Index for spatial heterogeneity in breast cancer

Abstract

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.