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UCLA Radiology
Bloch Equations & Relaxation
⊗I1
I6
I5
I4
I3
I2
⊗
⊗
⊗
⊗
⊗
⊗
~M
~B1(t)
MRI Systems II – B1
UCLARadiology
Lecture #3 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.
Mathematics of Hard RF Pulses
UCLA Radiology
Parameters & Rules for RF Pulses• 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
�B
~!
RF�⇥
↵
✓
Z
XY
~M
RF Flip Angle
UCLA Radiology
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. 374
B-fields induce precession!
↵
✓
Z
XY
~M
!1 = �B1
UCLA Radiology
Rules for RF Pulses
Phase
Flip Angle
Z
X YZ
X Y
Z
X Y
RF↵✓
RF90�
0� RF90�
90�
B-fields induce left-handed nutation!
UCLA Radiology
How to determine α?
RF�⇥B1,max
�
Rules: 1) Specify α 2) Use B1,max if we can 3) Shortest duration pulse
↵ = �
Z ⌧p
0Be
1 (t) dt
UCLA Radiology
How to determine α?
↵ = �
Z ⌧p
0Be
1 (t) dt
⌅ =�
⇥B1,max
=⇤/2
2⇤ · 42.57Hz/µT · 60µT = 0.098ms
RF�⇥B1,max
�
RF Phase
UCLA Radiology
Bulk Magnetization in the Lab Frame
↵
✓
Z
XY
~M
How do we mathematically account for α and θ?
UCLA Radiology
Change of Basis (θ)
RZ (✓) =
2
4cos ✓ sin ✓ 0
� sin ✓ cos ✓ 0
0 0 1
3
5
Rotate into a coordinate system where M falls along the y’-axis.
↵
✓
Z
XY
Y’
X’
~M
UCLA Radiology
Rotation by Alpha
RX (↵) =
2
41 0 0
0 cos ↵ sin ↵0 � sin ↵ cos ↵
3
5
Rotate M by α about x’-axis.
↵
✓
Z
XY
Y’
X’
~M
’
UCLA Radiology
Change of Basis (-θ)
RZ (�✓) =
2
4cos (�✓) sin (�✓) 0
� sin (�✓) cos (�✓) 0
0 0 1
3
5
↵
✓
Z
XY
Y’
X’
~M
Rotate back to the lab frame’s x-axis and y-axis.
UCLA Radiology
RF Pulse Operator
R�⇥ = RZ (�✓)RX (↵)RZ (✓)
=
2
4c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵
s✓s↵ �c✓s↵ c↵
3
5
↵
✓
Z
XY
Y’
X’
~M
This is the composite matrix operator for a hard RF pulse.
Types of RF Pulses
UCLA Radiology
Types of RF Pulses
• Excitation Pulses • Inversion Pulses • Refocusing Pulses • Saturation Pulses • Spectrally Selective Pulses • Spectral-spatial Pulses • Adiabatic Pulses
UCLA Radiology
Excitation Pulses• Tip Mz into the transverse plane • Typically 200µs 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
UCLA Radiology
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
UCLA Radiology
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
UCLA Radiology
Lecture #3 Summary - RF Pulses
R�⇥ = RZ (�✓)RX (↵)RZ (✓)
=
2
4c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵
s✓s↵ �c✓s↵ c↵
3
5 RF Pulse Operator
↵ = �
Z ⌧p
0Be
1 (t) dt Choosing the flip angle.
Circularly Polarized RF Fields~B1(t) =hcos(!RF t)ˆi� sin(!RF t)ˆj
i
↵
✓
Z
XY
~M
UCLA Radiology
Lecture #3 Summary - Rotating Frame
d
~
M
rot
dt
= ~Mrot
⇥ �⇣
~!
rot
�
+ ~Brot
⌘
Equation of Motion for the Bulk Magnetization in the Rotating Frame Without Relaxation
d
~
M
rot
dt
= ~Mrot
⇥ � ~Beff
Equation of Motion for the Bulk Magnetization in the Rotating Frame Without Relaxation
~Beff
⌘ ~!
rot
�
+ ~Brot
Definition of the “effective” B-field.~Brot
The applied B-field in the Rotating Frame.
~!rot
�“Fictitious field” that demodulates description of the bulk magnetization.
UCLA Radiology
Free Precession in the Rotating Frame without Relaxation
d ~Mrot
dt=
������
i0 j0 k0
Mx
0 My
0 Mz
0
0 0 0
������
~Beff
= ~!
rot
�
+ ~Brot
= ��B0k0
�
+B0k0
= 0
d ~Mrot
dt= ~M
rot
⇥ � ~Beff
dMx
0dt = 0
dMy
0
dt = 0
dMz
0dt = 0
UCLA Radiology
Forced Precession in the Rotating Frame without Relaxation
d
~
M
rot
dt
= ~Mrot
⇥ � ~Beff
= ~Mrot
⇥ �Be
1(t)i0
=
������
i0 j0 k0
~Mx
0 ~My
0 ~Mz
0
�Be
1(t) 0 0
������
dMx
0dt = 0
dMy
0
dt = �Be1(t)Mz0
dMz
0dt = ��Be
1(t)My0
To The Board…
UCLA Radiology
Bloch Equations & Relaxation
UCLA Radiology
1952 Nobel Prize in Physics
Felix Bloch b. 23 Oct 1905 d. 10 Sep 1983
Edward Purcell b. 30 Sep 1912 d. 07 Mar 1997
“for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith“
UCLA Radiology
Bloch Equations with Relaxation
• 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.
d~M
dt= ~M⇥ �~B� M
x
i +My
j
T2� (M
z
�M0) k
T1+Dr2 ~M
UCLA Radiology
Bloch Equations - Lab Frame
• Precession – Magnitude of M unchanged – Phase (rotation) of M changes due to B
• Relaxation – T1 changes are slow O(100ms) – T2 changes are fast O(10ms) – Magnitude of M can be ZERO
• Diffusion – Spins are thermodynamically driven to
exchange positions. – Bloch-Torrey Equations
{Precession Transverse
RelaxationLongitudinal Relaxation
Diffusion
{ { {
d~M
dt= ~M⇥ �~B� M
x
i +My
j
T2� (M
z
�M0) k
T1+Dr2 ~M
UCLA Radiology
Excitation and Relaxation
Z
XThe magnetization relaxes after excitation (forced precession).
UCLA Radiology
Bloch Equations – Rotating Frame
⇥ ⇤Mrot
⇥t= � ⇤M
rot
⇥ ⇤Beff
� Mx
0⇤i0 +My
0⇤j0
T2� (M
z
0 �M0)⇤k0
T1{“Precession” Transverse
RelaxationLongitudinal Relaxation
{ {
~Beff
⌘ ~!
rot
�
+ ~Brot
Effective B-field that M experiences in the
rotating frame. Fictitious field that demodulates the apparent effect of B0
Applied B-field in the rotating frame.
T1 Relaxation
UCLA Radiology
T1 and T2 Values
Tissue T1 [ms] T2 [ms]
gray matter 925 100
white matter 790 92
muscle 875 47
fat 260 85
kidney 650 58
liver 500 43
CSF 2400 180
TI=25ms TE=12ms
TI=200ms TE=12ms
TI=500ms TE=12ms
TI=1000ms TE=12ms
Each tissue as “unique” relaxation properties.
UCLA Radiology
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
UCLA Radiology
T1 Relaxation
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
Time [ms]
Long
itudi
nal M
agne
tizat
ion
[a.u
.]
White MatterGray Matter
Tissue T1 [ms] T2 [ms]gray matter 925 100white matter 790 92
36
UCLA Radiology
T1 Relaxation
Prepared Magnetization Decays (Mz0)
Return to Thermal Equilibrium (M0)
Mz (t) = M0z e
� tT1 +M0
⇣1� e�
tT1
⌘
{ {Net
Magnetization
{
Time [ms]
1.00
0.75
0.50
0.25
0.00
Frac
tion
of M
0
0 200 400 600 800 1000
M0z
M0
Free Precession in the Lab or Rotating Frame with Relaxation
T2 Relaxation
UCLA Radiology
T2 Relaxation
• Transverse or spin-spin relaxation – Molecular interaction causes spin dephasing
• T2 typically 10s to 100s of ms • T2 relatively independent of B0 • T2 always < T1 • T2 decreases with contrast agents • Long T2 is bright on T2 weighted image
UCLA Radiology
T2 Relaxation
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Time [ms]
Tran
sver
se M
agne
tizat
ion
[a.u
.]
White MatterGray Matter
Tissue T1 [ms] T2 [ms]gray matter 925 100white matter 790 92
40
UCLA Radiology
T2 Relaxation
0 200 400 600 800
100
75
50
25
00
Liver – 43ms Fat – 85msCSF – 180ms
Decay Time [ms]
Percent Signal [a.u.]
1.00
0.75
0.50
0.25
0.00
Fraction of Mxy
Mxy
(t) = M0xy
e�t/T2
Free Precession in the Rotating Frame with Relaxation
Matlab
Bloch Equation Simulations
UCLA Radiology
Rotating Frame Bloch Equations (Free Precession)
d�M
dt= �M
x
i+My
j
T2
� (Mz
�M0
) k
T1
UCLA Radiology
Rotating Frame Bloch Equations (Free Precession)
d�M
dt= �M
x
i+My
j
T2
� (Mz
�M0
) k
T1➠
2
4dM
x
dt
dM
y
dt
dM
z
dt
3
5 =
2
4� 1
T20 0
0 � 1T2
00 0 � 1
T1
3
5
2
4M
x
My
Mz
3
5+
2
400M0T1
3
5
UCLA Radiology
d�M
dt= �M
x
i+My
j
T2
� (Mz
�M0
) k
T1➠
2
4dM
x
dt
dM
y
dt
dM
z
dt
3
5 =
2
4� 1
T20 0
0 � 1T2
00 0 � 1
T1
3
5
2
4M
x
My
Mz
3
5+
2
400M0T1
3
5
d⇤M
dt= �⇤M+ ⇥
An affine transformation between two vector spaces consists of a translation followed by a linear transformation.
➠
http://en.wikipedia.org/wiki/Affine_transformation
Rotating Frame Bloch Equations
UCLA Radiology
Why Homogenous Coordinates?
Homogenous coordinates allow us to transform an affine (non-linear) equation in 3D to a linear equation in 4D.
http://en.wikipedia.org/wiki/Linear_transformation
d⇤M
dt= �⇤M+ ⇥
Affine Linear
↔ d�MH
dt= TH
�MH
Now we can use the machinery of linear algebra for writing out the Bloch Equation mechanics.
UCLA Radiology
Homogenous Coordinate Expressions
Cartesian Coordinates Homogeneous Coordinates
�M =
2
4Mx
My
Mz
3
5 �MH =
2
664
Mx
My
Mz
1
3
775
Augment������! �����Reduce
T =
2
4Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz
3
5 TH =
2
664
Txx Txy Txz Txt
Tyx Tyy Tyz Tyt
Tzx Tzy Tzz Tzt
0 0 0 1
3
775
UCLA Radiology
d�M
dt= �M
x
i+My
j
T2
� (Mz
�M0
) k
T1➠
➠
d�MH
dt= TH
�MH
2
664
dM
x
dt
dM
y
dt
dM
z
dt
1
3
775 =
2
664
� 1T2
0 0 00 � 1
T20 0
0 0 � 1T1
0 0 0 1
3
775
2
4M
x
My
Mz
3
5+
2
664
00M0T1
1
3
775
2
664
Mx
My
Mz
1
3
775
Rotating Frame Bloch Equations (Free Precession)
UCLA Radiology
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
B0 Fields
UCLA Radiology
Bulk Magnetization - Precession
2
4M
x
(t)M
y
(t)M
z
(t)
3
5=
2
4cos �B0t sin �B0t 0
� sin �B0t cos �B0t 0
0 0 1
3
5
2
4M0
x
M0y
M0z
3
5
~M(t) = Rz(�B0t) ~M0
UCLA Radiology
Bulk Magnetization - Precession
2
664
Mx
(0�)
My
(0�)
Mz
(0�)
1
3
775 =
2
664
cos �Bt sin �Bt 0 0
� sin �Bt cos �Bt 0 0
0 0 1 0
0 0 0 1
3
775
2
664
Mx
(0+)
My
(0+)
Mz
(0+)
1
3
775
Homogeneous coordinate expression for precession.
B0,H =
2
664
cos �Bt sin �Bt 0 0
� sin �Bt cos �Bt 0 0
0 0 1 0
0 0 0 1
3
775
RF Pulses
UCLA Radiology
RF Pulse Operator
↵
✓
R�⇥ = RZ (�✓)RX (↵)RZ (✓)
=
2
4c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵
s✓s↵ �c✓s↵ c↵
3
5
~M (0+) = RF↵✓
~M (0�)
UCLA Radiology
RF Pulse Homogeneous Operator
RF�⇥,H =
2
664
c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵ 0c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵ 0
s✓s↵ �c✓s↵ c↵ 00 0 0 1
3
775
UCLA Radiology
RF Pulse Homogeneous Operator
RF�⇥,H =
2
664
c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵ 0c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵ 0
s✓s↵ �c✓s↵ c↵ 00 0 0 1
3
775
~M+H = RF�
⇥,H~M�
H
UCLA Radiology
RF Pulse Homogeneous Operator
RF�⇥,H =
2
664
c2✓ + s2✓c↵ c✓s✓ � c✓s✓c↵ �s✓s↵ 0c✓s✓ � c✓s✓c↵ s2✓ + c2✓c↵ c✓s↵ 0
s✓s↵ �c✓s↵ c↵ 00 0 0 1
3
775
~M+H = RF�
⇥,H~M�
H
2
664
M+x
M+y
M+z
1
3
775 =
2
664
c2⇥ + s2⇥c� c⇥s⇥ � c⇥s⇥c� �s⇥s� 0c⇥s⇥ � c⇥s⇥c� s2⇥ + c2⇥c� c⇥s� 0
s⇥s� �c⇥s� c� 00 0 0 1
3
775
2
664
M�x
M�y
M�z
1
3
775
Relaxation
UCLA Radiology
Relaxation Operator
2
4M+
x
M+y
M+z
3
5 =
2
664
e� t
T2
e� t
T2
e� t
T1
3
775
2
4M�
x
M�y
M�z
3
5+
2
664
00
M0
✓1� e
� t
T1
◆
3
775
=
2
666664
e� t
T2
e� t
T2
e� t
T1 M0
✓1� e
� t
T1
◆
1
3
777775
2
664
M�x
M�y
M�z
1
3
775
UCLA Radiology
E (T1, T2, t,M0) =
�
⇧⇧⇤
E2 0 0 00 E2 0 00 0 E1 M0 (1� E1)0 0 0 1
⇥
⌃⌃⌅
⌅M+ = E(T1, T2, t,M0) ⌅M�
E1 = e�t/T1 E2 = e�t/T2
*In general we drop the sub-scripted H
Relaxation Operator
UCLA Radiology
B0, RF Pulse, & Relaxation Operators
�M+ = RF�⇥
�M�
⌅M+ = E(T1, T2, t,M0) ⌅M�
~M+ = B0,H~M�
UCLARadiology
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@UCLA 01.21.2015 function dB0=PAM_B0_op(gamma,B0,dt) if nargin==0 gamma=42.57e6; % 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=2*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; 0 0 1 0; 0 0 0 1]; endreturn
2
4M
x
(t)M
y
(t)M
z
(t)
3
5=
2
4cos �B0t sin �B0t 0
� sin �B0t cos �B0t 0
0 0 1
3
5
2
4M0
x
M0y
M0z
3
5
UCLARadiology
Matlab Example - Free Precession%% Filename: PAM_Lec02_B0_Free_Precession.m%% Demonstrate the precession of the bulk magnetization vector.%% DBE@UCLA 2015.01.06 %% Define some constantsgamma=42.57e6; % 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 simulatet=(0:nt-1)*0.01e-8; % Time vector [s] M0=[sqrt(2)/2 0 sqrt(2)/2 1]'; % Initial condition (I.C.) M=zeros(4,nt); % Initialize the magnetization arrayM(:,1)=M0; % Define the first time point as the I.C. %% Simulate precession of the bulk magnetization vectordB0=PAM_B0_op(gamma,B0,dt); % Calculate the homogenous coordinate transform for n=2:nt M(:,n)=dB0*M(:,n-1);end
%% Plot the resultsfigure; hold on; p(1)=plot(t,M(1,:)); % Plot the Mx component p(2)=plot(t,M(2,:)); % Plot the My component p(3)=plot(t,M(3,:)); % Plot the Mz component set(p,'LineWidth',3); % Increase plot thickness ylabel('Magnetization [AU]'); xlabel('Time [s]'); legend('M_x','M_y','M_z'); title('Bulk Magnetization Components as f(t)'); Time [s] ×10-8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Mag
netiz
atio
n [A
U]
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8Bulk Magnetization Components as f(t)
MxMyMz
~M(t) = Rz(�B0t) ~M0
UCLA Radiology
Hard RF Pulses
R90�
0�
R90�
0� =
2
41 0 00 0 10 �1 0
3
5
function dB1=PAM_B1_op(gamma,B1,dt,theta) % Define the incremental flip angle in time dtalpha=2*pi*gamma*B1*dt; % Change of basisR_theta=[ cos(theta) sin(theta) 0 0; -sin(theta) cos(theta) 0 0; 0 0 1 0; 0 0 0 1]; % Flip angle rotationR_alpha=[1 0 0 0; 0 cos(alpha) sin(alpha) 0; 0 -sin(alpha) cos(alpha) 0; 0 0 0 1]; % Homogeneous expression for RF MATRIXdB1=R_theta.'*R_alpha*R_theta; return
Z
X Y
UCLA Radiology
Hard RF Pulses
R90�
90�
R90�
90� =
2
40 0 �10 1 01 0 0
3
5
function dB1=PAM_B1_op(gamma,B1,dt,theta) % Define the incremental flip angle in time dtalpha=2*pi*gamma*B1*dt; % Change of basisR_theta=[ cos(theta) sin(theta) 0 0; -sin(theta) cos(theta) 0 0; 0 0 1 0; 0 0 0 1]; % Flip angle rotationR_alpha=[1 0 0 0; 0 cos(alpha) sin(alpha) 0; 0 -sin(alpha) cos(alpha) 0; 0 0 0 1]; % Homogeneous expression for RF MATRIXdB1=R_theta.'*R_alpha*R_theta; return
Z
X Y
UCLA Radiology
Thanks
Daniel B. Ennis, Ph.D. [email protected] 310.206.0713 (Office) http://ennis.bol.ucla.edu
Peter V. Ueberroth Bldg. Suite 1417, Room C 10945 Le Conte Avenue