Cell Invasion Models: How Complex do They Need to Be?

The invasion of the extracellular matrix (ECM) by cancer cells is a highly complex process, but many mathematical models describing it are quite simple. This raises the question how these simpler models differ from the more complex ones regarding their dependence on the input parameters and their qualitative results, and whether simpler models are able to capture and reproduce the results of complex models.To investigate these questions, three cell invasion models are examined: two simple to intermediate partial differential equation (PDE) models with six and eight parameters, respectively, and a complex hybrid model with 17 parameters. The hybrid model was originally developed for angiogenesis, featuring a Cellular Potts model and a finite element formulation, and is extended in this work to describe cell invasion as travelling waves.The models' parameter sensitivity with respect to initial ECM density, the ECM degradation rate and the cell proliferation probability, is examined using a variance-based method at various points in time. At first, the PDE models exhibit mainly first-order effects from the initial ECM density, and later in time from the ECM degradation rate. At all points in time, the hybrid model is affected the most by the proliferation probability, and by interaction effects between all three parameters.The data fitting abilities are tested by generating data using the hybrid model and estimating the corresponding PDE model parameters. This is done under consideration of various phenomena of the hybrid model, and with different numbers of parameters to be estimated. Using the fitted parameters, the shape approximations and the velocities of the travelling invading waves are compared between the models. The best approximations in shape are performed by the PDE model with eight parameters, the best approximations in wave speed are obtained using the model with six parameters.Overall, the main finding is that even though the model outputs are relatively similar and can be approximated adequately by the PDE models, the dependence of the hybrid model on its input parameters differs substantially from the one of the PDE models.     «

The invasion of the extracellular matrix (ECM) by cancer cells is a highly complex process, but many mathematical models describing it are quite simple. This raises the question how these simpler models differ from the more complex ones regarding their dependence on the input parameters and their qualitative results, and whether simpler models are able to capture and reproduce the results of complex models.To investigate these questions, three cell invasion models are examined: two simple to int...     »