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J Surg Oncol. 1999 Jul;71(3):189-95.

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

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Department of Colorectal Surgery, The Cleveland Clinic Foundation, Cleveland, Ohio, USA.



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.


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.


The coefficient of linear regression characterizing that relationship possesses noninteger value. It is 1. 585 (well-differentiated adenocarcinomas, standard deviation (SD) = 0.038, n = 100 sections), and 1.340 (moderately differentiated adenocarcinomas, SD = 0.044, n = 100 sections). While both nuclear area and nucleo-cytoplasmic ratio in those tissues remain within a similar range (Analysis of Variance (ANOVA), F(0) = 0.791 < F(alpha) = 3.84, P = 3 x 10(-3) and F(0) = 0.077 < F(alpha) = 3.84, P = 10(-6), respectively, for k = 20,000 cells, in which F(0) is a value of the test function, F(alpha) is a critical, limit value of the F-test at the constant confidence value alpha = 0.05), a difference of fractal dimension is significant (F(0) = 3.94 > F(alpha) = 0.693, n = 100 sections, P = 2 x 10(-3)). Also, variablity of fractal dimension between tumor sections is significant (moderately differentiated adenocarcinomas, F(0) = 1.9856 > F(alpha) = 1.4262, n = 100 sections, P = 0.189).


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.

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