Dynamical Monte Carlo for non-equilibrium systems with simultaneous events: application for Tumor Growth

The usual Dynamical Monte Carlo (DMC) method has the assumption that just one event may occur, in systems, in the shortest time scale. On the other hand one can find systems in which several events occur simultaneously, i.e., where more than one event takes place in the smallest time scale. In this work we study, as an example, the formulation of Dawson and Hillen for the evolution of a tumor system with active and quiescent cells. In this model, an active element generates simultaneously two individuals in the quiescent state. Mapping this model into one-event approach is possible. However, we develop here, a more comprehensive Markovian DMC theory, which can include, naturally, simultaneous events. Another remarkable feature of the usual one-event approach is the non-uniform time intervals: an increment depends on the rate and the number of elements responsible for each event type. In the regime of low transition rates, the system spends long time in the same configuration, and the waiting times are estimated in a way to overcome this time consumption with a large time step, making the algorithm very efficient. However, when the system reaches a small enough (critical) number of elements, the waiting times becomes poorly evaluated, leading to accumulation of biased errors, and consequently, deviating the system from its actual trajectory. We will address this issue here and solve it by rescaling the system size. Also, we will extend this approach to cases in which the systems are too large to simulate.