Efficient estimation of personalized biventricular mechanical function employing gradient‐based optimization

Figure 1. Left: finite element mesh of a biventricular geometry reconstructed from MR images separated into 3 material regions, namely, left ventricle free wall (blue), septum (green), and right ventricle free wall (right). Right: myocardial fiber orientation are assigned using the LDRB algorithm42 with an angle +α and −α prescribed on the endocardium and epicardium, respectively. Here showing the fiber architecture for α ranging from 30° to 80° with increments of 10°, where the absolute value of the fiber angle is used as color map

Figure 2. To the left, we see the model‐personalization pipeline. The image‐based geometry corresponds to some image frame taken at mid‐diastole. An estimate of the unloaded geometry was found by applying 1 iteration of the backward displacement method using a=1.291 kPa according to Asner et al,21 followed by an estimation of a by minimizing the fit of the end‐diastolic volumes (EDV). During systole, both cavity volumes and circumferential strain were used in the optimization to determine the amount of active contraction in terms of the active control, which are, respectively, γ and T a in the active strain and active stress formulation. To the right, we show a comparison of the unloaded geometry for CASE 3. The upper figure shows the resulting unloaded, zero pressure geometry in red and the the original image‐based geometry in transparent, while the bottom figure shows the unloaded geometry, inflated to the target pressure in the image‐based geometry, and the original image‐based geometry in transparent for comparison

Figure 3. Results of the gradient‐based minimization of model‐data mismatch for different choice of fiber angles. Left: simulated (color lines) and measured (black circles) pressure volume (PV) loops in the left ventricle (LV, top row) and right ventricle (RV, bottom row). Center: simulated (color lines) and measured (black circles) circumferential strain in the left ventricle (top row), right ventricle (middle row ), and septum (bottom row). Right: average values of cost functional for the volume (top row) and strain (bottom row), for each choice of fiber angle

Figure 4. Validation of the model‐personalization process using simulated and measured longitudinal strain that was not used in the optimization. Upper, middle, and lower panel show the longitudinal strain curves for different choice of fiber angles in the left ventricle (LV), right ventricle (RV), and septum, respectively. ED, end‐diastole; ES, end‐systole

Figure 5. To the left, average traces of the active strain parameter γ in (3) in the left ventricle (LV, top) and right ventricle (RV, bottom) for different choice of fiber angle. To the right, average traces of Cauchy fiber stress in the LV (top) and RV (bottom) for different choice of fiber angle. The fiber angles were defined symmetrically across the wall with a negative angle on the epicardium and a positive angle on the endocardium ranging from 30°to80° with increments of 10°. On the x−axis, we plot the normalized time with respect to end‐diastole (ED) and end‐systole (ES). Horizontal dotted lines indicate timings of opening of the aortic and mitral valve

Figure 6. Comparison of fiber stress and cardiac contraction using the active strain and active stress approach using 60° fiber angle. To the left, average traces of the active stress (top) parameter T a in (8), and the active strain (bottom) parameter in 3. The active stress parameter, with unit kPa, is plotted on a logarithmic scale for easier comparison with the active strain parameter. To the right, estimated Cauchy fiber stress using the active stress (top) and active strain formulation (bottom). On the x−axis, we plot the normalized time with respect to end‐diastole (ED) and end‐systole (ES). Horizontal dotted lines indicate timings of opening and closing of the aortic valve. LV, left ventricle; RV, right ventricle

Figure 7. Results obtained using different mesh resolutions as listed in Table 1. To the left and right, respectively, the active strain and fiber stress traces, in the left ventricle (LV, top) and right ventricle (RV, bottom) estimated using the coarse and refined geometry. ED, end‐diastole; ES, end‐systole

Figure 8. Snap shots of the end‐diastolic and end‐systolic configuration and the estimated fiber stresses shown as colormap. ED, end‐diastole; ES, end‐systole

Figure A1. To the left, we show the passive‐filling curves with volume in mL on the x−axis and pressure in kPa on the y−axis with different unloaded volume resulting from different material parameters used to estimate the unloaded geometries. Middle and right panels show average time traces of estimated active strain and Cauchy fiber stress, respectively, for different choices of unloaded configuration. Top row shows the results in the left ventricle (LV) while bottom row shows the results in the right ventricle (RV). Here ,the values a=0.5, 1.0, 2.0 and 4.0 kPa are used to unload the ventricles . ED, end‐diastole; ES, end‐systole