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The circular fractal model of adenocarcinomas and tumor aggressiveness

Tom 109 / 2016

Przemysław Waliszewski Banach Center Publications 109 (2016), 183-196 MSC: Primary 92B99; Secondary 92-08. DOI: 10.4064/bc109-0-12


There is no golden standard for the evaluation of tumor aggressiveness in pathology. This results in a high inaccuracy of both tumor grading and assessment of the progression risk as well as in unnecessary treatment of patients. Using some histological features of adenocarcinomas, a universal approach based on the circular fractal geometrical model and three global fractal dimensions of the Rényi family as the complexity measures of the spatial distribution of cancer cell nuclei in adenocarcinomas is proposed. This model is both geometrical and analytical. It can be used to select and calibrate the software for the image analysis. Both the global fractal dimension $D_0$ and the global information dimension $D_1$ allow the objective stratification of carcinomas into the classes of equivalence. The correlation coefficient between those two dimensions is about 0.95. The mean value of the global capacity fractal dimension $D_0$ 1.5820 defines a limit between low-grade prostate carcinomas with a well-preserved glandular structure and high-grade prostate carcinomas with more altered spatial distribution of cancer cell nuclei. This model has been validated by analyzing a large set of prostate carcinomas. According to the Bekenstein Bound, the amount of information present in the finite fractals representing the idealized distribution of cancer cell nuclei is limited, and changes linearly as $1:1.94:2.64$. A similar relationship can be seen in changes of entropy $S$ in a function of the $D_0$. Entropy also increases in a linear manner with the coefficient of correlation 0.904 and, therefore, determines the natural course of disease. Finally, the circular fractals CF($6+0$) and CF($6+1$) have the same topology. This means that the corresponding patterns of growth, albeit with different histological structure, are generated by the same dynamic forces, and may reveal the same aggressiveness. Therefore, they should be classified in the high-risk category. This novel, objective approach enables a stratification of prostate carcinomas into a category of low, intermediate or high aggressiveness. The subjective tumor grading should also be simplified.


  • Przemysław WaliszewskiDepartment of Urology, Pediatric Urology and Andrology
    Justus-Liebig University
    Rudolf-Buchheim-Strasse 7
    35392 Giessen, Germany

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