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Jérôme Pousin

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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Jérôme Pousin

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  1. Singular perturbations for an elastic model of the heart Jérôme Pousin 24 juin 2008

  2.  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

  3. 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

  4.  Theoretical context t(x) u(x) Superficial forces x Initial domain Deformed domain  Continuum mechanics : equilibrium of an elastic body

  5.  Theoretical context Equilibrium Stress tensor  Potential energy Stress vector Strain vector External energy Elastic energy

  6.  Elastic deformable template [Vincent, 2001] Incremental load Assumption : small displacements

  7.  Elastic deformable template Iterative local formulation Discretized expression

  8.  Computing 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

  9.  Computing a 3-D force field  Example : cube image

  10. Part 1. 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

  11.  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

  12.  Boundary regularization  Constitutive law for a fiber-collagen model [Ohayon, 1988] pression fibers Fiber direction  Asymptotic model (  0) [Destuynder, 1996] t1, t2

  13.  Boundary regularization Without boundary constraint With boundary constraint  Boundary constraint energy

  14.  Elastic deformable template Singular perturbation • Formaly take Dt=0 • Vanishing elasticity allows larger motion

  15. Part 1. 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

  16. Numerical results

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