AN “ASYMPTOTIC FRACTAL” APPROACH TO THE MORPHOLOGY OF MALIGNANT CELL NUCLEI

Abstract

To investigate quantitatively nuclear membrane irregularity, 672 nuclei from 10 cases of oral cancer (squamous cell carcinoma) and normal cells from oral mucosa were studied in transmission electron micrographs. The nuclei were photographed at ×1400 magnification and transferred to computer memory (1 pixel=35 nm). The perimeter of the profiles was analysed using the “yardstick method” of fractal dimension estimation, and the log-log plot of ruler size vs. boundary length demonstrated that there exists a significant effect of resolution on length measurement. However, this effect seems to disappear at higher resolutions. As this observation is compatible with the concept of asymptotic fractal, we estimated the parameters c, L and Bm from the asymptotic fractal formula Br=Bm {1+(r/L)c}−1, where Br is the boundary length measured with a ruler of size r, Bm is the maximum boundary for r→0, L is a constant, and c=asymptotic fractal dimension minus topological dimension (D−Dt) for r→∞. Analyses of variance showed c to be significantly higher in the normal than malignant cases (P<0.001), but log(L) and Bm to be significantly higher in the malignant cases (P<0.001). A multivariate linear discrimination analysis on c, log(L) and Bm re-classified 76.6% of the cells correctly (84.8% of the normal and 67.5% of the tumor). Furthermore, this shows that asymptotic fractal analysis applied to nuclear profiles has great potential for shape quantification in diagnosis of oral cancer.