Last modified: 2014-06-09

#### Abstract

1 Introduction

Tumor growth is a complex multiscale phenomenon. At the cellular level, it involves (among

others) cell division and blood vessel formation. At the subcellular level, variation in intracellular

concentrations play a role, whereas growth of the tumor is observed at a more macroscopic (tissue)

level. This multiscale nature leads to a large computational cost, especially when a stochastic

individual-based model is used at the cellular level and statistical noise in the simulation becomes

a major concern.

2 Microscopic and macroscopic models

Macroscopic continuum descriptions of tumor growth consist of a system of coupled partial dif-

ferential equations (PDEs), whereas a microscopic, particle-based description is usually speci_ed

in the form of a cellular automaton (CA) or a set of ordinary di_erential equations (ODEs) at-

tached to an individual cell [3]. The macroscopic description is inexpensive to simulate, but not

accurate enough to capture intracellular e_ects. Nevertheless, we are usually mainly interested in

the behavior of the tumor as a whole. Then, the stochastic, agent-based model is used to perform

a Monte Carlo simulation for a large population of N cells, leading to the classical Monte Carlo

statistical error of O(1=

p

N). This implies that one needs a large number of particles to obtain a

su_cient accuracy.

3 Variance reduction method and results

To reduce the variance, we start from an intermediate, mesoscopic kinetic description that can be

simulated deterministically. We then construct a second agent-based model that is an unbiased

stochastic discretization of this kinetic equation. By correlating the two stochastic simulations,

we compute the deviation of the microscopic description with respect to the mesoscopic one. This

deviation is then added to the deterministic mesoscopic simulation as a correction term. This

variance reduction method is based on [4]. We will discuss how this multiscale approach can

reduce the computational cost. We will illustrate the performance of the method via numerical

experiments, where we focus on the e_ciency of the algorithm and the accuracy of the results

[2, 1].

Acknowledgements

The author was funded by Agency for Innovation by Science and Technology in Flanders (IWT).

1

References

[1] Annelies Lejon and Giovanni Samaey. Macroscopic simulation of individual-based stochastic

models for biological processes. Springer. in preparation.

[2] Annelies Lejon and Giovanni Samaey. Variance-reduced simulation of indiviudal-based models

for tumor growth. in preparation.

[3] Markus R Owen, I Johanna Stamper, Munitta Muthana, Giles W Richardson, Jon Dobson,

Claire E Lewis, and Helen M Byrne. Mathematical modeling predicts synergistic antitumor

e_ects of combining a macrophage-based, hypoxia-targeted gene therapy with chemotherapy.

Cancer research, 71(8):2826{37, April 2011.

[4] Mathias Rousset and Giovanni Samaey. variance Simulating individual-based models of bac-

terial chemotaxis with asymptotic variance reduction. 2012.