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Infarcted regions of myocardium exhibit functional impairment ranging in severity from hypokinesis to dyskinesis. We sought to quantify the effects of injecting a calcium hydroxyapatite-based tissue filler on the passive material response of infarcted left ventricles. Three-dimensional (3D) finite element models of the left ventricle were developed using 3D echocardiography data from sheep with a treated and untreated anteroapical infarct, in order to estimate the material properties (stiffness) in the infarct and remote regions. This was accomplished by matching experimentally determined left ventricular volumes, and minimizing radial strain in the treated infarct, which is indicative of akinesia. The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction, and an elastance model for active fiber stress was incorporated. It was found that the passive stiffness parameter, C, in the treated infarct region is increased by nearly 345 times the healthy remote value. Additionally, the average myofiber stress in the treated left ventricle was significantly reduced in both the remote and infarct regions. Overall, injection of tissue filler into the infarct was found to render it akinetic and reduce stress in the left ventricle, which could limit the adverse remodeling that leads to heart failure.
Recent studies of the biomechanical response of the left ventricle (LV) to myocardial infarction (MI) have identified infarct expansion (i.e. stretching) as an important phenomenon that both initiates and sustains a progressive pathologic process that ultimately results in LV dilatation, loss of global contractile function, symptomatic heart failure and death. This maladaptive response is termed infarction-induced ventricular remodeling and is a complex process that is, in its initial phase, an almost completely mechanical problem manifest by abnormal myocardial stress patterns; however, with time these abnormal stress distributions lead to inherent biologic changes in the myocardium that become difficult to reverse by any means once established [1–5].
Awareness of the progressive nature of MI-induced LV remodeling and the relatively poor outcomes achieved with therapy for end-stage heart failure has lead to increasing interest in developing early post-MI therapies intended to limit adverse remodeling.
The finite element (FE) method is the most versatile approach for computing myocardial stress and strain distributions in the LV. FE models of the LV that incorporate myocardial contractility have been described and used to determine the effect of myocardial infarction on structure and function [6–8]. Additionally, the efficacy of various surgical procedures has been simulated with the FE method [9, 10].
FE models have also been used to investigate the effects of injecting biocompatible material into the wall of the LV to combat the progression of HF [11, 12]. Experimental studies by Ryan et al.  and Ifkovits et al.  have used large animal models to examine the effects of injecting tissue filler and hydrogels into the infarcted LV, respectively. A review of biomaterial injection therapy for treating HF has recently been published .
In the current study, using an ovine infarct model, we tested the hypothesis that eliminating dyskinesia by infarct stiffening reduces LV stress and limits adverse remodeling. A commercially available reactive tissue filler agent was used to stiffen the infarct. The material was injected into the infarct immediately and resulted in an akinetic infarct at 8 weeks after infarction. FE models were then created from contoured three-dimensional echocardiographic (3DE) images for an untreated (control) and treated animal, in order to estimate the material properties in the infarct and remote regions. This was accomplished by matching experimentally determined LV volumes, and minimizing radial strain in the treated infarct, which is indicative of akinesia. Two sheep were analyzed in order to demonstrate the methodology of model generation and numerical simulation from 3DE. A detailed description of the experimental measurements, taken from a large cohort of animals, can be found in the companion paper to this work, presented by Morita et al. 
The animals used in this work received care in compliance with the protocols approved by the Institutional Animal Care and Use Committee at the University of Pennsylvania in accordance with the guidelines for humane care (National Institutes of Health Publication 85-23, revised 1996).
The animal model has been described previously . Briefly, two adult 40 kg Dorset sheep underwent ligation of the left anterior descending and second diagonal coronary artery that resulted in an infarction of approximately 20% of the LV mass at the anteroapex. Due to the absence of collateral vessels, the demarcation between the infarct and healthy myocardium is quite distinct and visible during thoracotomy. The infarct region of one sheep was injected with 2.6cc of a calcium hydroxyapatite-based tissue filler agent (Radiesse®-Bioform Medical Inc., San Mateo, CA) distributed over 20 evenly spaced injections, resulting in an akinetic infarct . These injections were made directly into the myocardium. The second animal was used as a control (no tissue filler injection) resulting in a dyskinetic infarct. Further details about the injection procedure can be found in Morita et al. .
At 8 weeks post-myocardial infarction, epicardial real-time 3DE was performed using a Philips iE33 platform with a 2–7 MHz matrix array real-time 3D ultrasound probe (Philips X7-2, Philips, Bothell, WA). Full-volume 3D datasets were acquired and post-processed. After the 3D image was analyzed at ED and ES, on a dedicated workstation, the resulting contour points were used to calculate the chamber volumes and generate the geometric surface data for the FE models. The contour points for the treated and control animals are shown in Figure 1a and 1b. Mitral and aortic valve function was normal in both animals.
FE models were generated using early diastole as the initial unloaded reference state, since the LV pressure is lowest at this point and therefore stress is at a minimum. The infarct and remote regions were determined from the LV 3DE data points. In the case of the control animal, the boundary between the infarct and remote region was estimated by assessing the transition in wall thickness between the two regions . For the treated animal, the boundary between the remote and akinetic region was created by overlapping the endocardial contours at ED and ES in order to determine where the wall does not move, indicating akinesia. In each case, the boundary was defined using 3D spline curves. Surface meshes were then created from the contour data points to replicate the in vivo geometry (Rapidform, INUS Technology, Inc., Sunnyvale, CA).
Volumetric meshes were generated by filling the space between the endocardial and epicardial surfaces with 8-node tri-linear brick elements (Truegrid, XYZ Scientific Applications, Inc., Livermore, CA). Each 3D mesh consisted of 3 elements transmurally, 32 elements circumferentially, and 18 elements longitudinally, resulting in a total of 1728 elements (Figure 2a and 2b). The remote region was assigned different material properties than the infarct region, representing the heterogeneous nature of the infarcted LV. The inner endocardial surface was lined with a layer of extremely soft linearly elastic shell elements to create an enclosed volume for LV volume calculations at end-diastole and end-systole
Cardiac myofiber angles of −37°, 23° and 83° were assigned at the epicardium, midwall and endocardium respectively, in the remote region . At the infarct region fiber angles were assigned to be 0° in order to use experimentally determined infarct material parameters with respect to this direction . Nodes at the LV epicardial-basal edge were fully constrained, while the remaining basal nodes were restricted to displace in the circumferential-radial plane. The inner endocardial wall was loaded to the measured in vivo end-diastolic and end-systolic LV pressures.
The material response was assumed to be nearly incompressible, transversely isotropic, and hyperelastic for both passive  and active myocardium . An explicit FE solver (LS-DYNA, Livermore Software Technology Corporation, Livermore, CA) with a user-defined material subroutine was used to model the myocardium. The diastolic (passive) myocardial mechanics are described by the strain energy function, W, developed by Guccione et al. , which is transversely isotropic with respect to the local muscle fiber direction,
where C, bf, bt and bfs are diastolic myocardial material parameters. E11 is the fiber strain, E22 is cross-fiber in plane strain, E33 is radial strain and the rest are shear strains.
Systolic contraction was modeled by defining the second Piola-Kirchoff stress tensor as the sum of the passive stress derived from the strain energy function and an active fiber directional component, T0, which is a function of time, t, peak intracellular calcium concentration, Ca0, sarcomere length, l, and maximum isometric tension achieved at longest sarcomere length, Tmax ,
where S is the second Piola-Kirchoff stress tensor, p is the hydrostatic pressure introduced as the Lagrange multiplier needed to enforce incompressibility, J is the Jacobian of the deformation gradient tensor, C is the right Cauchy-Green deformation tensor and the Dev is the deviatoric projection operator,
is the deviatoric contribution of the strain energy function, W (Eq. 1). The strain energy function is decoupled into volumetric and deviatoric components due to the assumption of near incompressibility of the myocardium,
where U is the volumetric contribution.
The active fiber directional stress component is defined by a time-varying elastance model, which at end-systole, is reduced to ,
where m and b are constants, and the length-dependent calcium sensitivity, ECa50, is defined by,
where B is a constant, (Ca0)max is the maximum peak intracellular calcium concentration, l0 is the sarcomere length at which no active tension develops and lR is the stress-free sarcomere length. The material constants for active contraction were taken to be : Ca0 = 4.35 μmol/L, (Ca0)max = 4.35 μmol/L, B = 4.75 μm−1, l0 = 1.58 μm, m = 1.0489 sec μm−1, b = −1.429 sec, and lR,the sarcomere length in the unloaded configuration, was set at 1.85 μm. In accordance with biaxial stretching experiments  and FE analyses [8, 26], cross-fiber, in-plane stress equivalent to 40% of that along the myocardial fiber direction was added.
The diastolic material parameters, bf, bt and bfs, were assigned to be bf = 3.5, bt = 1.37, bfs = 1.24 for both cases. The passive stiffness parameters in the remote region, CR, and the infarct, CI, were determined by calibrating them such that the predicted end-diastolic LV volume matched the experimentally measured value. Similarly the remote active contraction parameter, Tmax_R, which represents the strength of contraction, was determined by matching the predicted end-systolic LV volume to the measured value. The infarct active contraction was set to zero, since there are no contracting myocytes in the infarct.
In the case of the treated infarct, the material parameter search was subject to an additional criterion. The passive stiffness parameters were adjusted until the radial component of strain was nominally zero in the infarct, indicating akinesia, as well as matching the measured chamber volumes. This approach was adopted from Dang et al. , who used FE models, extrapolated from two-dimensional echocardiographic data, to examine akinetic infarcts. In our study, we have utilized data from full 3DE images, which eliminates the need for extrapolation.
In the dyskinetic (control) case, the ED pressure was measured to be 9 mmHg and the ES pressure was 93 mmHg. The measured ED volume was 113.3 mL and the ES volume was 72.8 mL, which leads to an EF of 35%. In the akinetic (treated) case, the ED pressure was found to be 12 mmHg and the ES pressure was 97 mmHg. The ED volume was 48.4 mL and the ES volume was 22.8 mL, which is an EF of 53%. Bargaining
The deformed states at ED and ES, obtained for the akinetic LV, are shown in Figure 3a and 3b, while the overlap of the endocardial surfaces at ED and ES is shown in Figure 3c. The passive stiffness for the akinetic case was estimated to be CR=0.87 kPa in the remote region and CI=300 kPa for the infarct region. The active contraction parameter was estimated to be Tmax_R=130 kPa for the remote region and Tmax_I=0 kPa for the infarct region (as stated previously). The akinesia was estimated by computing the average radial strain in the infarct region, which was found to be err = 0.0014. This implies that on average the wall thickness in the infarct region changed by less than 0.14%.
In the dyskinetic LV case, the passive material properties were estimated to be CR=3 kPa and CI=150 kPa for the remote and infarct zones, respectively. The active contraction was found to be Tmax_R=500 kPa for the remote and Tmax_I=0 kPa for the infarct region. A summary of material properties described above is given in Table 1. Fiber stress distributions and regional average stress values for both cases are shown in Figure 4 and Table 2, respectively. The average ED fiber stress is nearly twice as high in the infarct of the untreated dyskinetic LV, while the average ES fiber stress is nearly 10 times higher in the infarct of the dyskinetic LV.
In the current study, we used contour data acquired from fully 3DE imaging to create FE models of two ovine hearts 8 weeks after large anteroapical infarction. One animal had infarct stiffening therapy with a reactive tissue filler agent and the other was an untreated control. The tissue filler initially limits infarct stretching and thinning by a mass effect but over time it stimulates increased collagen deposition that maintains tissue thickness and stiffness even after most of the injected material has been absorbed .
The use of 3DE (as opposed to 2DE) data allowed for more realistic estimates of LV volumes and regional myocardial material properties compared to our previously published work .
It was determined in Dang et al.  that in order for a non-contracting infarct to be akinetic the passive stiffness must be greater than 285 times the normal healthy value. In the current study, it was found that the tissue filler injection causes the passive stiffness parameter to be 345 times greater than normal myocardium. Ifkovits et al.  also showed an increase in stiffness due to the injection of hyaluronic acid hydrogels into infarcted tissue. Morita et al.  performed experimental biaxial tensile tests on injected infarct tissue, which showed that the peak strain in the longitudinal direction of the infarct was significantly reduced. This implies that the stiffness of the treated infarct has increased relative to the control case, which was also found in the current FE study.
In the control case, the passive stiffness was found to be 50 times higher in the infarct region than in the healthy remote region. This is the same order of magnitude increase as the values reported in similar studies of infarct modeling using the finite element method [28, 29]. In many cases the infarct is assumed to be 10 times stiffer than the remote, but it is possible for this ratio to be higher as there is variation between animals .
It can be seen in Table 2 that the average fiber stress is higher overall in the dyskinetic (untreated) case. The maximum difference between the dyskinetic and akinetic case is the end-systolic fiber stress in the infarct regions, where the injection of tissue filler reduced the stress by nearly 10 times. It can be seen in Figure 4 that there are stress concentrations at the boundary between the infarct and remote region, which could lead to the progressive impairment of contracting myocytes in the borderzone region. It can also be seen that the wall thickness in the infarct region is much greater in the treated LV, which also helps reduce stress. The effects of the injections were found to be transmural throughout the entire infarct region, by inducing collagen production within the infarct tissue . By eliminating the bulging that occurs during dyskinesis, the ejection fraction of the treated heart was 53%, which is close to a normally functioning heart . Thus both stress and ejection fraction are improved, relative to the untreated case.
There are limitations in the current study. First, the treated infarct was assumed to be homogeneous, rather than modeling the distribution of injected material and tissue response separately. Second, only two animals were analyzed in the study, in order to develop the modeling methodology; admittedly, the treatment animal was selected from a cohort of 11 animals in which it manifest the most benefit from injection of the material based on attenuation of LV dilatation . In the future, this method will be applied to a large cohort of animals, in order to determine the statistical relevance. Finally, the validation of the models was based on matching experimental LV chamber volume, and in the case of the treated animal, the radial strain in the infarct region. In the future, this will be improved by computing myocardial strain from 3D MRI, which will be compared to the FE simulation results.
In summary, we have presented a methodology for using 3DE images to generate FE models of an anteroapical infarct model. This was done to investigate the alterations to myocardial material properties, due to the injection of calcium hydroxyapatite-based tissue filler. It was found that the passive stiffness parameter in the treated infarct region is increased by near 345 times the healthy value, and the thickness of the infarct wall is also increased. Overall, the injection of tissue filler was found to cause akinesia in the infarct and globally reduce stress in the LV, which could limit adverse remodeling.
This work was supported by grants from the National Heart, Lung and Blood Institute of the National Institutes of Health, Bethesda, MD, (HL77921, HL86400, HL63954, HL73021 and HL99172). R. Gorman and J. Gorman are supported by individual Established Investigator Awards from the American Heart Association, Dallas, TX.