3/4/2011 1 MRI FYS-KJM 4740 Chap 1 Bloch equations and main principles FYS-KJEM 4740 Frédéric Courivaud (PhD) 2 z y x B0 M ωL Mz ωL Figure Error! No text of specified style in document.-1. The magnetic moment M rotates around The magnetic moment M rotates around the static B-field at the Larmor frequency ) ( B M M × = γ dt d FYS-KJM 4740 3 The Bloch equation ω 0 (rad/s) = γ (rad/s/Tesla) x B 0 (Tesla) γ hydrogen = 2.68 *10 8 rad/s/Tesla f L (MHz) = γ (MHz/Tesla) x B 0 (Tesla) γ hydrogen = 42.58 MHz/Tesla /2π Joseph Larmor z y x B0 M ωL Mz ωL Larmor equation FYS-KJM 4740 AtleBjørnerud 4 y x z ω 0 Laboratory frame B 0 y` x` z` B 0 Rotating frame Rotating frame of reference FYS-KJEM 4740 5 In MRI, short RF pulses are used to To change the direction of the magnetization M To get M to rotate around x or y axis, A linearly polarized magnetic field B1 is used during short time (pulse) to get M to rotate around B1 axis y` x` z B 1 M Flipping away the Magnetization from its equilibrium FYS-KJEM 4740 6
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3/4/2011
1
MRI
FYS-KJM 4740
Chap 1
Bloch equations and main
principles
FYS-KJEM 4740 Frédéric Courivaud (PhD) 2
z
y
x
B0
M
ωL
Mz
ωL
Figure Error! No text of specified style in document.-1. The magnetic moment M rotates around
The magnetic moment M rotates around the static B-field at the Larmor frequency
)( BMM
×= γdt
d
FYS-KJM 4740 3
The Bloch equation
ω0 (rad/s) = γ (rad/s/Tesla) x B0 (Tesla)
γhydrogen = 2.68 *108 rad/s/Tesla
fL (MHz) = γ (MHz/Tesla) x B0 (Tesla)
γhydrogen = 42.58 MHz/Tesla
/2π
Joseph Larmor
z
y
x
B0
M
ωL
Mz
ωL
Larmor equation
FYS-KJM 4740 AtleBjørnerud 4
y
x
zω0
Laboratory frame
B0
y`
x`
z`
B0
Rotating frame
Rotating frame of reference
FYS-KJEM 4740 5
In MRI, short RF pulses are used toTo change the direction of the magnetization
M
To get M to rotate around x or y axis,A linearly polarized magnetic field B1 is used during short time (pulse) to get M to rotate
around B1 axis
y`
x`
z
B1
M
Flipping away the Magnetization from its equilibrium
FYS-KJEM 4740 6
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2
Baseline, no phase coherence in transverse (X,Y) plane
B0
Zero net Mx and My components
Additional B1-field induces phase coherence in transverse (X,Y)
plane
B0
Net Mx,y component > 0
B1
The magnetic field due to the RF pulse, B1, is generatedby two circularly polarizedfields with opposing directionof rotation with angularfrequency +/- Ω
B1
time
Rotation
B1+ B1-
-Ω Ω
2B1cos(Ωt)
Ω
=
Ω
Ω
+
Ω−
Ω−
=+= −+
0
0
)cos(
2
0
)sin(
)cos(
0
)sin(
)cos(
111111
t
Bt
t
Bt
t
BBBB
z
y
x
B1
The RF-coil generates a magnetic field B1 along the x-axis
FYS-KJEM 4740 12
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3
z
y
x y’
z’
x’
Ω
Ω
The ‘rotating frame’ (x’, y’, z’-coordinates)
effBMM
×= γdt
d Beff= B0 + B1 + Ω/γ
Ω−
= 0
0
Ω
Bloch equation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 13
RF-eksitasjon med Larmor frekvens (rotating frame)
effBMM
×= γdt
d
Beff= B1
Ω= γB0
z
y
x y’
z’
x’
Ω
Ω
FYS-KJEM 4740 14
Using Matrix formalism
z’
y’
x’
B1
α1 M
ω1B0
−
=
z
y
x
x
x
M
M
M
B
Bdt
d
00
00
000
1
1γM
0
11
11
)0(
)0(
)0(
)cos()sin(0
)sin()cos(0
001
)( MRM ⋅=
−
=
z
y
x
M
M
M
tt
ttt
ωω
ωω
)cos()sin( '1'1' tBBtBAM xxy γγ +=
:
Using the boundary conditions My’=My’(0) and Mz’=Mz(0) at t=0, we get
11 Bγω −=
FYS-KJEM 4740 15
RF pulse duration is proportional to the
wanted flip angle, α.
tB1
= α / γ B1
z’
y’
x’
B1
α1
M
ω1
11 Bγω −=
RF pulse
FYS-KJEM 4740 Frédéric Courivaud (PhD) 16
2T
M
dt
dM xx −=
2T
M
dt
dM yy−=
02
1
*2
1B
TT∆+= γ
Relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 17
1
0
T
MM
dt
dM zz −−=
Relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 18
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)( 0MMRBMM
−−×= effdt
dγ
=
1
2
2
100
01
0
001
T
T
T
R
=
0
0 0
0
M
M
=
z
y
x
M
M
M
M
Relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 19
( )[ ] ( )110 /exp)0(/exp1)( TtMTtMtM zz −+−−=
)/exp()0()( 2TtMtM xyxy −=
Condition: relaxation during RF excitation is neglected
FYS-KJEM 4740 Frédéric Courivaud (PhD) 20
)( 0MMRM
−−=dt
d
)( 0MMRBMM
−−×= effdt
dγ
+
−−
−
−
=
10'
'
'
11
12
2
/
0
0
/10
/10
00/1
TMM
M
M
TB
BT
T
dt
dM
z
y
x
x
x
γ
γ
Summary with excitation and relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 21
Chap. 2
Slice-Selective RF excitation
Image formation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 22
Slice Selective RF pulse
FYS-KJEM 4740 Frédéric Courivaud (PhD) 23
Z
X
Y
sample volume voxel
B0
Use of field gradient pulses in the 3 directions
Gradients coding in space
Frédéric Courivaud (PhD) KJEM/FYS 4740 24
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5
z
x
y
δBG,z (z) = Gzz
δBG,z (y) = Gyy
δBG,z (x) = Gxx
Magnetic Field Gradients
FYS-KJEM 4740 Frédéric Courivaud (PhD) 25
Principles of Slice Selection
Frédéric Courivaud (PhD) KJEM/FYS 4740 26
Magnetfelt gradient
Stigningstid (ms)
Gradient styrke (mT/m)
Slew rate (mT/m/ms) = Gradient styrke
Stigningstid
FAVOURABLE SITUATION
28FYS-KJEM 4740 Frédéric Courivaud (PhD)
B1 applied for a certain duration (ms) B1 pulse frequency profile (HZ)
RF pulse generation
θ
B0
µµµµωωωω0
00 Bω ×=γprecession frequency ω :
( )zB0
g+×=γω
Spatial information included in the
precession frequency
Effect of Magnetic Field GradientExample: use of z field gradient
Applied field gradient (g)
B0
Z
Frédéric Courivaud (PhD) KJEM/FYS 4740 29
Ω−+
=++⋅+=
γ
γ10
1
0
zGB
B
z
ΩBrGBB 10eff
If we set: Ω=γ(B0 + Gzz1), then the effective field at z=z1 becomes:
=
0
0
1B
effB
What would happen if B0 + Gzz1 –Ω/γ >> B1 ??
Slice-selective excitation @ z=z1
Slice selective RF excitation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 30
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6
+
−−
−⋅−
⋅−
=
10'
'
'
11
12
2
/
0
0
/10
/1
0/1
TMM
M
M
TB
BT
T
dt
d
z
y
x
x
x
γ
γγ
γ
rG
rGM
Ω=ωL ;Beff=B1x
All effects together: excitation, precession and relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 31
+
−−
−⋅−
⋅−
=
10'
'
'
11
12
2
/
0
0
/10
/1
0/1
TMM
M
M
TB
BT
T
dt
d
z
y
x
x
x
γ
γγ
γ
rG
rGM
Precession around z-axis
Excitation around x-axis
Transverse (Mxy) relaxation
Longitudinal (Mz) relaxation
All effects together: excitation, precession and relaxation
FYS-KJEM 4740 Frédéric Courivaud (PhD) 32
Transverse magnetization, Mxy, Relaxation and precession
MT = Mx + jMy
MT = MT (0)exp − jγr ⋅ G∫ (t)dt( )exp −t
T2
B1x=0
derive!
Longitudinal magnetization, Mz
−+
−−=
11
0 exp)0(exp1)(T
tM
T
tMtM zz
derive!
As we have seen
before
FYS-KJEM 4740 Frédéric Courivaud (PhD) 33
Transverse Magnetization, Mxy, Excitation and Precession
MT = Mx + jMy T2=∞
01)( MBjMjdt
dMT
T γγ +⋅−= rG
Condition: Mz≈M0 (How can this be achieved?)
⋅−= ∫
t
t
T dttjtAM
1
')'(exp)( Grγ
General Solution:
FYS-KJEM 4740 Frédéric Courivaud (PhD) 34
Transversal magnetisering, Mxy, Eksitasjon og presesjon
Dersom vi sier at RF puls starter ved –T/2 og varer i T
sek:
T2=∞
dtdttjtBMjTM
T
T
T
t
T ∫ ∫−
⋅−=
2/
2/
2/
10 ')'(exp)(),2/( Grr γγ
Gz(t)
-T/2 0 T/2
t
FYS-KJEM 4740 Frédéric Courivaud (PhD) 35
( ) ( )∫−
−=2/
2/
10 exp)(2/exp),2/(
T
T
zzT dttzGjtBTzGjMjzTM γγγ
Phase dispertion Fourier transform of the B1 ”envelope”
(green shape)
For a constant gradient along the z-axis: G(t) = Gz
B1(t)
Gz(t)
-T/2 0 T/2
t
FYS-KJEM 4740 Frédéric Courivaud (PhD) 36
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7
( ) ( )∫−
−=2/
2/
10 exp)(2/exp),2/(
T
T
zzT dttzGjtBTzGjMjzTM γγγ
Slice profile = Fourier transform of B1(t)
The phase of MT(z) in the x-y- plane is a function of z
For a constant gradient along the z-axis: G(t) = Gz
FYS-KJEM 4740 Frédéric Courivaud (PhD) 37
Elimination of the phase dispertion in x-y plane use of an extra gradient of opposite polarity and half the length: -Gz
( )∫−
=T
T
k
k z
T dkjkzG
kBjMzTM exp
)(),( 1
0
This gives
k=γGzt and kT=γGzT/2.
B1(t)
Gz(t)
-T/2 0 T/2
FYS-KJEM 4740 Frédéric Courivaud (PhD) 38
We wish to have a ‘block’ excitation: MT(z) = M0sin(α)between –d/2 og d/2 og MT=0 resten