Embedding high-dimensional Bayesian optimization via generative modeling: Parameter personalization of cardiac electrophysiological models

Abstract

The estimation of patient-specific tissue properties in the form of model parameters is important for personalized physiological models. Because tissue properties are spatially varying across the underlying geometrical model, it presents a significant challenge of high-dimensional (HD) optimization at the presence of limited measurement data. A common solution to reduce the dimension of the parameter space is to explicitly partition the geometrical mesh. In this paper, we present a novel concept that uses a generative variational auto-encoder (VAE) to embed HD Bayesian optimization into a low-dimensional (LD) latent space that represents the generative code of HD parameters. We further utilize VAE-encoded knowledge about the generative code to guide the exploration of the search space. The presented method is applied to estimating tissue excitability in a cardiac electrophysiological model in a range of synthetic and real-data experiments, through which we demonstrate its improved accuracy and substantially reduced computational cost in comparison to existing methods that rely on geometry-based reduction of the HD parameter space.

Keywords: High-dimensional Bayesian optimization; personalized modeling; variational autoencoder.

Figure 1:

The workflow diagram of the presented parameter personalization framework which consists of following major components: 1) training of a generative model (blue rectangle), and 2) HD Bayesian optimization via a trained generative model (green rectangle).

Figure 2:

Comparison of DC (a), RMSE (b), and the number of model evaluations (c) between BO-VAE EI Post-1 (green bar) and three other groups of methods: FH and FS (yellow bars); BO-VAE using standard EI, EI isotropic, and EI Post-1 (red bars); and BO-PCA (purple bar).

Figure 3:

Examples of estimated tissue excitability with the BO-VAE, BO-PCA, FH, and FS methods. Progression of the FH optimization on the multi-scale hierarchy in (a) shows that only one node in level d3 consists of heterogeneous tissues (green nodes: homogeneous tissues; red nodes: heterogeneous tissues) which was refined along the multi-scale hierarchy to obtain an accurate solution as shown in example (c). By contrast the progression of the FH optimization in (b) shows that most of the nodes in level d2 consist of heterogeneous tissues. Since all of them could not be refined there is limited accuracy as reflected in example (c).

Figure 4:

Comparison of parameter estimation results by using a VAE trained on patient-specific vs. non-patient-specific training data. Left: accuracy in terms of DC and RMSE. Right: examples of estimated tissue excitability.

Figure 5:

(a) Average PCA reconstruction errors in terms of DC and RMSE on test data with increasing numbers of principle components, in comparison to those given by the VAE with two-dimensional and five-dimensional latent codes. (b). Comparison of the accuracy of BO-VAE and BO-PCA with different numbers of latent dimensions / principal components.

Figure 6:

Examples of estimated tissue excitability with BO-VAE and BO-PCA with two and five dimensional representations of the parameter space.

Figure 7:

Comparison of point selection during Bayesian optimization using the standard EI and the EI post-1 as acquisition functions. With EI post-1, the regions of higher q_α(z) was explored before the regions of lower q_α(z).

Figure 8:

Examples of tissue excitability estimated using BO-VAE with the standard EI and the EI Post-1. The former resulted in less optimal or incorrect solutions.

Figure 9:

Examples of the sample points selected during Bayesian optimization when various forms of ε(z) are used in augmenting the standard EI. Red cross represents the location of the optimum.

Figure 10:

(a)-(b): Scatter plots of two-dimensional latent codes colored by (a) infarct size and (b) infarct location. (c)-(d): t-SNE visualization of five-dimensional latent codes colored by (c) infarct size and (d) infarct location.

Figure 11:

Comparison of binary excitability corrupted with Gaussian noise (large-noise group) and binary excitability appended with boundary value between healthy and infarct core (border-zone group) in their (a) ground truth, (b) optimization result and (c) reconstruction by VAE trained on original uniform noise corrupted excitability data.

Figure 12:

Comparison of (a) the correlation coefficient and (b) the RMSE between the 120-lead ECG data simulated with tissue properties setup with the original setting (see Section 4.2) versus those setup with an additional Gaussian $\mathcal{N}(0,0.01)$ noise (large-noise group) and an additional transition border zone between the infarct and the healthy tissues (border-zone group). (c) Comparison of the accuracy in tissue properties reconstruction (VAE) and estimation (BO-VAE) in terms of DC and RMSE.

Figure 13:

Examples of estimated tissue excitability with the BO-VAE in the presence of multiple infarcts.

Figure 14:

Results of estimated tissue excitability from the BO-VAE and FH method in comparison to 3D infarcts delineated from in-vivo MRI images.

Figure 15:

Evaluation of the estimated tissue excitability in real data studies in comparison to in-vivo voltage maps on three patients. First, we compared the results obtained with BO-VAE (column: Data1) with those obtained with FH and FS methods. Second, we compared the results obtained with BO-VAE using individual ECG data vs. with the result obtained when using all ECG data at once (column: combined).

Figure 16:

Comparison of the accuracy (DC and RMSE) in parameter estimation using measurement data from 12-lead ECG and 120-lead ECG. Left: Summary statistics. Right: Examples of estimated tissue excitability.

Figure 17:

Comparison of the accuracy in parameter estimation of the presented BO-VAE when a VAE is trained on a heart described in Section 4.2 vs. when a VAE is trained on a heart described on Section 4.4; 18 test tissue property for optimization were taken from Sections 4.2. Left: Visual comparison of (a) ground truth parameters, (b) estimated parameters when VAE was trained on a heart from Section 4.2, and (c) estimated parameters when VAE was trained on a heart from Section 4.4. Right: Comparison of the accuracy in parameter estimation in terms of DC and RMSE.