Spatial encoding in Magnetic Resonance Imaging Jean-Marie BONNY
What s Qu est an image ce qu une? image? «a reproduction of a material object by a camera or a related technique» Multi-dimensional signal I(r) r = vector of spatial coordinates (x,y,z) T Signal intensity of the image I is scalar = Image with a single frame I is a vector = Multi-channel image
The first Qu est MR image ce qu une image? Made by Paul LAUTERBUR, Nature (1973) Zeumatography comes from the Greek word zeugma, or yoke, to signify the fact that the technique links chemical and spatial information
There Qu est are 3 spatial ce qu une dimensions image? In real world, images should be in 3D Human brain V = 3 x 3 x 3 mm 3
1D or 2D Qu est images ce qu une image? The concept of «profile» ( x) = I( x, y, z)dydz I Mulkern et al, NMR in Biomed icine (1999) PMID: 10195330
Multi-channel Qu est ce images qu une? image? MRI provides different contrasts Clark et al, Journal of the Science of Food and Agriculture (1998) DOI: 10.1002/(SICI)1097-0010(199811)78:3<349::AID-JSFA125>3.0.CO;2-X
The MRI Qu est challenges ce qu une image? More sensitivity Improvement of spatial and temporal resolutions More specificity Tissue discrimination through the signature given by the signal I
How to Qu est encode ce spatial qu une information image?? The target Amplitude M T of transverse magnetization over space M T ( r) r : Spatial Cartesian coordinates r = x y z Assumptions No relaxation Uncoupled spins
Signal Qu est without ce encoding qu une image? Free induction decay RF / ACQ Flip angle t
Signal Qu est without ce encoding qu une image? FID S( r, t) MT r, ( r) exp[ iφ( t )] S Complex signal because of quadrature detection φ φ because of reception gain reception field, amplifiers ( ) t ( r, t) = 2π [ f ( r, t' ) f ] 0 0 dt' Phase accumulated during t period On-resonance signal coming from the whole sample S = S( r, t) dr M dr T ( r)
Quadrature Qu est detection ce qu une image? Components of M T in the rotating frame Phase/off-resonance information Re ( S ) Im( S ) Hoult, Concepts in Magnetic Resonance (2000)
What is Qu est a gradient ce qu une? image? Mathematical operator Vector of partial derivatives with respect to each coordinate For a 3D function fun ( r) = fun( x, y, z) gradfun ( r) = fun( r) = fun fun x y fun z ( r) ( r) ( ) r
Qu est ce qu une image? fun = magnetic field (MF) B Linearly varying magnetic fields G x,y,z in each direction G is superposed to the static field (B 0 ) ( ) 0 B = B r ( ) Gr r.... 0 0 + = + + + = B z G y G x G B B z y x = z y x G G G G
Some Qu est examples ce qu une of linearly image varying? MF B ( r) = B + G. x + G. y G z 0 x y + z. G = 3 G = 0 z = 0 x y 1 0.8 3 2 1 0-1 -2-3 1 0.5 y 0.6 0.4 0.2 0-0.2-0.4-0.6 1 0 0.5-0.8 0-0.5-0.5-1 y -1-1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 x x
Some Qu est examples ce qu une of linearly image varying? MF B ( r) = B + G. x + G. y G z G = 3 G = 1 z = 0 x 0 x y + z. y 1 0.8 4 2 0-2 y 0.6 0.4 0.2 0-0.2-4 1 0.5-0.4-0.6 1 0 0.5-0.8 0-0.5-0.5-1 y -1-1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 x x
Qu est ce qu une image? Gradient of the magnetic field ( ) ( ) ( ) ( ) r = G = + + + + + + + + + = z y x z y x z y x z y x G G G z z G y G x G B y z G y G x G B x z G y G x G B B......... 0 0 0 G corresponds to the gradient of the MF The gradient coils generate G
Gradient Qu est coils ce / Principles qu une image? Hidalgo-Tobon, Concepts in Magnetic Resonance (2010)
Gradient Qu est coils ce / How qu une it looks image like??
Effect Qu est of currents ce qu une image? The Biot-Savart law (1820) MF generated by a current that circulates through wires db µ dl r I 3 4 r 0 = π I = current intensity B I
Time varying Qu est ce fields qu une image? Currents = independant functions of time I ( t) = I I I x y z ( t) ( t) ( ) t G ( t) = G G G x y z ( t) ( t) ( ) t x y z 0 t
Effects Qu est of MF ce gradients qu une image on NMR? signal Time AND spatially dependent frequency B ( r ) = B + 0 G. r ( r, t) = B0 G( t)r. B + Larmor equation f ( r, t) = B0 + G( t)r. Phase of the transverse magnetization φ γ 2π t ( r t) = 2π [ f ( r, t' ) f ] dt' = γ G( t' ), 0. rdt' 0 γ 2π t 0
The k variable Qu est ce qu une image? Gradient antiderivatives φ t ( r, t) γ G( t' ). rdt' = r. γ G( t' ) dt' = r.k( t) = 0 t 0 k ( t) = γ G( t' ) t 0 dt' = k k k x y z ( t) ( t) ( t) = t γ G 0 t γ G 0 t γ G 0 x y z ( t' ) ( t' ) dt' dt' ( t' ) dt' k looks like a «wave vector»
Effects Qu est of MF ce gradients qu une image on NMR? signal Signal coming from the whole sample S = T, dr ( t) S( r, t) dr M ( r) exp[ iφ( r t) ] Using the «k» formulation S T. S ( t) M ( r) exp[ ir k( t) ] dr ( ) ( ) [ ] 1 k M r exp ir. k dr = FT ( M ( r ) T T
The k-space Qu est ce qu une image? Rearranged acquired NMR signal Time-domain NMR signal is represented as function of k Reparameterization of acquired signal S ( k) FT 1 ( M ( r) ) T [ ( )] 1 k FT FT ( ( r) ) [ M ] T M ( r) FT S = T S ( k) FT M T ( r) k-space FT -1 Image
S ( k) M T ( r) k-space Image k y FT FT -1 y k x x
S ( k) M T ( r) k-space Image k y FT FT -1 y k x x Remember S is a complex number! z is orthogonal to the slide!
How riding Qu est about ce qu une the k-space image? (1) Change the gradients with time G ( t) = G G G x y z ( t) ( t) ( ) t (2) Calculate the wave-vector = integration ( t) = G( t' ) t k γ dt' 3D trajectory 0
How filling Qu est the ce k-space qu une? image? (1) Change the gradients = displacement along the 3D trajectory (2) Acquire the signal = fill the k-space with the (complex) value of NMR signal
How filling Qu est the ce k-space qu une? image? Many ways Multi-steps / Cartesian TR n.tr and many others!
Pure «phase» encoding
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
Pure phase Qu est encoded ce qu une imaging image? Cartesian trajectory over the k-space Gradients before the acquisition «Phase» encoding gradient pulses Encode the 3 directions Whole FID available All the chemical shift information is available! Acquisition time TA = Nx Ny Nz TR = 64.64.64. 1s = 73h
Spin Warp
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
90 RF ACQ k y G x k x G y G z
Spin warp Qu est ce qu une image? Cartesian trajectory over the k-space Gradients during the acquisition «Read» gradient pulses Encode one direction «Phase» encoding in the 2 other directions Acquisition time TA = Nx Ny Nz TR = 64.64. 1s = 1h8
Radial encoding
90 RF ACQ k y G x G y tanφ=g y /G x k x G z
90 RF ACQ k y G x k x G y G z
Radial Qu est encoding ce qu une image? Radial trajectory over the k-space Non-uniform density Gradients during the acquisition Only read-gradient pulses No phase encoding Acquisition time TA = Nφ Nθ TR = 64.64. 1s = 1h8
Echo planar
90 RF ACQ k y G x k x G y G z
Echo planar Qu est ce qu une image? (Quasi) Cartesian trajectory over the k-space Alternated read gradients during the acquisition Short phase encoding in-between the read gradients Blip Acquisition time TA = TR = 1s
Design of any trajectories t k( t) = γ G( t' ) dt' G ( t' ) 0 ( t) k = γ t Lee, Neuroimage (2010)
«Exotic» trajectories Khrapitchev, J Magn Reson (2006)
(Some) Limits
Spatial Qu est resolution ce qu une image? Signal-to-noise ratio V = voxel volume Voxel size SNR V = Δx. Δy. Δz Δx kmax Δx k 1 max High resolution = high = Strong gradients k max
Sampling Qu est ce qu une image? -100-80 -60-40 -20 0 20 40 60 80 100-100 -80-60 -40-20 0 20 40 60 80 100 Object 2 4 6 8 10 12 14 Spin warp encoding FOV = 128 mm 16 x 16 16 2 4 6 8 10 12 14 16 Spin warp encoding FOV = 128 mm 64 x 64
Aliasing Qu est ce qu une image? Field of view (FOV) should contain the object Solutions exist for reducing the object size Saturation band Selective pulse Surface coil
Off-resonance Qu est ce / Gradient qu une image imperfections? Off-resonance Frequency shift Ω( r) Trajectory perturbation Differences between experimental and theoretical gradient waveforms S T π * ( k( t ) M ( r) exp i r. k ( t) + 2 Ω( r) [ ( t) ] dr Trajectory perturbation All the off-resonance perturbations contribute
Off-resonance Qu est ce qu une image? -100-80 -60-40 -20 0 20 40 60 80 100-100 -80-60 -40-20 0 20 40 60 80 100 Object 2 4 6 8 10 12 14 Spin warp encoding FOV = 128 mm 16 x 16 No perturbations -50-40 -30-20 16 2 4 6 8 10 12 14 16-10 0 10 20 30 40 2 50-50 -40-30 -20-10 0 10 20 30 40 50 4 Off-resonance 6 8 10 12 With perturbations 14 16 2 4 6 8 10 12 14 16
Inhomogeneities Qu est ce qu une of B1 fields image? Quadrature surface coil 7 T Transmission Reception Resulting image Homogeneous phantom Wang et al, Magn Reson Med (2002) 48:362-369
Inhomogeneities Qu est ce qu une of B1 fields image? 1.5 Tesla 8 Tesla Novak et al., Magn. Reson. Imaging (2005)
Selective Qu est excitation ce qu une image? For reducing the number of directions to encode Band-limited pulse Pulse applied WITH a gradient pulse M T Off-resonance (khz)