cs.LGFeb 23, 2026
SaveShape-informed cardiac mechanics surrogates in data-scarce regimes via geometric encoding and generative augmentation
Organizations: MOX Laboratory, Department of Mathematics, Politecnico di Milano, Milano, Italy · Division 2.2 Process Simulation, Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin, Germany · Department of Mathematics, University of Trento, Trento, Italy
Abstract
High-fidelity computational models of cardiac mechanics provide mechanistic insight into the heart function but are computationally prohibitive for routine clinical use. Surrogate models can accelerate simulations, but generalization across diverse anatomies is challenging, particularly in data-scarce settings. We propose a two-step framework that decouples geometric representation from learning the physics response, to enable shape-informed surrogate modeling under data-scarce conditions. First, a shape model learns a compact latent representation of left ventricular geometries. The learned latent space effectively encodes anatomies and enables synthetic geometries generation for data augmentation. Second, a neural field-based surrogate model, conditioned on this geometric encoding, is trained to predict ventricular displacement under external loading. The proposed architecture performs positional encoding by using universal ventricular coordinates, which improves generalization across diverse anatomies. Geometric variability is encoded using two alternative strategies, which are systematically compared: a PCA-based approach suitable for working with point cloud representations of geometries, and a DeepSDF-based implicit neural representation learned directly from point clouds. Overall, our results, obtained on idealized and patient-specific datasets, show that the proposed approaches allow for accurate predictions and generalization to unseen geometries, and robustness to noisy or sparsely sampled inputs.
Figures & tables
Figure 1: Data processing, modeling, and training pipelines. Solid black lines/boxes show the pipeline of patient-specific left ventricular (LV) models; dashed gray lines/boxes that of the idealized LV models. They differ in that the idealized model cohort is not augmented by synthetically generated geometries. a) LV geometry extraction from 4-chamber patient-specific heart geometries, retrieving 44 LV geometries. b) Idealized LV models generated from varying long axis, diameter, and wall thickness , , and , respectively, retrieving 512 models. c) SDF-SM training: learning of signed-distance function to represent the geometry boundary. d) Geometry generation by SDF-SM . e) Mechanical forward problem setup. Left: Reference geometry , with Robin (spring) boundaries on the base ( ) and the epicardium ( ) as well as a Neumann boundary at the endocardium ( ). Right: Fiber field ; color indicates the fiber angle with respect to the circumference. f) SDF-SM encoding. g) PCA-SM encoding. h) Computation of universal ventricular coordinates (UVCs). i) Simulation of the forward model to generate displacement training data. Left: undeformed geometry. Right: deformed model with displacement field , color indicating its magnitude. j) Surrogate model, informed by cartesian coordinates , UVCs , a shape code (either from PCA-SM , SDF-SM , or an analytic descriptor of shape features, e.g. features from the idealized geometries), and possibly some parameters .
| Code | Split | CD [mm] | IoU | Dice | MLE | |
|---|---|---|---|---|---|---|
| Learned | Test | |||||
| Train | ||||||
| Original | Test | |||||
| Train |
Table 1: Reconstruction metrics for SDF-SM trained with learned latent codes versus original parametric features on idealized geometries.
Figure 2: Visualizations of the learned latent space for the idealized geometries dataset. a) Comparison between reconstructed latent codes and original geometric features for the idealized geometries. Each column corresponds to an individual geometry, with colors representing the z-normalized value of the associated original feature to ensure homogeneous color scaling across dimensions. b) Pairwise Pearson correlation matrices between the learned latent codes and the original geometric features for the idealized geometries. c) Three-dimensional visualization of the learned latent space for the idealized geometries. Points represent individual samples in the space of the first three latent codes and are colored according to the z-normalized values of the original geometric features, illustrating how variations in physical parameters are embedded in the latent representation.
| Cohort | Split | CD [mm] | IoU | Dice | |
|---|---|---|---|---|---|
| Healthy | Test | ||||
| Train | |||||
| Heart Failure | Test | ||||
| Train |
Table 2: Reconstruction metrics for SDF-SM by cohort and data split (mean ± std).
Figure 3: Performance metrics for SDF-SM reconstructed geometries.
Figure 4: a) Reconstructed meshes colored according to the distance [mm] to the closest point on the target mesh. First row: heart failure patients, second row: healthy patients. b) Reconstructed meshes (in purple) and original ones (in white). First row: heart failure patients; second row: healthy patients.
Figure 5: a) Correlation matrices computed after inference. Matrix on the left describes correlation among different components of the latent space. Matrix on the right describes correlation between codes of different geometries. Colored line at the bottom distinguishes the two cohorts of patients, while the color of the index indicates wether the geometry belongs to training or test set. b) Empirical distributions of pairwise correlations respectively between codes and geometries. Histograms show the density of the upper triangular elements of the corresponding correlation matrices (excluding the diagonal), so that each variable pair is counted once. Vertical dashed lines indicate correlation thresholds of .
| Cohort | Split | CD [mm] | |
|---|---|---|---|
| Healthy | Test | ||
| Train | |||
| Heart Failure | Test | ||
| Train |
Table 3: Reconstruction metrics for PCA-SM by cohort and data split (mean ± std).
Figure 6: Performance metrics for PCA-SM on reconstructed geometries.
Figure 7: SDF-SM performance on test geometries as a function of sample size and geometric noise level.
Figure 8: PCA-SM performance on test geometries as a function of sample size and geometric noise level.
Figure 9: and IoU for SDF-SM as a function of effective noise level on test geometries. Each point corresponds to a different combination. Error bars represent one standard deviation.
Figure 10: and CD for PCA-SM as a function of effective noise level on test geometries. Unlike SDF-SM (Fig. 9 ), points do not collapse onto a single curve, indicating that does not capture the full behavior of the optimization. Error bars represent one standard deviation.
Figure 11: Examples of five successfully generated synthetic left ventricular geometries.
Figure 12: Idealized geometries: Comparison of different input layers and shape codes ( : only Cartesian coordinates; : Cartesian coordinates plus UVCs; : Cartesian coordinates, UVCs, PCA-SM shape code; : Cartesian coordinates, UVCs, SDF-SM shape code; : Cartesian coordinates, UVCs, analytic shape code). No strain contributions to the loss function ( ), number of query points . a) RMSE on training, validation, and test set for the five different input layers. b) RMSE over training epochs, split into training and validation loss. c) Absolute difference in ground-truth and inferred displacement on ten selected test geometries.
Figure 13: Idealized geometries: RMSE on training, validation, and test set for different number of query points, . No strain contributions to the loss function ( ).
| Idealized Geometries | |||||||||||||||
| Query Points | Strain , | ||||||||||||||
| TRAIN | 0.680 | 0.558 | 0.445 | 0.432 | 0.420 | 0.427 | 0.433 | 0.441 | 0.428 | 0.437 | 0.403 | 0.441 | 0.425 | 0.440 | 0.399 |
| VALID | 0.674 | 0.553 | 0.444 | 0.429 | 0.417 | 0.424 | 0.430 | 0.436 | 0.425 | 0.433 | 0.400 | 0.438 | 0.422 | 0.438 | 0.396 |
| TEST | 0.668 | 0.552 | 0.446 | 0.435 | 0.422 | 0.430 | 0.435 | 0.442 | 0.429 | 0.437 | 0.406 | 0.441 | 0.426 | 0.443 | 0.401 |
Table 4: Average root mean squared errors (RMSE) over the respective geometry cohort of idealized models. Comparison of influence of number of query points and strain augmentation of the loss function.
Figure 14: Idealized geometries, investigation of effects of strain contributions to the loss function. No strain ( ) vs. strain loss, with chosen such that both loss contributions are of comparable magnitude; number of query points . a) Displacement RMSE on training, validation, and test set. b) Displacement RMSE over number of training epochs, split into training and validation loss.
Figure 15: Generated and real geometries: Augmented (real and generated geometries) vs. non-augmented (only real geometries) datasets. Comparison of different input layers and shape codes ( : only Cartesian coordinates plus UVCs; : Cartesian coordinates, UVCs, PCA-SM shape code; : Cartesian coordinates, UVCs, SDF-SM shape code). No strain contributions to the loss function ( ), number of query points . a) RMSE on training, validation, and test set for three different input layers, split into generated and real geometries. b) RMSE over training epochs, split into training and validation loss. c) Absolute difference in ground-truth and inferred displacement on all nine test geometries from the real cohort.
| Idealized Geometries | Gen. + Real Geometries | Real Geometries | |||||||||
| Shape | |||||||||||
| TRAIN | 4.772 | 2.856 | 0.637 | 0.609 | 0.425 | 3.926 | 1.318 | 1.221 | 2.914 | 0.722 | 0.561 |
| VALID | 4.960 | 2.979 | 0.796 | 0.620 | 0.422 | 3.876 | 1.609 | 1.711 | 7.498 | 8.468 | 8.383 |
| TEST | 4.631 | 2.749 | 0.730 | 0.620 | 0.426 | 4.004 | 1.818 | 2.388 | 5.790 | 7.395 | 6.653 |
Table 5: Average root mean squared errors (RMSE) over the respective geometry cohort of idealized, generated, and real geometries. Comparison of influence of shape code and data augmentation.
Figure 16: Views of anatomies from both cohorts pre and post alignment and orientation procedure.
Figure 17: Auto-decoder architecture visualization. The geometrical input is concatenated with the latent code associated to the geometry and with optional additional physical parameters .
Table 6: Parameters of the constitutive model Eq. ( 27 ) as well as boundary conditions in Eq. ( 25 ).
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 18: Examples of five discarded synthetic geometries (samples 6, 39, 184, 280, 361) exhibiting artifacts near the apex or base.
Figure 19: Mean shape and first 6 principal modes of variation obtained via PCA. The mean anatomy is shown on the left, while the first PCA modes are visualized on the right by displaying the shapes obtained by adding and subtracting along each mode.