Coarse-graining of perplexity for the spatial distribution of molecules

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 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.

Figure 1

A mouse kidney (sample 4-2) and examples of peaks on mass spectra at three different spots in the kidney.

Figure 2

Entropy and perplexity heat maps on kidney MSI data. (a),(d) HE staining of kidneys (samples 4-1 and 4-2) of four-month-old mice and ROIs (marked by rectangles). (b),(e) Entropy heat maps in ROIs. In the ROI of sample 4-1, the entropy changed between 5.0 and 9.6. In the ROI of sample 4-2, the entropy changed between 7.2 and 9.7. (c),(f) Perplexity heat maps in ROIs. In the ROI of sample 4-1, the perplexity changed between 149.0 and 846.5. In the ROI of sample 4-2, the perplexity changed between 32.2 and 755.3. Scale bar: 1 mm.

Figure 3

(a),(b) The HE staining and perplexity heat map of the four-month-old mouse kidney (sample 4-2) and spots selected for demonstrating slope calculations. The range of the perplexity was between 32.2 and 755.3. (c) Semilogarithmic plots of $ln\varepsilon$ and the perplexity. For spots a–c, slopes were 504.7, 631.5, and 410.2, respectively. (d) For spots d–f, slopes were 480.8, 482.5, and 488.0, respectively. (e) For spots g–j, slopes were 744.3, 715.8, 716.6, and 700.8, respectively. Scale bar: $1\mathrm{mm}$.

Figure 4

Heat maps of the slope $k$ are shown with overlays with HE staining. (a),(b) Heat maps of $k$ for samples 4-1 and 4-2. In (a) and (b), $[\min ,\max ]$ is [171.1,1105.4] and [104.4,902.6], respectively. (c),(d) Overlays of the heat map and HE staining of the kidneys. Scale bar: $1\mathrm{mm}$. (e),(f) Enlarged areas for the HE staining and heat map of $k$. Scale bar: $10\mu \mathrm{m}$.