Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation

D. E. Schiavazzi, G. Arbia, C. Baker, A. M. Hlavacek, T. Y. Hsia, A. L. Marsden, I. E. Vignon-Clementel, The Modeling of Congenital Hearts Alliance MOCHA
2015 International Journal for Numerical Methods in Biomedical Engineering  
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
more » ... entricle 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 0D 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. surgical procedures and treatments [1] . Modern cardiovascular simulation tools now incorporate the fluid dynamic response of vessels with deformable walls (see, e.g., [2, 3] ), account for the effects of the global circulation in patients (see, e.g., [4] ) and can be combined with optimization tools to improve surgical planning and medical device design [5] . Significant research has also focused on growth and remodeling of arterial vessels in response to altered mechanical loads [6, 7, 3] . Moreover, continued improvements in both the resolution of medical images and segmentation algorithms are progressively reducing the time and manual labor needed to create three-dimensional anatomic models. Despite these improvements, predictions from cardiovascular simulation are almost uniformly presented in terms of deterministic results, with few publications (see, e.g., [8]) reporting confidence levels or sensitivities associated with these predictions. Given the myriad uncertainties associated with cardiovascular simulations stemming from clinical and medical imaging data acquisition, physiologic and inter-patient variability, and modeling assumptions, including these sources of uncertainty and quantifying their propagation will increase trust in simulation results, enabling greater impact on clinical decision making. Recent methodological advances now allow for efficient solution of both inverse and forward problems in Uncertainty Quantification (UQ), where the inverse estimation problem requires sampling from an unknown distribution ρ(β) using m point realizations {ρ(β i ), i = 1, . . . , m} and the forward problem requires efficient computation of the distribution ρ(y), i.e., the stochastic response of the model y = G(β) with random inputs β distributed as ρ(β). Multiple approaches have been proposed to sample from unknown posterior distributions of random parameters, formulated as the Bayesian conjunction of likelihood and prior knowledge [9, 10, 11] . Due to its simple implementation and generality, the Markov Chain Monte Carlo (MCMC) method has been widely adopted in this context [12, 13] and various approaches have been proposed to reduce its random walk behavior, with consequent reduction in the number of simulations needed for convergence. Monte Carlo Sampling (MCS) was one of the first approaches introduced to solve the forward problem in UQ [14] . While appealing for high-dimensional problems, its convergence is typically slow for a moderate number of random inputs [15] and alternative techniques are generally preferred, especially when the calls to the deterministic solver are computationally expensive. Improvements in convergence can be achieved by the stochastic finite element method [16] , where intrusion into existing solvers allows simultaneous determination of physical unknowns and stochastic expansion coefficients. In this study, however, we focus on non-intrusive approaches, for their flexibility and ease of implementation. Collocation of a set of stochastic PDEs at the zeros of tensor product orthogonal polynomials (so-called stochastic collocation, SC) was proposed in [17, 18] and extended in [19, 20] to adaptive and anisotropic tensor spaces, respectively. It is also well known [21] that the use of families of polynomials orthogonal with respect to the probability measure of the random inputs can provide up to exponential convergence in characterizing the probability measure of a sufficiently smooth stochastic response. Convergence of polynomial representations has also been addressed in the context of so-called generalized polynomial chaos expansion (gPC) [22] . Non-intrusive propagation schemes have been applied to a stochastic response of interest characterized by sharp gradients or discontinuities. In this context, multi-element polynomial chaos approaches [23] have been proposed together with simplicial discretizations [24], fomulations based on Pade' approximants [25] and others. As an alternative UQ IN PEDIATRIC VIRTUAL SURGERY HEMODYNAMICS PREDICTIONS 3 to numerical integration on multi-variate grids, computation of the pseudo-spectral stochastic coefficients using sparsity promoting greedy heuristics has been proposed in [26] ; extensions to multiresolution representations are discussed in [27] . A few prior studies have begun to account for clinical data uncertainties in cardiovascular simulation. Sparse grid SC was used in [28] to solve the stochastic differential equations governing the propagation of blood pressure in a realistic one-dimensional CFD model of the human circulation. A robust optimization algorithm, combining the surrogate management framework (SMF [29]) with SC on adaptive sparse grids, was applied in [30] to parameterized models of bypass graft anastomoses, and in [31] to several cardiovascular simulation examples. A similar approach was also applied [32] to growth and remodeling predictions in arteries. A systematic application of SC is discussed in [33] where numerical experiments of increasing complexity were performed to quantify the impact of log-normally distributed input random inflow, boundary distal resistances, and vessel cross sectional area on blood pressures and flow rates at locations of interest. In the above studies, the PDFs of the stochastic inputs were selected a priori, i.e., they were only partially inferred by the process of acquiring data in clinical practice. This should instead, in our opinion, be the starting point of any quantification of confidence in the results of patient-specific cardiovascular simulations. For example, input distributions of resistance or compliance elements in 0D circulation networks should be inferred from distributions in the clinical data determined from the patient's pathology and physiology. This will enable improved quantification of PDFs and correlations of input random boundary conditions, in contrast to the typical assumption of independence and identical distribution. Finally, we note that strategies combining clinical data assimilation with physics-informed Navier-stokes constraint have been proposed in [34] . Another example is provided in [35] where visco-elastic support constants are inferred from clinical image data for a coupled fluid-structural model of the human aorta. While applicable to other problems and disciplines, the present study is motivated by the pressing need for uncertainty quantification in multiscale models of single ventricle palliation [36, 37] . Complex cardiac malformations, such as Hypoplastic Left Heart Syndrome or Tricuspid Atresia, are characterized by the existence of only one functional ventricle and treated with a staged surgical transition to the Fontan circulation [38] . As a direct transition to the Fontan circulation is contraindicated in the neonatal period due to high pulmonary vascular resistance, a three staged palliation is usually adopted, which allows the heart and lungs to progressively adapt to the new physiological paradigm: Norwood or BT shunt procedure (stage I), superior cavopulmonary connection (SCPC stage II, bi-directional Glenn or hemi-Fontan surgeries) and Fontan completion (stage III, lateral tunnel or extracardiac conduit). Numerical simulations were previously performed for a cohort of patients to understand the hemodynamic consequences of performing Stage II surgery with and without combined left pulmonary artery LPA arterioplasty on the resulting RPA/PA flow split ratio, average pulmonary pressure and mean pressure drop across the stenosis [39] . The boundary conditions for these simulations were determined using preoperative clinical data affected by uncertainty. In this study, we use the single ventricle example to present a framework for uncertainty quantification in cardiovascular simulations. For computational convenience, we break this question into two successive steps. The first step, formulated as an inverse problem, uses Bayesian inference and surrogate modeling to determine the sets of outlet boundary conditions matching the preoperative PDFs. The second step propagates these uncertainties through to UQ IN PEDIATRIC VIRTUAL SURGERY HEMODYNAMICS PREDICTIONS Output quantities of interest We consider a bounded three-dimensional region Γ ⊂ R 3 containing a subset of the human vasculature, with piecewise smooth boundary ∂Γ and parameterized in space and time by the
doi:10.1002/cnm.2737 pmid:26217878 fatcat:pwao4q3yxfgjtlbpnvfhh23g6a