Improved multifidelity Monte Carlo estimators based on normalizing flows and dimensionality reduction techniques.

We study the problem of multifidelity uncertainty propagation for computationally expensive models. In particular, we consider the general setting where the high-fidelity and low-fidelity models have a dissimilar parameterization both in terms of number of random inputs and their probability distributions, which can be either known in closed form or provided through samples. We derive novel multifidelity Monte Carlo estimators which rely on a shared subspace between the high-fidelity and low-fidelity models where the parameters follow the same probability distribution, i.e., a standard Gaussian. We build the shared space employing normalizing flows to map different probability distributions into a common one, together with linear and nonlinear dimensionality reduction techniques, active subspaces and autoencoders, respectively, which capture the subspaces where the models vary the most. We then compose the existing low-fidelity model with these transformations and construct modified models with an increased correlation with the high-fidelity model, which therefore yield multifidelity estimators with reduced variance. A series of numerical experiments illustrate the properties and advantages of our approaches.

Geometric Uncertainty in Patient-Specific Cardiovascular Modeling with Convolutional Dropout Networks.

We propose a novel approach to generate samples from the conditional distribution of patient-specific cardiovascular models given a clinically aquired image volume. A convolutional neural network architecture with dropout layers is first trained for vessel lumen segmentation using a regression approach, to enable Bayesian estimation of vessel lumen surfaces. This network is then integrated into a path-planning patient-specific modeling pipeline to generate families of cardiovascular models. We demonstrate our approach by quantifying the effect of geometric uncertainty on the hemodynamics for three patient-specific anatomies, an aorto-iliac bifurcation, an abdominal aortic aneurysm and a sub-model of the left coronary arteries. A key innovation introduced in the proposed approach is the ability to learn geometric uncertainty directly from training data. The results show how geometric uncertainty produces coefficients of variation comparable to or larger than other sources of uncertainty for wall shear stress and velocity magnitude, but has limited impact on pressure. Specifically, this is true for anatomies characterized by small vessel sizes, and for local vessel lesions seen infrequently during network training.

Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics.

Standard approaches for uncertainty quantification in cardiovascular modeling pose challenges due to the large number of uncertain inputs and the significant computational cost of realistic three-dimensional simulations. We propose an efficient uncertainty quantification framework utilizing a multilevel multifidelity Monte Carlo (MLMF) estimator to improve the accuracy of hemodynamic quantities of interest while maintaining reasonable computational cost. This is achieved by leveraging three cardiovascular model fidelities, each with varying spatial resolution to rigorously quantify the variability in hemodynamic outputs. We employ two low-fidelity models (zero- and one-dimensional) to construct several different estimators. Our goal is to investigate and compare the efficiency of estimators built from combinations of these two low-fidelity model alternatives and our high-fidelity three-dimensional models. We demonstrate this framework on healthy and diseased models of aortic and coronary anatomy, including uncertainties in material property and boundary condition parameters. Our goal is to demonstrate that for this application it is possible to accelerate the convergence of the estimators by utilizing a MLMF paradigm. Therefore, we compare our approach to single fidelity Monte Carlo estimators and to a multilevel Monte Carlo approach based only on three-dimensional simulations, but leveraging multiple spatial resolutions. We demonstrate significant, on the order of 10 to 100 times, reduction in total computational cost with the MLMF estimators. We also examine the differing properties of the MLMF estimators in healthy versus diseased models, as well as global versus local quantities of interest. As expected, global quantities such as outlet pressure and flow show larger reductions than local quantities, such as those relating to wall shear stress, as the latter rely more heavily on the highest fidelity model evaluations. Similarly, healthy models show larger reductions than diseased models. In all cases, our workflow coupling Dakota's MLMF estimators with the SimVascular cardiovascular modeling framework makes uncertainty quantification feasible for constrained computational budgets.

Uncertainty quantification of simulated biomechanical stimuli in coronary artery bypass grafts.

Coronary artery bypass graft surgery (CABG) is performed on more than 400,000 patients annually in the U.S. However, saphenous vein grafts (SVGs) implanted during CABG exhibit poor patency compared to arterial grafts, with failure rates up to 40% within 10 years after surgery. Differences in mechanical stimuli are known to play a role in driving maladaptation and have been correlated with endothelial damage and thrombus formation. As these quantities are difficult to measure in vivo, multi-scale coronary models offer a way to quantify them, while accounting for complex coronary physiology. However, prior studies have primarily focused on deterministic evaluations, without reporting variability in the model parameters due to uncertainty. This study aims to assess confidence in multi-scale predictions of wall shear stress and wall strain while accounting for uncertainty in peripheral hemodynamics and material properties. Boundary condition distributions are computed by assimilating uncertain clinical data, while spatial variations of vessel wall stiffness are obtained through approximation by a random field. We developed a stochastic submodeling approach to mitigate the computational burden of repeated multi-scale model evaluations to focus exclusively on the bypass grafts. This produces a two-level decomposition of quantities of interest into submodel contributions and full model/submodel discrepancies. We leverage these two levels in the context of forward uncertainty propagation using a previously proposed multi-resolution approach. The time- and space-averaged wall shear stress is well estimated with a coefficient of variation of <35%, but ignorance about the spatial distribution on the wall elastic modulus and thickness lead to large variations in an objective measure of wall strain, with coefficients of variation up to 100%. Sensitivity analysis reveals how the interactions between the flow and material parameters contribute to output variability.

Automated Tuning for Parameter Identification and Uncertainty Quantification in Multi-scale Coronary Simulations.

Atherosclerotic coronary artery disease, which can result in coronary artery stenosis, acute coronary artery occlusion, and eventually myocardial infarction, is a major cause of morbidity and mortality worldwide. Non-invasive characterization of coronary blood flow is important to improve understanding, prevention, and treatment of this disease. Computational simulations can now produce clinically relevant hemodynamic quantities using only non-invasive measurements, combining detailed three dimensional fluid mechanics with physiological models in a multiscale framework. These models, however, require specification of numerous input parameters and are typically tuned manually without accounting for uncertainty in the clinical data, hindering their application to large clinical studies. We propose an automatic, Bayesian, approach to parameter estimation based on adaptive Markov chain Monte Carlo sampling that assimilates non-invasive quantities commonly acquired in routine clinical care, quantifies the uncertainty in the estimated parameters and computes the confidence in local predicted hemodynamic indicators.

Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology.

When predicting physical phenomena through simulation, quantification of the total uncertainty due to multiple sources is as crucial as making sure the underlying numerical model is accurate. Possible sources include irreducible aleatoric uncertainty due to noise in the data, epistemic uncertainty induced by insufficient data or inadequate parameterization, and model-form uncertainty related to the use of misspecified model equations. In addition, recently proposed approaches provide flexible ways to combine information from data with full or partial satisfaction of equations that typically encode physical principles. Physics-based regularization interacts in nontrivial ways with aleatoric, epistemic and model-form uncertainty and their combination, and a better understanding of this interaction is needed to improve the predictive performance of physics-informed digital twins that operate under real conditions. To better understand this interaction, with a specific focus on biological and physiological models, this study investigates the decomposition of total uncertainty in the estimation of states and parameters of a differential system simulated with MC X-TFC, a new physics-informed approach for uncertainty quantification based on random projections and Monte-Carlo sampling. After an introductory comparison between approaches for physics-informed estimation, MC X-TFC is applied to a six-compartment stiff ODE system, the CVSim-6 model, developed in the context of human physiology. The system is first analyzed by progressively removing data while estimating an increasing number of parameters, and subsequently by investigating total uncertainty under model-form misspecification of non-linear resistance in the pulmonary compartment. In particular, we focus on the interaction between the formulation of the discrepancy term and quantification of model-form uncertainty, and show how additional physics can help in the estimation process. The method demonstrates robustness and efficiency in estimating unknown states and parameters, even with limited, sparse, and noisy data. It also offers great flexibility in integrating data with physics for improved estimation, even in cases of model misspecification.

Global sensitivity analysis with multifidelity Monte Carlo and polynomial chaos expansion for vascular haemodynamics.

Computational models of the cardiovascular system are increasingly used for the diagnosis, treatment, and prevention of cardiovascular disease. Before being used for translational applications, the predictive abilities of these models need to be thoroughly demonstrated through verification, validation, and uncertainty quantification. When results depend on multiple uncertain inputs, sensitivity analysis is typically the first step required to separate relevant from unimportant inputs, and is key to determine an initial reduction on the problem dimensionality that will significantly affect the cost of all downstream analysis tasks. For computationally expensive models with numerous uncertain inputs, sample-based sensitivity analysis may become impractical due to the substantial number of model evaluations it typically necessitates. To overcome this limitation, we consider recently proposed Multifidelity Monte Carlo estimators for Sobol' sensitivity indices, and demonstrate their applicability to an idealized model of the common carotid artery. Variance reduction is achieved combining a small number of three-dimensional fluid-structure interaction simulations with affordable one- and zero-dimensional reduced-order models. These multifidelity Monte Carlo estimators are compared with traditional Monte Carlo and polynomial chaos expansion estimates. Specifically, we show consistent sensitivity ranks for both bi- (1D/0D) and tri-fidelity (3D/1D/0D) estimators, and superior variance reduction compared to traditional single-fidelity Monte Carlo estimators for the same computational budget. As the computational burden of Monte Carlo estimators for Sobol' indices is significantly affected by the problem dimensionality, polynomial chaos expansion is found to have lower computational cost for idealized models with smooth stochastic response.

Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification.

Background: Non-invasive simulations of coronary hemodynamics have improved clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree, which ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. Objective: We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating vessel-specific coronary flows as well as cardiac function; and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. Methods: We assimilate patient-specific measurements of myocardial blood flow from clinical CT myocardial perfusion imaging to estimate branch-specific coronary artery flows. Simulated noise in the clinical data is used to estimate the joint posterior distributions of the model parameters using adaptive Markov Chain Monte Carlo sampling. Additionally, the posterior predictive distribution for the relevant quantities of interest is determined using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. This leads to improved correlations between high- and low-fidelity model outputs. Results: Our framework accurately recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement noise uncertainty. We observe substantial reductions in confidence intervals for estimated quantities of interest compared to single-fidelity Monte Carlo estimation and state-of-the-art multi-fidelity Monte Carlo methods. This holds especially true for quantities of interest that showed limited correlation between the low- and high-fidelity model predictions. In addition, the proposed multi-fidelity Monte Carlo estimators are significantly cheaper to compute than traditional estimators, under a specified confidence level or variance. Conclusions: The proposed pipeline for personalized and uncertainty-aware predictions of coronary hemodynamics is based on routine clinical measurements and recently developed techniques for CT myocardial perfusion imaging. The proposed pipeline offers significant improvements in precision and reduction in computational cost.

A probabilistic neural twin for treatment planning in peripheral pulmonary artery stenosis.

The substantial computational cost of high-fidelity models in numerical hemodynamics has, so far, relegated their use mainly to offline treatment planning. New breakthroughs in data-driven architectures and optimization techniques for fast surrogate modeling provide an exciting opportunity to overcome these limitations, enabling the use of such technology for time-critical decisions. We discuss an application to the repair of multiple stenosis in peripheral pulmonary artery disease through either transcatheter pulmonary artery rehabilitation or surgery, where it is of interest to achieve desired pressures and flows at specific locations in the pulmonary artery tree, while minimizing the risk for the patient. Since different degrees of success can be achieved in practice during treatment, we formulate the problem in probability, and solve it through a sample-based approach. We propose a new offline-online pipeline for probabilistic real-time treatment planning which combines offline assimilation of boundary conditions, model reduction, and training dataset generation with online estimation of marginal probabilities, possibly conditioned on the degree of augmentation observed in already repaired lesions. Moreover, we propose a new approach for the parametrization of arbitrarily shaped vascular repairs through iterative corrections of a zero-dimensional approximant. We demonstrate this pipeline for a diseased model of the pulmonary artery tree available through the Vascular Model Repository.

Predictive Modeling of Secondary Pulmonary Hypertension in Left Ventricular Diastolic Dysfunction.

Diastolic dysfunction is a common pathology occurring in about one third of patients affected by heart failure. This condition may not be associated with a marked decrease in cardiac output or systemic pressure and therefore is more difficult to diagnose than its systolic counterpart. Compromised relaxation or increased stiffness of the left ventricle induces an increase in the upstream pulmonary pressures, and is classified as secondary or group II pulmonary hypertension (2018 Nice classification). This may result in an increase in the right ventricular afterload leading to right ventricular failure. Elevated pulmonary pressures are therefore an important clinical indicator of diastolic heart failure (sometimes referred to as heart failure with preserved ejection fraction, HFpEF), showing significant correlation with associated mortality. However, accurate measurements of this quantity are typically obtained through invasive catheterization and after the onset of symptoms. In this study, we use the hemodynamic consistency of a differential-algebraic circulation model to predict pulmonary pressures in adult patients from other, possibly non-invasive, clinical data. We investigate several aspects of the problem, including the ability of model outputs to represent a sufficiently wide pathologic spectrum, the identifiability of the model's parameters, and the accuracy of the predicted pulmonary pressures. We also find that a classifier using the assimilated model parameters as features is free from the problem of missing data and is able to detect pulmonary hypertension with sufficiently high accuracy. For a cohort of 82 patients suffering from various degrees of heart failure severity, we show that systolic, diastolic, and wedge pulmonary pressures can be estimated on average within 8, 6, and 6 mmHg, respectively. We also show that, in general, increased data availability leads to improved predictions.

The effects of clinically-derived parametric data uncertainty in patient-specific coronary simulations with deformable walls.

Cardiovascular simulations are increasingly used for noninvasive diagnosis of cardiovascular disease, to guide treatment decisions, and in the design of medical devices. Quantitative assessment of the variability of simulation outputs due to input uncertainty is a key step toward further integration of cardiovascular simulations in the clinical workflow. In this study, we present uncertainty quantification in computational models of the coronary circulation to investigate the effect of uncertain parameters, including coronary pressure waveform, intramyocardial pressure, morphometry exponent, and the vascular wall Young's modulus. We employ a left coronary artery model with deformable vessel walls, simulated via an Arbitrary-Lagrangian-Eulerian framework for fluid-structure interaction, with a prescribed inlet pressure and open-loop lumped parameter network outlet boundary conditions. Stochastic modeling of the uncertain inputs is determined from intra-coronary catheterization data or gathered from the literature. Uncertainty propagation is performed using several approaches including Monte Carlo, Quasi Monte Carlo sampling, stochastic collocation, and multi-wavelet stochastic expansion. Variabilities in the quantities of interest, including branch pressure, flow, wall shear stress, and wall deformation are assessed. We find that uncertainty in inlet pressures and intramyocardial pressures significantly affect all resulting QoIs, while uncertainty in elastic modulus only affects the mechanical response of the vascular wall. Variability in the morphometry exponent used to distribute the total downstream vascular resistance to the single outlets, has little effect on coronary hemodynamics or wall mechanics. Finally, we compare convergence behaviors of statistics of QoIs using several uncertainty propagation methods on three model benchmark problems and the left coronary simulations. From the simulation results, we conclude that the multi-wavelet stochastic expansion shows superior accuracy and performance against Quasi Monte Carlo and stochastic collocation methods.

A computational bio-chemo-mechanical model of in vivo tissue-engineered vascular graft development.

Stenosis is the primary complication of current tissue-engineered vascular grafts used in pediatric congenital cardiac surgery. Murine models provide considerable insight into the possible mechanisms underlying this situation, but they are not efficient for identifying optimal changes in scaffold design or therapeutic strategies to prevent narrowing. In contrast, computational modeling promises to enable time- and cost-efficient examinations of factors leading to narrowing. Whereas past models have been limited by their phenomenological basis, we present a new mechanistic model that integrates molecular- and cellular-driven immuno- and mechano-mediated contributions to in vivo neotissue development within implanted polymeric scaffolds. Model parameters are inferred directly from in vivo measurements for an inferior vena cava interposition graft model in the mouse that are augmented by data from the literature. By complementing Bayesian estimation with identifiability analysis and simplex optimization, we found optimal parameter values that match model outputs with experimental targets and quantify variability due to measurement uncertainty. Utility is illustrated by parametrically exploring possible graft narrowing as a function of scaffold pore size, macrophage activity, and the immunomodulatory cytokine transforming growth factor beta 1 (TGF-ß1). The model captures salient temporal profiles of infiltrating immune and synthetic cells and associated secretion of cytokines, proteases, and matrix constituents throughout neovessel evolution, and parametric studies suggest that modulating scaffold immunogenicity with early immunomodulatory therapies may reduce graft narrowing without compromising compliance.

Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels.

Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual (GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multigrid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization preconditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and Incomplete LU preconditioners.

Bayesian inference of constitutive model parameters from uncertain uniaxial experiments on murine tendons.

Constitutive models for biological tissue are typically formulated as a mixture of constituents and the overall response is then assembled by superposition or compatibility. This ensures the stress response of the biological tissue to be in the range of a given constitutive relationship, guaranteeing that at least one parameter combination exists so that an experimental response can be sufficiently well captured. Another, perhaps more challenging, problem is to use constitutive models as a proxy to infer the structure/function of a biological tissue from experiments. In other words, we determine the optimal set of parameters by solving an inverse problem and use these parameters to infer the integrity of the tissue constituents. In previous studies, we focused on the mechanical stress-stretch response of the murine patellar tendon at various age and healing timepoints and solved the inverse problem using three constitutive models, i.e., the Freed-Rajagopal, Gasser-Ogden-Holzapfel and Shearer in order of increasing microstructural detail. Herein, we extend this work by adopting a Bayesian perspective on parameter estimation and implement the constitutive relations in the tulip library for uncertainty analysis, critically analyzing parameter marginals, correlations, identifiability and sensitivity. Our results show the importance of investigating the variability of parameter estimates and that results from optimization may be misleading, particularly for models with many parameters inferred from limited experimental evidence. In our study, we show that different age and healing conditions do not correspond to statistically significant separation among the Gasser-Ogden-Holzapfel and Shearer model parameters, while the phenomenological Freed-Rajagopal model is instead characterized by better indentifiability and parameter learning. Use of the complete experimental observations rather than averaged stress-stretch responses appears to positively constrain inference and results appear to be invariant with respect to the scaling of the experimental uncertainty.

Patient-specific parameter estimation in single-ventricle lumped circulation models under uncertainty.

Computational models of cardiovascular physiology can inform clinical decision-making, providing a physically consistent framework to assess vascular pressures and flow distributions, and aiding in treatment planning. In particular, lumped parameter network (LPN) models that make an analogy to electrical circuits offer a fast and surprisingly realistic method to reproduce the circulatory physiology. The complexity of LPN models can vary significantly to account, for example, for cardiac and valve function, respiration, autoregulation, and time-dependent hemodynamics. More complex models provide insight into detailed physiological mechanisms, but their utility is maximized if one can quickly identify patient specific parameters. The clinical utility of LPN models with many parameters will be greatly enhanced by automated parameter identification, particularly if parameter tuning can match non-invasively obtained clinical data. We present a framework for automated tuning of 0D lumped model parameters to match clinical data. We demonstrate the utility of this framework through application to single ventricle pediatric patients with Norwood physiology. Through a combination of local identifiability, Bayesian estimation and maximum a posteriori simplex optimization, we show the ability to automatically determine physiologically consistent point estimates of the parameters and to quantify uncertainty induced by errors and assumptions in the collected clinical data. We show that multi-level estimation, that is, updating the parameter prior information through sub-model analysis, can lead to a significant reduction in the parameter marginal posterior variance. We first consider virtual patient conditions, with clinical targets generated through model solutions, and second application to a cohort of four single-ventricle patients with Norwood physiology. Copyright © 2016 John Wiley & Sons, Ltd.

Hemodynamic effects of left pulmonary artery stenosis after superior cavopulmonary connection: a patient-specific multiscale modeling study.

OBJECTIVE: Currently, no quantitative guidelines have been established for treatment of left pulmonary artery (LPA) stenosis. This study aims to quantify the effects of LPA stenosis on postoperative hemodynamics for single-ventricle patients undergoing stage II superior cavopulmonary connection (SCPC) surgery, using a multiscale computational approach. METHODS: Image data from 6 patients were segmented to produce 3-dimensional models of the pulmonary arteries before stage II surgery. Pressure and flow measurements were used to tune a 0-dimensional model of the entire circulation. Postoperative geometries were generated through stage II virtual surgery; varying degrees of LPA stenosis were applied using mesh morphing and hemodynamics assessed through coupled 0-3-dimensional simulations. To relate metrics of stenosis to clinical classifications, pediatric cardiologists and surgeons ranked the degrees of stenosis in the models. The effects of LPA stenosis were assessed based on left-to-right pulmonary artery flow split ratios, mean pressure drop across the stenosis, cardiac pressure-volume loops, and other clinically relevant parameters. RESULTS: Stenosis of >65% of the vessel diameter was required to produce a right pulmonary artery:LPA flow split <30%, and/or a mean pressure drop of >3.0 mm Hg, defined as clinically significant changes. CONCLUSIONS: The effects of <65% stenosis on SCPC hemodynamics and physiology were minor and may not justify the increased complexity of adding LPA arterioplasty to the SCPC operation. However, in the longer term, pulmonary augmentation may affect outcomes of the Fontan completion surgery, as pulmonary artery distortion is a risk factor that may influence stage III physiology.