Abstract

We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Specifically, truncated Karhunen–Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.

References

1.
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
.10.1080/02656730801907937
2.
Debbage
,
P.
,
2009
, “
Targeted Drugs and Nanomedicine: Present and Future
,”
Curr. Pharm. Des.
,
15
(
2
), pp.
153
72
.10.2174/138161209787002870
3.
Swartz
,
M. A.
, and
Fleury
,
M. E.
,
2007
, “
Interstitial Flow and Its Effects in Soft Tissues
,”
Annu. Rev. Biomed. Eng.
,
9
, pp.
229
256
.10.1146/annurev.bioeng.9.060906.151850
4.
Loeve
,
M.
,
1977
, “Probability Theory I,”
Graduate Texts in Mathematics
, Vol.
45
,
Springer-Verlag
,
New York, Heidelberg, Berlin
.
5.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
,
1991
,
Stochastic Finite Elements: A Spectral Approach
,
Springer-Verlag
,
New York
.
6.
Ghanem
,
R.
,
1998
, “
Probabilistic Characterization of Transport in Heterogeneous Media
,”
Comput. Methods Appl. Mech. Eng.
,
158
(
3–4
), pp.
199
220
.10.1016/S0045-7825(97)00250-8
7.
Le Maître
,
O. P.
,
Reagan
,
M. T.
,
Najm
,
H. N.
,
Ghanem
,
R. G.
, and
Knio
,
O. M.
,
2002
, “
A Stochastic Projection Method for Fluid Flow: II—Random Process
,”
J. Comput. Phys.
,
181
(
1
), pp.
9
44
.10.1006/jcph.2002.7104
8.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2003
, “
Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos
,”
J. Comput. Phys.
,
187
(
1
), pp.
137
167
.10.1016/S0021-9991(03)00092-5
9.
Le Maitre
,
O.
,
Knio
,
O.
,
Najm
,
H.
, and
Ghanem
,
R.
,
2004
, “
Uncertainty Propagation Using Wiener–Haar Expansions
,”
J. Comput. Phys.
,
197
(
1
), pp.
28
57
.10.1016/j.jcp.2003.11.033
10.
Zhang
,
D.
, and
Lu
,
Z.
,
2004
, “
An Efficient, High-Order Perturbation Approach for Flow in Random Porous Media Via Karhunen–Loeve and Polynomial Expansions
,”
J. Comput. Phys.
,
194
(
2
), pp.
773
794
.10.1016/j.jcp.2003.09.015
11.
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
.10.1137/050645142
12.
Doostan
,
A.
,
Ghanem
,
R. G.
, and
Red-Horse
,
J.
,
2007
, “
Stochastic Model Reduction for Chaos Representations
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
37–40
), pp.
3951
3966
.10.1016/j.cma.2006.10.047
13.
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.
14.
Matthies
,
H. G.
, and
Keese
,
A.
,
2005
, “
Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
12–16
), pp.
1295
1331
.10.1016/j.cma.2004.05.027
15.
Graham
,
I. G.
,
Kuo
,
F. Y.
,
Nichols
,
J. A.
,
Scheichl
,
R.
,
Schwab
,
C.
, and
Sloan
,
I. H.
,
2015
, “
Quasi-Monte Carlo Finite Element Methods for Elliptic PDEs With Lognormal Random Coefficients
,”
Numerische Mathematik
,
131
(
2
), pp.
329
368
.10.1007/s00211-014-0689-y
16.
Tesei
,
F.
,
2016
, “
Numerical Approximation of Flows in Random Porous Media
,” Ph.D. thesis,
École Polytechnique Fédérale de Lausanne
,
Lausanne, Switzerland
.
17.
Elman
,
H.
,
2017
, “
Solution Algorithms for Stochastic Galerkin Discretizations of Differential Equations With Random Data
,”
Handbook of Uncertainty Quantification
,
Springer
,
Cham, Switzerland
, pp.
1
16
.
18.
Alexanderian
,
A.
,
Reese
,
W.
,
Smith
,
R. C.
, and
Yu
,
M.
,
2018
, “
Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties
,”
ASME
Paper No. IMECE2018-86216.10.1115/IMECE2018-86216
19.
Williams
,
D.
,
1991
,
Probability With Martingales
(Cambridge Mathematical Textbooks),
Cambridge University Press
,
Cambridge, UK
.
20.
Le Maitre
,
O. P.
, and
Knio
,
O. M.
,
2010
,
Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics
(Scientific Computation),
Springer
,
Dordrecht, The Netherlands
.
21.
Smith
,
R. C.
,
2013
,
Uncertainty Quantification: Theory, Implementation, and Applications
, Vol.
12
,
Siam
,
New Delhi, India
.
22.
Kress
,
R.
,
2014
, “
Linear Integral Equations
,”
Applied Mathematical Sciences
, 3rd ed., Vol.
82
,
Springer
,
New York
.
23.
Betz
,
W.
,
Papaioannou
,
I.
, and
Straub
,
D.
,
2014
, “
Numerical Methods for the Discretization of Random Fields by Means of the Karhunen–Loève Expansion
,”
Comput. Methods Appl. Mech. Eng.
,
271
, pp.
109
129
.10.1016/j.cma.2013.12.010
24.
Stein
,
M. L.
,
1999
, Interpolation of Spatial Data (Springer Series in Statistics),
Springer-Verlag
,
New York
.
25.
Spanos
,
P. D.
,
Beer
,
M.
, and
Red-Horse
,
J.
,
2007
, “
Karhunen–Loève Expansion of Stochastic Processes With a Modified Exponential Covariance Kernel
,”
J. Eng. Mech.
,
133
(
7
), pp.
773
779
.10.1061/(ASCE)0733-9399(2007)133:7(773)
26.
Cleaves
,
H. L.
,
Alexanderian
,
A.
,
Guy
,
H.
,
Smith
,
R. C.
, and
Yu
,
M.
,
2019
, “
Derivative-Based Global Sensitivity Analysis for Models With High-Dimensional Inputs and Functional Outputs
,” e-print
arXiv:1902.04630
.
27.
Alexanderian
,
A.
,
Zhu
,
L.
,
Salloum
,
M.
,
Ma
,
R.
, and
Yu
,
M.
,
2017
, “
Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Non-Intrusive Spectral Uncertainty Quantification Method
,”
ASME J. Biomech. Eng.
,
139
(
9
), p.
091006
.10.1115/1.4037102
You do not currently have access to this content.