A generalized multi-resolution expansion for uncertainty propagation with application to cardiovascular modeling.

Computational models are used in a variety of fields to improve our understanding of complex physical phenomena. Recently, the realism of model predictions has been greatly enhanced by transitioning from deterministic to stochastic frameworks, where the effects of the intrinsic variability in parameters, loads, constitutive properties, model geometry and other quantities can be more naturally included. A general stochastic system may be characterized by a large number of arbitrarily distributed and correlated random inputs, and a limited support response with sharp gradients or event discontinuities. This motivates continued research into novel adaptive algorithms for uncertainty propagation, particularly those handling high dimensional, arbitrarily distributed random inputs and non-smooth stochastic responses. In this work, we generalize a previously proposed multi-resolution approach to uncertainty propagation to develop a method that improves computational efficiency, can handle arbitrarily distributed random inputs and non-smooth stochastic responses, and naturally facilitates adaptivity, i.e., the expansion coefficients encode information on solution refinement. Our approach relies on partitioning the stochastic space into elements that are subdivided along a single dimension, or, in other words, progressive refinements exhibiting a binary tree representation. We also show how these binary refinements are particularly effective in avoiding the exponential increase in the multi-resolution basis cardinality and significantly reduce the regression complexity for moderate to high dimensional random inputs. The performance of the approach is demonstrated through previously proposed uncertainty propagation benchmarks and stochastic multi-scale finite element simulations in cardiovascular flow.

On a sparse pressure-flow rate condensation of rigid circulation models.

Cardiovascular simulation has shown potential value in clinical decision-making, providing a framework to assess changes in hemodynamics produced by physiological and surgical alterations. State-of-the-art predictions are provided by deterministic multiscale numerical approaches coupling 3D finite element Navier Stokes simulations to lumped parameter circulation models governed by ODEs. Development of next-generation stochastic multiscale models whose parameters can be learned from available clinical data under uncertainty constitutes a research challenge made more difficult by the high computational cost typically associated with the solution of these models. We present a methodology for constructing reduced representations that condense the behavior of 3D anatomical models using outlet pressure-flow polynomial surrogates, based on multiscale model solutions spanning several heart cycles. Relevance vector machine regression is compared with maximum likelihood estimation, showing that sparse pressure/flow rate approximations offer superior performance in producing working surrogate models to be included in lumped circulation networks. Sensitivities of outlets flow rates are also quantified through a Sobol׳ decomposition of their total variance encoded in the orthogonal polynomial expansion. Finally, we show that augmented lumped parameter models including the proposed surrogates accurately reproduce the response of multiscale models they were derived from. In particular, results are presented for models of the coronary circulation with closed loop boundary conditions and the abdominal aorta with open loop boundary conditions.

Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation.

The adoption of simulation tools to predict surgical outcomes is increasingly leading to questions about the variability of these predictions in the presence of uncertainty associated with the input clinical data. In the present study, we propose a methodology for full propagation of uncertainty from clinical data to model results that, unlike deterministic simulation, enables estimation of the confidence associated with model predictions. We illustrate this problem in a virtual stage II single ventricle palliation surgery example. First, probability density functions (PDFs) of right pulmonary artery (PA) flow split ratio and average pulmonary pressure are determined from clinical measurements, complemented by literature data. Starting from a zero-dimensional semi-empirical approximation, Bayesian parameter estimation is used to find the distributions of boundary conditions that produce the expected PA flow split and average pressure PDFs as pre-operative model results. To reduce computational cost, this inverse problem is solved using a Kriging approximant. Second, uncertainties in the boundary conditions are propagated to simulation predictions. Sparse grid stochastic collocation is employed to statistically characterize model predictions of post-operative hemodynamics in models with and without PA stenosis. The results quantify the statistical variability in virtual surgery predictions, allowing for placement of confidence intervals on simulation outputs.