[Mathematical modelling in systems biology. Simulation of the desmoplastic stromal reaction as an example]

A mathematical model of collagen fiber mesh formation was created to evaluate the possible role of chemotaxis and haptotaxis in the histomorphology of a desmoplastic stromal reaction (DSR). Fibroblasts were mathematicaly interpreted as mobile discrete objects, characterized by their velocity and position, both dependent on time. This resulted in cell migration paths, commonly termed "trajectories" which are modulated as stochastic process. The implementation of chemotactic effects requires knowledge of the concentration and distribution of the appropriate chemical substance in the scenario. A simplistic model assumption allows the calculation of a numerical solution of the resulting diffusion equation. Adding haptotaxis necessitates the simulation of the extracellular matrix (ECM). The fiber distribution is modeled as a vector field which contains information on both, fiber density and direction. The production of new fibers is based on ordinary differential equations coupled with the migratory behavior of the cells. Filters help smooth the trajectories. Appropriate visualization allows a direct comparison of the simulation results with histomorphology. Matches between computed data and their real counterparts indicate that the development of mathematical models is appropriate to describe and forecast the course of DSR. This makes systems biology a stepping stone to improving biomedical research.