Lecture 13: Field gradients
Homogeneous field
Pulsed field gradients (Gz)
Gradients
dephase transverse magnetization
(3 <3> (3> C3 C5) (3) C3 C5) C5> CS> (3
C3> <3)
= 0 units
Gradients
dephase transverse magnetization
CD
C3 C3 O
O
o
CD CD CD CD CD CO
Gz = 1 units
Gradients
dephase transverse magnetization
o o
CD
CD C3
O
o
(3) CD CD
o
Gz = 2 units
Gradients
dephase transverse magnetization
Gradient-induced dependence of phase
CD		CD		o		CD
CD		CD		o		CD
CD		CD				
CD		C3		CD		O
CD		O		CD		
CD		O		CD		O
CD		O		CD		CD
CD		CD		CD		CD
CD		CD		CD		CD
CD		CD		CD		CD
CD		CD		CD		<Z)
CD		CD				CD
CD		CD		O		
CD		CZ)		e>		CZ>
CD						CD
						
->• — jy cos(Q't) +    sin (n't)
= — j% cos(Q — jGzzt) + j^sin(Q — jGzzt)
Gradient-induced dependence of phase
Gz = AB0//\z n'(z) = Q — 'jGzZ
—jfy —»• — jfycosizi't) + j^sin(Q'£)
= -^ycos(Q -jGzzt) +^rSin(Q -^Gzzt)
<M+) = Tr{pCM>+}
= Ar72^2^0cigci(Q-^^)t 4fcBT
Performing phase correction and including relaxation
(M+) = AT^2^e-^ei(^-7fe)t
Pulsed field gradients (Gy)
AAAAAAAAA    A      A     A    A  A AA      ||| |
B
0
Gradient-induced dependence of phase
Gy —
y
—   cos(n;t) + j^x sin (Q't) = -^,cos(Q - jGvyt) + ^xsin(Q - jGyyt)
Gradient echoes
X
Wanted
Unwanted
Gz
Gradient-enhanced HSQC
Gz Cleaning echo imperfections Gz Cleaning INEPT imperfections
Gradient-enhanced HSQC
7W
± ± y I 4J I 4J |
3C or15N
Use of gradients
• Cleaning, filtering, selection similar use as phase cycling
• Translational diffusion measurement
• Imaging
GRADIENTS AND MAGNETIC RESONANCE IMAGING
Lars G. Hanson Copenhagen University Hospital Hvidovre
http: //eprints. drcmr. dk/37/l/MRI_English_a4. pdf
Slice selection by G
Slice selection by G
Selective pulse: amplitude modulation
Slice selection
o		cd				O
cd		ct>		o		cd
		c3				
(3)		c3		cd		cd
o		o		cd		c^d
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				<5d		
V2/				V2/		^2/
K
4,BT ^6
KeiS2t-R2t(Af(z)e-ikzZ
kz
Slice selection
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cd				cd		cd
cd		o		<£>		
cd		c3		cd		o
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cs>		cd		cd		<z>
		cd				cd
o		cd		O		
o		cd				cd
cd		^d				
V2/						^2/
7Gzz = Q : 7GZ* 7^ Q :
<M+) <M+)
,i(0)A
v
0
2
Axial slice selection by G
l
-yGzz = Q. :     <M+) = K (e"^ N(z) J^Wj jGzz ^ Q :     <M+) = K (e-Rzt) Af(z) (eK^-iGzz
0
Magnetization in the slice
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in the slice with
gradient G
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Regular patterns enhance signal
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Image reconstruction
• resembles diffraction methods (crystallography)
• wavelength of the phase patterns generated by gradients
• wavelength of the radio waves is irrelevant (but starts to interfere at high field, where it approaches the body dimensions)
• Q assumed to be identical
differences must be corrected to avoid artifacts
k space
k space
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ft			* •	* * t t
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-1		1	j		t i -	1	-1	i	-1	+1	-1	-1	+1
See Figure 15 in
http: //eprints. drcmr. dk/37/l/MRI_English_a4. pdf
ID imaging in the slice
Frequency encoding gradient
L
(M+)(kx) = f Ke'nt-R^Af(x)e-]kxX6x
o
oo
f Ke^t-R2tj^f(x-)e-\kxx6x J K'
—oo JV
Signal (M_|_)(fcx) is Fourier transform of spin density Af(x)
Spin density Af(x) can be reconstructed by Fourier transformation of the signal (M+)(kx)
Frequency encoding gradients
oo
(M+)(kx)  «  K' / Af(x)e-lkxX6x
—OO
AtAf = 1
N
kx = 7^x^ = n • Afcx a; = jAx A/ex  = 7G*A£ = 7 x
ATA/
A/-1
n=0
Better resolution than slice thickness
ID imaging in the slice
2D imaging in the slice
2D frequency encoding possible
Two frequency encoding gradients
oo oo
(M+)(kx,ky)   «  K' /   / Af(x,y)e-*(kxX+kyy^dxdy
—oo —oo
1 , 1
AtoAfo = — AtiAfi = —
NX Ny
A fcx  = jGxAt2 =
ATxA/2
Afcy     =     jGyAtx =      1 V
Ny&h
N(x,y) = —-^t—   E   E (M+)(kx,ky)e
nx=0 ny=0
Frequency and phase encoding gradients
Ak
y
y
[ ^tXAGy
MRI spin echo experiment
Phase encoding typical in MRI
MRI spin echo experiment
Frequency encoding in x
MRI spin echo experiment
Phase encoding in y
Frequency and phase encoding gradients
oo oo
(M+)(kx,ky)  «  K' I   / Af(x,y)e-^kxX+kyy^dix6y
—oo —oo
AtA / = —
X
kx  = lGxt = nx ■ Afcx x = jx£±x
ky    =    iGyty = ny • Afcy 7/ = jyA*/
AT^A/
Afcy    = jty/\.Gy
MRI spin echo experiment
Pre-phase gradient in x (—x —► x)
Frequency and phase encoding gradients
oo oo
(M+)(kx,ky)  «  K' I   / Af(x,y)e-^kxX+kyy^dix6y
—oo —oo
AtA / = —
X
kx  = lGxt = nx ■ Afcx    x = jx£±x
ky    =    iGyty = ny • Afcy       7/ = jyA*/
AT^A/
Axial slice selection by G
Sagittal slice selection by G
Coronal slice selection by G
3D gradient echo imaging
High resolution in all dimensions More time consuming
3D gradient echo imaging
Short (~ 10°) pulse to save time
=>• several measurements before return to equilibrium
Two phase encoding gradients
<M+)(fc) « K' jM(ř)e-'^óV
v
oo   oo oo
(M+)(kx,ky,kz)   fíä  K' /    /    / ÄÍ(x,y,z)e-^kxX+kyy+k^ÓxÓyÓ
—oo —OO —OO
kx = ^Gxtx = %Afcx x = Ax
fcy = 7Gyiy = riytAky y = Ay
fcz = 7G^t = nzt/\kz z = jz^z
Afcx = ^tx/\Gx
Afcy     = ^ty/\Gy
lGz
. r/ x A/cx A/cy A/c^ ^ ^ ^ Í2tt
M(x,y,z) = -^—EEE<M+>e
rix ny nz
/ Jx-nx i	i 3VnV i	
I, 1	1     Ny 1	1 N z J
Contrast and weighting
Contrast is more important than intensity
(M+XM = ^e,nt^(l-e-fliTR)e-^W(x)e-,;^^ if not started from thermodynamic equilibrium
(M+)(fe) oc / (l - e-ßiTR) e-^E ^(r) e"ife-W
Ti weighting: difference in i?i = 1/Ti, short TR and TE T2 weighting: difference in     = I/T2, long TR and TE spin density weighting: difference in A/\ long TR, short TE