On study of cell proliferation and diffusion using nonlinear transforms of heat equation solutions

Cell proliferation and diffusion can be modeled through reaction-diffusion systems describing the space-time evolution of a density variable. In this work, we present non-linear transformations of heat equation solutions to model cellular growth and diffusion using a Richards growth function. The solutions are obtained by using two-variable Hermite Polynomials. We estimated the parameters of the Richards function by comparing the model solutions with the experimentally observed data from scratch assays. To check robustness of the parameters estimated, we used a fractional Brownian Motion (fBM) field type noise with given Hurst exponent (H) for minimizing residual errors. We found that the parameters can be robustly estimated and match with previous estimations. Further, we also confirmed that the spatial-temporal patterns of cell density in the experiments can be well described by the solutions developed using the proposed nonlinear transforms method.