neuro & cardio imaging luc florack birs 2011, banff, canada · luc florack birs 2011, banff,...

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


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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Heart Function


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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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neuro & cardio imaging luc florack birs 2011, banff, canada · luc florack birs 2011, banff,...

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