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

Abstract

This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing "copy and paste" so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer's disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of computational anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.

Projective LDDMM: Mapping Molecular Digital Pathology with Tissue MRI

“…In the following two sections, we detail our means of representing histological data with a measure-based framework over physical space and feature space as introduced in [36]. We denote these measures, μ , borrowing the notation from [36] and describe the action of transformations on these measures to bring them to the 3D space of the Mai Paxinos Atlas and consequently our multiresolution resampling of them with a spatial kernel and feature map.…”

“…We model histology data at the microscopic scale following the generalized measure approach in [36] where each particle of tissue carries a weighted Dirac measure over histology image space and a Dirac measure over the feature space and ℱ = R 2 𝓁 . Weights reflect sampled tissue area cap-tured in each particle measure, defined at the finest scale ( µ 0 ) as cross-sectional area in the histology plane w i ∈ {2 µm 2 , 0}, computed with thresholding using Otsu’s method [54].…”

“…NFT densities (counts of NFTs per cross-sectional tissue area) were modeled as discrete particle measures and transported to the 3D space of the Mai Paxinos atlas via estimated transformations, ϕ n , φ (see Section 4.6). The flexibility afforded by a measure-based framework as in [36] is reflected in the diverse modes of resampling shown in Figure 6 both within volumes and over the surface of MTL subregions (see details in Supplementary Note S.4). Resampling via Gaussian kernels yields smoothed NFT densities computed within the dense metric (volume) of the brain at approximate resolutions of MRI.…”

Projective LDDMM: Spatially Reconstructing a Story of Rostrally-Dominant Tau in Alzheimer’s Disease

“…We model histology data at the microscopic scale following the generalized measure approach in [40] where each particle of tissue carries a weighted Dirac measure over histology image space and a Dirac measure over the feature space and ℱ = R 2ℓ . Weights reflect sampled tissue area captured in each particle measure, defined at the finest scale ( μ 0 ) as cross-sectional area in the histology plane w i ∈ {2 μm 2 , 0}, computed with thresholding using Otsu’s method [65].…”

“…As tau has exhibited stronger predisposition over A β for segregating to particular brain regions (ERC, CA1, subiculum) and layers (superficial) of cortex in AD [5], we use machine-learning based methods to detect and quantify neurofibrillary tangles (NFTs) from histological images. Modeling this data in a measure theoretic framework amenable to quantifying trends at different scales [40], we transport these detections to the 3D space via the correspondences yielded by Projective LDDMM.…”