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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

Browne, L. D. , Griffin, P. , Bashar, K. , Walsh, S. R. , Kavanagh, E. G. , and Walsh, M. T. , 2015, “ In Vivo Validation of the in Silico Predicted Pressure Drop Across an Arteriovenous Fistula,” Ann. Biomed. Eng., 43(6), pp. 1275–1286. [CrossRef] [PubMed]
Taylor, C. A. , Fonte, T. A. , and Min, J. K. , 2013, “ Computational Fluid Dynamics Applied to Cardiac Computed Tomography for Noninvasive Quantification of Fractional Flow Reserve scientific Basis,” J. Am. Coll. Cardiol., 61(22), pp. 2233–2241. [CrossRef] [PubMed]
Larrabide, I. , Aguilar, M. , Morales, H. , Geers, A. , Kulcsár, Z. , Rüfenacht, D. , and Frangi, A. , 2013, “ Intra-Aneurysmal Pressure and Flow Changes Induced by Flow Diverters: Relation to Aneurysm Size and Shape,” Am. J. Neuroradiol., 34(4), pp. 816–822. [CrossRef]
Zhang, Y. , Chong, W. , and Qian, Y. , 2013, “ Investigation of Intracranial Aneurysm Hemodynamics Following Flow Diverter Stent Treatment,” Med. Eng. Phys., 35(5), pp. 608–615. [CrossRef] [PubMed]
Vande Geest, J. P. , Di Martino, E. S. , Bohra, A. , Makaroun, M. S. , and Vorp, D. A. , 2006, “ A Biomechanics-Based Rupture Potential Index for Abdominal Aortic Aneurysm Risk Assessment: Demonstrative Application,” Ann. N.Y. Acad. Sci., 1085(1), pp. 11–21. [CrossRef] [PubMed]
Pluijmert, M. , Kroon, W. , Rossi, A. C. , Bovendeerd, P. H. M. , and Delhaas, T. , 2012, “ Why Sit Works: Normal Function Despite Typical Myofiber Pattern in Situs Inversus Totalis (sit) Hearts Derived by Shear-Induced Myofiber Reorientation,” PLoS Comput. Biol., 8(7), p. e1002611. [CrossRef] [PubMed]
Raghavan, M. , Vorp, D. A. , Federle, M. P. , Makaroun, M. S. , and Webster, M. W. , 2000, “ Wall Stress Distribution on Three-Dimensionally Reconstructed Models of Human Abdominal Aortic Aneurysm,” J. Vasc. Surg., 31(4), pp. 760–769. [CrossRef] [PubMed]
Huberts, W. , Bode, A. , Kroon, W. , Planken, R. , Tordoir, J. , Van de Vosse, F. , and Bosboom, E. , 2012, “ A Pulse Wave Propagation Model to Support Decision-Making in Vascular Access Planning in the Clinic,” Med. Eng. Phys., 34(2), pp. 233–248. [CrossRef] [PubMed]
Mulder, G. , 2011, “ Patient-Specific Modelling of the Cerebral Circulation for Aneurysm Risk Assessment,” Ph.D thesis, Eindhoven University of Technology, Eindhoven, The Netherlands. http://131.155.54.17/mate/pdfs/13267.pdf
Vande Geest, J. P. , Wang, D. H. J. , Wisniewski, S. R. , Makaroun, M. S. , and Vorp, D. A. , 2006, “ Towards a Noninvasive Method for Determination of Patient-Specific Wall Strength Distribution in Abdominal Aortic Aneurysms,” Ann. Biomed. Eng., 34(7), pp. 1098–1106. [CrossRef] [PubMed]
Speelman, L. , Bosboom, E. , Schurink, G. , Hellenthal, F. , Buth, J. , Breeuwer, M. , Jacobs, M. , and van de Vosse, F. , 2008, “ Patient-Specific AAA Wall Stress Analysis: 99-Percentile Versus Peak Stress,” Eur. J. Vasc. Endovasc. Surg., 36(6), pp. 668–676. [CrossRef] [PubMed]
Gasser, T. , Auer, M. , Labruto, F. , Swedenborg, J. , and Roy, J. , 2010, “ Biomechanical Rupture Risk Assessment of Abdominal Aortic Aneurysms: Model Complexity Versus Predictability of Finite Element Simulations,” Eur. J. Vasc. Endovasc. Surg., 40(2), pp. 176–185. [CrossRef] [PubMed]
Center for Devices and Radiological Health, 2014, “ Reporting of Computational Modeling Studies in Medical Device Submissions (Draft),” United States Food and Drug Administration, Report No. FDA-2013-D-1530-0002.
Council for Regulatory Environmental Modeling, 2009, “ Guidance on the Development, Evaluation, and Application of Environmental Models,” United States Environmental Protection Agency, Report No. EPA/100/K-09/003.
European Commission, 2015, “ Better Regulation Guidelines,” Regulatory Fitness and Performance Programme, Report No. SWD(2015).
Saltelli, A. , Ratto, M. , Andresa, T. , Campolongo, F. , Caribonia, J. , Gatelli, D. , Saisana, M. , and Tarantola, S. , 2008, Global Sensitivity Analysis, the Primer, Wiley, Chichester, UK.
Chen, P. , Quarteroni, A. , and Rozza, G. , 2013, “ Simulation-Based Uncertainty Quantification of Human Arterial Network Hemodynamics,” Int. J. Numer. Method Biomed. Eng., 29(6), pp. 698–721. [CrossRef] [PubMed]
Donders, W. P. , Huberts, W. , van de Vosse, F. N. , and Delhaas, T. , 2015, “ Personalization of Models With Many Model Parameters: An Efficient Sensitivity Analysis Approach,” Int. J. Numer. Methods Biomed. Eng., 31(10), p. e02727. [CrossRef]
Eck, V . G. , Feinberg, J. , Langtangen, H. P. , and Hellevik, L. R. , 2015, “ Stochastic Sensitivity Analysis for Timing and Amplitude of Pressure Waves in the Arterial System,” Int. J. Numer. Methods Biomed. Eng., 31(4 ), p. e02711. [CrossRef]
Huberts, W. , Donders, W. P. , Delhaas, T. , and van de Vosse, F. N. , 2014, “ Applicability of the Polynomial Chaos Expansion Method for Personalization of a Cardiovascular Pulse Wave Propagation Model,” Int. J. Numer. Methods Biomed. Eng., 30(12), pp. 1679–1704. [CrossRef]
Sankaran, S. , and Marsden, A. L. , 2011, “ A Stochastic Collocation Method for Uncertainty Quantification and Propagation in Cardiovascular Simulations,” ASME J. Biomech. Eng., 133(3 ), p. 031001. [CrossRef]
Xiu, D. , 2007, “ Efficient Collocational Approach for Parametric Uncertainty Analysis,” Commun. Comput. Phys., 2(2), pp. 293–309. http://www.global-sci.com/issue/abstract/readabs.php?vol=2&page=293&year=2007&issue=2&ppage=309
Crestaux, T. , Le Maître, O. , and Martinez, J.-M. , 2009, “ Polynomial Chaos Expansion for Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 94(7), pp. 1161–1172. [CrossRef]
Blatman, G. , 2009, “ Adaptive Sparse Polynomial Chaos Expansions for Uncertainty Propagation and Sensitivity Analysis,” Doctoral thesis, Blaise Pascal University, Aubière, France.
Choi, S.-K. , Grandhi, R. V. , Canfield, R. A. , and Pettit, C. L. , 2004, “ Polynomial Chaos Expansion With Latin Hypercube Sampling for Estimating Response Variability,” AIAA J., 42(6), pp. 1191–1198. [CrossRef]
Knio, O. M. , and Le Maître, O. P. , 2006, “ Uncertainty Propagation in CFD Using Polynomial Chaos Decomposition,” Fluid Dyn. Res., 38(9), pp. 616–640. [CrossRef]
Xiu, D. , and Karniadakis, G. E. M. , 2002, “ The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Blatman, G. , and Sudret, B. , 2011, “ Adaptive Sparse Polynomial Chaos Expansion Based on Least Angle Regression,” J. Comput. Phys., 230(6), pp. 2345–2367. [CrossRef]
Sudret, B. , 2008, “ Global Sensitivity Analysis Using Polynomial Chaos Expansions,” Reliab. Eng. Syst. Saf., 93(7), pp. 964–979. [CrossRef]
Blatman, G. , and Sudret, B. , 2010, “ Efficient Computation of Global Sensitivity Indices Using Sparse Polynomial Chaos Expansions,” Reliab. Eng. Syst. Saf., 95(11), pp. 1216–1229. [CrossRef]
Doostan, A. , and Owhadi, H. , 2011, “ A Non-Adapted Sparse Approximation of PDEs With Stochastic Inputs,” J. Comput. Phys., 230(8), pp. 3015–3034. [CrossRef]
Ma, X. , and Zabaras, N. , 2009, “ An Adaptive Hierarchical Sparse Grid Collocation Method for the Solution of Stochastic Differential Equations,” J. Comput. Phys., 228(8), pp. 1–59. [CrossRef]
Wan, X. , and Karniadakis, G. E. , 2005, “ An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Comput. Phys., 209(2), pp. 617–642. [CrossRef]
Blatman, G. , and Sudret, B. , 2010, “ An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis,” Probab. Eng. Mech., 25(2), pp. 183–197. [CrossRef]
Xiu, D. , and Karniadakis, G. E. , 2003, “ Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys., 187(1), p. 137167. [CrossRef]
Eck, V. , Donders, W. , Sturdy, J. , Feinberg, J. , Delhaas, T. , Hellevik, L. , and Huberts, W. , 2015, “ A Review and Guide to Uncertainty Quantification and Sensitivity Analysis for Cardiovascular Applications,” Int. J. Numer. Methods Biomed. Eng., 32(8), p. e02755. [CrossRef]
Dubreuil, S. , Berveiller, M. , Petitjean, F. , and Salaün, M. , 2014, “ Construction of Bootstrap Confidence Intervals on Sensitivity Indices Computed by Polynomial Chaos Expansion,” Reliab. Eng. Syst. Saf., 121, pp. 263–275. [CrossRef]
Sobol, I . M. , 1967, “ On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals,” USSR Comput. Math. & Math. Phys., 7(4), pp. 86–112. [CrossRef]
Budd, J. S. , Finch, D. , and Carter, P. , 1989, “ A Study of the Mortality From Ruptured Abdominal Aortic Aneurysms in a District Community,” Eur. J. Vasc. Surg., 3(4), pp. 351–354. [CrossRef] [PubMed]
de Putter, S. , Wolters, B. , Rutten, M. , Breeuwer, M. , Gerritsen, F. , and van de Vosse, F. , 2007, “ Patient-Specific Initial Wall Stress in Abdominal Aortic Aneurysms With a Backward Incremental Method,” J. Biomech., 40(5 ), p. 10811090. [CrossRef]
Fillinger, M. F. , Raghavan, M. , Marra, S. P. , Cronenwett, J. L. , and Kennedy, F. E. , 2002, “ In Vivo Analysis of Mechanical Wall Stress and Abdominal Aortic Aneurysm Rupture Risk,” J. Vasc. Surg., 36(3), p. 589597. [CrossRef]
Kok, A. M. , Nguyen, V . L. , Speelman, L. , Brands, P. J. , Schurink, G.-W. H. , van de Vosse, F. N. , and Lopata, R. G. , 2015, “ Feasibility of Wall Stress Analysis of Abdominal Aortic Aneurysms Using Three-Dimensional Ultrasound,” J. Vasc. Surg., 61(5), pp. 1175–1184. [CrossRef] [PubMed]
Raghavan, M. , and Vorp, D. A. , 2000, “ Toward a Biomechanical Tool to Evaluate Rupture Potential of Abdominal Aortic Aneurysm: Identification of a Finite Strain Constitutive Model and Evaluation of Its Applicability,” J. Biomech., 33(4), pp. 475–482. [CrossRef] [PubMed]
Speelman, L. , Bosboom, E. , Schurink, G. , Buth, J. , Breeuwer, M. , Jacobs, M. , and van de Vosse, F. , 2009, “ Initial Stress and Nonlinear Material Behavior in Patient-Specific AAA Wall Stress Analysis,” J. Biomech., 42(11), pp. 1713–1719. [CrossRef] [PubMed]
Grassmann, A. , Gioberge, S. , Moeller, S. , and Brown, G. , 2005. “ ESRD Patients in 2004: Global Overview of Patient Numbers, Treatment Modalities and Associated Trends,” Nephrol. Dial. Transplant., 20(12), pp. 2587–2593. [CrossRef] [PubMed]
Tordoir, J. , Canaud, B. , Haage, P. , Konner, K. , Basci, A. , Fouque, D. , Kooman, J. , Martin-Malo, A. , Pedrini, L. , Pizzarelli, F. , Tattersall, J. , Vennegoor, M. , Wanner, C. , Wee, P. T. , and Vanholder, R. , 2007, “ European Best Practice Guidelines on Vascular Access,” Nephrol. Dial. Transplant., 22(Suppl. 2), pp. ii88–ii117. [PubMed]
Moore, J. , Steinman, D. , Prakash, S. , Johnston, K. , and Ethier, C. , 1999, “ A Numerical Study of Blood Flow Patterns in Anatomically Realistic and Simplified End-to-Side Anastomoses,” ASME J. Biomech. Eng., 121(3), pp. 265–272. [CrossRef]
Van Canneyt, K. , Pourchez, T. , Eloot, S. , Guillame, C. , Bonnet, A. , Segers, P. , and Verdonck, P. , 2010, “ Hemodynamic Impact of Anastomosis Size and Angle in Side-to-End Arteriovenous Fistulae: A Computer Analysis,” J. Vasc. Access., 11(1), pp. 52–58. http://www.vascular-access.info/article/hemodynamic-impact-of-anastomosis-size-and-angle--in-side-to-end-arteriovenous-fistulae--a-computer-analysis-art006434 [PubMed]
Botti, L. , Van Canneyt, K. , Kaminsky, R. , Claessens, T. , Planken, R. N. , Verdonck, P. , Remuzzi, A. , and Antiga, L. , 2013, “ Numerical Evaluation and Experimental Validation of Pressure Drops Across a Patient-Specific Model of Vascular Access for Hemodialysis,” Cardiovasc. Eng. Technol., 4(4), p. 485499. [CrossRef]
Decorato, I. , Kharboutly, Z. , Vassallo, T. , Penrose, J. , Legallais, C. , and Salsac, A.-V. , 2013, “ Numerical Simulation of the Fluid Structure Interactions in a Compliant Patient-Specific Arteriovenous Fistula,” Int. J. Numer. Methods Biomed. Eng., 30(2), pp. 143–159. [CrossRef]
Ene-Iordache, B. , Semperboni, C. , Dubini, G. , and Remuzzi, A. , 2015, “ Disturbed Flow in a Patient-Specific Arteriovenous Fistula for Hemodialysis: Multidirectional and Reciprocating Near-Wall Flow Patterns,” J. Biomech., 48(10), pp. 2195–2200. [CrossRef] [PubMed]
Westerhof, N. , Lankhaar, J. W. , and Westerhof, B. E. , 2008, “ The Arterial Windkessel,” Med. Biol. Eng. Comput., 47(2), pp. 131–141. [CrossRef] [PubMed]
Geuzaine, C. , and Remacle, J.-F. , 2009, “ GMSH: A 3-D Finite Element Mesh Generator With Built-In Pre-and Post-Processing Facilities,” Int. J. Numer. Methods Eng., 79(11), pp. 1309–1331. [CrossRef]
Hulsen, M. A. , 2013, TFEM, A Toolkit for the Finite Element Method, User’s Manual, Eindhoven University of Technology, Eindhoven, The Netherlands.
Brooks, A. N. , and Hughes, T. J. , 1982, “ Streamline Upwind/Petrov–Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier–Stokes Equations,” Comput. Methods Appl. Mech. Eng., 32(1), pp. 199–259. [CrossRef]
Codina, R. , Oñate, E. , and Cervera, M. , 1992, “ The Intrinsic Time for the Streamline Upwind/Petrov–Galerkin Formulation Using Quadratic Elements,” Comput. Methods Appl. Mech. Eng., 94(2), pp. 239–262. [CrossRef]
Schiavazzi, D. E. , Arbia, G. , Baker, C. , Hlavacek, A. M. , Hsia, T. Y. , Marsden, A. L. , and Vignon-Clementel, I. E. , and The Modeling of Congenital Hearts Alliance MOCHA, 2016, “ Uncertainty Quantification in Virtual Surgery Hemodynamics Predictions for Single Ventricle Palliation,” Int. J. Numer. Methods Biomed. Eng., 32(3), p. e02737. [CrossRef]
Niederreiter, H. , 1988, “ Low-Discrepancy and Low-Dispersion Sequences,” J. Number Theory, 30(1), pp. 51–70. [CrossRef]
Mai, C. V. , and Sudret, B. , 2015, “ Hierarchical Adaptive Polynomial Chaos Expansions,” 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering, M. Papadrakakis , V. Papadopoulos , and G. Stefanou , eds., May, UNCECOMP, pp. 25–27. http://arxiv.org/abs/1506.00461

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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