We present a fast, patient-specific methodology for uncertainty quantification in electrophysiology, aimed at meeting the time constraints of clinical practitioners. We focus on computing the statistics of the activation map, given the uncertainties associated with the conductivity tensor modeling the fiber orientation in the heart. We use a fast parallel solution method implemented on a graphics processing unit for the eikonal approximation, in order to compute the activation map and to sample the random fiber field with correlation on the basis of geodesic distances. While this enables to perform uncertainty quantification studies with a manageable computational effort, the required time frame still exceeds clinically suitable time expectations. In order to reduce it further by 2 orders of magnitude, we rely on Bayesian multifidelity methods. In particular, we propose a low-fidelity model that is patient-specific and free from the additional training cost associated with reduced models. This is achieved by a sound physics-based simplification of the full eikonal model. The low-fidelity output is then corrected by the standard multifidelity framework. In practice, the complete procedure only requires approximately 100 new runs of our eikonal graphics processing unit solver for producing the sought estimates and their associated credible intervals, enabling a full online analysis in less than 5 minutes.
Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the accuracy of the estimates. This approach has recently gained widespread attention in uncertainty quantification [1]. This is partly due to the availability of optimal strategies for the estimation of the expectation of scalar quantities-of-interest [2]. In practice, the optimal strategy for the expectation is also used for the estimation of variance and sensitivity indices [3]. However, a general strategy is still lacking for vector-valued problems, nonlinearly statistically-dependent models, and estimators for which a closed-form expression of the error is unavailable. The focus of the present work is to generalize the standard multifidelity estimators to the above cases. The proposed generalized estimators lead to an optimization problem that can be solved analytically and whose coefficients can be estimated numerically with few runs of the highand low-fidelity models. We analyze the performance of the proposed approach on a selected number of experiments, with a particular focus on cardiac electrophysiology, where a hierarchy of physics-based low-fidelity models is readily available.
Extracting spatial heterogeneities from patient-specific data is challenging. In most cases, it is unfeasible to achieve an arbitrary level of detail and accuracy. This lack of perfect knowledge can be treated as an uncertainty associated with the estimated parameters and thus be modeled as a spatiallycorrelated random field superimposed to them. In order to quantify the effect of this uncertainty on the simulation outputs, it is necessary to generate several realizations of these random fields. This task is far from trivial, particularly in the case of complex geometries. Here, we present two different approaches to achieve this. In the first method, we use a stochastic partial differential equation, yielding a method which is general and fast, but whose underlying correlation function is not readily available. In the second method, we propose a geodesic-based modification of correlation kernels used in the truncated Karhunen-Loève expansion with pivoted Cholesky factorization, which renders the method efficient even for complex geometries, provided that the correlation length is not too small. Both methods are tested on a few examples and cardiac applications.
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