Research Papers

A Stochastic Collocation Method for Uncertainty Quantification and Propagation in Cardiovascular Simulations

[+] Author and Article Information
Sethuraman Sankaran, Alison L. Marsden

 University of California, San Diego, La Jolla, CA 92093–0411

J Biomech Eng 133(3), 031001 (Feb 04, 2011) (12 pages) doi:10.1115/1.4003259 History: Received July 29, 2009; Revised May 24, 2010; Posted December 15, 2010; Published February 04, 2011; Online February 04, 2011

Simulations of blood flow in both healthy and diseased vascular models can be used to compute a range of hemodynamic parameters including velocities, time varying wall shear stress, pressure drops, and energy losses. The confidence in the data output from cardiovascular simulations depends directly on our level of certainty in simulation input parameters. In this work, we develop a general set of tools to evaluate the sensitivity of output parameters to input uncertainties in cardiovascular simulations. Uncertainties can arise from boundary conditions, geometrical parameters, or clinical data. These uncertainties result in a range of possible outputs which are quantified using probability density functions (PDFs). The objective is to systemically model the input uncertainties and quantify the confidence in the output of hemodynamic simulations. Input uncertainties are quantified and mapped to the stochastic space using the stochastic collocation technique. We develop an adaptive collocation algorithm for Gauss–Lobatto–Chebyshev grid points that significantly reduces computational cost. This analysis is performed on two idealized problems – an abdominal aortic aneurysm and a carotid artery bifurcation, and one patient specific problem – a Fontan procedure for congenital heart defects. In each case, relevant hemodynamic features are extracted and their uncertainty is quantified. Uncertainty quantification of the hemodynamic simulations is done using (a) stochastic space representations, (b) PDFs, and (c) the confidence intervals for a specified level of confidence in each problem.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 3

(a) Stochastic space representation of mean shear stress over the abdominal aneurysm. Different depths of interpolation using conventional sparse grids are shown in the figure. (b) shows the onset of recirculation depicted using the vertical blood velocities.

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Figure 4

The figure shows wall shear stress plots at different radii corresponding to the extremes and mean of the stochastic space

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Figure 5

Contours of vz for the recirculating regions (with positive component of vz) (from left) ξ=0.02, ξ=0.09, ξ=0.14, and ξ=0.5

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Figure 6

The stochastic collocation points for a depth of interpolation 6 using (left) conventional sparse grid and (right) adaptive sparse grid

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Figure 7

The figure shows convergence of the PDF of shear stresses (cgs units) as the level of interpolation is increased. The relatively long tail in the PDF is a manifestation of the plateau in Fig. 3.

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Figure 8

Comparison of stochastic space representation of the wall shear stress in cgs units using (left) depth of interpolation 3 and (right) depth of interpolation 4

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Figure 9

Plots of the magnitude of blood velocity in the carotid artery at four points in the stochastic space (from left) ξ1,ξ2=(0.5,0), (0.5, 1), (0,0.5), and (0,1), where the first coordinate represents the stochastic velocity dimension and the second represents the stochastic radius dimension. The results were computed at t=0.923 s.

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Figure 10

The figure shows sensitivities of the wall shear stress to input uncertainties which is positive with velocity (left) ∂τ/∂ξ2 and negative with radius (right) ∂τ/∂ξ1. Contours illustrate similar order of magnitude of the sensitivity of wall shear stress to input parameters.

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Figure 11

A comparison of PDF of mean shear stress across the aneurysm using (left) a uniform PDF for velocity and (right) a Gaussian PDF for velocity

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Figure 12

The figure shows (left) an adaptive Chebyshev–Gauss–Lobatto grid and (right) a conventional sparse grid for the Carotid artery bifurcation problem. The adaptive method reduces the number of required simulations by roughly 50%.

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Figure 13

The figure shows mean flow velocities for the Fontan patient with uncertainty in flow-split. The plots show mean velocities at (top left) t=0.0, (top right) t=0.81, (bottom left) t=1.62, and (bottom right) t=2.43 s. The period of the respiratory cycle is 2.86 s

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Figure 14

The figure shows standard deviation of flow velocities for the Fontan patient with uncertainty in flow-split. The plots show standard deviations at (top left) t=0.0, (top right) t=0.81, (bottom left) t=1.62, and (bottom right) t=2.43 s.

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Figure 15

The figure shows (left) PDF of hemodynamic efficiency and (right) PDF of pressure difference between the IVC and the LPA using level-4 sparse grid collocation

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Figure 1

The figure shows (a) stochastic collocation points in a two-dimensional stochastic space using a level-4 depth of interpolation and (b) level-6 depth of interpolation. (c) shows the function g(⋅), which is a sum of two Gaussian distributions, N(2,0.22) and N(1,0.12). The PDF of g is shown in (d).

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Figure 2

A schematic of the stochastic collocation technique procedure for performing uncertainty analysis in cardiovascular simulations




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