M. Lustig, EECS UC Berkeley. Principles of MRI EE225E / BIO265

Transcription

1 Principles of MRI EE225E / BIO265

2 RF Excitation (Chap. 6) Energy is deposited into the system RF pulses used for: Excitation Contrast manipulation Refocussing (...more later) Saturation Tagging Transfer of magnetization

3 RF Excitation (Chap. 6) History: 80 s - 90 s Lots of Research Mid 90 s most problems figured out Mid 2000 new burst of research with multiple xmitters

4 Excitation Excitation is short ~2-3ms. Can neglect relaxation Simplified Block eq Only defines rotations!

5 Excitation

6 Rotating Frame In rotating frame at ω0, B+ω0/γ = 0 B e =[B 1x,B 1y, ~ G ~r] T Bloch equation for excitation: 2 4 Ṁ x Ṁ y Ṁ z 3 5 = G ~ ~r B1y ~G ~r 0 B 1x B 1y B 1x M x M y M z 3 5

7 RF Excitation Several Special cases: ~G ~r 6= 0 Gradient is on: excitation is spatially selective (next week) ~G ~r =0 Gradient is off: non-selective excitation (Today)

8 Non Selective Excitation 2 4 Ṁ x Ṁ y Ṁ z 3 5 = B 1y 0 0 B 1x B 1y B 1x M x M y M z 3 5 B e =[B 1x,B 1y, 0] T Magnetization precesses around Beff

9 RF Excitation Lab Frame Rotating Frame

10 RF Excitation

11 Time Varying B1 For time varying B1: Z t = B 1x ( )d Rotations about a common axis add! (not true in general) 0

12 Useful Rotations

13 Useful Rotations

14 Useful Rotations

15 What do these pulses do? = 2 = = =

16 Example: Non selective RF Typical Numbers: B1 = 0.1G ω1 = ɣb1 = 2π rad/sec f1 426Hz For a π/2 pulse: 1/4 rotation τ 1000/426 * 1/4 = 0.6 ms

17 What do these pulses do? = 2 = 2 B 1 =[ B 1, 0, 0] T B 1 =[B 1 cos( ),B 1 sin( ), 0] T = 2 B 1 =[0,B 1, 0] T

18 Slice Selective Example ~G =[0, 0, ~ Gz ] T Only spins near resonance frequency are excited.

19 Slice Selective Example pulse sequence: t = M xy (~r, ) Q: What is:?

20 Slice Selectivity as Rotations B1x, B1y are the same EVERYWHERE Gzz changes linearly with z Example: Gz=1G/cm, B1x=0.16G: G z z<< B 1 G z z B 1 G z z =0 G z z B 1 G z z>>b 1 z=-10cm z=-0.16cm z=0 cm z=+0.16cm z=+10cm

21 Slice Selectivity as Rotations B1x, B1y are the same EVERYWHERE Gzz changes linearly with z Example: Gz=1G/cm, B1x=0.16G: G z z<< B 1 G z z B 1 G z z =0 G z z B 1 G z z>>b 1 z=-10cm z=-0.16cm z=0 cm z=+0.16cm z=+10cm

22 Slice Selectivity Simple Cases: On-Resonance same as Non-Selective G z z =0) = Far off resonance, Gzz dominates no Mxy produced! Z 0 B 1 (t)dt G z z >> B 1 ) ~n [0, 0, 1]

23 Slice Selectivity M xy =sin M xy 0 M xy 0 z,! what happens here? In general, a hard problem Rotations fundamentally non-linear Many Solutions for special cases Including most interesting ones

24 Small Tip-Angle Excitation Pulses Basic idea: Tip angle is small Mz M0 throughout the excitation pulse M z = cos( )M 0 M 0 M xy = sin( )M 0 M 0

25 Bloch Equation - Selective RF 2 4 Ṁ x Ṁ y Ṁ z 3 5 = ~ G ~r B1y ~G ~r 0 B 1x B 1y B 1x M x M y M z 3 5 If M z M 0 the last equation decouples! 2 4 Ṁ x Ṁ y Ṁ z 3 5 = ~ G ~r B1y ~G ~r 0 B 1x M x M y M z 3 5

26 Bloch Equation - Small Tip Approximation apple Ṁx Ṁ y = apple Bloch Eq. Simplifies to: ~G ~r 0 B 1x M y 0 G ~ ~r B1y apple Mx M0 apple Ṁx Ṁ y = apple 0 ~ G~r ~G~r 0 apple Mx M y + apple B 1y B 1x M 0 Precession (like reception) Excitation Note: Mx, My, G, B1 are a function of time!

27 Small Tip-Angle Approximation Example: ISO-Center B1=B1x apple Ṁx Ṁ y = apple 0 M B 0 1x Mxy = Mx + imy is linear with B1! M xy = i B 1x M 0 t

28 Derivation apple Ṁx Ṁ y = apple 0 ~ G~r ~G~r 0 apple Mx M y + apple B 1y B 1x M 0

29 Derivation Solve like in the reception case! Integrating factor: e i R t G ~r ~ 1 d

30 Derivation Integrate from - to t=t to find Mxy(r,T) - t T

31 Derivation

32 Small Tip-Angle Approximation Solution for general Eq. at time t=t M xy (~r,t)=im 0 Z T 1 B 1 (t)e i2 ~ k(t) ~r dt,where ~ k(t) =2 Z T ~G( )d t k(t) is area of the remaining gradient

33 Example: Slice Selection M xy (z,t) =im 0 Z T M xy (z,t) =im 0 Z 0 W (k) = 2 B 1(k) G(k) K B 1 (t)e i2 k z(t)z dt This is not exactly a Fourier transform Would like: W (k)e i2 k zz dk z

34 Example: Slice Selection M xy (z,t) =im 0 Z T 0 B 1 (t)e i2 k z(t)z dt B1 Gz - First find kz(t) - Map B1(t) to kz(t) - Compute the integral (Fourier transform)

35 Example: Slice Selection B1 Gz kz - Plot B1(t) vs kz(t): 0 k z (T ) k z (0)

36 Example: Slice Selection 0 k z - B1 is not centered in k-space - We get the right magnitude Mxy Mxy z

37 Example: Slice Selection 0 k z - But we get phase across the slice... signal cancels! Mx(T) My(T)

38 Slice Refocussing B1 Gz kz 0 k z (T ) k z (0)

39 Graphical Interpretation M xy (t) =(i B 1 (t) t)m 0 - At each time-point new x-verse magnetization is created - The new magnetization exhibits precession

40 Graphical Interpretation M xy (t) =(i B 1 (t) t)m 0 k z (t) = 2 Z T t G z ( )d area of remaining gradient - Magnetization at position z precesses through angle: = 2 k z (t) - Magnetization excited at t, at position z will end up: (im 0 B 1 (t)dt)e i2 k z(t)z

41 Graphical Interpretation M xy (t) =(i B 1 (t) t)m 0 - Each magnetization increment is created/precesses independently! Result is the sum of all. - Sum up magnetizations from t=0 to t=t M xy (z,t) =im 0 Z T 0 B 1 (t)e i2 k z(t)z dt

42 Excitation k-space J. Pauly, D. Nishimura and A. Makovski A k-space analysis of small-tip-angle excitation JMR, 1989;81:43-56 * Slightly different conventions than we used.

43 RF and readout - RF burns magnetization in k-space - Receive reads the burned magnetization (weighted by object) uniform object B1 k-space z Gz kz A/D

44 Spin Bench simulation

45 k-space Weighting Example B1 Gz kz k z (T ) 0 k z (0)

46 Examples What does this do?

47 Examples

48 Examples

49 Multiple Excitations B1 Gz B1 Gz

50 RF Pulse Design Choose, B1(t) with a nice transform B 1 (t) / sinc(t) t[ms] rect(f) f[khz] Not prectical, since sinc is continuous indefinitely

51 Truncated sinc -N N B 1 (t) / sinc(t)rect(t/2n) rect(f)*2n sinc(2nf) Too much Ripple!

52 Windowed sinc -N N B 1 (t) / sinc(t)w(t/2n) rect(f)*2n W(2Nf) Hanning Window

53 Characterization of Pulse Shape Time-Bandwidth Product T(BW) = (2N)1 = 2N Total # of zero crossings TBW=2 TBW=4 TBW=8 TBW=12 rapid imaging slab and saturation

54 Slice Profile TBW=2 TBW=12

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64 TBW=4, flip = 10, slice = 10mm, duration =2ms Gx Gy Gz RF Time Mx My Mz ZPosition (mm) Magnitude Phase ZPosition (mm)

65 TBW=8, flip = 10, slice = 10mm, duration =2ms Gx Gy Gz RF Time area doubled Mx My Mz ZPosition (mm) Magnitude Phase ZPosition (mm) transition halved

66 TBW=8, flip = 10, slice = 5mm, duration =2ms Gx Gy Gz RF Time Mx area doubled again My Mz ZPosition (mm) Magnitude Phase ZPosition (mm) smaller slice

67 TBW=8, flip = 45, slice = 5mm, duration =2ms Gx Gy Gz RF Time Mx My Mz some Mx component ZPosition (mm) Magnitude Phase ZPosition (mm) Increased sidebands some phase

68 TBW=8, flip = 90, slice = 5mm, duration =2ms Gx Gy Gz RF Time Mx My Mz More Mx component ZPosition (mm) Magnitude profile still OK! Phase ZPosition (mm) Increased sidebands more phase

69 TBW=8, flip = 180, slice = 5mm, duration =2ms Gx Gy Gz RF Time Mx My Mz ZPosition (mm) Magnitude Phase Distorted Mz profile ZPosition (mm)

70 180 small-tip Pulse Gx Gy Gz RF Time Mx My Mz ZPosition (mm) Magnitude Phase Distorted Mz profile ZPosition (mm)

71 Gx Gy Gz RF 180 SLR Pulse not a sinc! higher bandwidth Time Mx My Mz ZPosition (mm) Magnitude Phase Nice Mz profile! ZPosition (mm)

72 small-tip 180 SLR 180

73 Spectral-Spatial Pulse Gx Gy Gz RF Time ZPosition (mm) water lipids water lipids ZPosition (mm) Magnitude Phase ZPosition (mm) ZPosition (mm) Magnitude ZPosition (mm) Phase Magnitude ZPosition (mm) Phase water lipids Frequency (Hz) Frequency (Hz) Frequency (Hz)