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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Swartz, M. A. , and Fleury, M. E. , 2007, “ Interstitial Flow and Its Effects in Soft Tissues,” Annu. Rev. Biomed. Eng., 9(1), pp. 229–256. [CrossRef] [PubMed]
Debbage, P. , 2009, “ Targeted Drugs and Nanomedicine: Present and Future,” Curr. Pharm. Des., 15(2), pp. 153–172. [CrossRef] [PubMed]
Salloum, M. , Ma, R. , Weeks, D. , and Zhu, L. , 2008, “ Controlling Nanoparticle Delivery in Magnetic Nanoparticle Hyperthermia for Cancer Treatment: Experimental Study in Agarose Gel,” Int. J. Hyperthermia, 24(4), pp. 337–345. [CrossRef] [PubMed]
Hilger, I. , Hergt, R. , and Kaiser, W. , 2005, “ Towards Breast Cancer Treatment by Magnetic Heating,” J. Mag. Mag. Mater., 293(1), pp. 314–319. [CrossRef]
Rand, R. , Snow, H. , Elliott, D. , and Snyder, M. , 1981, “ Thermomagnetic Surgery for Cancer,” Appl. Biochem. Biotechnol., 6(4), pp. 265–272. [CrossRef] [PubMed]
Rand, R. , Snow, H. , and Brown, W. , 1982, “ Thermomagnetic Surgery for Cancer,” J. Surg. Res., 33(3), pp. 177–183. [CrossRef] [PubMed]
Hase, M. , Sako, M. , Fujii, M. , Ueda, E. , Nagae, T. , Shimizu, T. , Hirota, S. , and Kono, M. , 1989, “ Experimental Study of Embolo-Hyperthermia for the Treatment of Liver Tumors by Induction Heating to Ferromagnetic Particles Injected Into Tumor Tissue,” Nihon Igaku Hoshasen Gakkai Zasshi, 49(9), pp. 1171–1173. https://www.ncbi.nlm.nih.gov/labs/articles/2587196 [PubMed]
Wust, P. , Gneveckow, U. , Johannsen, M. , Bohmer, D. , Henkel, T. , Kahmann, F. , Sehouli, J. , Felix, R. , Ricke, J. , and Jordan, A. , 2006, “ Magnetic Nanoparticles for Interstitial Thermotherapy–Feasibility, Tolerance and Achieved Temperatures,” Int. J. Hyperthermia, 22(8), pp. 673–685. [CrossRef] [PubMed]
Khaled, A.-R. A. , and Vafai, K. , 2003, “ The Role of Porous Media in Modeling Flow and Heat Transfer in Biological Tissues,” Int. J. Heat Mass Transfer, 46(26), pp. 4989–5003. [CrossRef]
Baxter, L. T. , and Jain, R. K. , 1990, “ Transport of Fluid and Macromolecules in Tumors (II): Role of Heterogeneous Perfusion and Lymphatics,” Microvasc. Res., 40(2), pp. 246–263. [CrossRef] [PubMed]
Wang, C.-H. , and Li, J. , 1998, “ Three-Dimensional Simulation of IgG Delivery to Tumors,” Chem. Eng. Sci., 53(20), pp. 3579–3600. [CrossRef]
Zhao, J. , Salmon, H. , and Sarntinoranont, M. , 2007, “ Effect of Heterogeneous Vasculature on Interstitial Transport Within a Solid Tumor,” Microvasc. Res., 73(3), pp. 224–236. [CrossRef] [PubMed]
Pishko, G. L. , Astary, G. W. , Mareci, T. H. , and Sarntinoranont, M. , 2011, “ Sensitivity Analysis of an Image-Based Solid Tumor Computational Model With Heterogeneous Vasculature and Porosity,” Ann. Biomed. Eng., 39(9), pp. 2360–2373. [CrossRef] [PubMed]
Sefidgar, M. , Soltani, M. , Raahemifar, K. , Bazmara, H. , Nayinian, S. M. M. , and Bazargan, M. , 2014, “ Effect of Tumor Shape, Size, and Tissue Transport Properties on Drug Delivery to Solid Tumors,” J. Biol. Eng., 8(1), p. 12. [CrossRef] [PubMed]
Xiu, D. , Lucor, D. , Su, C. , and Karniadakis, G. , 2002, “ Stochastic Modeling of Flow Structure Interactions Using Generalized Polynomial Chaos,” ASME J. Fluids Eng., 124(1), pp. 51–59. [CrossRef]
Xiu, D. , and Karniadakis, G. , 2002, “ The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Xiu, D. , and Karniadakis, G. , 2003, “ Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys., 187(1), pp. 137–167. [CrossRef]
Le Maître, O. , Knio, O. , Najm, H. , and Ghanem, R. , 2004, “ Uncertainty Propagation Using Wiener-Haar Expansions,” J. Comput. Phys., 197(1), pp. 28–57. [CrossRef]
Le Maître, O. , Najm, H. , Ghanem, R. , and Knio, O. , 2004, “ Multi-Resolution Analysis of Wiener-Type Uncertainty Propagation Schemes,” J. Comput. Phys., 197(2), pp. 502–531. [CrossRef]
Le Maître, O. , Mathelin, L. , Knio, O. , and Hussaini, M. , 2010, “ Asynchronous Time Integration for Polynomial Chaos Expansion of Uncertain Periodic Dynamics,” Discrete Contin. Dyn. Syst., 28(1), pp. 199–226. [CrossRef]
Le Maître, O. P. , and Knio, O. M. , 2010, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer, New York. [CrossRef]
Hosder, S. , Walters, R. , and Perez, R. , 2006, “ A Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations,” AIAA Paper No. 2006-891.
Babuška, I. , Nobile, F. , and Tempone, R. , 2007, “ A Stochastic Collocation Method for Elliptic Partial Differential Equations With Random Input Data,” SIAM J. Numer. Anal., 45(3), pp. 1005–1034. [CrossRef]
Alexanderian, A. , Winokur, J. , Sraj, I. , Srinivasan, A. , Iskandarani, M. , Thacker, W. , and Knio, O. , 2012, “ Global Sensitivity Analysis in Ocean Global Circulation Models: A Sparse Spectral Projection Approach,” Comput. Geosci., 16(3), pp. 757–778. [CrossRef]
Smolyak, S. A. , 1963, “ Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,” Dokl. Akad. Nauk SSSR, 4, pp. 240–243.
Liu, J. , 2001, “ Uncertainty Analysis for Temperature Prediction of Biological Bodies Subject to Randomly Spatial Heating,” J. Biomech., 34(12), pp. 1637–1642. [CrossRef] [PubMed]
Rabin, Y. , 2003, “ A General Model for the Propagation of Uncertainty in Measurements Into Heat Transfer Simulations and Its Application to Cryosurgery,” Cryobiology, 46(2), pp. 109–120. [CrossRef] [PubMed]
Turner, T. E. , Schnell, S. , and Burrage, K. , 2004, “ Stochastic Approaches for Modelling In Vivo Reactions,” Comput. Biol. Chem., 28(3), pp. 165–178. [CrossRef] [PubMed]
Deng, Z.-S. , and Liu, J. , 2002, “ Monte Carlo Method to Solve Multidimensional Bioheat Transfer Problem,” Numer. Heat Transfer, Part B, 42(6), pp. 543–567. [CrossRef]
Ghanem, R. , and Dham, S. , 1998, “ Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media,” Transp. Porous Media, 32(3), pp. 239–262. [CrossRef]
Saad, G. , and Ghanem, R. , 2009, “ Characterization of Reservoir Simulation Models Using a Polynomial Chaos-Based Ensemble Kalman Filter,” Water Resour. Res., 45(4), p. W04417.
Sobol, I. , 2001, “ Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates,” Math. Comput. Simul., 55(1–3), pp. 271–280. [CrossRef]
Sobol, I. , 1990, “ On Sensitivity Estimation for Nonlinear Mathematical Models,” Mat. Model., 2(1), pp. 112–118. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mm&paperid=2320&option_lang=eng
Sudret, B. , 2008, “ Global Sensitivity Analysis Using Polynomial Chaos Expansions,” Reliab. Eng. Syst. Saf., 93(7), pp. 964–979. [CrossRef]
Crestaux, T. , Maitre, O. L. , and Martinez, J.-M. , 2009, “ Polynomial Chaos Expansion for Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 94(7), pp. 1161–1172. [CrossRef]
Ma, R. , Su, D. , and Zhu, L. , 2012, “ Multiscale Simulation of Nanopartical Transport in Deformable Tissue During an Infusion Process in Hyperthermia Treatments of Cancers,” Nanoparticle Heat Transfer and Fluid Flow (Computational and Physical Processes in Mechanics and Thermal Science Series), Vol. 4, W. J. Minkowycz , E. Sparrow , and J. P. Abraham , eds., CRC Press, Boca Raton, FL.
Ghanem, R. G. , and Spanos, P. D. , 1991, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York. [CrossRef]
Mercer, J. , 1909, “ Functions of Positive and Negative Type, and Their Connection With the Theory of Integral Equations,” Philos. Trans. R. Soc. London Ser. A, 209(441–458), pp. 415–446. [CrossRef]
Xiu, D. B. , 2010, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, NJ.
Heiss, F. , and Winschel, V. , 2008, “ Likelihood Approximation by Numerical Integration on Sparse Grids,” J. Econometrics, 144(1), pp. 62–80. [CrossRef]
Sobol, I. , Tarantola, S. , Gatelli, D. , Kucherenko, S. , and Mauntz, W. , 2007, “ Estimating the Approximation Error When Fixing Unessential Factors in Global Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 92(7), pp. 957–960. [CrossRef]


Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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.

Grahic Jump Location
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)

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
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)

Grahic Jump Location
Fig. 10

Total sensitivity indices for the pressure field

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
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)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In