Complexity, Diversity, Homogeneity and Topology Measures Re-Define Biological Tumor Aggressiveness as a Function of Intercellular Connectivity

Abstract

Similarity between patterns of cellular self-organization in primary tumor and in its normal counterpart underlies the concept of biological tumor aggressiveness. That subjective parameter is supposed to correlate with both dynamics of local tumor growth and the risk of progression. Subjective evaluation during the microscopic examination is inaccurate. The co-application of fractal geometry and algebraic topology offers a novel perspective. A set of prostate carcinomas comprising all patterns of self-organization was classified according to complexity and entropy to investigate how complexity relates to topology of the patterns. Prostate carcinomas evolve from order to disorder along the increasing entropy as many physical dynamic complex systems do. Carcinomas can be classified into the classes of complexity equivalence according to both the values of the global fractal dimensions D0 or D1 and the Shannon entropy H. Topological measures, such as the bottleneck distance or the Wasserstein distance reveal different topology in very low- and low-aggressive prostate carcinomas (class C1, C2, C3) in comparison with high-aggressive carcinomas (class C6, C7). This is contingent on the appearance of infiltrating cancer cells. While the persistent homology entropy Hph remains similar in all classes, the persistent barcode entropy Hpb is significantly decreased in class C1 and increased in class C7 in comparison with the other ones. Patterns of self-organization can now be defined according to changes in intercellular connectivity by the set of complexity, diversity, homogeneity, and topology measures.