The animal model has been described previously [17
]. 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 [13
]. 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. [16
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 . Mitral and aortic valve function was normal in both animals.
Contour points created from the 3D echo images, which were used to generate the in vivo geometry for the (a) akinetic (treated) LV and (b) the dyskinetic LV.
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 [18
]. 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 (). 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
The volumetric mesh generated for the (a) akinetic (treated) LV and (b) the dyskinetic LV.
Cardiac myofiber angles of −37°, 23° and 83° were assigned at the epicardium, midwall and endocardium respectively, in the remote region [19
]. 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 [20
]. 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.
Diastolic and Systolic Material Properties
The material response was assumed to be nearly incompressible, transversely isotropic, and hyperelastic for both passive [21
] and active myocardium [22
]. 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. [21
], which is transversely isotropic with respect to the local muscle fiber direction,
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
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
). The strain energy function is decoupled into volumetric and deviatoric components due to the assumption of near incompressibility of the myocardium,
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 [23
are constants, and the length-dependent calcium sensitivity, ECa50
, is defined by,
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 [24
= 4.35 μmol/L, (Ca0)max
= 4.35 μmol/L, B
= 4.75 μm−1
= 1.58 μm, m
= 1.0489 sec μm−1
= −1.429 sec, and lR
,the sarcomere length in the unloaded configuration, was set at 1.85 μm. In accordance with biaxial stretching experiments [25
] and FE analyses [8
], 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. [27
], 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.