Molecular Computational Anatomy: Unifying the Particle to Tissue Continuum via Measure Representations of the Brain

Abstract

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 varifold measures of the microscale deterministic structure and function with the statistical ensembles of the spatially aggregated tissue scales. 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 functional 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—using mathematical structures as first introduced by Dirac as the class of generalized functions—to the tissue scales with methods originally introduced by Euler for fluids. Methods. We introduce two mathematical methods based on notions of generalized functions and statistical mechanics. We use geometric varifolds, a product measure on space and function, to represent functional states at the micro-scales—electrophysiology, molecular histology—integrated with a Boltzmann-like program to pass from deterministic particle descriptions to empirical probabilities on the functional states at the tissue scales. Results. Our space-function varifold representation provides a recipe for traversing from molecular to tissue scales in terms of a cascade of linear space scaling composed with nonlinear functional feature mapping. Following the cascade implies every scale is a geometric measure so that a universal family of measure norms can be introduced which quantifies the geodesic connection between brains in the orbit independent of the probing technology, whether it be RNA identities, Tau or amyloid histology, spike trains, or dense MR imagery. Conclusions. We demonstrate a unified brain mapping theory for molecular and tissue scales based on geometric measure representations. We call the consistent aggregation of tissue scales from particle and cellular scales, molecular computational anatomy.

A Universal Method for Crossing Molecular and Atlas Modalities using Simplex-Based Image Varifolds and Quadratic Programming

“…To build correspondences between these datasets of point measures, we unify the molecular scales with image-like functions as has been developed for building correspondences at tissue scales in MRI [4]. For this we represent the particles as “generalized functions” [15]. Since they carry gene or cell image data we call them image varifolds [27], linking to the rich literature on the geometric measure theory of varifolds.…”

“…To map the mRNA measures to atlases we follow [27] and define the space of image varifolds μ ∈ W * to have a norm , and transform the atlas coordinates onto the targets to minimize the norm. The space of varifold norms is associated to a reproducing kernel Hilbert space [34, 15] (see (7) below) defined by the inner-product of the space as .…”

“…To map the mRNA measures to atlases we follow [27] and define the space of image varifolds µ ∈ W * to have a norm ∥ • ∥ 2 W * , and transform the atlas coordinates onto the targets to minimize the norm. The space of varifold norms is associated to a reproducing kernel Hilbert space [34,15] (see (7) below) defined by the inner-product of the space as ⟨µ, ν⟩ W * , ∥µ∥ 2 W * = ⟨µ, µ⟩ W * . The mapping variational problem constructs φ : R d → R d and feature laws (π ℓ ) ℓ∈L on F to carry φ 1 • µ π τ0 onto µ τ minimizing the normed difference.…”

“…These come particularly from multiple labs in the magnetic resonance imaging (MRI) community [6,7,8,9,10,11,12,13,14]. More recently, Molecular CA [15] has emerged which unifies the dense tissue scales of MRI with high resolution micron scales of digital pathology imagery. These approaches are hierarchical [16], constructing what are termed image varifold representations, which are geometric measures of the brain existing at multiple scales, and therefore allowing for the simultaneous representation of both micron scale particle phenomena, such as the transcriptional or cell type data studied herein, and millimeter scale tissue phenomena, as traditionally studied in CA.…”

“…A subproblem covered by the mesh based image-varifold theory outlined in [27] is the mapping of molecular scale data to atlas coordinate systems. Image varifolds are geometric measures [15], which allow us to provide a single representation that supports molecular transcriptomics measurements, cell-based measurements, and tissue scale atlases. We explicate, here, the construction of a universal method rooted in this framework for transferring molecular scale data to tissue scale “cartoon” atlases, which are devoid of gene measurements, and rather, only contain a fixed partition into structures in their description.…”