Distribution of gland-like structures in human gallbladder adenocarcinomas possesses fractal dimension

Abstract

Background and Objectives: Epithelial cells form tissue patterns of higher order such as gland-like structures. A question arises whether distribution of those patterns in adenocarcinomas is subject to certain regularity. Methods: Due to the pilot nature of this study, gallbladder adenocarcinomas were preselected by histopathological, immunohistochemical, and morphometric analysis to ensure relative homogeneity of the patterns analyzed. A box-counting method was applied to investigate a relationship between a number of gland-like structures and a radius of the expanding family of the concentric circles. Results: The coefficient of linear regression characterizing that relationship possesses noninteger value. It is 1.585 (well-differentiated adenocarcinomas, standard deviation (SD) ‫ס‬ 0.038, n ‫ס‬ 100 sections), and 1.340 (moderately differentiated adenocarcinomas, SD ‫ס‬ 0.044, n ‫ס‬ 100 sections). While both nuclear area and nucleo-cytoplasmic ratio in those tissues remain within a similar range (Analysis of Variance (ANOVA), F 0 ‫ס‬ 0.791 < F ␣ ‫ס‬ 3.84, P ‫ס‬ 3 × 10 −3 and F 0 ‫ס‬ 0.077 < F ␣ ‫ס‬ 3.84, P ‫ס‬ 10 −6 , respectively, for k ‫ס‬ 20,000 cells, in which F 0 is a value of the test function, F ␣ is a critical, limit value of the F-test at the constant confidence value ␣ ‫ס‬ 0.05), a difference of fractal dimension is significant (F 0 ‫ס‬ 3.94 > F ␣ ‫ס‬ 0.693, n ‫ס‬ 100 sections, P ‫ס‬ 2 × 10 −3 ). Also, variablity of fractal dimension between tumor sections is significant (moderately differentiated adenocarcinomas, F 0 ‫ס‬ 1.9856 > F ␣ ‫ס‬ 1.4262, n ‫ס‬ 100 sections, P ‫ס‬ 0.189). Conclusions: There is fractal regularity in distribution of gland-like structures in human gallbladder adenocarcinomas. Fractal dimension is a holistic parameter which can be applied to evaluate tumor grading in a quantitative manner.

A simple method for estimating the fractal dimension from digital images: The compression dimension

“…One of the most common descriptions of a given case study in Science and Engineering is through a graphical representation in an image. Fractal analysis of digital images is of great value, for instance, in Medicine [1][2][3][4][5] or Botanics [6,7] and for the characterization of many other physical processes [8,9].…”