Using parametric model order reduction for inverse analysis of large nonlinear cardiac simulations

Abstract

Predictive high-fidelity finite element simulations of human cardiac mechanics commonly require a large number of structural degrees of freedom. Additionally, these models are often coupled with lumped-parameter models of hemodynamics. High computational demands, however, slow down model calibration and therefore limit the use of cardiac simulations in clinical practice. As cardiac models rely on several patient-specific parameters, just one solution corresponding to one specific parameter set does not at all meet clinical demands. Moreover, while solving the nonlinear problem, 90% of the computation time is spent solving linear systems of equations. We propose to reduce the structural dimension of a monolithically coupled structure-Windkessel system by projection onto a lower-dimensional subspace. We obtain a good approximation of the displacement field as well as of key scalar cardiac outputs even with very few reduced degrees of freedom, while achieving considerable speedups. For subspace generation, we use proper orthogonal decomposition of displacement snapshots. Following a brief comparison of subspace interpolation methods, we demonstrate how projection-based model order reduction can be easily integrated into a gradient-based optimization.We demonstrate the performance of our method in a real-world multivariate inverse analysis scenario. Using the presented projection-based model order reduction approach can significantly speed up model personalization and could be used for many-query tasks in a clinical setting.cardiac mechanics, inverse analysis, parametric model order reduction, proper orthogonal decomposition 1 | INTRODUCTION Cardiac solid mechanics simulations consist of solving large-deformation, materially nonlinear, elastodynamic coupled boundary-value problems. There exist different approaches to incorporate blood flow into the computational model. Three-dimensional fluid-structure interaction is resolved for example in References 1 and 2. As the exact fluid dynamics of blood within the heart are, however, usually not needed, the structural model is commonly coupled to | Model order reductionThe needed huge number of degrees of freedom (DOFs) and other challenges of solving nonlinear problems make the solution of cardiac models computationally expensive and limit the models' use in clinical practice. For example, using a single node with 24 cores, a simulation of one heartbeat, which takes about 1 s in reality, takes about 1 d to compute with our high-fidelity four chamber model. 9 The potential to reduce computation time motivates the use of reduced order models (ROMs). In this work, we solely consider model order reduction (MOR) of time-dependent parametric problems. In the following, different strategies in reduced order modeling are reviewed.An important category of cardiac ROMs is made up by simplified modeling. For these models, the same system of differential equations as for the full order model (FOM) is solved, but on a simplified analytical geometry. The displacements are commonly param...

Model order reduction of flow based on a modular geometrical approximation of blood vessels

“…We refer, e.g., to [ 30 , 31 ] and [ 32 ] for applications of POD to parabolic and fluid problems, respectively, and [ 31 ] for [ 30 ] for a comprehensive study of the properties of POD when applied to the solution of Ordinary Differential Equations (ODEs). In the context of cardiac simulations, this technique has been successfully employed both in fluid (e.g., in [ 33 ] to simulate blood flow in patient-specific coronary artery bypass grafts) and structural simulations (e.g., in [ 34 ], where POD is used to reduce the space of admissible displacements of the heart muscle). In the POD approach, the reduced basis is usually constructed by singular value decomposition (SVD) [ 35 , 36 ] out of the set of snapshots, which are obtained by sampling N s parameters in and by solving the corresponding FOM.…”

Model order reduction of flow based on a modular geometrical approximation of blood vessels

“…in [3] to simulate blood flow in patient-specific coronary artery bypass grafts) and structural simulations (e.g. in [46], where POD is used to reduce the space of admissible displacements of the heart muscle). In the POD approach, the reduced basis is usually constructed by singular value decomposition (SVD) [26,56] out of the set of snapshots, which are obtained by sampling N s parameters µ 1 , .…”

Challenges of order reduction techniques for problems involving polymorphic uncertainty

“…Dependencies on the parameter set μ can be incorporated through subspace interpolation. Several subspace interpolation methods were compared in Pfaller et al for cardiac problems with varying maximum active tension of cardiac muscle fibers σ 0 , which is a key determinant of cardiac function. The best approximation was achieved in Pfaller et al with the weighted concatenation method.…”

“…Several subspace interpolation methods were compared in Pfaller et al for cardiac problems with varying maximum active tension of cardiac muscle fibers σ 0 , which is a key determinant of cardiac function. The best approximation was achieved in Pfaller et al with the weighted concatenation method. In a scenario with K sample points of previous FOM evaluations, the matrix V ( μ ∗ ) is constructed at a new parameter set μ ∗ by considering the first q left singular vectors of the (weighted and) concatenated snapshot matrix based on snapshots D ( μ i ) with weights w i at sample point i .…”