On Some Probabilistic Aspects of Diffusion Models for Tissue Growth

Abstract

Understanding of tissue growth is in its nature multidisciplinary, since it varies from cancer diagnostics, image processing, fractal analysis to regular and non-regular heat flows. By medical sciences it was requested to better under- stood the tissue grow relation to mathematical modelling (stochastic geometry, fractal growth, diffusions). It is clear, that deterministic fractal is not an appropriate model for cancer growth. Stochastic fractal is more appropriate, however, a validation measure should be developed for better comparability with advanced stochastic geometry model, e.g. Quer- mass-interaction process. Moreover, relation temperature-geometry of the tissue is studied. We have partial results, where it is observed, that benign alterations and malignant tumors originating from glandular tissues (e.g.mammary, prostatic, pancreatic) are naturally modelled by non-standard diffusions. For standard diffusions, fair approximation is provided by analytical models based on convective heat transfer in infinite tissues volume (e.g. model given by Perl 1962, later ex- tended by (1)).