporated in the model are clearly understood.
The finite element approach can be used to analyze the effects of any cardiac operation that surgically remodels ventricular size or shape or changes regional stiffness. Ideally, the development of such simulations will refine our understanding of operations and pathologic processes that alter ventricular shape and material properties.
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