Toni Lassila

A reduced computational and geometrical framework for inverse problems in hemodynamics

The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

Model order reduction in fluid dynamics: challenges and perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map

Reduced basis approximations for geometrically parametrized advection-diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.

A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.

Generalized Reduced Basis Methods and n-Width Estimates for the Approximation of the Solution Manifold of Parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold.

A coupled 3D-1D numerical monodomain solver for cardiac electrical activation in the myocardium with detailed Purkinje network

We present a model for the electrophysiology in the heart to handle the electrical propagation through the Purkinje system and in the myocardium, with two-way coupling at the Purkinje-muscle junctions. In both the subproblems the monodomain model is considered, whereas at the junctions a resistor element is included that induces an orthodromic propagation delay from the Purkinje network towards the heart muscle. We prove a sufficient condition for convergence of a fixed-point iterative algorithm to the numerical solution of the coupled problem. Numerical comparison of activation patterns is made with two different combinations of models for the coupled Purkinje network/myocardium system, the eikonal/eikonal and the monodomain/monodomain models. Test cases are investigated for both physiological and pathological activation of a model left ventricle. Finally, we prove the reliability of the monodomain/monodomain coupling on a realistic scenario. Our results underlie the importance of using physiologically realistic Purkinje-trees with propagation solved using the monodomain model for simulating cardiac activation.

Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models

Computational inverse problems related to partial differential equations (PDEs) often contain nuisance parameters that cannot be effectively identified but still need to be considered as part of the problem. The objective of this work is to show how to take advantage of a reduced order framework to speed up Bayesian inversion on the identifiable parameters of the system, while marginalizing away the (potentially large number of) nuisance parameters. The key ingredients are twofold. On the one hand, we rely on a reduced basis (RB) method, equipped with computable a posteriori error bounds, to speed up the solution of the forward problem. On the other hand, we develop suitable reduction error models (REMs) to quantify in an inexpensive way the error between the full-order and the reduced-order approximation of the forward problem, in order to gauge the effect of this error on the posterior distribution of the identifiable parameters. Numerical results dealing with inverse problems governed by elliptic PDEs ...

Uncertainty quantification of wall shear stress in intracranial aneurysms using a data-driven statistical model of systemic blood flow variability

Adverse wall shear stress (WSS) patterns are known to play a key role in the localisation, formation, and progression of intracranial aneurysms (IAs). Complex region-specific and time-varying aneurysmal WSS patterns depend both on vascular morphology as well as on variable systemic flow conditions. Computational fluid dynamics (CFD) has been proposed for characterising WSS patterns in IAs; however, CFD simulations often rely on deterministic boundary conditions that are not representative of the actual variations in blood flow. We develop a data-driven statistical model of internal carotid artery (ICA) flow, which is used to generate a virtual population of waveforms used as inlet boundary conditions in CFD simulations. This allows the statistics of the resulting aneurysmal WSS distributions to be computed. It is observed that ICA waveform variations have limited influence on the time-averaged WSS (TAWSS) on the IA surface. In contrast, in regions where the flow is locally highly multidirectional, WSS directionality and harmonic content are strongly affected by the ICA flow waveform. As a consequence, we argue that the effect of blood flow variability should be explicitly considered in CFD-based IA rupture assessment to prevent confounding the conclusions.

A reduced-order strategy for solving inverse Bayesian shape identification problems in physiological flows

A reduced-order strategy based on the reduced basis (RB) method is developed for the efficient numerical solution of statistical inverse problems governed by PDEs in domains of varying shape. Usual discretization techniques are infeasible in this context, due to the prohibitive cost entailed by the repeated evaluation of PDEs and related output quantities of interest. A suitable reduced-order model is introduced to reduce computational costs and complexity. Furthermore, when dealing with inverse identication of shape features, a reduced shape representation allows to tackle the geometrical complexity. We address both challenges by considering a reduced framework built upon the RB method for parametrized PDEs and a parametric radial basis functions approach for shape representation. We present some results dealing with blood flows modelled by Navier-Stokes equations.

Improved Hybrid/GPU Algorithm for Solving Cardiac Electrophysiology Problems on Purkinje Networks

Cardiac Purkinje fibers provide an important pathway to the coordinated contraction of the heart. We present a numerical algorithm for the solution of electrophysiology problems across the Purkinje network that is efficient enough to be used in in silico studies on realistic Purkinje networks with physiologically detailed models of ion exchange at the cell membrane. The algorithm is on the basis of operator splitting and is provided with 3 different implementations: pure CPU, hybrid CPU/GPU, and pure GPU. Compared to our previous work, we modify the explicit gap junction term at network bifurcations to improve its mathematical consistency. Due to this improved consistency of the model, we are able to perform an empirical convergence study against analytical solutions. The study verified that all 3 implementations produce equivalent convergence rates, and shows that the algorithm produces equivalent result across different hardware platforms. Finally, we compare the efficiency of all 3 implementations on Purkinje networks of increasing spatial resolution using membrane models of increasing complexity. Both hybrid and pure GPU implementations outperform the pure CPU implementation, but their relative performance difference depends on the size of the Purkinje network and the complexity of the membrane model used.