Mathematical Biosciences and Engineering (MBE)

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

Pages: 37 - 57, Volume 10, Issue 1, February 2013      doi:10.3934/mbe.2013.10.37

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Giulio Caravagna - Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan, Italy (email)
Alex Graudenzi - Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan, Italy (email)
Alberto d’Onofrio - Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I-20141 Milan, Italy (email)

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:  Tumor, immune system, delay differential equation, Stochastic Hybrid Automata, piecewise deterministic Markov process, distributed delays.
Mathematics Subject Classification:  62P10, 92-08, 34A38, 65C05, 65L03.

Received: July 2012;      Accepted: September 2012;      Available Online: December 2012.