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Research Papers

Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method

[+] Author and Article Information
Alen Alexanderian

Department of Mathematics,
North Carolina State University,
Raleigh, NC 27695
e-mail: alexanderian@ncsu.edu

Liang Zhu, Ronghui Ma

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
Baltimore, MD 21250

Maher Salloum

Extreme Scale Data Science and Analytics,
Sandia National Labs,
Livermore, CA 94550

Meilin Yu

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
Baltimore, MD 21250
e-mail: mlyu@umbc.edu

Manuscript received February 3, 2017; final manuscript received May 16, 2017; published online July 14, 2017. Assoc. Editor: Ram Devireddy.

J Biomech Eng 139(9), 091006 (Jul 14, 2017) (11 pages) Paper No: BIO-17-1044; doi: 10.1115/1.4037102 History: Received February 03, 2017; Revised May 16, 2017

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen–Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

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Figures

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Fig. 1

MicroCT images of nanoparticle distribution (bright regions enclosed by the dash lines) in two tumors (grey regions): (a) regular nanoparticle distribution and (b) irregular nanoparticle distribution

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Fig. 2

Top the two-dimensional computational domain is given by a circle of radius Rtumor with the concentric inner circle of radius Rneedle removed. This models a tumor with an injection site at the center; and bottom: the pressure field with constant material properties.

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Fig. 3

Log-normal model for permeability values (at each spatial point in the tumor)

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Fig. 4

The normalized eigenvalues (top) and the ratio rNKL that quantifies variance saturation as a function of n (bottom), corresponding to correlation length ℓ=5 mm, which is the radius of the tumor model

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Fig. 5

Two realizations of the uncertain log-permeability field (top) and the corresponding pressure fields (bottom)

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Fig. 6

Two sets of realizations of the uncertain log-permeability fields and the corresponding concentration fields at 10 mins. From left to right, the correlation lengths ℓ are 5 mm, 2 mm, 1 mm, and 0.5 mm, respectively.

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Fig. 7

Left: tracking the convergence of PDFs of the PC representation of Pneedle; right: comparing the PDF of the third-order PC expansion of Pneedle (solid line) with its empirical distribution generated through Monte Carlo sampling (histogram)

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Fig. 8

Computing the relative L2 error of the PC representation of the pressure field with varying PC order

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Fig. 9

Mean (left) and standard deviation (middle) of pressure, and distribution of P(r,φ) for a fixed φ, and for increasing r (right)

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Fig. 10

Total sensitivity indices for the pressure field

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Fig. 11

Average sensitivity indices (left) and sensitivity indices for Pneedle (right)

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Fig. 12

Mean (left) and standard deviation (middle) of vr=−(κ/ϕμ)(∂P/∂r), and distribution of vr(r,φ) for a fixed φ, and for increasing r (right)

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