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Computational anatomy and diffeomorphometry: A dynamical systems model of neuroanatomy in the soft condensed matter continuum

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The nonlinear systems models of computational anatomy that have emerged over the past several decades are a synthesis of three significant areas of computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high‐dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models
Shows single particle flow for straight line, constant speed motion, pt = p0, vs(·) = k(·, ϕs(x1))p0, and action on the image, Iϕt
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Showing the effect of kernel size determining the scale of shape. Two particle “waltzing” solutions for kernel scales 1.0 (panels 1,3) and 2.5 (panels 2,4)
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Shows folding of a thin layer modeling cortical gray matter, for three particle momentum flow
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Shows the two particle curving solution . Left two panels show particles coming together along geodesics they turn “north” together; kernel σ = 3.5. Right two panels show two particles passing each other along geodesics are attracted within the zone of attraction of the kernel width, curving the trajectories; kernel σ = 3.5
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Showing landmark flowing with potential energy giving inputs that enters continuously along the path of the flow . Potential energy term U giving the g terms expressed by the distance squared between the green landmark and the target red landmark. Panel 1 is for a small kernel (0.01); panel 2 is for a larger kernel .05, note the trajectory slowly rotates, panel 3 shows large chaotic motions for kernel 0.1. See text in “Forces associated to Internal Potential Energy” section for more detail
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Dynamics model in CA organized by force, acceleration, velocity, and displacement. Input term (see (17)) represents energy‐derived forces from external measurements or internal energies of substructures. Images enter via the nonlinear observer Iϕ−1
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Example of rotation displacement vector field Oxx (panel 1) applied to the grid, ϕOgrid, (panel 2) and MIM face ϕO · I′(x):= I(O−1x)
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Showing space of face with bijective, 1–1, onto correspondence between them carrying the label maps. There is one face which is not bijective to others
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Showing atrophy in the medial temporal lobe. The top row shows all three structures: Amygdala, hippocampus, and hidden below is entorhinal cortex. The bottom row shows the entorhinal cortex which is hidden in the top row. The color bar red show 25% loss, with green showing no change relative to the first time point in the time series of scans
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Showing atrophy in the medial temporal lobe. The series of sections show the entorhinal cortex from two idividual (top and bottom rows), with the geodesic flow depicted across the four time points. Top row shows clear spread of disease as neutral green (no change) spreads to red (decrease) of localized volume indexed to the surfaces. The top subjects shows significantly more year over year loss than bottom
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Showing medial temporal lobe structures with both the MRI section and the solid models segmentations
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Panels show folding of the closed curve contour of entorhinal cortex boundary associated to a two‐dimnsional (2D) cortical section through the medial temporal lobe of a 36‐week old (cyan) folding onto the target (red). Blue line shows the cyan map BLUE:= φ · CY AN. Panel 6 shows the spline
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Top row: Single particle linear spline ψt: x ↦ x + tv0(x) (panels 1,2) and two particle spline (panels 3,4) with the grid crossing implying no inverse. Bottom row: Application of linear transformation splines to the images with the inverse catastrophe ψt · I = Iψt. Kernel σ = 3.5; panels 1,3 shows t = 0.5, panels 2,4 shows t = 1
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Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models
Laboratory Methods and Technologies > Imaging
Analytical and Computational Methods > Computational Methods

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