Differential Equations and Cellular Automata Models of the Growth of Cell Cultures and Transformation Foci

Two different modeling approaches are discussed in the study of in vitro cell cultures which, after exposure to a carcinogen, may develop transformation foci that may be considered the in vitro analogue of tumors. The most important variables that are measured in these tests are the number of foci found at the end of the experiment, starting from a different number of initial cells. It is shown that an approach based upon ordinary differential equations (ODEs) may fit the data, but in a fragile way, while a cellular automata (CA) approach provides a robust agreement. However, the story told here is not that of a conflict, but rather of a cooperation between the two modeling approaches: the results of the ODE study guided our exploration of the different alternatives in CA simulations, and provided checks during model development and testing. The CA model led us to consider the importance of the initial seeds, a point which has not been stressed in the previous literature, and to reinterpret published experimental data. It is shown that CA models, which retain cell individuality, can handle this aspect in a straightforward way, which would have been very difficult to introduce in methods based upon partial differential equations. It is also shown that quantitative modeling provides useful insights for the interpretation of experimental data as well as suggestions for further experiments.