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Jérôme Pousin

Singular perturbations for an elastic model of the heart

24 juin 2008

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Objectives

Extract the heart anatomy(3-D+t segmentation)

Combine complementary functional data

(Multimodal registration)

Difficulties

LVRV

- patient movement- moving deformable organ- acquisition geometry- the anatomy cannot be easily determined in all modalities

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Segmentation of the heart using an elastic deformable template

1. Elastic deformable template2. Boundary regularization3. Singular perturbation 4. Sketch of the proof

ICJ

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Deformed domain

Theoretical context

Continuum mechanics : equilibrium of an elastic body

x

Superficial forcesu(x)t(x)

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Initial domain

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Theoretical context

Equilibrium

Potential energy

Stress vector Strain vector

Elastic energy External energy

Stress tensor

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Incremental load

Elastic deformable template

Assumption : small displacements

[Vincent, 2001]

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~

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Iterative local formulation

Elastic deformable template

Discretized expression

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Computing a 3-D force field

Force field deriving from a potential image

Gradient vector flow [Xu, 1998]

Imposed conditionForce field null on the border of the object to be segmented

- Potential minimum on the object border

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Computing a 3-D force field

Example : cube image

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Part 1.Segmentation of the heart using an

elastic deformable template

1. Elastic deformable template2. Improving the model’s convergence 3. Boundary regularization4. Model inititialization 5. Results of segmentation

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Boundary regularization

Large isotropic middle layer

Thin peripheral layers

Three-layer model

[Streeter, 1969]

- Ratio of circumferential fibers to longitudinal fibers 10:1

Longitudinal

Circumferential

Longitudinal

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fibers

[Ohayon, 1988] Constitutive law for a fiber-collagen model

pression

Boundary regularization

Asymptotic model ( 0)

t1, t2

[Destuynder, 1996]

Fiber direction

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Boundary regularization

Boundary constraint energy

Without boundary constraint With boundary constraint

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Singular perturbation

Elastic deformable template

Formaly take t=0

Vanishing elasticity allows larger motion

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Part 1.Segmentation of the heart using an

elastic deformable template

1. Elastic deformable template2. Improving the model’s convergence 3. Boundary regularization4. Model inititialization 5. Results of segmentation

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Numerical results