Index for spatial heterogeneity in breast cancer.

Histopathological heterogeneity in cancer is a general concern. Breast carcinoma heterogeneity is now widely admitted as a source of histological grading imprecision and reproducibility problems. Classically, homogeneity is defined as equivalent to stationarity. A measure of heterogeneity based on asymptotic properties of spatial statistics is developed. Long-range dependences in heterogeneous spatial processes make estimation of the proposed heterogeneity measure unreliable. A robust estimator based on the wavelet transform is presented; this bypasses long-range dependences. The estimator extends previous works on one-dimensional stochastic processes to two dimensions as appropriate for histopathological analysis. As a side result, the estimator gives confidence intervals for the heterogeneity measure that enables the formulation and validation of testable hypothesis on the observed histopathological samples. This approach is applied to the characterization of breast cancer tumours. We show that the heterogeneity measure for various blocks of a single tumour is invariable, even when various blocks differ in size and in number of marked nuclei.

Breast carcinoma, intratumour heterogeneity and histological grading, using geostatistics.

Tumour progression is currently believed to result from genetic instability. Chromosomal patterns specific of a type of cancer are frequent even though phenotypic spatial heterogeneity is omnipresent. The latter is the usual cause of histological grading imprecision, a well documented problem, without any fully satisfactory solution up to now. The present article addresses this problem in breast carcinoma. The assessment of a genetic marker for human tumours requires quantifiable measures of intratumoral heterogeneity. If any invariance paradigm representing a stochastic or geostatistic function could be discovered, this might help in solving the grading problem. A novel methodological approach using geostatistics to measure heterogeneity is used. Twenty tumours from the three usual (Scarff-Bloom and Richardson) grades were obtained and paraffin sections stained by MIB-1 (Ki-67) and peroxidase staining. Whole two-dimensional sections were sampled. Morphometric grids of variable sizes allowed a simple and fast recording of positions of epithelial nuclei, marked or not by MIB-1. The geostatistical method is based here upon the asymptotic behaviour of dispersion variance. Measure of asymptotic exponent of dispersion variance shows an increase from grade 1 to grade 3. Preliminary results are encouraging: grades 1 and 3 on one hand and 2 and 3 on the other hand are totally separated. The final proof of an improved grading using this measure will of course require a confrontation with the results of survival studies.