Towards efficient uncertainty quantification in complex and large-scale biomechanical problems based on a Bayesian multi-fidelity scheme

Abstract

In simulation of cardiovascular processes and diseases patient-specific model parameters, such as constitutive properties, are usually not easy to obtain. Instead of using population mean values to perform “patient-specific” simulations, thereby neglecting the inter- and intra-patient variations present in these parameters, these uncertainties have to be considered in the computational assessment. However, due to limited computational resources and several shortcomings of traditional uncertainty quantification approaches, parametric uncertainties, modeled as random fields, have not yet been considered in patient-specific, nonlinear, large-scale, and complex biomechanical applications. Hence, the purpose of this study is twofold. First, we present an uncertainty quantification framework based on multi-fidelity sampling and Bayesian formulations. The key feature of the presented method is the ability to rigorously exploit and incorporate information from an approximate, low fidelity model. Most importantly, response statistics of the corresponding high fidelity model can be computed accurately even if the low fidelity model provides only a very poor approximation. The approach merely requires that the low fidelity model and the corresponding high fidelity model share a similar stochastic structure, i.e., dependence on the random input. This results in a tremendous flexibility in choice of the approximate model. The flexibility and capabilities of the framework are demonstrated by performing uncertainty quantification using two patient-specific, large-scale, nonlinear finite element models of abdominal aortic aneurysms. One constitutive parameter of the aneurysmatic arterial wall is modeled as a univariate three-dimensional, non-Gaussian random field, thereby taking into account inter-patient as well as intra-patient variations of this parameter. We use direct Monte Carlo to evaluate the proposed method and found excellent agreement with this reference solution. Additionally, the employed approach results in a tremendous reduction of computational costs, rendering uncertainty quantification with complex patient-specific nonlinear biomechanical models practical for the first time. Second, we also analyze the impact of the uncertainty in the input parameter on mechanical quantities typically related to abdominal aortic aneurysm rupture potential such as von Mises stress, von Mises strain and strain energy. Thus, providing first estimates on the variability of these mechanical quantities due to an uncertain constitutive parameter, and revealing the potential error made by assuming population averaged mean values in patient-specific simulations of abdominal aortic aneurysms. Moreover, the influence of correlation length of the random field is investigated in a parameter study using MC.