Magnetic Resonance Image Formation

1
Magnetic Resonance Image Formation
Daniel Bulte FMRIB Centre
2
MRI System Block Diagram
X amp
Y amp
Z amp
spectrometer
magnet
r.f. coil
gradient coil
3
Alignment of Spins in a Magnetic Field
M
M0
B0 field
4
Energy in a Magnetic Field(Zeeman Splitting,
Spin ½)
E1/2 ??hB0/2
E-1/2 ?hB0/2
mI ½
mI ?½
P1/2 0. 5000049
P-1/2 0.4999951
1.5T, T310K, P(E)?exp(?E/kT)
5
Larmor Frequency
mI ?½
mI ½
E1/2 ??hB0/2
E-1/2 ?hB0/2
Allowed transitions ?E ?hB0
h?0
?0 ?B0
6
Free Induction Decay
FT
M
time
frequency
FT
time
frequency
7
T1 Relaxation
Mz(t) M0 Mz(0) ? M0exp(-t/T1)
saturationrecovery
inversionrecovery
M0
M0
Mz
Mz
t
t
Mz(0) ?M0
Mz(0) 0
8
T1 Weighted Imaging
white matter
æ
ö
grey matter
T
T
T
b
ç

ln
1
a
b
1
1
è
ø
T

a
1
Optimal
TR
-
Contrast
T
T
a
b
1
1
TR
Optimal TR
9
T1 Weighted Image
T1/s
R1/s-1
white matter
0.7
1.43
grey matter
1
1
CSF
4
0.25
1.5T
SPGR, TR14ms, TE5ms, flip20º
10
T2 Relaxation
dMxy(t) ? Mxy(t)
dt
T2
Mxy(t) Mxy(0) exp(?t/T2)
Mxy
t
11
T2 Weighted Imaging
EchoAmplitude
grey
white
Contrast
TE
Optimum TE
12
T2 Weighted Image
T2/ms
CSF
500
80?90
grey matter
70?80
white matter
1.5T
SE, TR4000ms, TE100ms
13
Free Induction Decay
M
FT
time
frequency
FT
Note Signal only detected from Mxy component
time
frequency
14
1D Imaging Example
field
x
15
1D Imaging Example
90º pulse
field
x
16
1D Imaging Example
gradient on
field
x
17
1D Imaging Example
gradient on
field
x
18
1D Imaging Example
gradient on
field
x
19
1D Imaging Example
gradient on
field
x
20
1D Imaging Example
gradient on
field
x
21
1D Imaging Example
gradient on
field
x
22
1D Imaging Example
gradient on
field
x
23
1D Imaging Example
gradient on
field
x
24
1D Imaging Example
gradient on
field
x
25
1D Imaging Example
gradient on
field
x
26
1D Imaging Example
gradient on
field
x
27
1D Imaging Example
gradient on
field
x
28
1D Imaging Example
gradient on
field
x
29
1D Imaging Example
gradient on
field
x
30
1D Imaging Example
gradient on
field
x
31
1D Imaging Example
Signal from left-hand tube
Signal from right-hand tube
Measured signal from both tubes
32
1D Imaging Example
S(t)
S(?)
?
t
Fourier Transform
33
2D Back Projection Imaging
y
Oblique
x
34
The Fourier Transform
FFT
35
The Fourier Transform
FFT
Jean-Baptiste-Joseph Fourier
36
1D Fourier Transform
?(x) a0 b0 a1 cos(?x/xmax) a-1
cos(??x/xmax)
b1 sin(?x/xmax) b-1 sin(??x/xmax)
a2 cos(2?x/xmax) a-2 cos(?2?x/xmax)
b2 sin(2?x/xmax) b-2 sin(?2?x/xmax)
37
1D Fourier Transform
a0
b0
cumulative sum
a0
b0
38
1D Fourier Transform
a1
b1
cumulative sum
a1 cos(?x/xmax)
b1 sin(?x/xmax)
39
1D Fourier Transform
a2
b2
cumulative sum
a2 cos(2?x/xmax)
b2 sin(2?x/xmax)
40
1D Fourier Transform
a63
b63
cumulative sum
a63 cos(63?x/xmax)
b63 sin(63?x/xmax)
41
1D Fourier Transform
a0
a32
a63
a?32
a?63
b0
b32
b63
b?32
b?63
42
2D Extension
43
2D Extension
a?63,63
a63,63
a?63,0
a63,0
a?63,?63
a63,?63
?(x,y) ? an,m cos(n?x/xmax m?y/ymax)
? bn,m sin(n?x/xmax m?y/ymax)
44
2D Fourier Transform
?(x,y)
S(kx,ky)
y
ky
x
kx
?(x,y) ?? S(kx,ky) exp2?i(kxx kyy) dkxdky
S(kx,ky) ?? ?(x,y) exp?2?i(kxx kyy) dxdy
45
Full k Space Coverage
ky
kx
46
Only Centre of k Space
ky
kx
47
Only Edges of k Space
ky
kx
48
Why Use k Space?
? ?B0
Larmor equation
x0
x??1cm
x??1cm
?(x,y) ?B0 ?Gxx ?Gyy
?(x,y,t) 2? ??B0dt 2? ??Gxxdt 2? ??Gyydt
phase
?S(x,y,t) ?(x,y) expi ?(x,y,t)
elemental signal
S(t) ???(x,y) expi ?(x,y,t) dxdy
total signal
49
A Few Substitutions
S(t) ???(x,y) expi ?(x,y,t) dxdy
total signal
From
kx(t) ??Gxdt
ky(t) ??Gydt
In rotating frame
S(t) ???(x,y) exp2?i(kxxkyy) dxdy
total signal
To
This is the standard Fourier Equation!
50
How to Picture k Space
kx(t) ??Gxdt
ky(t) ??Gydt
1) kx and ky are measures of the x and y gradient
history
2) kxky0 following excitation of Mz into
transverse plane
51
Gradient-Echo Sequence
r.f
Gx
Gy
1st phase encode step
52
Gradient-Echo Sequence
r.f
Gx
Gy
2nd phase encode step
53
Gradient-Echo Sequence
r.f
Gx
Gy
centre phase encode step
54
Gradient-Echo Sequence
r.f
Gx
Gy
last phase encode step
55
Gradient-Echo Sequence
r.f
Gx
Gy
all phase encode steps
56
Gradient-Echo EPI Sequence
r.f
Gx
Gy
snap shot sequence
57
Interleaved EPI
r.f
Gx
Gy
n
1st interleave
2nd interleave
etc...
58
Fast Spin Echo
180?
180?
180?
90?
r.f
Gx
Gy
n
1st interleave
2nd interleave
etc...
59
Fast Spin Echo
180?
180?
180?
180?
180?
180?
90?
r.f
Gx
Gy
n
1st interleave
echo train length (ETL) 6
60
k Space Symmetry
cid
aib
When data are phased in to pure cosine and
sine terms
k space is hermitian conjugate symmetric
a?ib
c?id
61
Phase Correcting Images
real part
real part
imaginary part
imaginary part
pre-correction
post-correction
62
Partial k Space Acquisitions
measured data
missing data
cid
aib
p.e. direction
Half NEX acquisition
use centre lines to generate phase correction
a?ib
c?id
cut scan time almost in half
but SNR worse by ?2
read direction
63
Fractional NEX
r.f
Gx
Gy
limited number of phase encode steps
64
Partial k Space Acquisitions
measured data
missing data
cid
aib
p.e. direction
Min TE acquisition
use centre lines to generate phase correction
a?ib
c?id
cut min TE almost in half
but SNR worse by ?2
read direction
65
Minimum TE
r.f
r.f
Gx
Gx
Gy
Gy
min full TE
min (partial) TE
66
Field of View and Resolution
?kx 1/FOV
?kxmax
kxmax
kxmax 1/(2?x)
?kx
67
Rectangular Field of View
ky
?kxmax
kxmax
kx
?kx
68
Slice Selection
time
frequency
?0
G
69
2D Multi Slice Imaging
z1
z3
ky
z2
z4
kx
2D
Multi-Slice 2D
70
3D Imaging
ky
ky
kz
kx
2D
True 3D
kx
71
3D Imaging
r.f
ky
Gx
Gy
kz
True 3D
kx
72
Thank you

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