Effects of delayed immune-response in tumor immune-system interplay

Tumors constitute a wide family of diseases kinetically characterized by the co-presence of multiple spatio-temporal scales. So, tumor cells ecologically inte rplay with other kind of cells, e.g. endothelial cells or immune system effectors, producing and exchanging various chemical signals. As such, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model agents at low concentrations, and mean-field equations mode l chemical signals. In previous works we proposed a hybrid version of the well-known Panetta-Kirschner mean-field model of tumor cells, effector cells and Interleukin-2. Our hybrid model suggested -at variance of the inferences from its original formulation- that immune surveillance, i.e. t umor elimination by the immune system, may occur through a sort of side-effect of large stochastic o scillations. However, that model did not account that, due to both chemical transportation and cellular differentiation/division, the tumorinduced recruitment of immune effectors is not instantaneous but, instead, it exhibits a lag period. To capture this, we here integrate a mean-field equation for I nterleukins-2 with a bi-dimensional delayed stochastic process describing such delayed interpla y. An algorithm to realize trajectories of the underlying stochastic process is obtained by coupling the Piecewise Deterministic Markov process (for the hybrid part) with a Generalized Semi-Markovian clock structure (to account for delays). We (i) relate tumor mass growth with delays via simulations and via parametric sensitivity analysis techniques, (ii) we quantitatively determine probabilistic eradication ti mes, and(iii) we prove, in the oscillatory regime, the existence of a heuristic stochasti c bifurcation resulting in delay-induced tumor eradication, which is neither predicted by the mean-field no r by the hybrid non-delayed models.