Behaviour Prediction and Simulation of Partition-Based Biological Models – Some Open Issues of Lattice Paradigms

The state space partition problem for hybrid systems, closely related to prediction of systems' behaviour, consists of finding the conditions under which a given quantised system is deterministic. Some biological processes present a hybrid nature, involving the interaction between the continuous evolution of internal cell variables, such as concentrations, and abrupt changes of discrete cells characteristics. In these classes of models, the basic modelling paradigm, implying a choice of the state space structure, is a lattice, with elements generally represented by single cells. This paper comparatively presents two approaches from the literature, both with a hybrid nature but involving two distinct lattice types. The description of their behaviour prediction mechanisms are overviewed and simulation of an abstract quasi-random invasion process, inspired from the second model of a brain tumor growth, is finally discussed, emphasising the difficulties in solving the general state space partition problem. Also a tumor-growth limiting model is proposed, as a dynamic nutrient inhibitor. Some aspects driving to a general lattice-based model of a growing process are finally discussed.