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DTI lecture 100710
- 1. Introduction to Diffusion Tensor Imaging Why DTI? Diffusion – what it is, how it affects MR signal Tensor – how we represent diffusion Imaging – how we measure it in MRI
- 2. Stroke/ischemia Alzheimer’s Disease Multiple Sclerosis Brain maturation studies Ischemia and stroke Neoplasm Preoperative planning Traumatic brain injury Congenital anomalies and diseases of white matter Encephalopathies Neurodegenerative diseases Spinal Cord Injury Epilepsy Dementia, schizophrenia, depression Developmental disorders Autism Aging Why diffusion? http://www.vh.org/Providers/Textbooks/ BrainAnatomy/Ch5Text/Section18.html http://eclipse.nichd.nih.gov/nichd/DTMRI/mri/ Conceptually: in vivo histology
- 3. Why diffusion? Diffusion is EXTREMELY SENSITIVE to differences and changes in tissue microstructure Myelination/Demyelination Axon damage/loss Inflammation/Edema Necrosis It is NOT a biomarker of white matter integrity It is NOT just about white matter Gray matter Cardiac tissue
- 4. Example DTI image “Fractional Anisotropy” map “map” is a computed parameter, unlike an “image” which is acquired signal Also called a “tractogram” since it clearly shows major white matter fiber tracts
- 5. What is Diffusion? stochastic movement of particles in a solvent, driven by the thermal molecular motion of the solvent… … and also applies to motion of the solvent itself (Einstein, 1905) time τ∆ NOTE: In the limit N→∞, use the Central Limit Theorem to assume “step size” ∆ is fixed and equal to the average of individual displacements ∆i.
- 6. 1D Fick’s Law - what the flux? t = t0 + τ x t = t0 0 2x − ∆ 0x − ∆ 0x + ∆0x 0 2x + ∆ t = t0 + τ x t = t0 0 2x − ∆ 0x − ∆ 0x + ∆0x 0 2x + ∆ What is the flux (J) through x0 after one time interval τ ? C1(x)C2(x) dx dC J τ 2 2 1 ∆ −= Adolf Fick, 1855: Flux is proportional to the particle concentration gradient (conservation of mass)
- 7. The Diffusion Coefficient 3D Fick’s Law Note the minus sign: flux goes from high to low concentration del operator replaces partial derivative factor of 6, not 2 (why?) D is the diffusion coefficient This is the expression for isotropic diffusion τ62 ∆=D CDJ ∇−= CJ ∇ ∆ −= τ6 2
- 8. Isotropic Diffusion (water) water ink Dtr 6= r1 t1 r2 t2 τ62 ∆=D
- 9. Diffusion in Tissue (Anisotropic) t ink r2 r3 r1 diffusion ellipsoid tDr 11 2= tDr 22 2= tDr 33 2= x y z laboratory frame DON’T try this at lab!!!!!
- 10. The Diffusion Tensor zzyzxz yzyyxy xzxyxx DDD DDD DDD x y z r2 r3 r1 3 2 1 00 00 00 D D D diagonalization Lab frame Intrinsic frame
- 11. Tensor Invariants Eigenvalues: diagonalization (iterative QR factorization) Eigenvectors xx xy xz xy yy yz xz yz zz D D D D D D D D D { }1 2 3D D D { }321 eee
- 12. Tensor Invariants Shape invariants: analytical calculation directly from tensor coeffs xx xy xz xy yy yz xz yz zz D D D D D D D D D ( )1 3 av xx yy zzD D D D= + + 1 2 2 2 2 1 3 xx yy xx zz yy zz surf xy xz yz D D D D D D D D D D + + = − − − 1 2 3 2 2 2 xx yy zz xx yz vol xy xz yz zz xy yy xz D D D D D D D D D D D D D − = + − −
- 13. Scalar Anisotropy Indices ( ) avND D DADC DDDADC = ++= 321 3 1 ( ) ( ) ( ) 2 2 2 3 2 2 2 1 2 3 2 2 2 1 1 2 3 mag surf ND D D D FA DDD DDDDDD FA −= ++ −+−+− =
- 14. FA vs. ADC FA (x 10 -4 ) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Probability 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 WM GM CSF MD (x 10 -3 mm2 /s) 0 500 1000 1500 2000 2500 3000 3500 4000 Probability 0.000 0.001 0.002 0.003 0.004 0.005 WM GM CSF FA and ADC are very useful clinically, but are very different. Tensor has a LOT of information! Q. Which metric would you use to detect brain cancer?
- 15. Vector anisotropy measures We can use eigenvector information from the tensor as well Represent direction of primary eigenvector as color on a scalar map Or render the primary eigenvectors as “fibers” for astonishing* 3D visualizations Red = R/L Green = A/P Blue = S/I *but how “real” is it? Many PhD theses have asked….
- 16. Diffusion tensor coefficients Diffusion tensor invariants Scalar anisotropy indices Vector anisotropy indices
- 17. Effect of Diffusion on MRI signal Signal attenuation! Diffusion term
- 18. Diffusion weighted MRI ∆ δ G G echoπα ( ) ( )( )2 0 δexpδ 3 M G D M γ= − × ∆ − × ∆ δ ( ) ( )2 δδ 3 b Gγ= × ∆ − (boxcar gradients) “b-value” Consider simplified diffusion experiment…
- 19. MR Measurement of Diffusion Tensor j T j j j p G G q r = × ur 0 xx xy xz j j j j xy yy yz j xz yz zz j D D D p p q r D D D q D D D r j b S S e − × × × = ( ) 22 2 γ δ Δ δ 3b G= − ( ) 1 6 j N N = ≥ Kjth diffusion- weighted image Diffusion magnitude Diffusion direction Gz Gy Gx ... ... ...
- 20. Solving for D 20 0 xx xy xz j j j j xy yy yz j xz yz zz j D D D p p q r D D D q D D D r j b S S e − × × × = 1. Acquire T2W image (b = 0 s/mm2 ) 3. Choose a diffusion gradient orientation2. Choose a b-value 4. Acquire image (Sj) 5. Repeat steps 1 – 3, j = 1 … N times 6. Solve for D…. How?
- 21. Let’s do some linear algebra… [ ] ⋅ ⋅⋅−= zj yj xj zzzyzx yzyyyx xzxyxx zjyjxjj DDD DDD DDD bSS α α α αααexp0 ( ) ( ) ( ) ( ) ( ) ( ) ⋅ ⋅= yz xz xy xx yy xx T jxy jxy jxy jzz jyy jxx j D D D D D D b S S α α α α α α 2 2 2 ln 2 2 2 0 1661 xNxNx xAY ⋅=
- 22. B-matrix formalism 22 ( )yzyzxzxzxyxyzzzzyyyyxxxx DDDDDDb αααααα 222222 +++++⋅ ( )yzyzxzxzxyxyzzzzyyyyxxxx DbDbDbDbDbDb 222 +++++= ∑∑= = = 3 1 3 1i j ijij Db The “b-matrix” The b-matrix formalism summarizes total attenuating effect of all gradient waveforms in all directions (including imaging gradients)
- 23. T2W (b = 0 s/mm2 ) Y, -ZY, Z-X, Y X, Y-X, Z+X, Z
- 24. 24 SVD DIAG T2W (b = 0 s/mm2 ) … DWI (j = 1, 2, 3 … N) Dij
- 25. N=27 N=55 N=13N=6 N NEX # DWI 6 8 56 13 4 56 27 2 56 55 1 56 #DWI = (N + 1) x NEX If TR = 4 sec, then acq time = 56*4sec = 3.7 minutes Tradeoff: N vs NEX
- 26. Rotational invariance Hasan et al, JMRI 2001 Jones MRM 2004
- 27. 27 Empirical Image Quality increasing N, decreasing NEX increasingb-value
- 28. How low can you go? High b-values mean more attenuation, lower SNR Lower b-values mean higher SNR, room for more N At very low b-values, imaging gradients’ diffusion effects are no longer negligible Lower b-values also do not probe same diffusion scale, less clinically interesting b=100 s/mm2 b=500 s/mm2 (N = 6, 8 NEX)
- 29. Echo-Planar Imaging (EPI) Advantages Minimal motion artifacts NEX N Disadvantages Eddy current artifacts T2* limits spatial resolution Geometric distortion (susceptibility) 29
- 30. DT-MRI Alexander SLFSLF CRCR CCCC CINGCING Partial Volume Effects on Anisotropy
- 31. DT-MRI Alexander Mapping Complex Diffusion Based Upon Q-Space Theory – Model Independent ODF – orientation density function (Tuch et al., Neuron 2003) Diffusion Spectrum Imaging (DSI) (Tuch et al. Neuron 2003, Wedeen et al. 2005) High Angular Diffusion Imaging (HARDI), Q-Ball (Frank 2002; Tuch et al. Neuron 2003)
Hinweis der Redaktion
- Before I go over the diffusion enhancements, let me give a bit of background on diffusion imaging and why it is desirable Heres a highly invasive technique for imaging the corpus callosum And heres diffusion fiber tractography of the structure
- The particles will move forward or backwards with equal probability, so half the particles undergo displacement -D and the other half undergo displacement +D . After a single time interval tau we only need consider particles within distance D of a given point x0 to calculate the flux half are displaced left and the other half are displaced right. Therefore, only half the particles between x0 and x0 D will cross x0 in one step.
- THIS IS ONLY A CARTOON! demonstrate concept injecting ink not part of methods and materials if we repeat thought experiment in brain tissue, diffusing molecules will encounter obstructions from tissue microstructure (unlike free water) isotropic: degenerate case uynder this formalism where D1 = D2 = D3 asymmetric diffusion called “anisotropic”, Diffusion is strongly anisotropic in WMFT NOTE: - ellipsoid true in limit of infinitesimal point - oblique diffusion has components along three axes
- we actually measure directly in DT-MRI : diffusion tensor coefficients, averaged over entire voxel. (volume element) Diagonalize -> reduce to intrinsic reference frame of ellipsoid (lose orientation)
- The shape invariants have the advantage that they can be constructed analytically from the tensor coefficients, unlike the eigenvalues which are usually obtained using an iterative computation.
- The shape invariants have the advantage that they can be constructed analytically from the tensor coefficients, unlike the eigenvalues which are usually obtained using an iterative computation.
- to visualize-> construct vector (2d) and scalar (1d) indices to characterize the anisotropy of tissue in that voxel focus on scalar indices in this research RA : VISUALIZATION gold standard, linear, 0-sqrt(2) FA : more sensitive to low aniso, high aniso saturates (0-1) ND/D definitions are equivalent - index VALUE is the same --- EXTRA INFO -- - vector indices sensitive to noise, therefore formidable acquisition requirements (time, hardware). - don’t need vectors : construct scalar indices from any set of tensor invariants
- MR signal is attenuated by diffusion effects: apply gradient For every diffusion-weighted gradient we apply, we obtain one such equation. 6 unknowns in the diffusion tensor (due to symmetry), need at least N = 6 diffusion weighted gradients. more than 6 gradients -> use least-squares method like SVD to find the best fit -- EXTRA info -- [ typically maximize Gd for shortest TE] static spins see no net dephasing - 1800 pulse reverses polarity of first gradient. Diffusing spins, undergo net dephasing - experience different local magnetic susceptibility as sensitize any MR pulse sequence by inserting diffusion-weighting magnetic field gradients
- example of data set in Experiment 1 - N = 6, b = 500 s/mm2 diffusion-weighted images are attenuated from T2 baseline signal varies with orientation of the gradient
- fiber tracts’ orientation not known a-priori equal weighting of anisotropy in all directions needed for max sensitivity -> even spacing of diffusion-weighting gradients (“acquisition scheme”) for this project. we developed acquisition schemes for N = 6, 13, 27, and 55 [ shown from same angle for consistency ] N = 6: standard isocahedral encoding [X, Z], [Y, Z], and [X, Y] N = 27, 55: Perl algorithm tiling unit hemisphere by equal solid angle disks uses circular tiles as an approximation, cannot cover the surface precisely algorithm performs poorly for N < 27 N = 13: modeled by electrostatic repulsion each gradient treated as point charge on surface of hemisphere iterative computation - solve for lowest energy configuration
- all images are RA, gold standard commonly used
- experiment 14 (N = 6) exp. 15 - similar results < 500 are worse, interference with imaging gradients