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11: Echo formation and spatial encoding

11: Echo formation and spatial encoding. What makes the magnetic resonance signal spatially dependent ? How is the position of an MR signal identified ? Slice selection What is echo formation and how is it achieved ? Echo formation Gradient echo sequence

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11: Echo formation and spatial encoding

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  1. 11: Echo formation and spatial encoding • What makes the magnetic resonance signal spatially dependent ? • How is the position of an MR signal identified ? Slice selection • What is echo formation and how is it achieved ? Echo formation Gradient echo sequence • How is a two-dimensional MR image encoded ? • After this course you • Understand the principle of slice selection • Are familiar with dephasing and rephasing of transverse magnetization and how it leads to echo formation • Understand the principle of spatial encoding in MRI • Can describe the basic imaging sequence and the three necessary elements • Understand the principle of image formation in MRI and how it impacts spatial resolution

  2. 11-1. What do we know about magnetic resonance so far ?Adding a 3rd magnetic field RF coils measure signal from entire body (no spatial information) • B0: Static Magnetic Field • Creates equilibrium magnetization • 0.1 T to 12 T • Earth’s field is 0.5 10-4 T • B1: Radiofrequency Field (RF) • 0.05mT, on resonance • Detection of MR signal (RF coils) • So far • 1) Excite spins using RF field at L • 2) Record time signal (Known as FID) • 3) Mxy decays, Mz grows (T2 and T1 relaxation) Static Magnetic Field Precessional Frequency L=B0 How to encode spatial position ? e.g. G=(Gx,0,0) wL=f(x) Magnetic field B along z varies spatially with x, y, and/or z: Gx Gy Gz

  3. How is the gradient field created ?One coil for each spatial dimension: Gx, Gy, Gz • G: Gradient Field • 10-50 mT/m in ~100µs • Used to determine spatial position of signal (frequency) Example: z-gradient coil principle (Helmholtz pair) B0 Gz Created by a set of 3 additional coils (gradient coil) One gradient coil All three gradient coils NB. Why are MRI scans so loud ? Lorentz-force of Bz (3T) on rapidly switched current in gradient coil (wire) (~100A in ~100µs)

  4. How is slice-selection achieved ?Only magnetization on-resonance is excited On-resonance: Frequency wRF of RF field B1 matches the precession frequency of magnetization Moving Frequency wRF alters position of slice : Frequency y wRF wRF=gGzz Position z • NB. Not to confuse: • (x,y) refers to spatial dimensions • Mxy M or M refers to transverse magnetization (in magnetization space) • (coordinate systems are different, but share z) x z z0 wRF=gB0+gGzz0

  5. 11-2. What is the basic principle of encoding spatial information ? frequency encoding - 1D example Detected signal = sum of all precessing magnetization: Spatial-varying resonance frequency gB(x) during detection Bz(x)= B0 + Gxx gBz(x) What does this resemble ? = Inverse Fourier Transformation ! x For 2D object: =Radon Transform FT of S(t) (or S(k)) → M(x) Rotating frame: B(x) = Gxx Reconstruction as in CT (in principle)

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

  7. 11-3. When is the signal maximal in the presence of G ?Echo formation: Dephasing and rephasing x S(t)<S(0) → “dephasing” y TE Magnetization in-phase → maximal signal (echo formation) Echo formation: equal area Gradient Gy(t) Magnetization in-phase initially Phase of magnetization f(t)=Gyyt t Dephasing Rephasing t t=TE/2: S(t)=maximal (constant Gy) when t=t: echo formation

  8. Echo formationVectors tips (or the hare and tortoise) running on a circle Fast magnetization (Hare) TE/2 t=0 TE=“echo time” 180o turn t = TE/2 TE/2 t=TE Slow magnetization (tortoise) TE/2 t=0 v=wR 180o turn t = TE/2 R TE/2 t=TE

  9. Is it important when a gradient is applied ? gradient applied at different time has the same effect on magnetization phase Gradient applied sequentially Gradient applied simultaneously TE TE Gradient Gz(t) Gradient Gx(t) t 2t t 2t Question: Is there a difference in effect on echo ? Application of two orthogonal gradients simultaneously or sequentially generates the same phase for Mxy

  10. What are the basic elements of the Gradient echo sequence ? a° • 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)=Mz RF Slice Select (Gz) Slice  read (frequency encode) gradient TE Freq. Encode (Gx) =max (at t=t) Signal Data sampling Sampled Signal First half of echo also measured (more signal)

  11. a° TE 11-4. How is the 2nd dimension encoded ? gradient echo imaging sequence RF Slice Select (Gz) Slice  read gradient  phase encode gradient Freq. Encode (Gx) DGy Phase Encode (Gy) Signal Data sampling Repeated every TR seconds With Gy incremented by DGy Sampled Signal 3. Frequency encoding Echo formation 2. Phase encoding 1. Excitation Slice selection

  12. y imaginary x real How does the phase encoding gradient encode the 2nd spatial dimension ? Consider two-dimensional object Phase encode step 1 voxel magnetization M Step 2 (twice the gradient strength) After applying phase encode gradient (Gy for t seconds) Gradient Gy(t) Gradient Gx(t) t Phase of voxel magnetization eif:

  13. How is incrementing the phase step-by-step (phase encoding)equivalent to frequency encoding ? Phase f of a single pixel in x,y plane: Step 3 Signal of the single voxel: kx≡gGxt ky≡gGyt Signal of the entire object : Step 2 Dt Step 1 What does this resemble ? Readout (Gxt) time Gy(t) NB. Step n: Gy=nDGv MR image generation: FT of the signal Gx(t)

  14. Ghosting and ringing artifacts How is the spatial information encoded in MRI ?scanning k-space (Fourier or reciprocal space) sequentially Fourier or reciprocal space (kx, ky) Phase Encode Phase Direction (y) Gx(t) Sampled Signal Maximum kx (or ky)  Resolution (Nyquist) Increment Dk  Field-of-view ~ ms Acq. For k-space line every TR=1s: 2562 image matrix > 4 min Uniform resolution and sensitivity (Limited by voxel magnetization) Frequency Direction (x) MR scans are long and motion-sensitive One line of k-space acquired per TR Subject moved head during acquisition center of k-space (kx,ky=0)

  15. What are some effects of incomplete sampling ?of Fourier space (k-space) 16 x 16 32 x 32 512 x 512 8 x 8 x,y Time of acquisition of center of k-space point (kx,ky=0) determines contrast of image: kx,ky Discrete FFT (periodicity + time shift) = 64 x 64 128 x 128 256 x 256

  16. Summary: Spatial encoding with gradientsPhase encoding, echo formation + 2DFT Dephasing at point (x,y) in space: m(t)=m(0)e-if with f = Gyyt kyy (r,t)=g(B0 + G(t)·r) • Magnetization at time points specified: • 1: (0,0,Mz) • rotated by RF pulse by a0 about x: • 2: (0,Mzsina,Mzcosa)(0,My,…) • [now only consider Mxy] • Precesses with B = -gGyy-gGxx • 3: My(sin[-g(nDGyy+Gxx)t], cos[-g(nDGyy+Gxx)t] • My [sin(-fx), cos(fx)], with fx =gGxxt(rotation by angle –fx) • inverting gradient, i.e. B = +gGxx: • after another t, rotates by angle +fx • maximal signal at TE=2t (DGy=0) • 4: My (0,1,…)= (0,Mzsina,Mzcosa) → Echo formation m(t)=m(t)e-iFwith F = Gxxt  kxx Echo formation:F(TE)=0 flip angle a RF (B1) 1 2 3 4 nDGy Gy time t TE Gx *Precession matrix (for B||z) Signal S(t,t)  M(t) =∫ ∫ m(x,y,t)dxdy = ∫ ∫m(x,y,0) eg(nDGyy)tegGxx(t+t)dxdy S(kx,ky) ∫ ∫m(x,y) e-kxxe-kyydxdy MRI measures the 2DFourier transformation of the object (measuring the 2nd dimension requires time!)

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