Non-Intrusive Parametrized-Background Data-Weak Reconstruction of Cardiac Displacement Fields from Sparse MRI-like Observations

TL;DR

The paper tackles reconstructing 3D myocardial displacement fields from sparse MRI-like observations using a non-intrusive Parametrized-Background Data-Weak (PBDW) framework. It builds a background space from solution snapshots and augments it with a data-driven update space, while enabling efficient sensor selection via an H-size minibatch wOMP and memory-optimized vectorial implementations. The offline-online decomposition yields four-order-of-magnitude speedups over full finite-element simulations, with noise-free relative errors near 1e-4 and robust performance under 10% Gaussian noise or realistic slice sparsity (few percent to few tens of percent error in L2 norms). Validation on a 3D LV model with scar demonstrates strong reconstruction accuracy and practical runtime suitable for clinical workflows, provided model mismatch is accounted for in the offline basis. The work advances non-intrusive data-model fusion for cardiac mechanics, delivering fast, interpretable reconstructions compatible with diverse commercial solvers and imaging protocols, and outlines avenues for extending to time-dependent dynamics and uncertainty quantification.

Abstract

Personalized cardiac diagnostics require accurate reconstruction of myocardial displacement fields from sparse clinical imaging data, yet current methods often demand intrusive access to computational models. In this work, we apply the non-intrusive Parametrized-Background Data-Weak (PBDW) approach to three-dimensional (3D) cardiac displacement field reconstruction from limited Magnetic Resonance Image (MRI)-like observations. Our implementation requires only solution snapshots -- no governing equations, assembly routines, or solver access -- enabling immediate deployment across commercial and research codes using different constitutive models. Additionally, we introduce two enhancements: an H-size minibatch worst-case Orthogonal Matching Pursuit (wOMP) algorithm that improves Sensor Selection (SS) computational efficiency while maintaining reconstruction accuracy, and memory optimization techniques exploiting block matrix structures in vectorial problems. We demonstrate the effectiveness of the method through validation on a 3D left ventricular model with simulated scar tissue. Starting with noise-free reconstruction, we systematically incorporate Gaussian noise and spatial sparsity mimicking realistic MRI acquisition protocols. Results show exceptional accuracy in noise-free conditions (relative L2 error of order O(1e-5)), robust performance with 10% noise (relative L2 error of order O(1e-2)), and effective reconstruction from sparse measurements (relative L2 error of order O(1e-2)). The online reconstruction achieves four-order-of-magnitude computational speed-up compared to full Finite Element (FE) simulations, with reconstruction times under one tenth of second for sparse scenarios, demonstrating significant potential for integration into clinical cardiac modeling workflows.

Figures (13)

• Figure 1: Computational domain representing an idealised . On the left we indicate the boundaries of the computational domain and different colours refer to an idealised scar region with different material parameters; on the right we indicate the rule-based fiber directions $\bm{f}_0$.
• Figure 2: Performance of in noiseless test case. Left: norm of the observation vector $\bm{y}^\mathrm{obs}$ in mm; middle: norm of the state reconstruction $\bm{u}^*$ in mm; right: absolute value of the difference between the reconstruction and the ground truth in mm.
• Figure 3: This visualization illustrates the sensors selected by the algorithm. Each sensor measures the average displacement field within a voxel of fixed dimension. The images display the chosen voxels along with their measured values. For context, the background shows the entire displacement field. Left: $x$-component of the ground truth $\bm{u}_\mathrm{true}$ in mm; middle: $y$-component of the ground truth $\bm{u}_\mathrm{true}$ in mm; right: $z$-component of the ground truth $\bm{u}_\mathrm{true}$ in mm.
• Figure 4: Performance of in noisy test case. Left: norm of the observation vector $\bm{y}^\mathrm{obs}$ in mm; middle: norm of the state reconstruction $\bm{u}^*$ in mm; right: absolute value of the difference between the reconstruction and the ground truth in mm.
• Figure 5: Performance of in noisy test case. Left: observation vector $\bm{y}^\mathrm{obs}$, restricted to the $x$-component, in mm; middle: $x$-component of the state reconstruction $\bm{u}^*$ in mm; right: absolute value of the difference between the reconstruction and the ground truth in mm.

Theorems & Definitions (3)

• Remark 1: Non-intrusive implementations
• Remark 2: versus
• Remark 3: Computational complexity