Gaussian Process Regressions for Inverse Problems and Parameter Searches in Models of Ventricular Mechanics

Figure 1. Automatic processing of cine-MRI images from the Sunnybrook Cardiac MRI database and fitting of idealized geometric model. (A) Complete processing of a representative short axis view frame from patient I-01 in the database. This Cine-MRI modality showed sufficient in-plane resolution, but significantly lower detail in the long axis view (e.g., compare first and second row of the first column). An atlas-based image processing algorithm was employed to extract LV boundaries for each patient. Segmented pixels are shown marked in red in the second and third column. Finally, an idealized 6-parameter model of LV geometry was fitted to the segmentation results, partially correcting for the artifacts introduced by the low resolution in the long axis (see model cross-section rendered in white in the third column). (B) Geometric meaning of idealized LV model parameters. Radial and axial coordinates are indicatd by ρ and ζ, respectively. Rb = outer radius at base, L = wall thickness at base, Z = distance from center of the ventricle to apex of outer wall, H = wall thickness at apex, Ψ0 = truncation angle. Not shown is the e parameter, which governs the curvature of the LV external and…

Figure 2. Design of training sets for the 2 statistical models: LV unloading (A–C) and infarct shape effects (D–F). (A) Pairs of LV cross-sections representing extreme geometries limiting parameter space dimensions. Gray (black) cross-sections correspond to extreme negative (positive) variations of one of the geometric parameters, with the remaining 5 parameters kept at mid-range values. (B) Projection of the 6-D parametric space onto a 3-D cube obtained by neglecting the last 3 dimensions (H, e, and Ψ0). Spherical glyphs indicate locations of 600 sampling points chosen via latin hypercube sampling from a normal distribution centered on the average LV geometry and with a doubled standard deviation compared to that of the complete Sunnybrook database. (C) Cross-section of the parameter space for LV unloading showing combined variations of Rb and Z parameters. (D) Similar to (A), but showing pairs of FE meshes including infarct regions with extreme shapes. The lightest tone of gray indicates the healthy region, the darkest tone indicates the infarct, and the intermediate one marks the refined border zone. (E) 3-D projection of the 4-D parameter space defining infarct shape obtained…

Figure 3. Kinematics of LV motion during a normalized cardiac cycle described by changes over time to the 6 parameters of an idealized model of LV geometry. Traces show averages (marked solid line) over the 4 patient groups (N, normal; HYP, hypertrophic; HF-NI, heart failure without infarct; HF-I, heart failure with infarct) plus or minus standard deviations (shown as semi-transparent overlays). Each subplots shows results for one of the geometric parameters. Results were obtained by custom procedure to fit idealized model to segmentation results. See text for more detail.

Figure 4. Generation of training dataset for the unloading problem. (A) Passive inflation and subsequent parameterization results for selected sample of 3 out of 600 left ventricle geometries considered to build the training datasets for the unloading problem. Idealized geometries, chosen via latin hypercube sampling to probe the parameter space, were discretized and subjected to passive inflation using 3 different sets of material properties. Shown in the panel are results for 2 sets of material properties (WU and WHOG) and 2 loading pressures (1 and 5 kPa). Shown also are color-coded distributions of strain expressed as the first invariant of the Green-Lagrange tensor. (B) Violin plots depicting changes in geometric parameter distributions upon inflation for all the 533 LV geometries included in the training datasets, and for the 3 sets of material properties. Black tone plots indicate distributions of geometric parameters at the BoD configuration. Lightest gray tone plots correspond to distributions synthesized via latin hypercube sampling from a normal distribution constructed based on the BoD configuration, but allowing 2-times larger variations. White segments close to the center of…

Figure 5. Unloading via kriging and comparison to the fixed point iteration method. (A) Unloading procedure is shown applied to a representative case (NI-14, unloading pressure P = 1 kPa, and WHOG material properties) for which a statistical model trained on 75 arbitrary ventricles matched best unloading results via fixed point iteration method. While the fixed point iteration method required meshing of the ventricles in the loaded configuration and iterative updates (middle row), the statistical method allowed to infer the unloaded geometry directly from the 6 parameters describing the end-diastolic (loaded) configuration (bottom row). Top row is similar to bottom row, but shows result obtained after training a statistical model on results from the full parametric study of 500+ LVs. The rightmost column shows overlapped cross-sections of unloaded LVs obtained via the fixed point iteration method (dashed boundary) and 2 statistical models (solid gray tones). (B) Similar to (A), but applied to another representative case (I-07, unloading pressure P = 2 kPa, and WU material properties) for which the statistical learning method (with ntrain = 75) yielded the worst overlap to fixed point…

Figure 6. Accuracy of unloading via GP regression compared to unloading via fixed point iteration method. (A) Barplots of average Dice scores comparing beginning of diastole (BoD), end of diastole (EoD), the best-matching diastolic configuration (OptD), and unloaded configurations obtained via kriging with different ntrain to unloaded configurations predicted via the fixed point iteration method. Subplots show results for combinations of considered material properties (WU , WHOG, or WHOW) and unloading pressures (P = 1 kPa or 2 kPa). (B) Similar to (A), but unloaded configurations are estimated prescribing average midwall strain at end diastole (λff10% on the left, λff15% in the right column). In both panels the dashed white lines drawn on kriging-related bars indicate lower 10th-percentile Dice score for each subcategory.

Figure 7. Statistical model of infarct shape and location effects on simulated SV. (A) Color-coded distribution of SV as predicted by kriging on 6 slices of the 4-D parametric hyperspace. Each plot shows combined effects of variations of 2 parameters on simulated SV, as shown by scale bar (values outside the range are truncated). Darker (lighter) color tones indicate stronger (weaker) impairment due to infarct. Dots represent projections of the probing points onto the slice plane. (B) 5-fold cross-validation to assess performance of the statistical model for varying training sizes ntrain. Relative error on simulated SV predictions approached 6% for the maximum training set size (ntrain = 40).

Figure 8. Comparison between 2 select simulations (out of the 40 considered). (A) On the left, the INF16 model has a smaller basal infarct (volume of 8.8 ml, Long. = 0.43, ΔCirc. = 1.22, ΔLong. = 0.40, Depth = 0.40). On the right, INF30 presents a larger transmural lesion (volume of 16 ml, Long. = 0.79, ΔCirc. = 1.57, ΔLong. = 0.46, Depth = 1.0). (B) Simulated PV loops showing smaller SV for the largest lesion INF30 (dashed line), as expected.