EP-PINNs: Cardiac Electrophysiology Characterisation Using Physics-Informed Neural Networks

Figures

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  Figure 1. Numerical solutions to the Aliev-Panfilov monodomain system for: (a) Planar wave. (b) Centrifugal wave. (c) Spiral wave. (d) Centrifugal wave in the presence of a square heterogeneity in D. (e) Spiral wave in the presence of a square heterogeneity in D.

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  Figure 2. Network architectures and training schemes for EP-PINNs used in this study. Neurons are represented by σ. In forward simulations only network parameters θ were estimated, whereas in the inverse setting one or more EP parameters (λ) were also calculated. (A) Architecture used in all forward simulations and all inverse simulations (1D and 2D) with homogeneous EP parameters. (B) Architecture used in all 2D forward simulations and all inverse simulations where D was a spatially-varying field. (C) Used training schemes. 1 was used in in silico 1D problems, whereas training scheme 2 was used in 2D problems and for optical mapping experimental data. The number of NN neurons and layers varied across different experiments, as did the learning rates (lr) and number of iterations (iter) in training scheme 2. Further details about the architecture and parameterisation of the NNs and training schemes can be found in Supplementary Table 2.

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  Figure 3. Effect of the number of sampled experimental points (A) and experimental noise (B) on EP-PINNs V estimates. The error in estimation is measured using the root mean square error (RMSE, see Equation 7) in forward mode in 1D. Representative V(t) plots sampled at a random spatial location are shown as insets for some of the probed conditions.

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  Figure 4. EP-PINNs 2D forward solutions to the Aliev-Panfilov monodomain system for the same conditions as the GT data depicted in Figure 1. (a) Planar wave. (b) Centrifugal wave. (c) Spiral wave. (d) Centrifugal wave in the presence of a square heterogeneity in D. (e) Spiral wave in the presence of a square heterogeneity in D.

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  Figure 5. Error in global EP parameter estimates by EP-PINNs in the inverse setting in 1D in the presence of different amounts of experimental noise. We show the relative error for a, b and D, when estimated separately and in pairs.

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  Figure 6. 1D inverse EP-PINNs solution for a detailed canine left atrial model in the conditions of: (A) no remodelling. (B) moderate AF remodelling. (C) severe AF remodelling. Representative V(t) plots are shown throughout, accompanied by the models 90% APD and EP-PINNS estimates for b, a parameter inversely proportional to APD.

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  Figure 7. EP-PINNs inverse solution in homogeneous conditions in 2D. (a) Relative error for global estimates of: a and D and b and D, when estimated separately or simultaneously. (b–d) Corresponding representative V maps for: planar wave (b), centrifugal wave (c), and spiral wave (d). Compare (b–d) to the corresponding GT in Figures 1a-c and the forward solutions in Figures 4a–c.

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  Figure 8. EP-PINNs 2D inverse solution in the presence of heterogeneities in D. Maps showing representative V and D estimates for: (a,c) centrifugal wave and (b,d) spiral wave. (e) Error for global estimates of a and b and RMSE for estimates of D across the 2D domain, for all estimated parameter combinations. Compare (a,b) to the corresponding GT in Figures 1d,e and the forward solutions in Figures 4d,e.

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  Figure 9. 1D inverse EP-PINNs solution for experimental optical mapping data, in the presence of (A) ICaL channel blocker nifedipine and (B) IKr channel blocker E-4031. (Δb refers to the change in model parameter b in the presence of the drug when compared to the baseline value. Each AP was acquired in separate datasets and juxtaposed in the figure to allow visual comparisons.