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) = 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), F0 = 0.791 < Fα = 3.84, P = 3 × 10−3 and F0 = 0.077 < Fα = 3.84, P = 10−6, respectively, for k = 20,000 cells, in which F0 is a value of the test function, Fα is a critical, limit value of the F‐test at the constant confidence value α = 0.05), a difference of fractal dimension is significant (F0 = 3.94 > Fα = 0.693, n = 100 sections, P = 2 × 10−3). Also, variablity of fractal dimension between tumor sections is significant (moderately differentiated adenocarcinomas, F0 = 1.9856 > Fα = 1.4262, n = 100 sections, P = 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. J. Surg. Oncol. 1999:71:189–195. © 1999 Wiley‐Liss, Inc.

Fractal Dimension for a Cancer Invasion Model

“…The determination of fractal dimensions in the biomedical field has been especially successful in cancer-related work that has been carried out for determining the mammographic density of breast carcinoma, 12 MCF-7 breast cancer cells, 13 benign and malignant breast epithelial cell nuclei, 14 mastopathic and mammary cancer tissue, 15 dysplastic lesions of cervix uteri, 16 gallbladder adenocarcinomas, 17 retinoid-induced differentiation of cancer cells 18 malignant epithelial lesions of the mouth, 19 and has also been used for the study of the role of fractals, chaos and cancer. 20 The box-counting method, the sausage method and a local method have been implemented for the calculation or estimation of the fractal capacity dimension.…”

Nuclear fractal dimension in oral squamous cell carcinoma: a novel method for the evaluation of grading, staging, and survival

“…FD is a shape index, and specifically, it can be defined as a measure of irregularly shaped objects. Previous studies have examined the application of fractal analysis for the characterization of tumors and have demonstrated that FD can describe the complex pathological texture of several neoplasms, such as carcinoma of the gallbladder , lung , uterus , breast and oral cavity . It has been previously showed that the increase of abnormalities observed on epithelial‐connective tissue interface of the floor of the mouth (characterized by severe dysplasia and invasive carcinoma) was accompanied by an increase of FD .…”

Oncologic Mathematics

“…Fractals have also been applied to diagnosis and histopathological analysis of tumors. [38][39][40] In Figure 6A, we generated a simple progression of cellular diversity based on the antigenic change and genetic mutation shown in Figure 3. It must be emphasized that the changes in cells are caused under the same specific conditions as the first set of changes.…”