Post on 21-Jun-2020
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NEURO & CARDIO IMAGINGfrom a mathematical perspective
LUC FLORACKBIRS 2011, Banff, Canada
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Imaging Science & Technology Eindhoven (IST/e)
• 2-dimensional images one can look at ➞ complex non-visual data• calls for cross-divisional “biomatical” approach
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Imaging Science & Technology Eindhoven (IST/e)
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Brain Connectomics: Grand Challenge
• connectivity + tractography = “connectome” • macro scale
Anna Vilanova Bram Platel
mm↔
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Heart Function: Grand Challenge
• operational definition of “(ab)normal” myocardial contractility • diagnosis of cardiac pathology • assessment of stem cell therapy
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Brain Connectomics
• diffusion weighted MRI: DTI, HARDI, etc.
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Brain Connectomics
• diffusion weighted MRI: DTI, HARDI, etc.
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Methodology
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Methodology
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Methodology
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Methodology
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Methodology
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Methodology
r ∼ √(Dt)
• isotropic medium ~ isotropic diffusion ~ spherical gauge figure
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Methodology
• fibrous tissue ~ anisotropic diffusion ~ ellipsoidal gauge figure
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Methodology
• DTI ~ ellipsoidal gauge figure ~ Riemann metric (6 d.o.f.’s)
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Methodology
• tractography = geodesic path problem on a Riemannian manifold
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Methodology
• analogy with GRT: DTI image ~ inhomogeneous gravitational field
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Methodology
• there are infinitely many geodesics through any given point• distant points may be connected by multiple geodesics• conjecture: neural tracts ⊂ geodesics• probabilistic criterion: scale invariant connectivity measure
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Work in Progress
• Constantin Carathéodory (1873-1950): “der Königsweg”• general formalism for the calculus of variations• velocity driven geodesic congruences versus momentum driven distance fronts
“Solutions to the Hamilton-Jacobi Equations in the Euclidean Plane” -- Sem Florack, pencil on paper (2011)
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Work in Progress
Neda Sepasian
• Constantin Carathéodory (1873-1950): “der Königsweg”• general formalism for the calculus of variations• velocity driven geodesic congruences versus momentum driven distance fronts
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Work in Progress
Neda Sepasian
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Heart Function
• cine & tagging MRI
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Methodology
• velocity gradient L=∇v ➞ deformation tensor F ➞ strain tensor E• dF/dt = LF: exact solution via non-commutative multiplicative calculus
F(t) = exp(L(s)ds)
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Work in Progress
• strain tensor atlas of “normals” + “deviations” ➞ pathology• strain tensor evolution for quantitative stem cell therapy assessment
Hans van Assen
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The End
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