1 TT Liu, BE280A, UCSD Fall 2007 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2007 MRI Lecture 2 TT Liu, BE280A, UCSD Fall 2007 Gradients Spins precess at the Larmor frequency, which is proportional to the local magnetic field. In a constant magnetic field B z =B 0 , all the spins precess at the same frequency (ignoring chemical shift). Gradient coils are used to add a spatial variation to B z such that B z (x,y,z) = B 0 +Δ B z (x,y,z) . Thus, spins at different physical locations will precess at different frequencies. TT Liu, BE280A, UCSD Fall 2007 Simplified Drawing of Basic Instrumentation. Body lies on table encompassed by coils for static field B o , gradient fields (two of three shown), and radiofrequency field B 1 . MRI System Image, caption: copyright Nishimura, Fig. 3.15 TT Liu, BE280A, UCSD Fall 2007 Z Gradient Coil B(mT) L Credit: Buxton 2002
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1
TT Liu, BE280A, UCSD Fall 2007
Bioengineering 280APrinciples of Biomedical Imaging
Fall Quarter 2007MRI Lecture 2
TT Liu, BE280A, UCSD Fall 2007
GradientsSpins precess at the Larmor frequency, which isproportional to the local magnetic field. In a constantmagnetic field Bz=B0, all the spins precess at the samefrequency (ignoring chemical shift).
Gradient coils are used to add a spatial variation to Bzsuch that Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins atdifferent physical locations will precess at differentfrequencies.
TT Liu, BE280A, UCSD Fall 2007
Simplified Drawing of Basic Instrumentation.Body lies on table encompassed by
coils for static field Bo, gradient fields (two of three shown),
Rotating Frame of ReferenceReference everything to the magnetic field at isocenter.
TT Liu, BE280A, UCSD Fall 2007
Interpretation
∆x 2∆x-∆x-2∆x 0
∆Bz(x)=Gxx
!
exp " j2#1
8$x
%
& '
(
) * x
%
& '
(
) *
!
exp " j2#2
8$x
%
& '
(
) * x
%
& '
(
) *
!
exp " j2#0
8$x
%
& '
(
) * x
%
& '
(
) *
FasterSlower
3
TT Liu, BE280A, UCSD Fall 2007Fig 3.12 from Nishimura
kx=0; ky=0 kx=0; ky≠0
TT Liu, BE280A, UCSD Fall 2007
Phase with time-varying gradient
TT Liu, BE280A, UCSD Fall 2007
K-space trajectoryGx(t)
tt1 t2
ky
!
kx(t1)
!
kx(t2)
Gy(t)
t3 t4kx
!
ky (t4 )
!
ky (t3)
TT Liu, BE280A, UCSD Fall 2007 Nishimura 1996
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TT Liu, BE280A, UCSD Fall 2007
K-space trajectoryGx(t)
tt1 t2
ky
Gy(t)
kx
TT Liu, BE280A, UCSD Fall 2007
Spin-WarpGx(t)
t1
ky
Gy(t)
kx
TT Liu, BE280A, UCSD Fall 2007
Spin-WarpGx(t)
t1 ky
Gy(t)
kx
TT Liu, BE280A, UCSD Fall 2007
Spin-Warp Pulse Sequence
Gx(t)
ky
kx
Gy(t)
RF
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TT Liu, BE280A, UCSD Fall 2007
Gradient Fields
!
Gx x + Gy y + Gzz =r G "
r r
!
r G "Gx
ˆ i + Gyˆ j + Gz
ˆ k
!
Bz(r r ,t) = B
0+
r G (t) "
r r
Define
!
r r " xˆ i + yˆ j + z ˆ k
So that
Also, let the gradient fields be a function of time. Thenthe z-directed magnetic field at each point in thevolume is given by :
TT Liu, BE280A, UCSD Fall 2007
Static Gradient Fields
!
M(t) = M(0)e" j#0te
" t /T2
In a uniform magnetic field, the transverse magnetizationis given by:
In the presence of non time-varying gradients we have
!
M (r r ) = M (
r r ,0)e
" j#Bz (r r )t
e" t /T2 (
r r )
= M (r r ,0)e
" j# (B0+r
G $r r )t
e" t /T2 (
r r )
= M (r r ,0)e
" j%0te" j#
r G $
r r te" t /T2 (
r r )
TT Liu, BE280A, UCSD Fall 2007
Time-Varying Gradient FieldsIn the presence of time-varying gradients the frequencyas a function of space and time is:
!
"r r ,t( ) = #B
z(r r ,t)
= #B0
+ #r G (t) $
r r
="0
+ %"(r r ,t)
TT Liu, BE280A, UCSD Fall 2007
PhasePhase = angle of the magnetization phasorFrequency = rate of change of angle (e.g. radians/sec)Phase = time integral of frequency
!
"#r r ,t( ) = $ "%(
r r ,& )
0
t
' d&
= $ (v
G (r r ,& ) )
r r
0
t
' d&
!
"r r ,t( ) = # $(
r r ,% )
0
t
& d%
= #$0t + '"
r r ,t( )
Where the incremental phase due to the gradients is
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TT Liu, BE280A, UCSD Fall 2007
Phase with constant gradient
!
"#r r ,t
3( ) = $ "%(r r ,& )
0
t3
' d&
!
"#r r ,t2( ) = $ "%(
r r ,& )
0
t2
' d&
= $"%(r r )t2
if "% is non - time varying.
!
"#r r ,t
1( ) = $ "%(r r ,& )
0
t1
' d&
TT Liu, BE280A, UCSD Fall 2007
Time-Varying Gradient FieldsThe transverse magnetization is then given by
!
M(r r ,t) = M(
r r ,0)e
" t /T2 (r r )
e# (
r r ,t )
= M(r r ,0)e
" t /T2 (r r )
e" j$0t exp " j %$
r r ,t( )d&
o
t
'( )= M(
r r ,0)e
" t /T2 (r r )
e" j$0t exp " j(
r G (&) )
r r d&
o
t
'( )
TT Liu, BE280A, UCSD Fall 2007
Signal EquationSignal from a volume
!
sr(t) = M(r r ,t)
V" dV
= M(x,y,z,0)e# t /T2 (
r r )
e# j$0t exp # j%
r G (&) '
r r d&
o
t
"( )z"
y"
x" dxdydz
For now, consider signal from a slice along z and dropthe T2 term. Define
!
m(x,y) " M(r r ,t)
z0#$z / 2
z0 +$z / 2
% dz
!
sr(t) = m(x,y)e" j#0t exp " j$
r G (%) &
r r d%
o
t
'( )y'
x' dxdy
To obtain
TT Liu, BE280A, UCSD Fall 2007
Signal EquationDemodulate the signal to obtain
!
s(t) = ej" 0t sr(t)
= m(x,y)exp # j$r G (% ) &
r r d%
o
t
'( )y'
x' dxdy
= m(x,y)exp # j$ Gx (%)x + Gy (%)y[ ]d%o
t
'( )y'
x' dxdy
= m(x,y)exp # j2( kx (t)x + ky (t)y( )( )y'
x' dxdy
!
kx (t) ="
2#Gx ($ )d$0
t
%
ky (t) ="
2#Gy ($ )d$0
t
%
Where
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TT Liu, BE280A, UCSD Fall 2007
MR signal is Fourier Transform
!
s(t) = m(x,y)exp " j2# kx (t)x + ky (t)y( )( )y$
x$ dxdy
= M kx (t),ky (t)( )= F m(x,y)[ ]
kx (t ),ky (t )
TT Liu, BE280A, UCSD Fall 2007
Recap• Frequency = rate of change of phase.• Higher magnetic field -> higher Larmor frequency ->
phase changes more rapidly with time.• With a constant gradient Gx, spins at different x locations
precess at different frequencies -> spins at greater x-valueschange phase more rapidly.
• With a constant gradient, distribution of phases across xlocations changes with time. (phase modulation)
• More rapid change of phase with x -> higher spatialfrequency kx
TT Liu, BE280A, UCSD Fall 2007
K-space
!
s(t) = M kx (t),ky (t)( ) = F m(x,y)[ ]kx ( t ),ky ( t )
!
kx (t) ="
2#Gx ($ )d$0
t
%
ky (t) ="
2#Gy ($ )d$0
t
%
At each point in time, the received signal is the Fouriertransform of the object
evaluated at the spatial frequencies:
Thus, the gradients control our position in k-space. Thedesign of an MRI pulse sequence requires us toefficiently cover enough of k-space to form our image.
TT Liu, BE280A, UCSD Fall 2007
K-space trajectoryGx(t)
t
!
kx(t) =
"
2#G
x($ )d$
0
t
%
t1 t2
kx
ky
!
kx(t1)
!
kx(t2)
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TT Liu, BE280A, UCSD Fall 2007
UnitsSpatial frequencies (kx, ky) have units of 1/distance.Most commonly, 1/cm
Gradient strengths have units of (magneticfield)/distance. Most commonly G/cm or mT/m