Spatial Pattern Analysis using Chaos Theory: A Nonlinear Deterministic Approach to the Histological Texture of Tumours

Abstract

Signals from some dynamical systems look like stochastic processes although their latent mechanism is deterministic. When such systems show sensitive dependence on small changes in the initial conditions, they are denoted as deterministic chaos. Motivated by recent advances in statistical inference methods for chaotic systems and by the concept of spatial chaos, we present a deterministic approach to the study of epithelial tissue texture. Methods for estimation of the autocorrelation function, for evaluation of the power spectrum, for attractor reconstruction, for estimation of the correlation dimension, and for the generation of surrogate data sets are outlined. In our biological example, these methods are applied to 20 cases of mastopathy and compared to 20 cases of mammary cancer. The input signals for the analysis were estimates of epithelial volume fraction measured at low magnification at 5100 equally spaced positions per case along an arbitrarily directed axis. The results suggest the existence of a low-dimensional deterministic attractor in mastopathic tissue texture, which is replaced by coloured noise in the majority of mammary carcinomas. Biological mechanisms for this finding and scale effects are discussed, and some methodological aspects and possible extensions of our approach are outlined.