Lecture k-space. k-space illustrations. Zeugmatography 3/7/2011. Use of gradients to make an image echo. K-space Intro to k-space sampling

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1 Lecture K-space Intro to k-space sampling (chap 3) Frequenc encoding and Discrete sampling (chap 2) Point Spread Function K-space properties K-space sampling principles (chap 3) Basic Contrast mechanism (chap 4) K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms k-space k t k = γg(τ ) dτ = k z Limiting discussion to a slice (2D plane), magnetization distribution is given b the 2-dimensional Fourier transform of the spin distribution across the slice M T (t) = ρ(r) ep( jk r)dr slice ρ(r) is obtained from the inverse Fourier transform of M T (t)under the influence of a known gradient configuration ρ(,) = 1 k M T (k, )ep( j( k + ) )dk d k-space = visualization of the distribution of spatial frequencies in the image. k-space = Fourier transform of the MR image k-space illustrations FT {FT} -1 {FT} -1 FT ρ(, ) = 1 M T ( k, k ) ep( j ( k + k ) k M T (t) k t k = γ = G( τ ) dτ k k z dk dk FYS-KJEM FYS-KJEM Use of gradients to make an image echo Zeugmatograph G G Combination of G and G to rotate the total gradient orientation reconstruction b back projection Relationship between a three-dimensional object, its two-dimensional projection along the Y-ais, and four one-dimensional projections at 45 intervals in the XZ-plane. The arrows indicate the gradient directions. Lauterbur PC. Image formation b induced local interactions: eamples of emploing nuclear magnetic resonance. Nature 1973; 242: FYS-KJEM FYS-KJEM

2 What describes waves (signal) K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms Difference of phases between the 2 sums Frequenc information Sum of waves with different frequencies Same frequencies as above FYS-KJEM FYS-KJEM SE magnetization evolution The phase angle of a spin in a slice at a tis given b: t ω (,, t) dt = γgnt + γgt G pulsed field gradient along Definition of k: Gradient on t k i = γ G i ( t) dt (in the direction i) surface t The total transverse magnetisation is a function of k, and the position in the slice: M T (k, ) Image reconstruction: ρ(,) = 1 k k M T (k, )ep( j( k + ) )dk d 2D Fourier Transform FYS-KJEM FYS-KJEM Spin Echo: freq. encoding t= t=/2 t= Phase encoding t= t=/2 t= echo (frequenc encoding) G (frequenc encoding) G dephasing samplingof the signal which contains frequenc information (-ais) FYS-KJEM Spatial information in direction FYS-KJEM G 2

3 Frequenc and Phase encoding signal acquisition profile Phase encoding: needs multiple echo acquisitions FYS-KJEM FYS-KJEM Digital signal sampling Discrete sampling K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms FYS-KJM FYS-KJEM Discrete sampling PSF () = +T read / 2 T read / 2 A.e ( iγg..t ) dt T read MR-signal (M T ) U(t) Sampling interval: U(t) = 1 if t [-T read /2, T read /2] and elsewhere Signal sampling is modulated b a Block Function U(t) T read = N.t s In frequenc domain, this translates to: + PSF() = FFT( U(t) ) = U(t)e ( iγg..t ) dt PSF() = A iγg e( iγg..t ) T [ ] read / 2 Tread / 2 PSF() = A sin(γg..t read / 2) γg sin γgt read 2 PSF() = T read γg T read 2 periodic function (See eq. 2-24) FYS-KJEM FYS-KJM

4 PSF() P Field of View (FOV) ρ(,) = 1 M T (k, )ep( j( k + ) )dk d ρ(,) = 1 M T(k )ep( j( k ) )dk k k δfunction (point object) image representation of the point object We can calculate the Full Width at Half Height (FWHH) π = γ G T read k, ma = γg T read / 2 T read = N.t s λ ma = = = FOV k, min γg t s G = = f s γ FOV t s γ FOV (smallest wavelength) wavelength G is automaticall calculated b the scanner FOV is entered b the user (see Eq. 2-28) λ = / k FYS-KJM FYS-KJM FOV K-space properties Same definition can be done for the direction FOV = =, min γ( G, n G, n 1 )T πphase difference at the edge of the FOV Consider a square matri: N 2 = N.N Resolution (): δ = γg N t Field of view (): λ,ma,min s = = = FoV k γg t s Maimum frequenc in read-out () direction ± ω = ±γg FoV / 2 ma Min sampling rate (): 1 s / t γg FoV / G,n G,n 1 = G πn G, ma = (See Eq. 2-33) γt FOV We need to have G N 2 = G, ma Field of view (): πn λ,ma = FoV = γg T _ ma Sampling rate (): N = γg _ ma T FoV FYS-KJM FYS-KJEM K-space vs image space Back-folding / direction S(r) F(k) Increase 1/t s, N. ρ(,) FT FoV {FT} -1 k Object FoV N k t s k _ma FoV -ω ma = -γg FoV /2 ω ma = γg FoV /2 1 ρ(, ) = FYS-KJEM 474 M T ( k, k ) ep( j( k + k ) ) k dk dk N k k t k = γ = G( τ) dτ k k z 23 FYS-KJEM 474 Image FoV discarded discarded 24 4

5 Back-folding / direction FFT 1D: Truncation Artefact Object FoV Image FoV Fold-Over artefact FYS-KJEM FYS-KJEM FFT 2D: Truncation artefact Truncation artifact Ringing- (or truncation) artifacts in regions with high spatial frequencies (edges) in a phantom. The artifacts are more evident in the right image due to a lower matri (N=112, vs N=256 in the left image). FYS-KJEM FYS-KJEM Fold-Over artefact K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms FYS-KJEM FYS-KJEM

6 Frequenc and Phase encoding Phase encoding: needs multiple echo acquisitions Pulse sequence introduction Field gradients Spatial coding frequenc encoding (gradient on during signal acquisition) Phase encoding (gradient on before signal acquisition, repeated at different amplitudes) FYS-KJEM FYS-KJM Spatiall encoded Echoes Spin-Echo: SE Use of gradient in a spin-echo eperiment to induce spatiall dependent dephasing and rephasing Gradient Echo: GRE Gradient induced echo Spin Echo: freq. encoding (frequenc encoding) t= t=/2 t= 9 18 dephasing echo G FYS-KJM samplingof the signal which contains frequenc information (-ais) FYS-KJEM Gradient Echo 9 Gradient Echo M T 2 * relaation (FID) G t M z z z echo z z k = γ T G _ rew dt + T +T read / 2 T G _ rew dt = G T _ rewt + G read _ r 2 = FYS-KJEM FYS-KJEM

7 Gradient Echo (GRE) Travelling in k-space 1 sample RF G k G FYS-KJM FYS-KJM Gradient Echo 9 Gradient Echo 9 G k G k FYS-KJEM FYS-KJEM Gradient Echo Gradient Echo 9 9 G k G k repeated N s t s FYS-KJEM FYS-KJEM

8 Spin-Echo k-space travelling t= t=/2 t= 9 18 echo Repeated Phase encoding slice selection 18º rf pulse read-out phase encode t=/2 Acquisition of a profile (frequenc encoding) G t= k Spatial information in direction G FYS-KJEM FYS-KJEM Repeated Phase encoding Repeated Phase encoding (frequenc encoding) G (frequenc encoding) G Spatial information in direction G Spatial information in direction G FYS-KJEM FYS-KJEM Repeated Phase encoding Repeated Phase encoding (frequenc encoding) G (frequenc encoding) G Spatial information in direction G Spatial information in direction G FYS-KJEM FYS-KJEM

9 Repeated Phase encoding Acquisition of profile Repeated Phase encoding acquisition of all profiles (frequenc encoding) G (frequenc encoding) G Spatial information in direction G Spatial information in direction G FYS-KJEM FYS-KJEM Repeated acquisition of profiles k profiles +128 the field of view: FOV phase K FOV depends on the - gradient strengths -sampling of a profile frequenc -127 FYS-KJEM FYS-KJEM Signal intensit distribution in the selected slice phase K k frequenc encoding 2D FT frequenc -127 m(, ) = 1 k M T ( k, )ep[ i( k + ) ]dk d ω FYS-KJEM frequenc ω FYS-KJEM

10 Image generation IMAGE ω frequenc ω FYS-KJEM K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms FYS-KJEM and TR Basic Contrast 1 9 o Longitudinal and Transverse Relaation: TR FYS-KJEM FYS-KJEM Longitudinal and Transverse Relaation: at the same moment 9 o magnetisation 1 short T1 9 o 9 o signal FYS-KJEM FYS-KJEM

11 short T1 long T1 1 9 o 9 o short T2 1 9 o 9 o FYS-KJEM FYS-KJEM long T1 repetition 9 o 9 o 9 o TR 9 o 1 long T2 1 FYS-KJEM FYS-KJEM echo 9 o 9 o o 9 o Short FYS-KJEM FYS-KJEM

12 Longer 2 different tissues with different T1/T2 1 9 o 9 o 1 9 o 9 o FYS-KJEM FYS-KJEM T1 contrast brain eample T1 contrast 1 9 o White matter 1 9 o 9 o WM Gre matter GM FYS-KJEM FYS-KJEM T1 contrast T1 contrast 1 9 o 9 o WM 1 9 o 9 o WM GM GM FYS-KJEM FYS-KJEM

13 T1 contrast T1-weighted images TR 4 ms 1 9 o 9 o WM GM 1 ms 3 ms 4 ms FYS-KJEM FYS-KJEM Fat bone marrow bright i.v.contrast T1 weighted images: white matter TR short(se) < 6 ms (canbeas lowas 1.5ms) gre matter muscle bod fluids gre short < 25 ms bone air black TR < 6 ms < 25 ms FYS-KJEM FYS-KJEM T1-w knee PD (ρ*, proton densit) & T2 contrast 9 o 9 o 1 * Denoted rho contrast in the compendium FYS-KJEM FYS-KJEM

14 PD & T2 contrast PD & T2 contrast 1 9 o 9 o 1 9 o 9 o FYS-KJEM FYS-KJEM PD contrast long TR, short T2 contrast: increases 1 9 o 1 9 o FYS-KJEM FYS-KJEM T2 contrast, long TR, long PD & T2 weighted images TR 25 ms 9 o 1 2 ms 6 ms 1 ms 14 ms FYS-KJEM FYS-KJEM

15 Spine / summar T2 weighted images: TR long long > 18 ms > 8 ms T1 weighted T2 weighted FYS-KJEM FYS-KJEM Table of relaation s From: Greg J. Stanisz, Magnetic Resonance in Medicine 54: (25) FYS-KJM