Basics of diffusion-weighted MRI Fiber Mapping from Rank-2 - - PowerPoint PPT Presentation

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Imaging Translational Water Diffusion with Magnetic Resonance for Fiber Mapping in the Central Nervous System

• Basics of diffusion-weighted MRI
• Fiber Mapping from Rank-2 Tensor Model
• High-Angular-Resolution Diffusion Imaging

Matrix Acquisition Model fitting Diffusion parameter images 3D fiber tract mapping

• Example structures in brain and spinal cord
• Challenges and limitations
• Acknowledgements

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Magnets, Magnetic Resonance, and Imaging

Black: Horizontal-solenoid magnet field Gray: Spatial gradients in magnetic field White: Perpendicular rf magnetic field Subject stationary Timing determines contrast

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Image Acquisition

t G k γ =

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Magnetic Resonance Imaging of 1H in Human Head (THM) at 3 Tesla (128 MHz)

Proton density and T1 weighted Spin echo image acquisition TR 3700 ms, TE 15 ms T2 weighted Spin echo image acquisition TR 3700 ms, TE 90 ms

Gray Matter (cortex) White Matter (sub-cortex) Cerebral-Spinal Fluid

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aS, glial cell aT, axonal cell

• G. Stanisz, A. Szafer, G. Wright, R. M. Hendelman, 1997

EM, Bovine optic nerve a) parallel, b) transverse x 160,000 EM Waxman, et al., The Axon, Oxford UP, 1995

Axon, ~ 0.2 - 20 μm Microtubules, ~ 0.024 μm Neurofilaments, ~ 0.010 μm

White Matter and Tissue Structure

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• 3
• 2
• 1

4000 8000 12000 16000

Diffusion weighting – b / s mm-2 Log (Signal)

Water Tissue

Microstructure properties that affect diffusion:

• Cell size and density
• Cell orientation (anisotropy)
• Membrane permeability
• Intracellular viscosity
• Extracellular viscosity

Inglis, et al., Magn. Reson. Med. 2001; 45; 580-587 Chin, et a, Magn. Reson. Med. 2002; 47; 455-460

Tissue Microstructure and MR Measures of Water Diffusion

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Spin Echo Method for Diffusion Weighted MRI

90° 180° Acquire Rf Gradient

δ Δ G Stejskal-Tanner Equation:

Gradient, G has strength and direction

( )

( )

( )

( )

G 2 q where , e S 3 G b where , e S e S S

1 D 3 q 4 2 2 2 D b D 3 G

2 2 2 2 2

γδ π = = δ − Δ δ γ = = =

− δ − Δ π − − δ − Δ δ γ −

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Diffusion-weighting Gradient Strength and Orientation Dependence

x-gradient diffusion weighting z-gradient diffusion weighting

z y x

3 9 15 21 27 G/cm Increasing diffusion weighting

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Diffusivity in the direction defined by the unit vector , along which the gradient is applied, is given by,

Diffusion Tensor Imaging and Displacement Profiles

( )

u ˆ D b u ˆ D u ˆ b

e S e S S

T

− −

= =

( )

u ˆ D u ˆ u ˆ D

T

=

Diffusivity in each voxel can be described by an ellipsoidal displacement profile, such as the following, with major and minor axes (eigenvectors). Diffusion tensor imaging assumes a rank-2, symmetric, positive-definite tensor model for

• diffusivities. In this case, the Bloch-Torrey equation for magnetization can be written as,

u ˆ

+ + + + +

∇ ⋅ ⋅ ∇ + − ⋅ γ − ω − = ∂ ∂ M D T M M g r i M i t M

2

Using a spin-echo measurement method, the diffusion-dependent part of the measured signal, that results from solving this equation, can be written as; Basser et al. J. Magn. Reson. B 1994;103: 247-254

u ˆ g g =

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Anisotropic Diffusion Tensor

• Diffusion is highly anisotropic in fibrous structures.
• MR is sensitive to the molecular diffusion in the direction of the gradient applied.

( ) ( )

bD trace S ln S ln − =

D, Cartesian tensor for rank 2

Rank-2 Diffusion tensor image

• f an excised rat brain at 17.6T

(off-diagonal x 10)

• Each voxel is described by a cartesian tensor of

rank 2 (32 = 9 elements), but only 6 are unique (real, symmetric matrix), i.e. Diffusion has antipodal symmetry.

• Allows measurement of anisotropy
• Allows determination of fiber directions.

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Scalar Measures of Diffusion (orientation independent)

Fractional anisotropy, FA;

( )

( )

3 2 1

3 1 D Tr 3 1 D λ + λ + λ = =

( ) ( ) ( )

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

D D D 2 3 FA λ + λ + λ − λ + − λ + − λ =

Mean diffusivity;

Basser, NMR Biomed. 1995;8:333-344

FA: ~0.5 ~1.0

3 2 1

, , λ λ λ

Eigenvalues of ;

D

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Injured excised fixed SD rat spinal cord SE DWI HARDI acquisition, multiple slice TR=3000ms, TE=27.7ms Δ=17.8ms, δ=2.4ms b = 0, NA 24 b = 1250 s/mm2 in 21 directions, NA 8 FOV 4.8 x 4.8 x 12 mm3 (0.2 mm slices) Matrix 96 x 96 x 60 Resolution 50 x 50 x 200 micron3 Total time, ~ 12 hours

So FA <D> Tr(D)color EVcolor

MR Microscopy of Injured Rat Spinal Cord at 600 MHz (14.1 T, 5.2 cm)

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( )

s t ds ) s ( r d r r =

Using the principle eigenvector, ε1, at all locations in the image to suggest the direction of tracts, a fiber trajectory, , along an arc length, s, may be calculated by solving a Frenet equation,

Fiber Tract Mapping Algorithm

( )

)) s ( r ( s t

1

r r r ε =

Basser, et al., MRM 2000;44:625-632

( )

• r

r r r =

where the tangent vector, , is assumed to the equal to the principle eigenvector along the path,

( )

s t r

Therefore the Frenet equation can be solved with the initial condition,

) s ( r r

The trajectory is terminated when the principle eigenvector can no longer be assumed to represents the tract direction (low anisotropy).

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a) Specific regions of interest are defined within the three dimensional MR image. b) Starting from these user supplied initial conditions (ROI’s), fiber tracing is initiated in both directions (antipodal symmetry) along the direction defined by the principle eigenvector. c) Then the tract is continued until the anisotropy falls below a pre-specified threshold value (e.g. fractional anisotropy) since it is assumed that fibers do not exist below this level of anisotropy.

Fiber Tract Mapping Implementation

a) b) c)

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Fiber Tracking Results

MR Microscopy at 750 MHz (17.6 T, 89 cm) Normal (L) and injured (R) excised fixed rat spinal cord MR Microscopy at 750 MHz (17.6 T, 89 cm) Normal fixed rat brain

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Rank-2 DT-MRI assumes that there is single fiber orientation within the voxel. What happens when there is directional heterogeneity?

• Fiber direction is uncertain
• Anisotropy is reduced

Possible Improvement : High Angular Resolution Diffusion Imaging (HARDI) and modeling with a higher rank Cartesian tensor (>rank 2). Then diffusion measurement can be performed with gradients along many directions making it possible to directly measure distribution of diffusivities. (Tuch et.al., Proc. ISMRM, 1999.) Idealize Voxel

A Problem with rank-2 Diffusion-Tensor MRI

Typically voxel size, 100 x 100 x 100 micron3, which in white matter might contain ~ 25 to 125,000,000 axons

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Extention to Generalized Diffusion Tensor Imaging

Derive a new expression for signal attenuation: Rewrite Bloch-Torrey equation in terms of a rank-l Cartesian tensor:

+ = = = + + + +

∇ + − ⋅ γ − ω − = ∂ ∂

∑ ∑ ∑

M u ... u u D ... T M M g r i M i t M

2 i i 3 1 i i i ... i i 3 1 i 3 1 i 2

l 2 l 1 l 2 1 1 2 l i 2 i 3 1 l i 1 i l i ... 2 i 1 i 3 1 1 i 3 1 2 i

u ... u u D ... b 0 e

S S

∑ ∑ ∑ −

= = =

=

Ozarslan and Mareci, Magn. Reson. Med 2003;50:955-965 Ozarslan, Vemuri and Mareci, Magn. Reson. Med. 2005;53:866-876

Reformulate DTI by incorporating Cartesian tensors of higher rank

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Diffusion Displacement Probabilities

Therefore, the water displacement probability function is given by the Fourier integral, l 2 l 1 l 2 1 1 2

i i 3 1 i i i ... i i 3 1 i 3 1 i

u ... u u D ... ) u ˆ ( D

∑ ∑ ∑

= = =

=

( ) ( ) ( )

[ ]

u ˆ D 3 q 4 exp S q S

2 2

δ − Δ π − = r

The generalized diffusion tensor defines the rate of diffusion along each direction. Assuming mono-exponential attenuation, the normalized signal can be written as,

( )

( )

( )

R q 2 exp S q S q d , R P r r r r r ⋅ π − = Δ

∫

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Generalized Diffusion Tensor Imaging

Simulation results Simulation results

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Effect of Noise on Calculated Voxel Structure

(a) Simulated system of two crossing fiber bundles. (b) Probability surfaces calculated using the expansion of the probability on the surface of a sphere. (c-f) Surfaces in the framed area of panel b recalculated under increasing levels of noise added to the signal values. These panels represent images with signal-to-noise ratios (SNRs) between 50:1 and 12.5:1. Ozarslan, et al., NeuroImage 2006, in press

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Generalized Diffusion Tensor Imaging

• f Excised Rat Spinal Cord at 14.1 Tesla (600 MHz)

Multiple-slice spin echo acquisition 46 gradient directions, 1 non-weighted image Matrix 72 x 72 x 40, resolution 60 x 60 x 300 μm3 Total acquisition time, 9 hrs, 47 mins

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Structures in Optic Chiasm of the Excised Rat Brain

Ozarslan, et al., NeuroImage 2006, in press

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Calculated Voxel Structures in Excised Rat Brain

Ozarslan, et al., NeuroImage 2006, in press

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Challenges and Limitations

Challenges

• Fiber mapping and probabilistic mapping

– Modeling – Optimize data acquisition with modeling – Smoothing – Segmentation – Calculate most probable paths

• Relate structure to function

– Fiber structure to pathology – Fiber structure to neuronal processing

Limitation

• MR image measurement

– Signal strength – Time for measurement – Motion Mark Griswold, Univ. Wurzburg in collaboration with Siemens Medical

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Acknowledgements

Research Group Sara Berens Neuroscience Min Sig Hwang Biomed Engineering Tom Mareci Biochemistry Evren Ozarslan (now at NIH) Hector Sepulveda Biomed Engineering Nelly Volland Biomed Engineering Collaborators Doug Anderson Neuroscience Steve Blackband Neuroscience Paul Carney Pediatrics Tim Shepherd Neuroscience Baba Vemuri

• Comp. Info. Sci.& Eng.

Bob Yezierski Orthodontics/Neuroscience AMRIS Facility Staff Barbara Beck Kelly Jenkins Jim Rocca Dan Plant Xeve Silver Raquel Torres Grant Support NIH: R01 NS042075 R01 NS004752 P41 RR16105

• Dept. of Defense