B.Hargreaves - RAD 229

Signal Calculations

• Bloch Equations and Matrix Calculations

• Extended Phase Graphs

• Examples (both)

113

B.Hargreaves - RAD 229

Bloch Equation Matrix Simulations

• Basic Bloch Equation

• Bloch Equation with B1 / rotating frame

• Basic matrix simulations / Hard Pulse Approx.

• Many-spin simulations: Brute force

• Bloch Equation with Exchange

• Bloch-Torrey Equations (McNab?)

114

B.Hargreaves - RAD 229

Bloch Equation

• Basic Bloch Equation:

• In Matrix form with:

• Becomes:

115

dM

dt= M ⇥B � M

xy

T2+

M0 �Mz

T1

M =

2

4M

x

My

Mz

3

5

dM

dt=

2

4�1/T2 �B

z

��By

��Bz

�1/T2 �Bx

�By

��Bx

�1/T1

3

5M +

2

400

M0/T1

3

5

B.Hargreaves - RAD 229

Relaxation

• Over time period τ

116

E1 = e�⌧/T1

E2 = e�⌧/T2

M 0 =

2

4E2 0 00 E2 00 0 E1

3

5M +

2

400

M0(1� E1)

3

5

B.Hargreaves - RAD 229

RF Rotations• Over time period τ

117

M 0= R

x

M =

2

41 0 0

0 sin↵ cos↵0 cos↵ � sin↵

3

5M

M 0= RyM =

2

4� sin↵ 0 cos↵

0 1 0

cos↵ 0 sin↵

3

5M

↵ = �B1⌧

B.Hargreaves - RAD 229

RF Rotations (Arbitrary B1 phase)

118

R� =

2

4cos

2 ↵+ sin

2 ↵ cos� cos↵ sin↵(1� cos�) � sin↵ sin�cos↵ sin↵(1� cos�) sin

2 ↵+ cos

2 ↵ cos� cos↵ sin�sin� sin↵ � sin� cos↵ cos↵

3

5

� = tan�1(By

/Bx

)

• Note that Rφ is just Rz(φ)Rx(α) Rz(-φ)

B.Hargreaves - RAD 229

Gradient / ΔB0 Rotations

119

M 0= RzM =

2

4cos ✓ � sin ✓ 0

sin ✓ cos ✓ 0

0 0 1

3

5M

✓ = �(G ·�!r +�B0)

B.Hargreaves - RAD 229

Perfect Spoiling

120

M 0 =

2

40 0 00 0 00 0 1

3

5M

• Explicitly set transverse magnetization to zero

RF

Gz

TRTE

B.Hargreaves - RAD 229

Example: Excitation/Recovery

M1 M2 M1 M2 M1 M2

α α α

1

Recall Prior Example!

B.Hargreaves - RAD 229

Overlapping RF/Gradients?

• z rotations and relaxation commute

• RF rotations do not commute with others

• Hard-Pulse Approximation:

• If rotations/relaxation are small they commute

• Break pulses into very short segments

• Aside: Basis for Variable Rate Selective Excitation

122

B.Hargreaves - RAD 229

Matrix Propagation

• Rotations / Relaxation: M’ = AM+B

• M1 = A1M0+B1

• M2 = A2M2+B2 ..... Mn (n “operations”)

123

A =1Y

i=n

Ai

B = A3A2B1 +A3B2 +B3

A = A3A2A1B =nX

i=1

0

@i+1Y

j=n

Aj

1

ABi

Example (n=3)

B.Hargreaves - RAD 229

Steady States

•Propagation over 1 TR: Mn+1 = AMn+B

•Steady State: Mn+1 = Mn

•Combine: Mss = AMss+B = (I-A)-1B

124

B.Hargreaves - RAD 229

Short-TR IR Sequence (Repeated)

• Inversion Recovery Sequence:

• TR = 1s, TI = 0.5s, TE=50ms

• What is the signal for T1=0.5s, T2=100ms?

125

180º 180º

RF

TI

TE

TR

90º

B.Hargreaves - RAD 229

Example

• 90 Excitation pulse

• Time samples of 4µs

• 3 sync cycles

• 2ms duration

• Area of 5.9 µΤ*ms

• BW ~ 3 kHz

• 2.3 mT/m gradient (1kHz/cm)126

B.Hargreaves - RAD 229

Simulation

• Loop over z

• Define Rz

• Loop over t

• M’=RzRx(t)M

• Plot M over time and space

127

B.Hargreaves - RAD 229

Example (Off-Resonance)

• 90 Excitation pulse

• BW ~ 3 kHz

• 2.3 mT/m gradient (1kHz/cm)

• 2kHz off-resonance??

128

B.Hargreaves - RAD 229

Excitation Recovery (Real Pulse)

• Simulate full pulse and position

• Perfect spoiling (“keep only Mz” matrix)

• Matrix propagation to calculate steady-state at each position

129

α=30

0 Mxy 0 Mxy

α=30 α=30

B.Hargreaves - RAD 229

Exchanging Systems

• Ma and Mb are magnetizations in “exchanging” pools:

130

M =

2

6666664

Ma

x

Ma

y

Ma

z

M b

x

M b

y

M b

z

3

7777775 dM

dt=

2

4�1/T2 �B

z

��By

��Bz

�1/T2 �Bx

�By

��Bx

�1/T1

3

5M +

2

400

M0/T1

3

5

Single-Pool Bloch Equation:

dM

dt=

2

6666664

�1/T a

2 � 1/⌧a

�a

Bz

��a

By

1/⌧b

0 0��

a

Bz

�1/T a

2 � 1/⌧a

�a

Bx

0 1/⌧b

0�a

By

��a

Bx

�1/T a

1 � 1/⌧a

0 0 1/⌧b

1/⌧a

0 0 �1/T b

2 � 1/⌧b

�b

Bz

��b

By

0 1/⌧a

0 ��b

Bz

�1/T b

2 � 1/⌧b

�b

Bx

0 0 1/⌧a

�b

By

��b

Bx

�1/T b

1 � 1/⌧b

3

7777775M+

2

6666664

00

Ma0 /T

a1

00

M b0/T

b1

3

7777775

B.Hargreaves - RAD 229

Summary

• Bloch equation: Rotations and Relaxation

• Consider M as a 3x1 vector

• Rotations ~ Simple multiplier

• Relaxation ~ M’ = AM+B

• Propagate Effects like “Operators”

• Brute force simulations by looping:

• Time, Position, Frequency, etc

131