Singular perturbations for an elastic model of the heart
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Singular perturbations for an elastic model of the heart . Jérôme Pousin. 24 juin 2008.  Objectives. RV. LV.  Extract the heart anatomy (3-D+t segmentation)  Combine complementary functional data (Multimodal registration).  Difficulties. patient movement

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Objectives

RV

LV

  •  Extract the heart anatomy

    • (3-D+t segmentation)

  • Combine complementary functional data

    • (Multimodal registration)

 Difficulties

  • 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 template

2. Boundary regularization

3. Singular perturbation

4. Sketch of the proof

ICJ


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templateTheoretical context

t(x)

u(x)

Superficial forces

x

Initial domain

Deformed domain

 Continuum mechanics : equilibrium of an elastic body


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templateTheoretical context

Equilibrium

Stress tensor

 Potential energy

Stress vector

Strain vector

External energy

Elastic energy


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templateElastic deformable template

[Vincent, 2001]

Incremental load

Assumption : small displacements


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templateElastic deformable template

Iterative local formulation

Discretized expression


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templateComputing a 3-D force field

 Force field deriving from a potential image

- Potential minimum on the object border

 Gradient vector flow [Xu, 1998]

 Imposed condition

Force field null on the border of the object to be segmented


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templateComputing a 3-D force field

 Example : cube image


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Part 1. template

Segmentation of the heart using an elastic deformable template

1. Elastic deformable template

2. Improving the model’s convergence

3. Boundary regularization

4. Model inititialization

5. Results of segmentation


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template Boundary regularization

 Three-layer model

- Ratio of circumferential fibers to longitudinal fibers 10:1

[Streeter, 1969]

Longitudinal

Circumferential

Longitudinal

Large isotropic

middle layer

Thin peripheral layers


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template Boundary regularization

 Constitutive law for a fiber-collagen model

[Ohayon, 1988]

pression

fibers

Fiber direction

 Asymptotic model (  0)

[Destuynder, 1996]

t1, t2


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template Boundary regularization

Without boundary constraint

With boundary constraint

 Boundary constraint energy


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templateElastic deformable template

Singular perturbation

  • Formaly take Dt=0

  • Vanishing elasticity allows larger motion


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Part 1. template

Segmentation of the heart using an elastic deformable template

1. Elastic deformable template

2. Improving the model’s convergence

3. Boundary regularization

4. Model inititialization

5. Results of segmentation



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