The Fibonacci constant and limits of tissue self-organization; local complexity measures in evaluation of the risk of metastasis formation

The classification of patients with prostate carcinomas into the WHO prognostic groups with a heterogeneous morphological spectrum and the averaged likelihood of tumor progression is based on the subjective evaluation of similarity between patterns of local tumor growth and normal prostatic tissue; an ambiguous procedure with a significant intra- and interobserver variability. We propose foundations for a classification based on the quantitative measures of both complexity of the spatial distribution of epithelial cells and intercellular interactions.

We find that the Fibonacci constant is a limit for the ratio of the integer dimension of the Euclidean space, in which growth, proliferation and self-organization of cells occurs and the global spatial capacity fractal dimension characterizing the spatial distribution of epithelial cells. The global fractal dimensions $D_0$ or $D_1$ define the coefficient of cellular expansion and relate complexity of tissue system with the capacity for intercellular interactions. They also subordinate precisely each carcinoma into the classes of equivalence. Mapping of carcinoma images with the local fractal dimensions identifies cancer cells with enhanced metastatic potential. The significant risk of metastasis formation appears if the mean value of the local fractal dimension is larger than 1.7000, that is in the classes of equivalence C4–C7. Those parameters should be applied in the quantitative models of tumor progression risk assessment.