Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum Via Measure Representations of the Brain

Objective The objective of this research is to unify the molecular representations of spatial transcriptomics and cellular scale histology with the tissue scales of Computational Anatomy for brain mapping. Impact statement We present a unified representation theory for brain mapping based on geometric measures of the micro-scale phenotypes of molecular disease simultaneously with the connectomic scales of complex interacting brain circuits. Introduction Mapping across coordinate systems in computational anatomy allows us to understand structural and functional properties of the brain at the millimeter scale. New measurement technologies in digital pathology and spatial transcriptomics allow us to measure the brain molecule by molecule and cell by cell based on protein and transcriptomic identity. We currently have no mathematical representations for integrating consistently the tissue limits with the molecular particle descriptions. The formalism derived here demonstrates the methodology for transitioning consistently from the molecular scale of quantized particles – as first introduced by Dirac as the class of generalized functions – to the continuum and fluid mechanics scales appropriate for tissue. Methods We introduce two methods based on notions of generalized function geometric measures and statistical mechanics. We use generalized functions expanded to include functional geometric descriptions - electrophysiology, transcriptomic, molecular histology – to represent the molecular biology scale integrated with a Boltzman like procedure to pass from the sparse particles to empirical probability laws on the functional state of the tissue. Results We demonstrate a unified mapping methodology for transferring molecular information in the transcriptome and histological scales to the human atlas scales for understanding Alzheimer’s disease. Conclusions We demonstrate a unified brain mapping theory for molecular and tissue scales based on geometric measure representations.