On the relationship between tumor structure and complexity of the spatial distribution of cancer cell nuclei: a fractal geometrical model of prostate carcinoma.

Background: A risk of the prostate cancer patient is defined by both the objective and subjective criteria, that is, PSA concentration, Gleason score, and pTNM-stage. The subjectivity of tumor grading influences the risk assessment owing to a large inter- and intra-observer variability. Pathologists propose a central prostate pathology review as a remedy for this problem; yet, the review cannot eliminate the subjectivity from the diagnostic algorithm. The spatial distribution of cancer cell nuclei changes during tumor progression. It implies changes in complexity measured by the capacity dimension D0, the information dimension D1, and the correlation dimension D2.

Methods: The cornerstone of the approach is a model of prostate carcinomas composed of the circular fractals CF(4), CF(6 + 0), and CF(6 + 1). This model is both geometrical and analytical, that is, its structure is well-defined, the capacity fractal dimension D0 can be calculated for the infinite circular fractals, and the dimensions D0, D1, D2 can be computed for their finite counterparts representing distribution of cell nuclei. The model enabled both the calibration of the software and the validation of the measurements in 124 prostate carcinomas. The ROC analysis defined the cut-off D0 values for seven classes of complexity.

Results: The Gleason classification matched in part with the classification based on the D0 values. The mean ROC sensitivity was 81.3% and the mean ROC specificity 75.2%. Prostate carcinomas were re-stratified into seven classes of complexity according to their D0 values. This increased both the mean ROC sensitivity and the mean ROC specificity to 100%. All homogeneous Gleason patterns were subordinated to the class C1, C4, or C7. D0 = 1.5820 was the cut-off D0 value between the complexity class C2 and C3 representing low-risk cancers and intermediate-risk cancers, respectively.

Conclusions: The global fractal dimensions eliminate the subjectivity in the diagnostic algorithm of prostate cancer. Those complexity measures enable the objective subordination of carcinomas to the well-defined complexity classes, and define subgroups of carcinomas with very low malignant potential (complexity class C1) or at a large risk of progression (complexity ass C7).