The measured end-diastolic pressure (6.1 mmHg) was applied to the endocardial wall of the FE model while the passive myocardial stiffness was varied, such that VED
was accurately predicted at 104.9 mL with the measured value being 105 mL. The parameters CR
were calibrated to be 0.017 kPa, while CI
was defined to be 10 times stiffer at 0.17 kPa. The calibrated end-diastolic state was used as the reference state for the systolic strain calculation, which was computed using a multiplicative decomposition of the deformation gradient, as described in Sun et al. [1
The minimum MSE of 11.13, consisting of 720 systolic strain and 2 LV volume data points, was reached in 10 iterations. The optimized contractility parameters for this sheep were found to be Tmax_R = 180 kPa, Tmax_B = 115.4 kPa, and Tmax_I = 0.0 kPa. This shows a 36% decrease in contractile function in the BZ compared to the remote region, and confirms that the infarct has no contractile function. Using the measured end-systolic pressure (91.5 mmHg), the volume VES was predicted to be 67.4 mL, which is only 4.6% higher than the measured value of 64.4 mL.
The predicted systolic strains, using the optimized material parameters, were in generally good agreement with the in-vivo measured strains. Of particular interest is the agreement in the circumferential component of strain, because it aligns very closely to the fiber direction at mid-wall. The mean and standard deviation of the circumferential strain, by region, is given in and . It can be seen that the experimental and finite element strains are within one standard deviation of each other, in each region. In fact, the difference is well below one standard deviation in the remote and BZ regions.
Experimental circumferential strain shown by region, determined from tagged MR images.
FE calculated circumferential strain shown by region, determined from the optimization method.
The root mean square (RMS) error for the circumferential strain component between the 116 pairs of measured and predicted strains in the remote region was 0.673, in the BZ the RMS error was 0.249 with 18 pairs of strain points, and in the infarct region the RMS was 0.167 with 10 pairs of strain points. An example of the circumferential strain agreement at a mid-ventricular slice is shown in . The mean and standard deviation of the experimental strain in this slice is −0.112 ± 0.078 and for the FE strain is −0.118 ± 0.080, which are in very close agreement. An example of the longitudinal strain in the same mid-ventricular slice is shown in . It can be seen that the strain is also positive in the infarct region, indicating longitudinal stretching.
Figure 3 A comparison of the experimental and finite element (a) circumferential strain and (b) longitudinal strain in a slice near mid-ventricle. On average, there is good agreement between the experimental and numerical strains in the remote, borderzone, and (more ...)
The total stress in the myofiber direction at end-systole, as given by Equ. (2)
, was elevated in the BZ region relative to the remote region.. This is indicated by the peak stresses seen in . The region of the BZ that is most affected corresponds to sector 11 (shown in ). At this sector the circumferential strain is elevated relative to the remote region, which indicates that there is less shortening going on in the muscle fibers at this point (i.e. they are being extended). This stress concentration could potentially lead to further expansion of the infarct and BZ regions, through impairment of the myocytes.
Stress (kPa) in the myofiber direction at end-systole. The black lines represent the boundaries of the borderzone. The stress is elevated in the borderzone region, which is indicated by the red color contour.
For the optimization of 3 material parameters, 7 experimental points were selected using a D-optimal method for each iteration. The entire optimization process involving 10 iterations required about 3 hours on a Dell Precision T7400 with dual 3.2 GHz Quad Core Xeon processors.