Applications of Spin Echo and Gradient Echo: Diffusion and Susceptibility Contrast

Transcription

1 Applications of Spin Echo and Gradient Echo: Diffusion and Susceptibility Contrast Chunlei Liu, PhD Department of Electrical Engineering & Computer Sciences and Helen Wills Neuroscience Institute University of California, Berkeley, CA R.F. TE TE 90º º Review of Spin Echo and Gradient Echo 90º 180º Spin Echo G z G y G x TR Readout Readout R.F. θ TE θ Grad Echo G z G y G x TR Readout Readout 1

2 Outline Spin echo: diffusion contrast and quantification Diffusion-weighted imaging (DWI) Diffusion-tensor imaging (DTI) Diffusion fiber tractography Gradient echo: magnetic susceptibility contrast and quantification T2* weighting and blood oxygen level dependent (BOLD) contrast Susceptibility weighted imaging (SWI) Quantitative susceptibility mapping (QSM) 2

3 It s all about phase!!!! Spin Echo 1.1 Diffusion-Weighted Imaging 3

4 Water in Brain and Muscle 75% in skeletal muscle 78% in brain Water molecules are at constant random movement; described by a diffusion coefficient D. x 2 2Dt R.F. G z G y 90º 180º Diffusion Encoding with Single-Shot EPI G G x Start End b = γ 2 G 2 δ 2 ( δ 3) m b = m 0 exp( bd) large diffusion coefficient small diffusion coefficient x 2 2Dt m(0) m(b) 4

5 Spin Echo Without Diffusion Encoding R.F. 90º x 180º x Spin Echo G z G y G x Equilibrium Static spin sees the same Field inhomogeneity: field inhomogeneity: Excitation Dephasing Refocusing Rephasing x z M 0 y M 0 x z m y x z m y M 0 y y y x x x No Diffusion: running at constant speed 90 o 180 o Spin Echo Spin Echo Without Diffusion Encoding spin1 spin2 spin3 5

6 No Diffusion: running at constant speed 90 o 180 o Spin Echo Spin Echo Without Diffusion Encoding spin1 spin2 spin3 Spin Echo With Diffusion Encoding R.F. 90º x 180º x Spin Echo G z G y G x Equilibrium Excitation Dephasing Refocusing Static spins: Rephasing Moving spins: Dephasing x z M 0 y M 0 x z m y x z m y M 0 y y y y x x x x 6

7 With Diffusion: running when drunk 90 o G 180 o G Spin Echo Spin Echo With Diffusion Encoding With Diffusion: running when drunk 90 o G 180 o G Spin Echo Spin Echo With Diffusion Encoding 7

8 Derive Diffusion Signal with Bloch Equation Fick s Second Law t C D 2 C C is spin density Each spin carries magnetic moment. Magnetization is proportional to spin density. M i M j M M t T T M M B k D M 2 1 This equation can be solved using standard methods for solving partial differential equations. For a spin echo sequence, the solution for transverse magnetization is given by b G ( ) 3 m( b) m (0)exp bd Derive Diffusion Signal with Statistics 90 o G 180 o G Spin Echo p() r 1 Gr () t dt left Probability Distribution Function of diffusion 1 e 4 Dt 2 r 2Dt 2 Gr () t dt right B 0 +G r j( 2 1) M M 0 e p( r) dr M 0exp( b D) b: b-value D: diffusion coefficient 8

9 Diffusion-Weighted Imaging b = 0 No diffusion weighting b = 1000 s/mm 2 Diffusion weighting D (mm 2 /s) Computed diffusion coefficient Need a minimal of 2 measurements at 2 different b-values to computed D. Diffusion coefficient is commonly referred to as apparent diffusion coefficient (ADC) ADC of free water at room temperature: 2.2x10-3 mm 2 /s ADC of brain tissue around 1.0x10-3 mm 2 /s Why Single-Shot EPI? G y G x K-Space ifft DWI measures molecular diffusion ~ 10 µm during imaging window; Bulk motion (body motion, breathing, cardiac, brain pulsation) ~ 1 mm, introducing more phase than diffusion; varies from TR to TR. 9

10 G y If Acquire One k-space Line per TR Single-Shot G x m r, b = m(r, 0)e bd e jφ 1(r,TR 1 ) m r, b = m(r, 0)e bd e jφ 2(r,TR 1 ) m r, b = m(r, 0)e bd e jφ 3(r,TR 1 ) How to address it? Inconsistent k-space data causing aliasing Spatial varying signal cancellation Spin Echo 1.2 Diffusion-Tensor Imaging and Tractography 10

11 R.F. 90º 180º Anisotropic Diffusion: Orientation Dependent G z G y G x Diffusion encoding gradients can be applied in either one of the three axis or a combination of axis. The gradients are represented by a vector (Gx Gy Gz) D = 0.55x10-3 mm 2 /s D = 1.5x10-3 mm 2 /s log(s) (a) right splenium corpus callosum (b) right splenium corpus callosum (1 1 0) (1-1 0) b(s/mm 2 ) Mathematical Models of Diffusion Isotropic Anisotropic Scalar D similar molecular displacements in all directions Dxx Dxy Dxz Dxy Dyy Dyz Dxz Dyz D zz greater molecular displacement along cylinders than across 11

12 Diffusion Tensor Signal Model Scalar Diffusion Tensor Diffusion b = γ 2 G 2 δ 2 ( δ 3) m b = m 0 exp( bd) b ij = γ 2 G i G j δ 2 ( δ 3) m b ij = m 0 exp( b ij D ij ) m ( b) m (0)exp bi i Di i Isotropic Diffusion Cerebral Spinal Fluid Instead of a diffusion coefficient, we have a diagonal diffusion tensor, a 3x3 matrix D xx D yy D zz Probability Density Function 12

13 Anisotropic Diffusion m ( b) m (0)exp bi i Di i Instead of a diffusion coefficient, we have a diffusion tensor, a 3x3 matrix D D D D D D D D D xx xy xz xy yy yz xz yz zz Probability Density Function Bloch Equation with Diffusion Term Fick s Second Law C D ijij C t C is spin density; Einstein summation rule. Each spin carries magnetic moment. Magnetization is proportional to spin density. M M B M i M j M M k M t T T Dij ij 2 1 This equation can be solved using standard methods for solving partial differential equations. For a spin echo sequence, the solution for transverse magnetization is given by b G G b b ij i j ( ), ij ji m( b) m(0)exp b D, D D symmetric positive definite ij ij ij ji 13

14 Probability Distribution Function of anisotropic diffusion 1 p() r (4 D t) 32 T 2r Dr t covariance matrix: Σ=2Dt e Statistical Interpretation for Anisotropic Diffusion For diffusion-weighted spin-echo sequence, echo amplitude is M M e p( r) dr M exp( b D ) j( 21) 0 0 b11 b12 b13 D11 D12 D13 b b b b, D D D D b31 b32 b 33 D31 D32 D 33 Tensor Product b : D b D b D ij ij ij ij i, j ij ij Example G G b G b b b : D b( D D D D ) , ( ) b( D 2 D D )

15 Determine Diffusion Tensor Experimentally D is a symmetric tensor. It has six unknowns. A minimal of six non-colinear measurements are required to determine a diffusion tensor. Different measurements are achieved by varying the diffusion encoding gradients including both amplitude and direction. m (0) s ln bijdij b D 2b D 2b D b D 2b D b D mb ( ) D 11 (1) (1) (1) (1) (1) (1) (1) s b11 2b12 2b13 b22 2b23 b D (2) (2) (2) (2) (2) (2) (2) s b11 2b12 2b13 b22 2b23 b D M M M M M M M D22 ( n) ( n) ( n) ( n) ( n) ( n) ( n) s b11 2b12 2b13 b22 2b D 23 b33 23 D 33 Rows have to be independent. (1 1 0) (1 0 1) (0 1 1) (1-1 0) (1 0-1) One Simple Encoding Scheme (0 1-1) x z y R.F. 90º 180º G z G y G x (1 1 0) 15

16 Eigen Decomposition D is coordinate system dependent. If the subject rotates in the magnet, the measured diffusion tensor will be different. Eigen decomposition defines rotation invariant quantities. D = UΛU T 1 2 3, U1, U2, 3 U = U U U U are eigenvectors Λ 0 2 0, 1, 2, 3 are eigenvalues mean diffusivity 3 Matlab function: eig(). U 2 U 3 U 1 1 Diffusion Ellipsoid Fractional Anisotropy (FA) Fractional Anisotropy (FA): a measure of diffusion anisotropy, 0<FA<1 FA 3(( ) ( ) ( ) ) ( ) y z x FA Color-coded FA: based on the orientation of the eigenvector corresponding to the largest eigenvalue 16

17 Fractional Anisotropy (FA) DTI Fiber Tractography x y z y z x Fiber Tractography: a representation of 3D white matter fiber structure. Summary Diffusion-weighted imaging is created by applying diffusion encoding gradients Tissue contrast is based difference in diffusion coefficient Diffusion-tensor imaging measures the orientation dependent diffusion coefficient Major eigenvector of a diffusion tensor is parallel to white matter fiber 17

18 Gradient Echo 2.1 T2*-Weighting and BOLD R.F. θ TE θ G z G y G x TR Readout Readout Magnitude and Phase of Gradient Echo abs(image) angle(image) Phase is due to offset in Larmor frequency. Different voxel has different frequency, consequently, accumulates different phase angle over time. This frequency offset is mainly due to field inhomogeneity caused by magnetic susceptibility variations. 18

19 What is Magnetic Susceptibility? Magnetic susceptibility is a physical quantity that measures the extent to which a material is magnetized by an applied magnetic field. M magnetization vector H Magnetic field vector χ volume magnetic susceptibility (unitless in SI units) B magnetic flux density vector, or magnetic induction μ 0 vacuum permeability applied H applied H M B M = χ H B = μ 0 (1+χ) H Paramagnetic vs. Diamagnetic H H Paramagnetic χ > 0 M B M = χ H B = μ 0 (1+χ) H H H Diamagnetic χ < 0 M B M = χ H B = μ 0 (1+χ) H 19

20 Magnetic Susceptibility in MRI B 0 = 0 H 0 magnetization m susceptibility H 0 m B B B B 0 B 0 is perturbed by local magnetization induced by susceptibility RF Susceptibility Induced Tissue Contrast Echo 1 Echo 2 Echo n 20

21 Susceptibility Induces Field Inhomogeneity θ TE Readout t Ideal Case Inhomogeneity Image Recon m k = m r e i2πk r dr t is k dependent m k = m r e iγδb(r)(te+t) e i2πk r dr m r = m k e i2πk r dk If t << TE m r = m(r) e iγδb(r)te phase Sampling m r n = m(r n ) e iγδb(r n)te k max sinc k max r n T 2 decay Voxel Effect of Field Inhomogeneity f B 2 R 2 2 FWHM 2 B 2 Full Width at Half Maximum FWHM (Hz) or B (Tesla) R2 Frequency 21

22 Magnitude: T2* Decay S() t S e tt 0 2 OR R2* Mapping and Contrast T2* 25 ms R2* 0 40 Hz White matter Blood vessels Globus pallidus Red nucleus Substatia nigra 22

23 Blood-Oxygen-Level-Dependent Signal Diamagnetic Hb Paramagnetic Hbr Ogawa S. et al, Phase Images of Gradient Echo 23

24 What Is in the Phase? Sources of phase Phase wraps Receiver coil Objects outside the FOV Objects inside the FOV background background tissue Phase Unwrapping Extensively researched; perfect solution still lacking Examples: 3D SRNCP or BP-ASL H. Abdul-Rahamn, et al, "Fast And Robust Three-Dimensional Best Path Phase Unwrapping Algorithm", Applied Optics, Vol. 46, No. 26, pp , 2007 FSL PRELUDE, University of Oxford 24

25 Laplacian Phase Unwrapping Totally automatic; Fast; Guarantee continuity Remove phase originated from sources outside FOV Li W. et al, NeuroImage 2011; 55: Filter Background: Sphere mean value property Harmonic function Mean phase over a sphere S Phase at the center 25

26 Summary of Phase Processing unwrapping Filtering 2.3 Susceptibility Weighted Imaging (SWI) 26

27 Susceptibility Weighted Imaging GRE Phase GRE Magn SWI = Magn*(PhaseMask) 4 SWI MIP SWI X Unwrapping Filtering Phase Mask Susceptibility Weighted Imaging Courtesy of Juergen Reichenbach 27

28 Pitfalls of SWI high sensitivity (micro-lesions) venous blood, iron, calcium low specificity (hypointense) qualitative, not quantitative Bo Phase is orientation dependent Bo Phase Is this bleeding? SWI 2.4 Quantitative Susceptibility Mapping (QSM) 28

29 Susceptibility Source Magnetic Field Change B m H 0 m B 0 Susceptibility Source Magnetic Field Change B m H 0 m B 0 29

30 Susceptibility Source Magnetic Field Change? B m B 0 Find the Demagnetizing Field h Magnetic flux density distribution in a first order approximation B (1 χ)( H h) H 0 B 0 0 H ( χ ) h 0 h? h = FT 1 k k z k 2χ k H 0 30

31 Magnetic Field Observed By a Spin Magnetic flux density seen by a spin B = μ 0 ( χ)(h 0 + h) H 0 Susceptibility inclusion B h Magnetic susceptibility Applied field vector Demagnetizing field vector Magnetic Field Observed By a Spin B z = μ χ H 0z + h z = μ χ H 0 FT 1 k z 2 k 2χ k H 0 μ 0 H χμ 0H 0 μ 0 FT 1 k z 2 k 2χ k H 0 δb z (r) = B z μ 0 H 0 = FT 1 ( 1 3 k z 2 k 2)χ k μ 0H 0 δb z (k) = ( 1 3 k z 2 k 2)χ k B 0 For simplicity, write δb z as δb B 0 = μ 0 H 0 31

32 Step 3: Solve A Deconvolution Problem Measurements Unknown Convolution Deconvolution k z Dividing by zero! k y k x 32

33 Quantitative Susceptibility Mapping Filter Background Phase Solve Inverse Problem Raw Phase Unwrapped Phase Tissue Phase Susceptibility (ppm) 3 T Tissue Phase ppm Susceptibility 33

34 Susceptibility Is Orientation Dependent B 0 Magnetic Susceptibility Is Anisotropic Susceptibility at Orientation # Magnetic susceptibility is orientation dependent, MRM 2010; 63:

35 Susceptibility Tensor Imaging H 0 k x k z k k y T 1 ( ) ˆ ( ) ˆ ˆ ˆ T k χ k H k 0 H χ( k) H H k Ht 2 3 k Susceptibility tensor is symmetric; 6 unknowns, MRM 2010; 63: Susceptibility Tensor Imaging = 35

36 Summary Magnetic susceptibility causes field perturbation Field perturbation results in frequency shift that can be measured by gradient echo phase images High-passed filtered phase is used to generate SWI images Background phase can be removed with sphere mean value filter The relation between field perturbation and susceptibility is a convolution QSM solves the deconvolution problem STI treats susceptibility as a tensor instead of a scalar 36