Reconstruction of 3-D Dense Cardiac Motion from Tagged MRI - - PowerPoint PPT Presentation

1

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)

2

Carnegie Mellon

Outline

• Introduction
• Methodology: Prior knowledge + MRI data

– Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization

• Results and Conclusions

3

Carnegie Mellon

N slices M frames per slice

• Y. Sun, Y.L. Wu, K. Sato, C. Ho, and J.M.F. Moura,
• Proc. Annual Meeting ISMRM 2003

sparse displacements

2-D Cardiac MRI Images

dense displacements

4

Carnegie Mellon

3-D Reconstruction: myocardial fiber model

Use a fiber based model to find the correspondence between transversal slices.

5

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

6

Carnegie Mellon

Outline

• Introduction
• Methodology: Prior knowledge + MRI data

– Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization

• Results and Conclusions

7

Carnegie Mellon

Endocardium

• 60º

+60º

Mid-wall Epicardium

Prior Knowledge: myocardial anatomy

Streeter, in Handbook of Physiology Volume 1: the Cardiovascular System, American Physiological Society, 1979

Multiple-layer view:

8

Carnegie Mellon

a(t) da(t) a(t)+da(t) a(0)+da(0) a(0) da(0) Displacement: u(t)=a(t)-a(0)

) ( ) ( ) ( ) ( a a a a d t t d ∂ ∂ =

Motion of a small segment

          = ) ( ) ( ) ( ) (

3 2 1

t a t a t a t a

Notations are column vectors, ex:

Prior Knowledge: fiber dynamics

9

Carnegie Mellon

Deformation Gradient Matrix

          = ∂ ∂ =

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1

) ( ) ( ) (

a t a a t a a t a a t a a t a a t a a t a a t a a t a

t t a a F           + = ∂ ∂ + = + =

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1

) ( ) ( ) (

a t u a t u a t u a t u a t u a t u a t u a t u a t u

t t d I a u I F I

Deformation gradient F(t) is a function of displacement u(t).

10

Carnegie Mellon

Strain

) ( 2 1 I F F S − =

T

[ ]

I F F I I F I F S − + = − + + + ≈ ) ( 2 1 2 1

T T

d d

• When strain is small, it is approximated as
• Strain is the displacement per unit length, and

is written mathematically as

Ref: Y.C. Fung, A First Course in Continuum Mechanics, 3rd ed., Prentice-Hall, New Jersey, 1994

(Note: S is symmetric)

[ ] [ ]

F F F F I F I F I S d d d d d d

T T T

+ + = − + + = 2 1 ) ( ) ( 2 1

11

Carnegie Mellon

Linear Strain Energy Model

• S is symmetric, so we vectorize the entries at

upper triangle.

          =

33 23 22 13 12 11

S S S S S S S

) (u Cs s e e

T

= =

[ ]

T

S S S S S S

23 13 12 33 22 11

, , , , , = s

• Let C describe the material properties. It can be

shown the linear strain energy is

• The entire energy of the heart:

∑ ∑ ∑ ∑

∀ ∀ ∀ ∀

= =

fibers segments fibers segments

) ( ) ( Cs s u U

T

e E

12

Carnegie Mellon

Constrained Energy Minimization

• Internal energy:

continuum mechanics governs the fibers to move as smooth as possible.

∑ ∑

∀ ∀

=

fibers segments T

) ( Cs s U

int

E

) ( ) ( ) ( ) , (

2 1

U U U U

con ext int

E E E E λ γ γ λ + + =

2

) 1 ( ) ( ) ( + − = t t Eext I I U

• External energy: pixel

intensities of fibers should be kept similar across time.

13

Carnegie Mellon

2-D Displacement Constraints

) ( ) ( ) ( ) , (

2 1

U U U U

con ext int

E E E E λ γ γ λ + + =

D: 2-D displacements of the taglines ӨU: picks the entries of U corresponding to D 2-D displacement constraints: ӨU=D λ: Lagrange multiplier

2 2 1

) ( ) ( ) , ( D ΘU U U U − + + = λ γ γ λ

ext int

E E E

14

Carnegie Mellon

Outline

• Introduction
• Methodology: Prior knowledge + MRI data

– Myocardial Fiber Based Structure – Continuum Mechanics – Constrained Energy Minimization

• Results and Conclusions

15

Carnegie Mellon

4 slices 10 frames per slice

Data Set

256×256 pixels per image

• Y. Sun, Y.L.

Wu, K. Sato, C. Ho, and J.M.F. Moura, Proc. Annual Meeting ISMRM 2003

Transplanted rats with heterotropic working hearts. MRI scans performed

• n a Bruker AVANCE

DRX 4.7-T system

16

Carnegie Mellon

Whole left ventricle endocardium mid-wall epicardium

Fiber Based Model

17

Carnegie Mellon

3-D Reconstruction of the Epicardium

18

Carnegie Mellon

Conclusions

• Take into account the

myocardial fiber based structure.

• Adopt the continuum

mechanics framework.

• Implement constrained

energy minimization algorithms.