Fund BioImag : Echo formation and spatial encoding 1.What makes the magnetic resonance signal spatially dependent ? 2.How is the position of an MR signal identified ? Slice selection 3.What is echo formation and how is it achieved ? Echo formation Gradient echo sequence 4.How is a two-dimensional MR image encoded ? After this course you 1.Understand the principle of slice selection 2.Are familiar with dephasing and rephasing of transverse magnetization and how it leads to echo formation 3.Understand the principle of spatial encoding in MRI 4.Can describe the basic imaging sequence and the three necessary elements 5.Understand the principle of image formation in MRI and how it impacts spatial resolution

Fund BioImag What do we know about magnetic resonance so far ? Adding a 3 rd magnetic field So far 1) Excite spins using RF field at  L 2) Record time signal (Known as FID) 3) M xy decays, M z grows (T 2 and T 1 relaxation) RF coils measure signal from entire body (no spatial information) How to encode spatial position ? L=B0L=B0 B 0 : Static Magnetic Field Creates equilibrium magnetization 0.1 T to 12 T » Earth’s field is T B 1 : Radiofrequency Field (RF) 0.05mT, on resonance Detection of MR signal (RF coils) Precessional Frequency Static Magnetic Field  L =f(x) Magnetic field B along z varies spatially with x, y, and/or z: GxGx GyGy GzGz e.g. G=(G x,0,0)

Fund BioImag How is the gradient field created ? One coil for each spatial dimension: G x, G y, G z Created by a set of 3 additional coils (gradient coil) G: Gradient Field mT/m in ~100µs Used to determine spatial position of signal (frequency) NB. Why are MRI scans so loud ? Lorentz-force of B z (3T) on rapidly switched current in gradient coil (wire) (~100A in ~100µs) Example: z-gradient coil principle (Helmholtz pair) One gradient coil All three gradient coils B0B0 GzGz

Fund BioImag On-resonance: Frequency  RF of RF field B 1 matches the precession frequency of magnetization How is slice-selection achieved ? Only magnetization on-resonance is excited Frequency Position z  RF =  G z z  RF Moving Frequency  RF alters position of slice :  RF =  B 0 +  G z z 0 y z x z0z0 NB. Not to confuse: (x,y) refers to spatial dimensions M xy M or M  refers to transverse magnetization (in magnetization space) (coordinate systems are different, but share z) NB. Not to confuse: (x,y) refers to spatial dimensions M xy M or M  refers to transverse magnetization (in magnetization space) (coordinate systems are different, but share z)

Fund BioImag For 2D object: =Radon Transform What is the basic principle of encoding spatial information ? frequency encoding - 1D example  B z (x) Spatial-varying resonance frequency  B(x) during detection Detected signal = sum of all precessing magnetization: x B z (x) = B 0 + G x x Rotating frame: B(x) = G x x What does this resemble ? = Inverse Fourier Transformation ! FT of S(t) (or S(k)) → M(x) Reconstruction as in CT (in principle)

Fund BioImag Spatial Encoding: 1D example w/o encoding w/ encoding Constant Magnetic Field Gradient: Varying Magnetic Field Fourier transformation (Frequency Decomposition) Paul Lauterbur Peter Mansfield Physiology and Medicine 2003 Two point-like objects x

Fund BioImag Dephasing Rephasing TE y x When is the signal maximal in the presence of G ? Echo formation: Dephasing and rephasing Phase of magnetization  (t)=G y yt Gradient G y (t) Magnetization in-phase → maximal signal (echo formation) S(t)=maximal (constant G y ) when t=  : echo formation  =TE/2: S(t)<S(0) → “dephasing” Magnetization in-phase initially  Echo formation: equal area t

Fund BioImag o turn t = TE/2 180 o turn t = TE/2 TE/2 t=0 t=TE t=0 t=TE Fast magnetization (Hare) Slow magnetization (tortoise) TE/2 Echo formation Vectors tips (or the hare and tortoise) running on a circle TE=“echo time” R v=  R

Fund BioImag Is it important when a gradient is applied ? gradient applied at different time has the same effect on magnetization phase Gradient G x (t) Gradient G z (t) TE Gradient applied simultaneously Gradient applied sequentially Question: Is there a difference in effect on echo ?  Application of two orthogonal gradients simultaneously or sequentially generates the same phase for M xy 

Fund BioImag What are the basic elements of the Gradient echo sequence ? Slice Select (G z ) Freq. Encode (G x ) RF Signal °° TE Data sampling Sampled Signal Slice  read (frequency encode) gradient NB. Why echo formation ? Gradient switching → Finite rise time  Dephasing of magnetization (signal decays like FID in presence of gradient )  Rephasing (negative) gradient leads to echo formation M(0)=M z First half of echo also measured (more signal) =max (at t=  )

Fund BioImag How is the 2 nd dimension encoded ? gradient echo imaging sequence Slice Select (G z ) Freq. Encode (G x ) RF Signal °° TE Data sampling Sampled Signal Repeated every TR seconds With G y incremented by  G y Slice  read gradient  phase encode gradient Phase Encode (G y ) GyGy 1. Excitation Slice selection 2. Phase encoding 3. Frequency encoding Echo formation

Fund BioImag Phase encode step 1 Step 2 (twice the gradient strength) Gradient G y (t) Gradient G x (t) Phase of voxel magnetization e i  : y x imaginary real How does the phase encoding gradient encode the 2 nd spatial dimension ?  Consider two-dimensional object voxel magnetization M  After applying phase encode gradient (G y for  seconds)

Fund BioImag Step 3 Step 2 Step 1 Phase  of a single pixel in x,y plane: Readout (G x t) G x (t) G y (t) How is incrementing the phase step-by-step (phase encoding) equivalent to frequency encoding ? time Signal of the single voxel: kx≡Gxtkx≡Gxt ky≡Gyky≡Gy Signal of the entire object : What does this resemble ? MR image generation: FT of the signal MR image generation: FT of the signal NB. Step n: G y =n  G v tt

Fund BioImag Fourier or reciprocal space (k x, k y ) G x (t) Phase Encode Acq. Sampled Signal Phase Direction (y) Frequency Direction (x) One line of k-space acquired per TR MR scans are long and motion-sensitive How is the spatial information encoded in MRI ? scanning k-space (Fourier or reciprocal space) sequentially For k-space line every TR=1s: image matrix > 4 min Subject moved head during acquisition Ghosting and ringing artifacts Maximum k x (or k y )  Resolution (Nyquist) Increment  k  Field-of-view center of k-space (k x,k y =0) Uniform resolution and sensitivity (Limited by voxel magnetization) Uniform resolution and sensitivity (Limited by voxel magnetization) ~ ms

Fund BioImag x 8512 x x 1632 x x x x 256 What are some effects of incomplete sampling ? of Fourier space (k-space) Time of acquisition of center of k- space point (k x,k y =0) determines contrast of image: = Discrete FFT (periodicity + time shift) k x,k y x,y

Fund BioImag Summary: Spatial encoding with gradients Phase encoding, echo formation + 2DFT time GyGy Dephasing at point (x,y) in space: m(  )=m(0)e -i  with  = G y y   k y y  GxGx Signal S( ,t)  M  (t) = ∫ ∫ m(x,y,t)dxdy = ∫ ∫ m(x,y,0) e  (n  Gyy)  e  Gxx(  +t) dxdy S(k x,k y )  ∫ ∫ m  (x,y) e -k x x e -k y y dxdy m(t)=m(  )e -i   with  = G x xt  k x x Echo formation:  (TE )=0 TE RF (B 1 ) 1234 Magnetization at time points specified: 1: (0,0,M z ) rotated by RF pulse by  0 about x: 2: (0,M z sin ,M z cos  )  (0,M y,…) [now only consider M xy ] Precesses with B = -  G y y -  G x x 3: M y (sin[-  (n  G y y+G x x)  ], cos[-  (n  G y y+G x x)  ]  M y [sin(-  x ), cos(  x )], with  x =  G x x  (rotation by angle –  x ) inverting gradient, i.e. B = +  G x x: after another , rotates by angle +  x  maximal signal at TE=2  (  Gy=0) 4: M y (0,1,…)= (0,M z sin ,M z cos  ) → Echo formation cos  Btsin  Bt 0 -sin  Btcos  Bt *Precession matrix (for B||z) MRI measures the 2D Fourier transformation of the object (measuring the 2 nd dimension requires time!)  (r,t)=  (B 0 + G (t)·r) nGynGy flip angle 