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# International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019027857
pages 275-294

## AN EFFICIENT NUMERICAL METHOD FOR UNCERTAINTY QUANTIFICATION IN CARDIOLOGY MODELS

Xindan Gao
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai, P.R. China, 200240
Wenjun Ying
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, SAR, China
Zhiwen Zhang
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, SAR, China

### ABSTRACT

Mathematical models of cardiology involve conductivity and massive parameters describing the dynamics of ionic channels. The conductivity is space dependent and cannot be measured directly. The dynamics of ionic channels are highly nonlinear, and the parameters have unavoidable uncertainties because they are estimated using repeated experimental data. Such uncertainties can impact model dependability and credibility since they spread to model parameters during model calibration. It is necessary to study how the uncertainties influence the solution compared to the deterministic solution and to quantify the difference resulting from uncertainty. In this paper, the generalized polynomial chaos method and stochastic collocation method are used to solve the corresponding stochastic partial differential equations. Numerical results are shown to demonstrate that each parameter has different effects on the model responses. More importantly, a quadratic convergence of the expectation is exhibited in the numerical results. The amplitude of standard deviation of the stochastic solution can be controlled by the parameter uncertainty. More precisely, the standard deviation of the stochastic solution is positively linear to the standard deviation of the random parameter. We utilized monodomain equations, which are representative mathematical models to demonstrate the results with the most widely used ionic models, the Hodgkin-Huxley model and Fitz-Hugh Nagumo model.

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