Quantifying uncertainty of a mathematical model of drug transport in tumors

: pp. 567–578
Received: October 28, 2021
Accepted: March 23, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 567–578 (2022)

Ibn Zohr University, Department of Science Computing
Ibn Zohr University, Department of Science Computing
Cadi Ayyad University, Department of Mathematics

This paper presents a numerical simulation in the two-dimensional for a system of PDE governing drug transport in tumors with random coefficients, which is described  as a random field.  The continuous stochastic field is approximated by a finite number of random variables via the Karhunen–Loève expansion and transform the stochastic problem into a determinate one with a parameter in high dimension.  Then we apply a finite difference scheme and the Euler–Maruyama Integrator in time.  The Monte Carlo method is used to compute corresponding simple averages.  We compute the error estimate using the Central Limits Theorem (CLT) and the error estimate for the finite difference method.  Some numerical results are simulated to illustrate the theoretical analysis.  We also propose a comparison between the stochastic and determinate solving processes of our system where we show the efficiency of our adopted method.

  1. Deb M. K., Babuška I. M., Oden J. T.  Solution of stochastic partial differential equations using Galerkin finite element techniques.  Computer Methods in Applied Mechanics and Engineering. 190 (48), 6359–6372 (2001).
  2. Gunzburger M. D., Webster C. G., Zhang G.  Stochastic finite element methods for partial differential equations with random input data.  Acta Numerica. 23, 521–650 (2014).
  3. Babuska I., Tempone R., Zouraris G.  Galerkin fiite element approximations of stochastic elliptic differential equations.  SIAM Journal on Numerical Analysis. 42 (2), 800–825 (2004).
  4. Barth A., Shwab C., Zollinger N.  Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients.  Numerische Mathematik. 119, 123–161 (2011).
  5. Cliffe K. A., Giles M. B., Scheicl R., Teckentrup A. L.  Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients.  Computing and Visualization in Science. 14, 3 (2011).
  6. Barth A., Lang A.  Multilevel Monte Carlo method with applications to stochastic partial differential equations.  International Journal of Computer Mathematics. 89 (18), 2479–2498 (2012).
  7. Xiu D., Hesthaven J. S.  High-order collocation methods for differential equations with random inputs.  SIAM Journal on Scientific Computing. 27 (3), 1118–1139 (2005).
  8. Babuška I. M., Nobile F., Tempone R.  A stochastic collocation method for elliptic partial differential equations with random input data.  SIAM Journal on Numerical Analysis. 45 (3), 1005–1034 (2007).
  9. Nobile F., Tempone R., Webster C. G.  A sparse grid stochastic collocation method for partial differential equations with random input data.  SIAM Journal on Numerical Analysis. 46 (5), 2309–2345.
  10. Sandeep S., Sinek J. P., Frieboes H. B., Ferrari M., Fruehauf J. P., Cristini V.  Mathematical modeling of cancer progression and response to chemotherapy.  Expert Review of Anticancer Therapy. 6 (10), 1361–1376 (2006).
  11. Sinek J. P., Sanga S., Zheng X., Frieboes H. B., Ferrari M., Cristini V.  Predicting drug pharmacokinetics and effect in vascularized tumors using computer simulation.  Journal of Mathematical Biology. 58, 485 (2008).
  12. El-Kareh A. W., Secomb T. W.  A Mathematical model for Cisplatin Cellular Pharmacodynamics.  Neoplasia. 5 (2), 161–169 (2003).
  13. Ghanem R., Spanos P. D.  Stochastic Finite Elements: A Spectral Approach. Dover Publications (2003).
  14. Xiu D.  Numerical Methods for Stochastic Computations: A Spectral Method Approach.  Princeton University Press, Princeton, NJ, USA (2010).
  15. Troger V., Fischel J. L., Formento P., Gioanni J., Milano G.  Effects of prolonged exposure to cisplatin on cytotoxicity and intracellular drug concentration.  European Journal of Cancer. 28 (l), 82–86 (1992).
  16. Levasseur L. M., Slocum H. K., Rustum Y. M., Greco W. R.  Modeling of the time-dependency of in vitro drug cytotoxicity and resistance.  Cancer Research. 58 (24), 5749–5761 (1998).