Distributed delays in a hybrid model of  tumor-Immune system interplay

Giulio Caravagna; Alex Graudenzi; Alberto d’Onofrio

doi:10.3934/mbe.2013.10.37

Article Contents

2013, Volume 10, Issue 1: 37-57. Doi: 10.3934/mbe.2013.10.37

This issue Previous Article Model of tumour angiogenesis -- analysis of stability with respect to delays Next Article Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells

Distributed delays in a hybrid model of tumor-Immune system interplay

1.
Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan
2.
Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I-20141 Milan

Received: July 2012

Revised: September 2012

Published: December 2012

Abstract / Introduction Related Papers Cited by

Abstract

A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect.
We here present a hybrid model with a generic delay kernel accounting that, due to many complex phenomena such as chemical transportation and cellular differentiation, the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model with two well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process.
Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.

Keywords:

Mathematics Subject Classification: 62P10, 92-08, 34A38, 65C05, 65L03.

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