Carnegie Mellon Reconstruction of 3-D Dense Cardiac Motion from Tagged MRI Sequences Hsun - H s ien Chang and José M.F. Moura Dept. of Electrical and Computer Engineering Yijen Wu , Kazuya Sato , and Chien Ho Pittsburgh NMR Center for Biomedical Research Carnegie Mellon University, Pittsburgh, PA, USA Work supported by NIH grants (R01EB/AI-00318 and P4EB001977) 1
Carnegie Mellon Outline • Introduction • Methodology: Prior knowledge + MRI data – Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization • Results and Conclusions 2
Carnegie Mellon 2-D Cardiac MRI Images M frames per slice N slices sparse displacements dense displacements Y. Sun, Y.L. Wu, K. Sato, C. Ho, and J.M.F. Moura, 3 Proc. Annual Meeting ISMRM 2003
Carnegie Mellon 3-D Reconstruction: myocardial fiber model Use a fiber based model to find the correspondence between transversal slices. 4
Carnegie Mellon 3-D Reconstruction: fiber deformation model Use continuum mechanics to describe the motion of fibers. Fit the model to MRI data by constrained energy minimization 5
Carnegie Mellon Outline • Introduction • Methodology: Prior knowledge + MRI data – Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization • Results and Conclusions 6
Carnegie Mellon Prior Knowledge: myocardial anatomy +60º Endocardium Multiple-layer view: Mid-wall -60º Epicardium Streeter, in Handbook of Physiology Volume 1: the Cardiovascular System , American Physiological Society, 1979 7
Carnegie Mellon Prior Knowledge: fiber dynamics Motion of a small segment a ( t )+ d a ( t ) d a ( t ) a ( t ) Displacement: u ( t )= a ( t )- a (0) Notations are column d a (0) vectors, ex: a ( t ) 1 = a ( t ) a ( t ) a (0)+ d a (0) a (0) 2 a ( t ) ∂ 3 a ( t ) = a ( ) a ( 0 ) d t d ∂ a ( 0 ) 8
Carnegie Mellon Deformation Gradient Matrix ∂ ∂ ∂ a ( t ) a ( t ) a ( t ) 1 1 1 ∂ ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) a ( ) t 1 2 3 = = ∂ ∂ ∂ a ( t ) a ( t ) a ( t ) F ( t ) 2 2 2 ∂ ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) a ( 0 ) 1 2 3 ∂ ∂ ∂ a ( t ) a ( t ) a ( t ) 3 3 3 ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) 1 2 3 ∂ ∂ ∂ u ( t ) u ( t ) u ( t ) 1 1 1 ∂ ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) u ( t ) 1 2 3 = + = + = + ∂ ∂ ∂ u ( t ) u ( t ) u ( t ) I F I I d ( t ) 2 2 2 ∂ ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) a ( 0 ) 1 2 3 ∂ ∂ ∂ u ( t ) u ( t ) u ( t ) 3 3 3 ∂ ∂ ∂ a ( 0 ) a ( 0 ) a ( 0 ) 1 2 3 Deformation gradient F( t ) is a function of displacement u( t ) . 9
Carnegie Mellon Strain • Strain is the displacement per unit length, and 1 = − is written mathematically as T S ( F F I ) 2 [ ] [ ] 1 1 = + + − = + + T T T S ( I F ) ( I F ) I F F F F d d d d d d 2 2 Ref: Y.C. Fung, A First Course in Continuum Mechanics , 3rd ed., Prentice-Hall, New Jersey, 1994 • When strain is small, it is approximated as [ ] 1 1 ≈ + + + − = + − T T S F I F I I ( F F ) I d d 2 2 (Note: S is symmetric) 10
Carnegie Mellon Linear Strain Energy Model • S is symmetric, so we vectorize the entries at upper triangle. [ ] = T s S , S , S , S , S , S S S S 11 22 33 12 13 23 11 12 13 = S S S 22 23 S 33 • Let C describe the material properties. It can be = = T e s Cs e ( u ) shown the linear strain energy is • The entire energy of the heart: ∑ ∑ ∑ ∑ = = T E ( U ) e ( u ) s Cs ∀ ∀ ∀ ∀ fibers segments fibers segments 11
Carnegie Mellon Constrained Energy Minimization λ = γ + γ + λ U U U U E ( , ) E ( ) E ( ) E ( ) 1 int 2 ext con ∑ ∑ = T = − + 2 E ( U ) s Cs ( U ) I ( ) I ( 1 ) E ext t t int ∀ ∀ fibers segments • External energy: pixel • Internal energy: continuum mechanics intensities of fibers governs the fibers to should be kept similar move as smooth as across time. possible. 12
Carnegie Mellon 2-D Displacement Constraints λ = γ + γ + λ U U U U E ( , ) E ( ) E ( ) E ( ) 1 int 2 ext con D : 2-D displacements of the taglines Ө U : picks the entries of U corresponding to D 2-D displacement constraints: Ө U = D λ : Lagrange multiplier λ = γ + γ + λ − 2 Θ U E ( U , ) E ( U ) E ( U ) D 1 int 2 ext 13
Carnegie Mellon Outline • Introduction • Methodology: Prior knowledge + MRI data – Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization • Results and Conclusions 14
Carnegie Mellon Data Set 10 frames per slice 4 slices 256 × 256 pixels per image � Transplanted rats with heterotropic working Y. Sun, Y.L. hearts. Wu, K. Sato, C. � MRI scans performed Ho, and J.M.F. Moura, Proc. on a Bruker AVANCE Annual Meeting DRX 4.7-T system ISMRM 2003 15
Carnegie Mellon Fiber Based Model Whole left ventricle epicardium endocardium mid-wall 16
Carnegie Mellon 3-D Reconstruction of the Epicardium 17
Carnegie Mellon Conclusions • Take into account the myocardial fiber based structure . • Adopt the continuum mechanics framework. • Implement constrained energy minimization algorithms. 18
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