K-space. Spin-Warp Pulse Sequence. At each point in time, the received signal is the Fourier transform of the object s(t) = M( k x

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1 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2015 MRI Lecture 4 k (t) = γ 2π k y (t) = γ 2π K-space At each point in time, the received signal is the Fourier transform of the object s(t) = M( k (t),k y (t)) = F[ m(, y) ] k (t ),k y (t ) evaluated at the spatial frequencies: t 0 t 0 G (τ)dτ G y (τ)dτ Thus, the gradients control our position in k-space. The design of an MRI pulse sequence requires us to efficiently cover enough of k-space to form our image. RF Spin-Warp Pulse Sequence G (t) Readout (or Frequency) Gradient G y (t) Phase Encode Gradient G y (t) k y Phase Encodes 1

2 Sampling and Windowing k y k Y < 1/(FOV Y ) k X < 1/(FOV X ) W ky Δk y k W k W ky >1/δ >1/δ y Δk W k Goal : FOV = FOV y = 25.6 cm δ = δ y = 0.1 cm Readout Gradient : 1 FOV = γ 2π G rδt Pick Δt = 32 µsec Eample 1 1 G r = = γ FOV 2π Δt ( 25.6cm) T 1 s 1 = T/cm = G/cm ( )( s) G r t 1 ADC Δt Eample Consider the k-space trajectory shown below. ADC samples are acquired at the points shown with Δt = 10 µsec. The desired FOV (both and y) is 10 cm and the desired resolution (both and y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Label the waveform with the gradient amplitudes required to achieve the desired FOV and resolution. Also, make sure to label the time ais correctly. Assume γ 2π ( ) = 4000Hz / G 1 Gauss = Tesla PollEv.com/be280a 2

3 Gibbs Artifact Apodization rect(k ) Hanning Window h(k )=1/2(1+cos(2πk ) image image sinc() * = * = 0.5sinc()+0.25sinc(-1) +0.25sinc(+1) Images from Images from readout Aliasing and Bandwidth Faster FOV f Slower B=γG r FOV Lowpass filter in the readout direction to prevent aliasing. GE Medical Systems

4 G (t) G r FOV rect FOV sinc( FOV k ) FOV FOV sinc FOV γ 2π G t r t 1 t γg r FOV 2π f γg r FOV 2 2π f = γg r FOV 2 2π f * = FOV sinc FOV γ 2π G rt 2π 2π rect γg r γg r FOV f γg r FOV 2π f RF Signal Aliasing and Bandwidth cosω 0 t sinω 0 t FOV LPF LPF ADC ADC I Q 2FOV/3 Temporal filtering in the readout direction limits the readout FOV. So there should never be aliasing in the readout direction. f RF Ecitation * t t From Buton

5 Free Induction Decay (FID) Relaation An ecitation pulse rotates the magnetization vector away from its equilibrium state (purely longitudinal). The resulting vector has both longitudinal M z and tranverse M y components. Due to thermal interactions, the magnetization will return to its equilibrium state with characteristic time constants. T 1 spin-lattice time constant, return to equilibrium of M z T 2 spin-spin time constant, return to equilibrium of M y Longitudinal Relaation T1 Values dm z dt = M z M 0 T 1 Gray Matter muscle White matter After a 90 degree pulse M z (t) = M 0 (1 e t /T 1 ) Due to echange of energy between nuclei and the lattice (thermal vibrations). Process continues until thermal equilibrium as determined by Boltzmann statistics is obtained. kidney liver fat The energy ΔE required for transitions between down to up spins, increases with field strength, so that T 1 increases with B. Image, caption: Nishimura, Fig

6 Transverse Relaation T2 Relaation dm y dt = M y T 2 y y y z z z Each spin s local field is affected by the z-component of the field due to other spins. Thus, the Larmor frequency of each spin will be slightly different. This leads to a dephasing of the transverse magnetization, which is characterized by an eponential decay. T 2 is largely independent of field. T 2 is short for low frequency fluctuations, such as those associated with slowly tumbling macromolecules. After a 90 degree ecitation M y (t) = M 0 e t /T 2 T2 Relaation T2 Values Runners Net signal Tissue T 2 (ms) gray matter 100 white matter 92 muscle 47 fat 85 kidney 58 liver 43 CSF 4000 Table: adapted from Nishimura, Table 4.2 Solids ehibit very short T 2 relaation times because there are many low frequency interactions between the immobile spins. On the other hand, liquids show relatively long T 2 values, because the spins are highly mobile and net fields average out. Credit: Larry Frank 6

7 Eample Bloch Equation dm dt = M γb M i + M j y ( M M z 0)k T 2 T 1 T 1 -weighted! Density-weighted! T 2 -weighted! Precession Transverse Relaation Longitudinal Relaation Questions: How can one achieve T2 weighting? What are the relative T2 s of the various tissues? i, j, k are unit vectors in the,y,z directions. Precession " dm dt% " 0 B 0 0% " $ ' $ ' $ $ dm y dt' = γ $ B ' $ # $ dm z dt& ' # $ 0 0 0& '# $ Useful to define M M + jm y dm dt = d dt( M + im y ) = jγb 0 M M M y M z % ' ' &' M jm y " dm $ $ dm y # $ dm z Matri Form with B=B 0 dt% " 1/T 2 γb 0 0 %" ' $ ' $ dt' = $ γb 0 1/T 2 0 ' $ dt& ' # $ 0 0 1/T 1 &' # $ M M y M z % " 0 % ' $ ' ' + $ 0 ' &' # $ M 0 /T 1 &' Solution is a time-varying phasor M(t) = M(0)e jγb 0t = M(0)e jω 0t Question: which way does this rotate with time? 7

8 Z-component solution M z (t) = M 0 + (M z (0) M 0 )e t /T 1 Saturation Recovery If M z (0) = 0 then M z (t) = M 0 (1 e t /T 1 ) M M + jm y Transverse Component ( ) dm dt = d dt M + im y = j( ω 0 +1/T 2 )M Inversion Recovery M(t) = M(0)e jω 0t e t /T 2 If M z (0) = M 0 then M z (t) = M 0 (1 2e t /T 1 ) Summary 1) Longitudinal component recovers eponentially. 2) Transverse component precesses and decays eponentially. Summary 1) Longitudinal component recovers eponentially. 2) Transverse component precesses and decays eponentially. Source: Fact: Can show that T 2 < T 1 in order for M(t) M 0 Physically, the mechanisms that give rise to T 1 relaation also contribute to transverse T 2 relaation. 8

9 Static Inhomogeneities In the ideal situation, the static magnetic field is totally uniform and the reconstructed object is determined solely by the applied gradient fields. In reality, the magnet is not perfect and will not be totally uniform. Part of this can be addressed by additional coils called shim coils, and the process of making the field more uniform is called shimming. In the old days this was done manually, but modern magnets can do this automatically. Field Inhomogeneities In addition to magnet imperfections, most biological samples are inhomogeneous and this will lead to inhomogeneity in the field. This is because, each tissue has different magnetic properties and will distort the field. Signal Decay 0 TE Some inhomogeneity, Some dephasing time More inhomogeneity, More dephasing, Decrease in MR signal s r (t) = V Static Inhomogeneities The spatial nonuniformity in the field can be modeled by adding an additional term to our signal equation. M( r,t) dv y z = M(, y,z,0)e t /T 2 ( r ) e jω 0t e jω E ( )t ep jγ r ( o ) t G (τ) r dτ ddydz The effect of this nonuniformity is to cause the spins to dephase with time and thus for the signal to decrease more rapidly. To first order this can be modeled as an additional decay term of the form s r (t) = y z M(, y,z,0)e t /T 2 ( r ) e t / T 2 #( r ) e jω 0 t ep jγ ( o ) t G (τ) r dτ ddydz 9

10 The overall decay has the form. where T 2 * decay ep t /T 2 * r ( ( )) 1 T 2 * = 1 T " T 2 Due to random motions of spins. Not reversible. Due to static inhomogeneities. Reversible with a spin-echo sequence. T 2 * decay Gradient echo sequences ehibit T 2 * decay. RF G z (t) G (t) G y (t) ADC Slice select gradient Slice refocusing gradient TE = echo time Gradient echo has ep(-te/t 2 *) weighting Spin Echo Spin Echo Discovered by Erwin Hahn in τ 180º τ The spin-echo can refocus the dephasing of spins due to static inhomogeneities. However, there will still be T 2 dephasing due to random motion of spins. There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Hahn Image: Larry Frank Source: 10

11 Spin Echo Spin Echo Pulse Sequence τ 180º τ RF G z (t) τ τ Phase at time τ τ ϕ(τ) = Phase after 180 pulse 0 ω E ( r )dt = ϕ(τ + ) = ω E ( r )τ ω E ( r )τ Phase at time 2τ ϕ(2τ) = ω E ( r )τ + ω E ( r )τ = 0 Image: Larry Frank G (t) G y (t) ADC ep( t TE / T 2 #) ep( t /T 2 ) TE = echo time Image Contrast Different tissues ehibit different relaation rates, T 1, T 2, and T 2*. In addition different tissues can have different densities of protons. By adjusting the pulse sequence, we can create contrast between the tissues. The most basic way of creating contrast is adjusting the two sequence parameters: TE (echo time) and TR (repetition time). Spin-echo Image Gradient-Echo Image 11

12 Saturation Recovery Sequence 90 TE 90 TE 90 TR Gradient Echo TR [ ]e TE /T 2 * (,y ) I(, y) = ρ(, y) 1 e TR /T 1 (,y) T1-Weighted Scans Make TE very short compared to either T 2 or T 2 *. The resultant image has both proton and T 1 weighting. I(, y) ρ(, y) 1 e TR /T 1 (,y) [ ] Spin Echo TE I(, y) = ρ(, y) 1 e TR /T 1 (,y) [ ]e TE /T 2(,y ) TR T1-Weighted Scans T2-Weighted Scans Make TR very long compared to T 1 and use a spin-echo pulse sequence. The resultant image has both proton and T 2 weighting. I(, y) ρ(, y)e TE /T 2 12

13 T2-Weighted Scans Proton Density Weighted Scans Make TR very long compared to T 1 and use a very short TE. The resultant image is proton density weighted. I(, y) ρ(, y) Eample T 1 -weighted! Density-weighted! T 2 -weighted! Tissue Proton Density T1 (ms) T2 (ms) Csf Gray White PollEv.com/be280a a) Which has the longest T1? b) Which has the shortest T1? c) Which has the longest T2? d) Which has the shortest T2? e) Which might be pure water? f) Which has the most firm jello? Hanson

14 FLASH sequence θ TE θ TE θ 3 4 PollEv.com/be280a a) Which is the most T1 weighted? b) Which is the most T2 weighted? c) Which is the most PD weighted? Hanson 2009 TR TR Gradient Echo I(,y) = ρ(,y) 1 e TR /T1(,y) [ ]sinθ 1 e TR/T 1 (,y) cosθ [ ] ep( TE /T 2 Signal intensity is maimized at the Ernst Angle θ E = cos 1 ( ep( TR /T 1 )) FLASH equation assumes no coherence from shot to shot. In practice this is achieved with RF spoiling. ) FLASH sequence 180 Inversion Recovery TI TE TR I(, y) = ρ(, y) 1 2e TI /T 1 (,y ) + e TR /T 1 (,y) [ ]e TE /T 2 (,y ) Intensity is zero when inversion time is # TI = T 1 ln 1+ ep( TR /T 1) & $ % 2 ' ( θ E = cos 1 ( ep( TR /T 1 )) 14

15 Inversion Recovery Inversion Recovery Biglands et al. Journal of Cardiovascular Magnetic Resonance :66 doi: / x