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Carnegie Mellon
Reconstruction of 3-D Dense Cardiac Motion from Tagged MRI Sequences
Hsun- Hsien Chang and José M.F. MouraDept. of Electrical and Computer Engineering
Yijen Wu, Kazuya Sato, and Chien HoPittsburgh NMR Center for Biomedical Research
Carnegie Mellon University, Pittsburgh, PA, USA
Work supported by NIH grants (R01EB/AI-00318 and P4EB001977)
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Outline
• Introduction• Methodology: Prior knowledge + MRI data
– Myocardial Fiber Based Structure
– Continuum Mechanics
– Constrained Energy Minimization
• Results and Conclusions
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Nslices
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
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3-D Reconstruction: myocardial fiber model
Use a fiber based model to find the correspondence between transversal slices.
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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
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Carnegie Mellon
Outline
• Introduction
• Methodology: Prior knowledge + MRI data– Myocardial Fiber Based Structure
– Continuum Mechanics
– Constrained Energy Minimization
• Results and Conclusions
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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:
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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)
)0()0(
)()( a
aa
a dt
td∂∂=
Motion of a small segment
=
)(
)(
)(
)(
3
2
1
ta
ta
ta
ta
Notations are column vectors, ex:
Prior Knowledge: fiber dynamics
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Deformation Gradient Matrix
=∂∂=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
)0(
)(
)0(
)(
)0(
)()0(
)(
)0(
)(
)0(
)()0(
)(
)0(
)(
)0(
)(
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
)0(
)()(
a
ta
a
ta
a
taa
ta
a
ta
a
taa
ta
a
ta
a
ta
tt
aa
F
+=∂∂+=+=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
)0(
)(
)0(
)(
)0(
)()0(
)(
)0(
)(
)0(
)()0(
)(
)0(
)(
)0(
)(
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
)0(
)()(
a
tu
a
tu
a
tua
tu
a
tu
a
tua
tu
a
tu
a
tu
ttd I
au
IFI
Deformation gradient F(t) is a function of displacement u(t).
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Strain
)(2
1IFFS −= T
[ ] IFFIIFIFS −+=−+++≈ )(2
1
2
1 TT dd
• 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)
[ ] [ ]FFFFIFIFIS dddddd TTT ++=−++=2
1)()(
2
1
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Linear Strain Energy Model
• S is symmetric, so we vectorize the entries at upper triangle.
=
33
2322
131211
S
SS
SSS
S
)(uCss ee T ==
[ ]TSSSSSS 231312332211 ,,,,,=s
• Let C describe the material properties. It can be shown the linear strain energy is
• The entire energy of the heart:
∑ ∑∑ ∑∀ ∀∀ ∀
==fibers segmentsfibers segments
)()( CssuU TeE
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Constrained Energy Minimization
• Internal energy: continuum mechanics governs the fibers to move as smooth as possible.
∑ ∑∀ ∀
=fibers segments
T)( CssUintE
)()()(),( 21 UUUU conextint EEEE λγγλ ++=
2)1()()( +−= ttEext IIU
• External energy: pixel intensities of fibers should be kept similar across time.
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2-D Displacement Constraints
)()()(),( 21 UUUU conextint EEEE λγγλ ++=
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
21 )()(),( DΘUUUU −++= λγγλ extint EEE
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Outline
• Introduction
• Methodology: Prior knowledge + MRI data– Myocardial Fiber Based Structure
– Continuum Mechanics
– Constrained Energy Minimization
• Results and Conclusions
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4slices
10 frames per sliceData 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 on a Bruker AVANCE DRX 4.7-T system
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Whole left ventricle
endocardium mid-wall epicardium
Fiber Based Model
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3-D Reconstruction of the Epicardium
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Conclusions
• Take into account the myocardial fiber based structure.
• Adopt the continuum mechanics framework.
• Implement constrained energy minimizationalgorithms.
