Bloch Equations & Relaxation UCLA. Radiology

Transcription

1 Bloch Equations & Relaxation

2 MRI Systems II B1 I 1 I ~B 1 (t) I 6 ~M I I 5 I 4

3 Lecture # Learning Objectives Distinguish spin, precession, and nutation. Appreciate that any B-field acts on the the spin system. Understand the advantage of a circularly polarized RF B-field. Differentiate the lab and rotating frames. Define the equation of motion in the lab and rotating frames. Know how to compute the flip angle from the B1-envelope function. Understand how to apply the RF hard pulse matrix operator.

4 Mathematics of Hard RF Pulses

5 Parameters & Rules for RF Pulses B RF pulses have a flip angle (α) - RF fields induce left-hand rotations All B-fields do this for positive γ RF pulses have a phase (θ) - Phase of 0 is about the x-axis - Phase of 90 is about the y-axis Z ~M RF ~! X Y

6 RF Flip Angle

7 Flip Angle Amount of rotation of the bulk magnetization vector produced by an RF pulse, with respect to the direction of the static magnetic field. Liang & Lauterbur, p. 74 Z ~M X Y! 1 = B 1 B-fields induce precession!

8 Rules for RF Pulses RF Phase Flip Angle Z Z X Y Z X Y RF 90 0 RF B-fields induce left-handed nutation! X Y

9 How to determine α? B1,max RF = Z p 0 B e 1 (t) dt Rules: 1) Specify α ) Use B1,max if we can ) Shortest duration pulse

10 How to determine α? B1,max RF = B1,max = = Z p 0 B e 1 (t) dt / 4.57Hz/µT 60µT =0.098ms

11 RF Phase

12 Bulk Magnetization in the Lab Frame Z ~M X Y How do we mathematically account for α and θ?

13 Change of Basis (θ) Z ~M X X Y Y R Z ( ) = 4 cos sin 0 sin cos Rotate into a coordinate system where M falls along the y -axis.

14 Rotation by Alpha Z ~M X X Y Y R X ( ) = cos sin 0 sin cos 5 Rotate M by α about x -axis.

15 Change of Basis (-θ) Z ~M X X Y Y R Z ( ) = 4 cos ( ) sin( ) 0 sin ( ) cos ( ) Rotate back to the lab frame s x-axis and y-axis.

16 RF Pulse Operator Z ~M X X Y Y R = R Z ( ) R X ( ) R Z ( ) c +s c c s c s c s s = 4 c s c s c s +c c c s s s c s c 5 This is the composite matrix operator for a hard RF pulse.

17 Types of RF Pulses

18 Types of RF Pulses Excitation Pulses Inversion Pulses Refocusing Pulses Saturation Pulses Spectrally Selective Pulses Spectral-spatial Pulses Adiabatic Pulses

19 Excitation Pulses Tip Mz into the transverse plane Typically 00µs to 5ms Non-uniform across slice thickness Imperfect slice profile Non-uniform within slice Termed B1 inhomogeneity Non-uniform signal intensity across FOV 90 Excitation Pulse Small Flip Angle Pulse

20 Inversion Pulses Typically, 180 RF Pulse non-180 that still results in -MZ Invert MZ to -MZ Ideally produces no MXY Hard Pulse Constant RF amplitude Typically non-selective Soft (Amplitude Modulated) Pulse Frequency/spatially/spectrally selective Typically followed by a crusher gradient 180 Inversion Pulse

21 Refocusing Pulses Typically, 180 RF Pulse Provides optimally refocused MXY Largest spin echo signal Refocus spin dephasing due to imaging gradients local magnetic field inhomogeneity magnetic susceptibility variation chemical shift Typically followed by a crusher gradient 180 Refocusing Pulse

22 Lecture # Summary - RF Pulses ~B 1 (t) = h cos(! RF t)î i sin(! RF t)ĵ Circularly Polarized RF Fields R = R Z ( ) R X ( ) R Z ( ) c +s c c s c s c s s = 4 c s c s c s +c c c s s s c s c 5 RF Pulse Operator Z = Z p 0 B e 1 (t) dt X ~M Y Choosing the flip angle.

23 Lecture # Summary - Rotating Frame d ~ M rot dt = ~ M rot ~! rot + ~ B rot Equation of Motion for the Bulk Magnetization in the Rotating Frame Without Relaxation ~B eff ~! rot + ~ B rot Definition of the effective B-field. ~! rot Fictitious field that demodulates description of the bulk magnetization. ~B rot The applied B-field in the Rotating Frame. d ~ M rot dt = ~ M rot ~ B eff Equation of Motion for the Bulk Magnetization in the Rotating Frame Without Relaxation

24 Free Precession in the Rotating Frame without Relaxation ~B eff = ~! rot + ~ B rot = B 0ˆk 0 + B 0ˆk0 =0 d ~ M rot dt = ~ M rot ~ B eff dm x 0 dt =0 dm y 0 dt =0 d ~ M rot dt = î 0 ĵ 0 ˆk0 M x 0 M y 0 M z dm z 0 dt =0

25 Forced Precession in the Rotating Frame without Relaxation d ~ M rot dt = ~ M rot ~ B eff = ~ M rot B e 1(t)î 0 = î 0 ĵ 0 ˆk0 ~M x 0 My ~ 0 Mz ~ 0 B1(t) e 0 0 dm x 0 dt =0 dm y 0 dt = B e 1(t)M z 0 dm z 0 dt = B e 1(t)M y 0

26 To The Board

27 Bloch Equations & Relaxation

28 195 Nobel Prize in Physics for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith Felix Bloch b. Oct 1905 d. 10 Sep 198 Edward Purcell b. 0 Sep 191 d. 07 Mar 1997

29 Bloch Equations with Relaxation d ~ M dt = ~ M ~ B M xî+m yĵ T (M z M 0 ) ˆk T 1 + Dr ~ M Differential Equation Ordinary, Coupled, Non-linear No analytic solution, in general. Analytic solutions for simple cases. Numerical solutions for all cases. Phenomenological Exponential behavior is an approximation.

30 Bloch Equations - Lab Frame d ~ M dt = ~ M ~ B M xî+m yĵ T (M z M 0 ) ˆk T 1 { Precession { Transverse Relaxation { Longitudinal Relaxation Precession Magnitude of M unchanged Phase (rotation) of M changes due to B Relaxation T1 changes are slow O(100ms) T changes are fast O(10ms) Magnitude of M can be ZERO Diffusion Spins are thermodynamically driven to exchange positions. Bloch-Torrey Equations + Dr ~ M { Diffusion

31 Excitation and Relaxation Z X The magnetization relaxes after excitation (forced precession).

32 Bloch Equations Rotating Frame M rot t = M rot B eff M x 0 i 0 + M y 0 j 0 T (M z0 M 0 ) k 0 T 1 { Precession { Transverse Relaxation { Longitudinal Relaxation ~B eff ~! rot + ~ B rot Effective B-field that M experiences in the rotating frame. Applied B-field in the rotating frame. Fictitious field that demodulates the apparent effect of B0

33 T1 Relaxation

34 T1 and T Values Tissue T1 [ms] T [ms] gray matter white matter muscle fat kidney liver CSF TI=5ms TE=1ms TI=00ms TE=1ms TI=500ms TE=1ms TI=1000ms TE=1ms Each tissue as unique relaxation properties.

35 T1 Relaxation Longitudinal or spin-lattice relaxation Typically 100s to 1000s of ms T1 increases with increasing B0 T1 decreases with contrast agents Short T1s are bright on T1-weighted image

36 T1 Relaxation 6 Tissue T1 [ms] T [ms] gray matter white matter Longitudinal Magnetization [a.u.] Time [ms] White Matter Gray Matter

37 T1 Relaxation Free Precession in the Lab or Rotating Frame with Relaxation M z (t) =M 0 z e t T 1 + M 0 1 e t T 1 { Net Magnetization { Prepared Magnetization Decays (Mz 0 ) { Return to Thermal Equilibrium (M0) 1.00 M 0 Fraction of M Mz Time [ms]

38 T Relaxation

39 T Relaxation Transverse or spin-spin relaxation Molecular interaction causes spin dephasing T typically 10s to 100s of ms T relatively independent of B0 T always < T1 T decreases with contrast agents Long T is bright on T weighted image

40 T Relaxation 40 Tissue T1 [ms] T [ms] gray matter white matter Transverse Magnetization [a.u.] Time [ms] White Matter Gray Matter

41 T Relaxation Fraction of Mxy Percent Signal [a.u.] Free Precession in the Rotating Frame with Relaxation M xy (t) =Mxye 0 t/t Liver 4ms Fat 85ms CSF 180ms Decay Time [ms]

42 Matlab

43 Bloch Equation Simulations

44 Rotating Frame Bloch Equations (Free Precession) dm dt = M xî +M yĵ (M z M 0 ) ˆk T T 1

45 Rotating Frame Bloch Equations (Free Precession) dm dt = M xî +M yĵ (M z M 0 ) ˆk T T 1 dm x dt dm y dt dm z dt 5 = 4 1 T T T M x M y M z M 0 T 1 5

46 Rotating Frame Bloch Equations dm dt = M xî +M yĵ (M z M 0 ) ˆk T T 1 dm x dt dm y dt dm z dt 5 = 4 1 T T T 1 d M dt = M+ 5 4 M x M y M z 5 + An affine transformation between two vector spaces consists of a translation followed by a linear transformation M 0 T 1 5

47 Why Homogenous Coordinates? Homogenous coordinates allow us to transform an affine (non-linear) equation in D to a linear equation in 4D. Affine Linear d M dt = M+ dm H dt = T H M H Now we can use the machinery of linear algebra for writing out the Bloch Equation mechanics.

48 Homogenous Coordinate Expressions Cartesian Coordinates Homogeneous Coordinates M M x x M= 4 M y 5 M H = 6 M y 7 4 M M z 5 z 1 Augment! T = 4 T xx T xy T xz T yx T yy T yz T zx T zy T zz Reduce 5 T H = 6 4 T xx T xy T xz T xt T yx T yy T yz T yt T zx T zy T zz T zt

49 Rotating Frame Bloch Equations (Free Precession) dm dt = M xî +M yĵ (M z M 0 ) ˆk T T 1 dm x dt dm y dt dm z dt = T T T M x M x y M z y M z M 0 T dm H dt = T H M H

50 Advantages/Disadvantages + 1:1 Correlation with pulse diagram + Simple to implement (Matlab!) + Not ad hoc + Provides understanding in complex systems - Masks understanding in simple systems - Reduction to algebraic expression is cumbersome - Discrete (not continuous) - Perfect simulations are very difficult - Must consider assumptions - Image Prep vs. Imaging

51 B0 Fields

52 Bulk Magnetization - Precession M x (t) M y (t) M z (t) 5 = 4 cos B 0 t sin B 0 t 0 sin B 0 t cos B 0 t M 0 x M 0 y M 0 z 5 ~M(t) =R z ( B 0 t) ~ M 0

53 Bulk Magnetization - Precession B 0,H = 6 4 cos Bt sin Bt 0 0 sin Bt cos Bt M x (0 ) M y (0 ) M z (0 ) = 6 4 cos Bt sin Bt 0 0 sin Bt cos Bt M x (0+) M y (0+) M z (0+) Homogeneous coordinate expression for precession.

54 RF Pulses

55 RF Pulse Operator R = R Z ( ) R X ( ) R Z ( ) c +s c c s c s c s s = 4 c s c s c s +c c c s s s c s c 5 ~M(0 + )=RF ~ M(0 )

56 RF Pulse Homogeneous Operator RF,H = 6 4 c +s c c s c s c s s 0 c s c s c s +c c c s 0 s s c s c

57 RF Pulse Homogeneous Operator RF,H = 6 4 c +s c c s c s c s s 0 c s c s c s +c c c s 0 s s c s c ~M + H =RF,H ~ M H

58 RF Pulse Homogeneous Operator RF,H = 6 4 c +s c c s c s c s s 0 c s c s c s +c c c s 0 s s c s c ~M + H =RF,H ~ M H 6 4 M + x M + y M + z = 6 4 c +s c c s c s c s s 0 c s c s c s +c c c s 0 s s c s c M x M y M z 1 7 5

59 Relaxation

60 Relaxation Operator 4 M + x M + y M + z 5 = = e e t T t T e e t T t T e e t T M x M y M z 5 + t T 1 M 0 1 e t T M 0 1 e M x M y M z 1 t T

61 Relaxation Operator E (T 1,T, t, M 0 )= E E E 1 M 0 (1 E 1 ) E 1 = e t/t 1 E = e t/t M + =E(T 1,T, t, M 0 ) M *In general we drop the sub-scripted H

62 B0, RF Pulse, & Relaxation Operators ~M + = B 0,H ~ M M + = RF M M + =E(T 1,T, t, M 0 ) M

63 Matlab Example - B0 % This function returns the 4x4 homogenous coordinate expression for % precession for a particular gyromagnetic ratio (gamma), external % field (B0), and time step (dt). % % SYNTAX: db0=pam_b0_op(gamma,b0,dt) % % INPUTS: gamma - Gyromagnetic ratio [Hz/T] % B0 - Main magnetic field [T] % dt - Time step or vector [s] % % OUTPUTS: db0 - Precessional operator [4x4] % % DBE@ function db0=pam_b0_op(gamma,b0,dt) if nargin==0 gamma=4.57e6; % Gyromagnetic ratio for 1H B0=1.5; % Typical B0 field strength dt=ones(1,100)*1e-6; % 100 1µs time steps end db0=zeros(4,4,numel(dt)); % Initialize the array for n=1:numel(dt) dw=*pi*gamma*b0*dt(n); % Incremental precession (rotation angle) % Precessional Operator (left handed) db0(:,:,n)=[ cos(dw) sin(dw) 0 0; -sin(dw) cos(dw) 0 0; ; ]; end return 4 cos B 0 t sin B 0 t 0 sin B 0 t cos B 0 t

64 Matlab Example - Free Precession %% Filename: PAM_Lec0_B0_Free_Precession.m % % Demonstrate the precession of the bulk magnetization vector. % % DBE@ %% Define some constants gamma=4.57e6; % Gyromagnetic ratio for 1H [MHz/T] B0=1.5; % B0 magnetic field strength [T] dt=0.01e-8; % Time step [s] nt=500; % Number of time points to simulate t=(0:nt-1)*0.01e-8; % Time vector [s] M0=[sqrt()/ 0 sqrt()/ 1]'; % Initial condition (I.C.) M=zeros(4,nt); M(:,1)=M0; % Initialize the magnetization array % Define the first time point as the I.C. %% Simulate precession of the bulk magnetization vector db0=pam_b0_op(gamma,b0,dt); % Calculate the homogenous coordinate transform for n=:nt M(:,n)=dB0*M(:,n-1); end ~M(t) =R z ( B 0 t) ~ M Bulk Magnetization Components as f(t) M x M y M z 0.4 %% Plot the results figure; hold on; p(1)=plot(t,m(1,:)); p()=plot(t,m(,:)); p()=plot(t,m(,:)); % Plot the Mx component % Plot the My component % Plot the Mz component % Increase plot thickness set(p,'linewidth',); -0.4 ylabel('magnetization [AU]'); -0.6 xlabel('time [s]'); legend('m_x','m_y','m_z'); title('bulk Magnetization Components as f(t)'); Time [s] 10-8 Magnetization [AU]

65 Hard RF Pulses function db1=pam_b1_op(gamma,b1,dt,theta) Z % Define the incremental flip angle in time dt alpha=*pi*gamma*b1*dt; % Change of basis R_theta=[ cos(theta) sin(theta) 0 0; -sin(theta) cos(theta) 0 0; ; ]; % Flip angle rotation R_alpha=[ ; 0 cos(alpha) sin(alpha) 0; 0 -sin(alpha) cos(alpha) 0; ]; % Homogeneous expression for RF MATRIX db1=r_theta.'*r_alpha*r_theta; return X R 90 0 = R Y

66 Hard RF Pulses function db1=pam_b1_op(gamma,b1,dt,theta) Z % Define the incremental flip angle in time dt alpha=*pi*gamma*b1*dt; % Change of basis R_theta=[ cos(theta) sin(theta) 0 0; -sin(theta) cos(theta) 0 0; ; ]; % Flip angle rotation R_alpha=[ ; 0 cos(alpha) sin(alpha) 0; 0 -sin(alpha) cos(alpha) 0; ]; % Homogeneous expression for RF MATRIX db1=r_theta.'*r_alpha*r_theta; return X R = R Y

67 Thanks Daniel B. Ennis, Ph.D (Office) Peter V. Ueberroth Bldg. Suite 1417, Room C Le Conte Avenue