Numerical methods for the simulation of a kinetic multiscale model for tumor growth.
Multiscale phenomena play an important role in our current society. In biological contexts, we can model cellular motion and more specific, we can model the evolution of tumors. In a similar way, the behavior of people can be modeled, which is useful within the context of evacuations. Those multiscale systems are characterized by the fact that they contain both slow and fast evolving elements. Simulating those systems over a long time in an accurate way requires a lot of computational power.
Multiscale systems can be modeled and simulated in many ways. On the one hand, we can simulate each single cell individually (agent-based model) what we call the microscopic level. This is characterized by a high level of detail, which implies a high computational cost. Therefore, one often considers the evolution of the density (macroscopic level) rather than the set of individual cells. Finally, there are also so-called kinetic equations containing more detail than a continuum model, but less than an agent-based model.
A first goal of the thesis is to accelerate the simulation of multiscale kinetic equations. Due to the multiscale character, simulating such a system can be computationally expensive when using classical techniques. By using projective integration, we can circumvent the stiffness related to the multiscale character. In the thesis we developed a higher order version of projective integration, leading to an additional increase in efficiency compared to the existing projective integration methods.
Moreover, multiscale systems often contain uncertain factors, because the systems are not fully understood. Therefore, we choose to not model just one specific scenario, but instead the evolution of the probability distribution.
In this thesis, we have chosen to specifically develop algorithms to lower the impact of the uncertainties (noise) without a large additional cost. In the tumor growth model, proposed in the thesis, the noise is mainly caused by random motion on the one hand and cell divisions on the other hand. At first, we developed an algorithm to filter caused by random motion. This is done by means of an simple approximate model (control process) which covers the evolution of the random motion. Moreover, the control process can be simulated deterministically. The complete model can then be used to incorporate the influence of cell divisions.
In general using more cells yields a more accurate result. However, simulating a high number of particles during the whole simulation time would imply a high computational cost. To achieve accurate results at a limited cost, we add (clone) extra particles locally to determine the locations of cell divisions more accurately.