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.

This study introduces a reaction-diffusion model of atrophy spread across the rhinal cortex during early stages of Alzheimer's disease. Our finite elements model of atrophy spread is motivated by histological evidence of a spatio-temporally specific pattern of neurofibrillary tau accumulation, and evidence of grey matter atrophy correlating with sites of neurofibrillary tau accumulation. The goal is to estimate disease-related parameters such as the origin of atrophy, the speed at which atrophy spreads, and the stage of the disease. We solve a constrained optimization problem using the adjoint state method and gradient descent to match modeled cortical thickness to observed cortical thickness as calculated from 3T MRI scans. Simulation testing shows that disease-related parameters can be estimated accurately with as little as 2 years of annual observations, depending on the stage of the disease. Case studies of 3 subjects suggests that we can pinpoint the origin of atrophy to the anterior transentorhinal cortex, and that the speed of atrophy spread is less than 1 mm per year. In the future, this type of modeling could be useful to stage the progression of the disease prior to the onset of clinical symptoms.

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