An ED pressure of 10.85 mmHg and an ES pressure of 99.55 mmHg is applied to the endocardium in the FE model. Displacements at the base are constrained in the longtitudinal direction, with the epicardial–basal edge being constrained in all directions. Because VED
depends only on diastolic material parameters, CR
) is calibrated to 0.95 kPa so that the predicted value of VED
equals the measured value ED
To determine the optimal size of the BZ width dn, the optimization is carried out with dn ranging from 1cm to 7cm by an incremental change of 1cm.
shows the convergence of OBJ with dn
as a parameter. In , the value of OBJ converges quickly; within ~ 6 iterations for all cases. shows that, except for the case when dn
= 1cm, adopting a linear variation of Tmax_BZ
with distance in the FE model produces a lower value of OBJ upon convergence when compared to that obtained from the FE model having a homogeneous Tmax_BZ
]. In that model, the BZ is defined as the steep transition in wall thickness between the infarct and the remote and has a width dn
~ 2cm. Our result therefore implies that the measured in–vivo
systolic strain can be better predicted by modeling Tmax
as a linearly–varying parameter within the BZ. The figure shows that the optimal BZ width for the sheep LV model at 14 weeks postmyocardial infarction is ~ 3cm. Though the change in OBJ is small, we show later that the impact on myofiber stress is substantial.
Optimization results: (a) convergence with BZ width dn as a parameter (b) converged value of OBJ vs. dn.
The resultant optimal distribution of Tmax
in the border zone (i.e for dn
= 3cm) is shown in . In the figure, Tmax
varies linearly from Tmax_I
= 0kPa to Tmax_R
= 186.3kPa within the BZ. In comparision, the optimal values found from assuming a homogenous distribution of Tmax
within the BZ are Tmax_R
= 190.1kPa and Tmax_BZ
= 60.3kPa [1
Contractility in the border zone. dn = 3cm. Tmax_R = 186.3kPa.
shows a comparison of myofiber stress distribution at ES between (a) the optimal linearly–varying Tmax_BZ
FE model and (b) the optimal homogeneous Tmax_BZ
FE model from Sun et al.
]. Between them, the myofiber stress distribution Sf f
is almost identical. Because the stress–strain relation is monotonic and the two models are optimized to have the same strain distribution measured from tagged MRI, we expect the stress distribution of the two models to be similar i.e. regions having high myofiber stresses should correspond to regions where the myofiber undergoes large strain.
Comparison of stress in myofiber direction at ES. (a) Linearly–varying Tmax_BZ with dn = 3cm and Tmax_R = 186.3kPa. (b) Homogeneous Tmax_BZ with Tmax_R = 190.1kPa and Tmax_BZ = 60.3kPa. Unit of fringe levels in kPa.
By contrast, the peak myofiber stress found in the linearly–varying Tmax_BZ model (92.5kPa) is about 15% higher than that found in the homogeneous Tmax_BZ model (80.1kPa). Thus, despite the small difference in OBJ between these two cases, the magnitude of difference in the myofiber stress is substantial.