Spatial-Molecular Information Analysis Based on Histopathological Geometry: KL-Divergence Decomposition of Cell Distribution and Location-Dependent Expression States in PD-1 Immunohistochemistry

Abstract

Spatial biology enables molecular expression to be analyzed together with the location of cells in tissue. However, it is often difficult to distinguish whether cells preferentially accumulate near a histopathological boundary from whether their molecular expression states vary with that location. To establish informational scientific framework, we define boundary distance as R and a binary molecular expression state as G, and compare the observed joint distribution P\left(R,G\right) with Q\left(R\right)P\left(G\right), where Q\left(R\right) is a null distribution determined by tissue geometry. The resulting Kullback-Leibler divergence decomposes into D_{R}, which quantifies deviation of the spatial distribution of cells from the geometric null, and I\left(R;G\right), which measures how strongly molecular expression state depends on boundary distance. Idealized simulations and semi-synthetic validation using geometry reconstructed from a PD-1 IHC image confirmed that these two effects can be generated separately and recovered by the corresponding terms. Plug-in estimators showed positive finite-sample bias, supporting permutation-based inference. We then applied the framework to five publicly available PD-1 lung cancer IHC images from the Human Protein Atlas. In pooled analysis, I\left(R;G\right)=0.0302 nats and was significant in a stratified within-image label-permutation test (p=0.00040). This framework provides a quantitative way to distinguish where cells are located from how their molecular expression states vary with tissue location. Larger cohorts, independent datasets, additional cancer types, and other molecular markers will be required to establish generalizability and clinical utility.