Deep learning-based reduced order models in cardiac electrophysiology

Abstract

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms-such as deep feedforward neural networks and convolutional autoencoders-to obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

Fig 1. DL-ROM architecture.

DL-ROM architecture used during the training phase. In the red box, the DL-ROM to be queried for any new selected couple (t, μ) during the testing phase. The FOM solution u(t; μ) is provided as input to block (A) which outputs ${\tilde{\mathbf{u}}}_n(t;\mathit{\mu })$. The same parameter instance associated to the FOM, i.e. (t; μ), enters block (B) which provides as output u_n(t; μ) and the error between the low-dimensional vectors (dashed green box) is accumulated. The intrinsic coordinates u_n(t; μ) are given as input to block (C) returning the ROM approximation $\tilde{\mathbf{u}}(t;\mathit{\mu })$. Then the reconstruction error (dashed black box) is computed.

Fig 2. Test 1: Comparison between FOM and DL-ROM solutions for a testing-parameter instance.

FOM solution (left), DL-ROM solution with n = 3 (center) and relative error ϵ_k (right) for the testing-parameter instance μ_test = (6.25, 6.25) cm at $\tilde{t}=100$ ms (top) and $\tilde{t}=356$ ms (bottom). The maximum of the relative error ϵ_k is 10⁻³ and it is associated to the diseased tissue.

Fig 3. Test 1: Comparison between the FOM and DL-ROM APs at six points P₁, …, P₆.

Left: FOM solution evaluated for μ_test = (6.25, 6.25) cm at $\tilde{t}=400$ ms together with the points P₁, …, P₆. Right: APs evaluated for μ_test = (6.25, 6.25) cm at points P₁, …, P₆. The DL-ROM, with n = 3, is able to to sharply reconstruct the AP in almost all the points and the main features are captured also for the points close to the scar.

Fig 4. Test 1: Variability of the FOM and DL-ROM solutions over the parameter space.

FOM (right) and DL-ROM (left) AP variability over $\mathcal{P}$ at P = (7.46, 6.51) cm. The DL-ROM sharply reconstructs the FOM variability over $\mathcal{P}$.

Fig 5. Test 1: FOM, DL-ROM and POD-Galerkin ROM CPU computational times.

Left: CPU time required to solve the FOM, by DL-ROM at testing time with n = 3 and by the POD-Galerkin ROM at testing time with N_c = 6 vs. N. The DL-ROM provides the smallest testing computational time for each N considered. Right: FOM, POD-Galerkin ROM and DL-ROM CPU computational times to compute $\tilde{\mathbf{u}}(\overline{t};{\mathit{\mu }}_{test})$ vs. $\overline{t}$ averaged over the testing set. Thanks to the fact that the DL-ROM can be queried at any time istance it is extremely efficient in computing $\tilde{\mathbf{u}}(\overline{t};{\mathit{\mu }}_{test})$ with respect to both the FOM and the POD-Galerkin ROM.

Fig 6. Test 2: Comparison between FOM and DL-ROM solutions for a testing-parameter instance.

FOM solution (left), DL-ROM one (center) with n = 5, and relative error ϵ_k (right) at $\tilde{t}=141.2$ ms (top) and $\tilde{t}=157.2$ ms (bottom), for the testing-parameter instance μ_test = 0.9625 cm. The relative error ϵ_k is below 0.1% at both time instants.

Fig 7. Test 2: Trend of the relative error over time.

Relative error ${ϵ}_k^s$ vs. $\tilde{t}$ with n = 5 for the testing-parameter instance μ_test = 0.9625 cm (the red band indicates the IQR). The error distribution is almost uniform over time.

Fig 8. Test 2: Comparison between FOM and DL-ROM solutions for different testing-parameter instances.

FOM solution (left), DL-ROM one (center) with n = 5, and relative error ϵ_k (right) at $\tilde{t}=153.2$ ms, for the testing-parameter instance μ_test = 0.6125 cm (top) and μ_test = 0.9125 cm (bottom). In both cases the relative error ϵ_k is below 1%.

Fig 9. Test 2: Trend of the relative error over the parameter space.

Relative error ${ϵ}_k^s$ vs. μ_test with n = 5 for the time instance $\tilde{t}=147$ ms (the violet band indicates the IQR). The maximum error is associated to μ_test = 0.7875 cm, the testing-parameter instance between μ_train = 0.775 cm (the last value for which re-entry does not arise) and μ_train = 0.8 cm (the first value for which re-entry is elicited).

Fig 10. Test 2: POD-Galerkin ROM solutions for different testing-parameter instances.

POD-Galerkin ROM solution (left) and relative error ϵ_k (right) for N_c = 2 (top) and N_c = 4 (bottom) at $\tilde{t}=157.2$ ms, for μ_test = 0.9625 cm.

Fig 11. Test 2: FOM, DL-ROM and POD-Galerkin ROM APs at P₁ and P₂.

AP obtained through the FOM, the DL-ROM and the POD-Galerkin ROM with N_c = 4, for the testing-parameter instance μ_test = 0.9625 cm, at P₁ = (0.64, 1.11) cm and P₂ = (0.69, 1.03) cm. The POD-Galerkin ROM approximations are obtained by imposing a POD tolerance ε_POD = 10⁻⁴ and 10⁻³, resulting in error indicator (15) values equal to 5.5 × 10⁻³ and 7.6 × 10⁻³, respectively.

Fig 12. Test 2: Trend of the error indicator versus the CPU testing computational time.

Error indicator ϵ_rel vs. CPU testing computational time for different values of N_c and ε_POD. The DL-ROM outperforms the POD-Galerkin ROM in terms of both efficiency and accuracy.

Fig 13. Test 3: FOM solutions for different testing-parameter instances.

FOM solutions for μ = 12.9 ⋅ 0.0739 mm²/ms (left) and μ = 12.9 ⋅ 0.1991 mm²/ms (right) at $\tilde{t}=276$ ms. By increasing the value of μ the wavefront propagates faster.

Fig 14. Test 3: Comparison between FOM and DL-ROM solutions for a testing-parameter instance at different time instances.

FOM solution (left) and DL-ROM one (right), with n = 10, at $\tilde{t}=42.1$ ms (top) and $\tilde{t}=276$ ms (bottom), for the testing-parameter instance μ_test = 12.9 ⋅ 0.1435 mm²/ms.

Fig 15. Test 3: Comparison between FOM and DL-ROM solutions for a testing-parameter instance at different time instances.

FOM solution (left) and DL-ROM one (right), with n = 10, at $\tilde{t}=42.1$ ms (top) and $\tilde{t}=276$ ms (bottom), for the testing-parameter instance μ_test = 12.9 ⋅ 0.3243 mm²/ms.

Fig 16. Test 3: FOM, DL-ROM and POD-Galerkin ROM APs for a testing-parameter instance.

FOM and DL-ROM CPU computational times. Left: FOM, DL-ROM and POD-Galerkin ROM APs for the testing-parameter instance μ_test = 12.9 ⋅ 0.31 mm²/ms. For the same n, the DL-ROM is able to provide more accurate results than the POD-Galerkin ROM. Right: CPU time required to solve the FOM, by DL-ROM with n = 10 and by the POD-Galerkin ROM with N_c = 6 at testing time vs N. The DL-ROM is able to provide a speed-up equal to 41.

Fig 17. Test 4: Comparison between FOM and DL-ROM solutions for different testing-parameter instances.

FOM solution (left), DL-ROM solution with n = 4 (center) and relative error ϵ_k (right) for the testing-parameter instances μ_text = (12.9 · 0.1125 cm²/ms, 12.9 · 0.0563 cm²/ms, 0.1875) (top) and μ_text = (12.9 · 0.1475 cm²/ms, 12.9 · 0.0737 cm²/ms, 0.2375) (bottom) at $\tilde{t}=319.7$ ms. The maximum of the relative error ϵ_k is about 10⁻³.

Fig 18. Test 4: Comparison between FOM and DL-ROM solutions for different testing-parameter instances.

APs obtained through the FOM and the DL-ROM with n = 4. Left: ${\mathit{\mu }}_{test}^1=(12.9·0.1125{\text{cm}}^2/\text{ms},12.9·0.0737{\text{cm}}^2/\text{ms},0.1625)$ and ${\mathit{\mu }}_{test}^2=(12.9·0.1125{\text{cm}}^2/\text{ms},12.9·0.0737{\text{cm}}^2/\text{ms},0.2375)$; right: ${\mathit{\mu }}_{test}^1=(12.9·0.0775{\text{cm}}^2/\text{ms},12.9·0.0387{\text{cm}}^2/\text{ms},0.2125)$ and ${\mathit{\mu }}_{test}^2=(12.9·0.1825{\text{cm}}^2/\text{ms},12.9·.0912{\text{cm}}^2/\text{ms},0.2125)$. The DL-ROM approximation accurately reproduces the APD variability and the different depolarization patterns.