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Research Papers

Application of an Adaptive Polynomial Chaos Expansion on Computationally Expensive Three-Dimensional Cardiovascular Models for Uncertainty Quantification and Sensitivity Analysis

[+] Author and Article Information
Sjeng Quicken

Department of Biomedical Engineering,
School for Cardiovascular Diseases (CARIM),
Maastricht University,
Universiteitssingel 50,
Maastricht 6229 ER, The Netherlands
e-mail: s.quicken@maastrichtuniversity.nl

Wouter P. Donders

Department of Biomedical Engineering,
School for Mental Health and
Neuroscience (MHENS),
Maastricht University,
Universiteitssingel 50,
Maastricht 6229 ER, The Netherlands
e-mail: w.donders@maastrichtuniversity.nl

Emiel M. J. van Disseldorp

Department of Biomedical Engineering,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: e.m.j.v.disseldorp@tue.nl

Kujtim Gashi

Department of Biomedical Engineering,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: k.gashi@tue.nl

Barend M. E. Mees

Department of Vascular Surgery,
Maastricht University Medical Center,
P.O. Box 5800,
Maastricht 6202 AZ, The Netherlands
e-mail: barend.mees@mumc.nl

Frans N. van de Vosse

Department of Biomedical Engineering,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: f.n.v.d.vosse@tue.nl

Richard G. P. Lopata

Department of Biomedical Engineering,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: r.lopata@tue.nl

Tammo Delhaas

Department of Biomedical Engineering,
School for Cardiovascular Diseases (CARIM),
Maastricht University,
Universiteitssingel 50,
Maastricht 6229 ER, The Netherlands
e-mail: tammo.delhaas@maastrichtuniversity.nl

Wouter Huberts

Department of Biomedical Engineering,
School for Cardiovascular Diseases (CARIM),
Maastricht University,
Universiteitssingel 50,
Maastricht 6229 ER, The Netherlands
e-mail: wouter.huberts@maastrichtuniversity.nl

1Corresponding author.

Manuscript received April 8, 2016; final manuscript received September 6, 2016; published online November 4, 2016. Assoc. Editor: Alison Marsden.

J Biomech Eng 138(12), 121010 (Nov 04, 2016) (11 pages) Paper No: BIO-16-1139; doi: 10.1115/1.4034709 History: Received April 08, 2016; Revised September 06, 2016

When applying models to patient-specific situations, the impact of model input uncertainty on the model output uncertainty has to be assessed. Proper uncertainty quantification (UQ) and sensitivity analysis (SA) techniques are indispensable for this purpose. An efficient approach for UQ and SA is the generalized polynomial chaos expansion (gPCE) method, where model response is expanded into a finite series of polynomials that depend on the model input (i.e., a meta-model). However, because of the intrinsic high computational cost of three-dimensional (3D) cardiovascular models, performing the number of model evaluations required for the gPCE is often computationally prohibitively expensive. Recently, Blatman and Sudret (2010, “An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis,” Probab. Eng. Mech., 25(2), pp. 183–197) introduced the adaptive sparse gPCE (agPCE) in the field of structural engineering. This approach reduces the computational cost with respect to the gPCE, by only including polynomials that significantly increase the meta-model’s quality. In this study, we demonstrate the agPCE by applying it to a 3D abdominal aortic aneurysm (AAA) wall mechanics model and a 3D model of flow through an arteriovenous fistula (AVF). The agPCE method was indeed able to perform UQ and SA at a significantly lower computational cost than the gPCE, while still retaining accurate results. Cost reductions ranged between 70–80% and 50–90% for the AAA and AVF model, respectively.

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References

Figures

Grahic Jump Location
Fig. 1

A graphical representation of the total and main effects of a meta-model with a maximal univariate polynomial order of two, with three parameters: X1, X2, and X3. Univariate polynomials with increasing polynomial degree in either X1, X2, or X3 are found on any of the three axes. Multivariate polynomials are defined according to Eq. (3), were αi is equal to the order of the univariate polynomial of Xi. The zeroth order polynomial is located in the origin. Polynomials needed to compute the sensitivity indices are represented by black dots. Hence, the main indices are found along the axes and the total indices are found in all coordinates where ϕαi>0.

Grahic Jump Location
Fig. 2

Graphical example of the forward and backward step of the agPCE algorithm for a model with two input parameters. The example starts at the forward step where polynomials of degree two are added to the sparse basis. The increase in R2 for each test basis: A{d}* is determined with respect to A{d}. If ΔR2>ε1, the term is added to the meta-model, finally resulting in A{d}+. Then, in the backward step, polynomials with a degree smaller than two are removed. If the decrease in R2 for a test polynomial, A{d}* with respect to A{d}+ is smaller than ε2, the polynomial is removed from the meta-model. This finally results in a new A{d}. The visualization of the meta-model follows the conventions of Fig. 1, applied to two dimensions.

Grahic Jump Location
Fig. 3

Front view of the parameterized AAA geometry. The boundary surfaces of the geometry are fixed in space.

Grahic Jump Location
Fig. 4

Side view (left) and bottom view (right) of the parameterized AVF geometry. A indicates the arterial inlet, B indicates the arterial outlet, and C indicates the venous outlet. B and C are truncated by two-element Windkessels. The arrows denote the orientation of the blood flow.

Grahic Jump Location
Fig. 5

The obtained Q2 values for the 2453 meta-models were successfully obtained for the AAA problem, grouped by each target Qtrgt2 (also indicated by the gray horizontal lines). The highest obtained Q2 per group is indicated in black. The reference value is the value of Q2 of the full gPCE of polynomial degree three (Q2=0.9984).

Grahic Jump Location
Fig. 6

The obtained Q2 values for the 757 meta-models were successfully obtained for the AVF problem, grouped by each target Qtrgt2 (also indicated by the gray horizontal lines). The highest obtained Q2 per group is indicated in black. The reference value is the value of Q2 of the full gPCE of polynomial degree three (Q2=0.9101).

Grahic Jump Location
Fig. 7

The percentage of meta-models that were successfully obtained with the algorithm with respect to all possible combinations for both the AAA model (top row) and the AVF model (bottom row)

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