The Fibonacci Constant, the Wasserstein Distance, and Biological Tumor Aggressiveness in Prostate Cancer
Waliszewski Przemyslaw · 2023
Both epithelial and mesenchymal cells interact with each other at various levels of hierarchical organization forming tissue; a complex dynamic system. The Fibonacci constant sets a limit for self-organization of epithelial cells into tissue structures of the higher order, such as glands. It also determines the optimal use of space available for growth. Patterns of self-organization of cancer cells can be stratified according to the novel parameter, the function of cellular expansion. This function is related to the spatial global capacity fractal dimension D0. That stratification enables a selection of the objective reference set of images for a neural network. Neither normal nor malignant epithelial cells fill the available space in the perfect manner. The first ones self-organize approaching values far from the Fibonacci constant. The second ones self-organize until they reach the value zero. The spatial distribution of cancer cell nuclei can be compared between different prostate carcinomas using a topological measure, the Wasserstein distance. In that way, topological similarity can be quantified in the objective manner. However, values of the Wasserstein distance overlap between the classes of complexity. This may end up in a significant inaccuracy during the automated evaluation of biological tumor aggressiveness by neural networks.