Coarse-graining of the perplexity for the spatial distribution of molecules

Abstract

Biological tissue consists of various molecules. Instead of focusing on a particular molecule, we consider the Shannon entropy which is calculated from the abundance of different molecules at each spot in the tissue. The spatial distribution of the Shannon entropy is of interest. In this paper, we first obtain the heat map of perplexity, whose logarithm is the entropy. To characterize the spatial variety of molecules, we propose a scalar $k$ that is concerned with the coarse-graining of the perplexity heat map. To verify the usefulness of the number, experiments with MSI (mass spectrometry imaging) were performed for mouse kidneys. We found that $k$ has large values in the renal pelvis area, cortex area, veins, and arteries in the mouse kidney, whereas fractal dimensions fail to distinguish those regions.