Efficient Ventricular Parameter Estimation Using AI-Surrogate Models

Figure 1. (left) AI-surrogate depicting its inputs and outputs (Maso Talou et al., 2020); (right) Example of the geometry of the ventricle at diastasis for g = 0.

Figure 2. Mean displacement of the ventricular wall (denoted μu^) for different combinations of the constitutive parameters c1 and c2, and an intra-ventricular pressure load of p = 0.9 kPa. For details of mean displacements for other loads, refer to the Supplementary Material.

Figure 3. Relative error of the two-parameter estimation. Estimates of the constitutive parameters c1 (left) and c2 (right) are presented using the full-field tracking and contour matching approaches with a load of p = 0.9 kPa in the absence of noise in the observations. For details of estimations for other loads, refer to the Supplementary Material.

Figure 4. Mean and standard deviation of the relative error for the two-parameter estimation of c1 (left) and c2 (right) constitutive parameters using the full-field tracking and contour matching approaches with a load of p = 0.9 kPa and the presence of Gaussian noise (σ = 1mm ≈ 2 pixels) in the observations. For details of estimations for other loads, refer to the Supplementary Material.

Figure 5. Mean and standard deviation of the relative error for the two-parameter estimation of c1 (left) and c2 (right) constitutive parameters using full-field tracking and contour matching approaches, a pressure trace with 10 time-points, and independent Gaussian noise (σ = 1mm ≈ 2 pixels) in the observations.

Figure 6. Relative error in estimating intra-ventricular pressure p using full-field tracking and contour matching approaches with a load of p = 0.9 kPa in the absence of noise in the observations. For details of the estimation of other loads, refer to the Supplementary Material.

Figure 7. Mean and standard deviation of the relative errors for recovering ground truth intra-ventricular pressure using the full-field tracking and contour matching approaches and an intra-ventricular pressure of p = 0.9 kPa with Gaussian noise (σ = 1 mm ≈ 2 pixels) in the observations. Refer to the Supplementary Material for details of the relative errors for other intra-ventricular pressures.

Figure 8. Loss function for the full-field tracking and contour matching approaches with respect to p in the presence (right) and absence (left) of noise in the observations. The surface plot in the p-p^ plane and the opaque manifold represent the loss function value L(p) when p^ is the ground truth (optimal) value. Level curves in L-p^ and L-p planes correspond to parallel cuts of the opaque manifold for fixed values of p and p^, respectively (blue to red shades indicate values of p and p^ that correspond to the ticks on their respective axes). Note that for a given p^, L(p) is convex with the minimum p=p^ resulting in precise and well-behaved formulations for gradient-based optimisers.

Figure 9. Loss function for the full-field tracking and contour matching approaches with respect to c1 and c2 in the presence (right) and absence (left) of noise in the observations. The first and second rows present the loss function values when c1 = 3.5 kPa, c2 = 22.0 and c1 = 2.5 kPa, c2 = 10.0 are the ground truth parameters, respectively. Surface plots in c1-c2 plane and the opaque manifolds represent the loss function value L(c1,c2). Level curves in L-c1 and L-c2 planes correspond to parallel cuts of the opaque manifold for fixed values of c2 and c1, respectively (blue to red shades indicate values of c1 and c2 that correspond to the ticks on their respective axes). Note that the objective functions are more sensitive to perturbations of c1 and c2 for softer materials (as highlighted by the blue level curves, which represent lower values of c1 and c2).