title: The Quantitative Criteria Based on the Fractal Dimensions, Entropy, and Lacunarity for the Spatial Distribution of Cancer Cell Nuclei Enable Identification of Low or High Aggressive Prostate Carcinomas.
abstract: BACKGROUND
  Tumor grading, PSA concentration, and stage determine a risk of prostate cancer patients with accuracy of about 70%. An approach based on the fractal geometrical model was proposed to eliminate subjectivity from the evaluation of tumor aggressiveness and to improve the prediction. This study was undertaken to validate classes of equivalence for the spatial distribution of cancer cell nuclei in a larger, independent set of prostate carcinomas.
  
  
  METHODS
  The global fractal capacity D 0, information D 1 and correlation D 2 dimension, the local fractal dimension (LFD) and the local connected fractal dimension (LCFD), Shannon entropy H and lacunarity λ were measured using computer algorithms in digitalized images of both the reference set (n = 60) and the test set (n = 208) of prostate carcinomas.
  
  
  RESULTS
  Prostate carcinomas were re-stratified into seven classes of equivalence. The cut-off D 0-values 1.5450, 1.5820, 1.6270, 1.6490, 1.6980, 1.7640 defined the classes from C1 to C7, respectively. The other measures but the D 1 failed to define the same classes of equivalence. The pairs (D 0, LFD), (D 0, H), (D 0, λ), (D 1, LFD), (D 1, H), (D 1, λ) characterized the spatial distribution of cancer cell nuclei in each class. The co-application of those measures enabled the subordination of prostate carcinomas to one out of three clusters associated with different tumor aggressiveness. For D 0 < 1.5820, LFD < 1.3, LCFD > 1.5, H < 0.7, and λ > 0.8, the class C1 or C2 contains low complexity low aggressive carcinomas exclusively. For D 0 > 1.6980, LFD > 1.7644, LCFD > 1.7051, H > 0.9, and λ < 0.7, the class C6 or C7 contains high complexity high aggressive carcinomas.
  
  
  CONCLUSIONS
  The cut-off D 0-values defining the classes of equivalence were validated in this study. The cluster analysis suggested that the number of the subjective Gleason grades and the number of the objective classes of equivalence could be decreased from seven to three without a loss of clinically relevant information. Two novel quantitative criteria based on the complexity and the diversity measures enabled the identification of low or high aggressive prostate carcinomas and should be verified in the future multicenter, randomized studies.
authors.affiliation: Department of Urology, Alb Fils KlinikenGoeppingen, Germany; The Bȩdlewo Institute for Complexity ResearchPoznań, Poland.
authors.forename: Przemyslaw
authors.initials: P
authors.lastname: Waliszewski
keywords: Renyi dimensions; complexity; entropy; fractals; grading; lacunarity; prostate carcinoma; tumor aggressiveness