UNIVERSITY OF FLORIDA McKnight Brain Institute 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 1
UNIVERSITY OF FLORIDA McKnight Brain Institute 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 2
UNIVERSITY OF FLORIDA McKnight Brain Institute Image Acquisition = γ k G t 3
UNIVERSITY OF FLORIDA McKnight Brain Institute Magnetic Resonance Imaging of 1 H in Human Head (THM) at 3 Tesla (128 MHz) Gray Matter (cortex) White Matter (sub-cortex) Cerebral-Spinal Fluid Proton density and T 1 weighted T 2 weighted Spin echo image acquisition Spin echo image acquisition TR 3700 ms, TE 15 ms TR 3700 ms, TE 90 ms 4
UNIVERSITY OF FLORIDA White Matter and Tissue Structure McKnight Brain Institute Axon, ~ 0.2 - 20 μ m Microtubules, ~ 0.024 μ m Neurofilaments, ~ 0.010 μ m x 160,000 EM Waxman, et al., The Axon , Oxford UP, 1995 a S , glial cell EM, Bovine optic nerve a T , axonal cell a) parallel, b) transverse G. Stanisz, A. Szafer, G. Wright, R. M. Hendelman, 1997 5
UNIVERSITY OF FLORIDA Tissue Microstructure and MR Measures of Water Diffusion McKnight Brain Institute Inglis, et al., Magn. Reson. Med. 2001; 45; 580-587 Chin, et a, Magn. Reson. Med. 2002; 47; 455-460 Microstructure properties that affect diffusion: • Cell size and density • Cell orientation (anisotropy) Diffusion weighting – b / s mm -2 • Membrane permeability 0 • 0 4000 8000 12000 16000 Intracellular viscosity • Extracellular viscosity -1 Log (Signal) Tissue -2 Water -3 6
UNIVERSITY OF FLORIDA Spin Echo Method for Diffusion Weighted MRI McKnight Brain Institute 90° 180° Acquire Rf Δ Gradient, G has strength and direction G Gradient δ Stejskal-Tanner Equation: ( ) 2 2 2 − γ δ Δ − δ G 3 D = S S e 0 ( ) − b D 2 2 2 = = γ δ Δ − δ S e , where b G 3 0 ( ) 2 2 ( ) − π Δ − δ − 4 q 3 D 1 = = π γδ S e , where q 2 G 0 7
UNIVERSITY OF FLORIDA Diffusion-weighting Gradient McKnight Brain Institute Strength and Orientation Dependence x-gradient diffusion weighting y z-gradient diffusion weighting x z 3 9 15 21 27 G/cm Increasing diffusion weighting 8
UNIVERSITY OF Diffusion Tensor Imaging and Displacement Profiles FLORIDA McKnight Brain Institute Basser et al. J. Magn. Reson. B 1994;103: 247-254 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, ∂ M + = − ω − γ ⋅ − + ∇ ⋅ ⋅ ∇ i M i r g M M T D M + + + + 0 2 ∂ t Using a spin-echo measurement method, the diffusion-dependent part of the measured signal, that results from solving this equation, can be written as; ( ) T − − ˆ ˆ ˆ b u D u b D u = = S S e S e 0 0 g = ˆ ˆ g u u Diffusivity in the direction defined by the unit vector , along which the gradient is applied, is given by, ( ) T = ˆ ˆ ˆ D u u D u Diffusivity in each voxel can be described by an ellipsoidal displacement profile, such as the following, with major and minor axes (eigenvectors). 9
UNIVERSITY OF FLORIDA Anisotropic Diffusion Tensor McKnight Brain Institute • Diffusion is highly anisotropic in fibrous structures. • MR is sensitive to the molecular diffusion in the direction of the gradient applied. ( ) ( ) = − ln S ln S 0 trace bD D , Cartesian tensor for rank 2 • Each voxel is described by a cartesian tensor of rank 2 (3 2 = 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. Rank-2 Diffusion tensor image of an excised rat brain at 17.6T (off-diagonal x 10) 10
UNIVERSITY OF FLORIDA Scalar Measures of Diffusion (orientation independent) McKnight Brain Institute Basser, NMR Biomed. 1995;8:333-344 λ λ λ , , D Eigenvalues of ; 1 2 3 ( ) 1 1 ( ) = = λ + λ + λ D Tr D Mean diffusivity; 1 2 3 3 3 Fractional anisotropy, FA ; ( ) ( ) ( ) 2 2 2 λ − + λ − + λ − D D D 3 1 2 3 = FA 2 2 2 2 λ + λ + λ 1 2 3 FA: 0 ~0.5 ~1.0 11
UNIVERSITY OF FLORIDA MR Microscopy of Injured Rat Spinal Cord McKnight Brain Institute at 600 MHz (14.1 T, 5.2 cm) SE DWI HARDI acquisition, multiple slice TR=3000ms, TE=27.7ms S o Δ =17.8ms, δ =2.4ms b = 0, NA 24 b = 1250 s/mm 2 in 21 directions, NA 8 FOV 4.8 x 4.8 x 12 mm 3 (0.2 mm slices) FA <D> Matrix 96 x 96 x 60 Resolution 50 x 50 x 200 micron 3 Total time, ~ 12 hours Tr(D) color EV color Injured excised fixed SD rat spinal cord 12
UNIVERSITY OF FLORIDA Fiber Tract Mapping Algorithm McKnight Brain Institute Basser, et al., MRM 2000;44:625-632 Using the principle eigenvector, ε 1 , at all locations in the image to suggest the direction of tracts, a r fiber trajectory, , along an arc length, s , may be calculated by solving a Frenet equation, r ( s ) r r d r ( s ) ( ) = t s ds r ( ) t s where the tangent vector, , is assumed to the equal to the principle eigenvector along the path, r r r ( ) = ε t s ( r ( s )) 1 Therefore the Frenet equation can be solved with the initial condition, r r ( ) = r 0 r o The trajectory is terminated when the principle eigenvector can no longer be assumed to represents the tract direction (low anisotropy). 13
UNIVERSITY OF FLORIDA Fiber Tract Mapping Implementation McKnight Brain Institute 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. a) b) c) 14
UNIVERSITY OF FLORIDA Fiber Tracking Results McKnight Brain Institute MR Microscopy at 750 MHz (17.6 T, 89 cm) Normal fixed rat brain MR Microscopy at 750 MHz (17.6 T, 89 cm) Normal (L) and injured (R) excised fixed rat spinal cord 15
UNIVERSITY OF A Problem with rank-2 Diffusion-Tensor MRI FLORIDA McKnight Brain Institute Rank-2 DT-MRI assumes that there is single fiber orientation within the voxel. Typically voxel size, 100 x 100 x 100 micron 3 , which in white matter might contain ~ 25 to 125,000,000 axons What happens when there is directional heterogeneity? • Fiber direction is uncertain • Anisotropy is reduced Idealize Voxel 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.) 16
UNIVERSITY OF FLORIDA Extention to Generalized Diffusion Tensor Imaging McKnight Brain Institute 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 Rewrite Bloch-Torrey equation in terms of a rank- l Cartesian tensor: ∂ M + = − ω − γ ⋅ − i M i r g M M T + + + 0 2 ∂ t 3 3 3 2 ∑ ∑ ∑ + ∇ ... D u u ... u M + i i ... i i i i 1 2 l 1 2 l = = = i 1 i 1 i 1 1 2 l Derive a new expression for signal attenuation: 3 3 3 − b ∑ ∑ ... ∑ D u u ... u i i ... i i i i 1 2 l 1 2 l = = = i 1 i 1 i 1 = S S 0 e 1 2 l 17
UNIVERSITY OF FLORIDA Diffusion Displacement Probabilities McKnight Brain Institute The generalized diffusion tensor defines the rate of diffusion along each direction. 3 3 3 ∑ ∑ ∑ = ˆ D ( u ) ... D u u ... u i i ... i i i i 1 2 l 1 2 l = = = i 1 i 1 i 1 1 2 l Assuming mono-exponential attenuation, the normalized signal can be written as, [ ] r ( ) S q ( ) ( ) 2 2 = − π Δ − δ ˆ exp 4 q 3 D u S 0 Therefore, the water displacement probability function is given by the Fourier integral, r ( ) ( ) ( ) r r r S q r Δ = ∫ − π ⋅ P R , d q exp 2 q R S 0 18
UNIVERSITY OF FLORIDA McKnight Brain Institute Generalized Diffusion Tensor Imaging Simulation results Simulation results 19
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