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3/25/2014
EE 519 Advanced Topics in Medical Imaging 1
FUNDAMENTALS OF
MAGNETIC RESONANCE
IMAGING (MRI)
EE 519 Advanced Topics in
Medical Imaging
(Spring 2009)SIGNAL
LOCALIZATION
EE 519 Advanced Topics in
Medical Imaging2
3/25/2014
EE 519 Advanced Topics in Medical Imaging 2
• In practice, the objects we deal with are
heterogeneous, and it becomes necessary to
differentiate signals from different parts of
the object.
• There are basically two types of spatial
localization method:
– selective excitation (or reception),
– spatial encoding
• Central to localization methods of both types
is the use of a gradient field.
• To facilitate the understanding of the role of
RF pulses and gradient fields in a general
imaging scheme we shall describe the basic
concepts of
– slice-selective excitation, and
– spatial information encoding.
and then discuss multidimensional imaging
using mathematical formalism known as k-
space.
EE 519 Advanced Topics in
Medical Imaging3
Slice selection
• To selectively excite spins in a slice,
two things are essential:
– a gradient field,
– a shaped RF pulse.
Line equation:
EE 519 Advanced Topics in
Medical Imaging4
x
y
)sin,(cos θθµ =s
r
tyx =+ θθ sincos
θ
t
rr
tr s =⋅ µrr
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EE 519 Advanced Topics in Medical Imaging 3
Slice equation
EE 519 Advanced Topics in
Medical Imaging5
0srs =⋅ rrµ
2/0 ssrs ∆<−⋅ rrµ slice thicknessis explicitly specified.
x
y
z
θs0
s∆
(Gençer, 2009)
rr
sµr
Slice definition:
OR
Special cases
EE 519 Advanced Topics in
Medical Imaging6
z=z0
transverseslice
y=y0coronalslice
x=x0sagittalslice
(Gençer, 2009)
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EE 519 Advanced Topics in Medical Imaging 4
EE 519 Advanced Topics in
Medical Imaging7
Slice selection gradients
• An RF pulse can only be frequency
selective and spins at different locations
will be excited in the same way if they
resonate at the same frequency.
• To make an RF pulse spatially selective,
it is necessary to make the spin
resonance frequency position
dependent, or more desirably, to vary
linearly along the slice select direction
.
• One should augment the homogeneous
B0 field with a linear gradient during the
excitation period. Such a gradient field
is called a slice-selection gradient.
EE 519 Advanced Topics in
Medical Imaging8
sµr
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EE 519 Advanced Topics in Medical Imaging 5
Slice selection gradients
EE 519 Advanced Topics in
Medical Imaging9
(Bernstein, 2004)
Linear gradient field
• A linear gradient field
– points along z-direction,
– has an amplitude that varies linearly along
a particular gradient direction .
• Denote the desired slice-selection
gradient as
where
• If the amplitude of the gradient field
varies linearly along , then spin
resonance frequency varies linearly
along .EE 519 Advanced Topics in
Medical Imaging10
Gµr
222
),,(
zyxSS
GSSzyxSS
GGGG
GGGGG
++=
== µrr
Gµr
Gµr
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EE 519 Advanced Topics in Medical Imaging 6
Need to match gradient direction
to slice-select direction
• Expressing in terms of slice-orientation
angles as
• We have
• specifies the necessary gradient
vector for selecting a slice in an arbitrary
direction.
EE 519 Advanced Topics in
Medical Imaging11
),( φθGµr
)cos,sinsin,cos(sin θφθφθµ =s
r
θφθφθ
cos
sinsin
cossin
ssz
ssy
ssx
GG
GG
GG
===
SSGr
Slice-selective RF pulses
• The next step is to translate the
desired frequency selectivity
established by the slice-selection
gradient to the temporal waveform of
an RF pulse.
• RF pulse is of the form
• So, the question now is how to select
and .
EE 519 Advanced Topics in
Medical Imaging12
tie rfetBtB ω−= )()( 11
rr
)(1 tBe
rfω
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EE 519 Advanced Topics in Medical Imaging 7
The Fourier Transform Approach
EE 519 Advanced Topics in
Medical Imaging13
Define a spatial selection function )(zps
∆−=
∆<−
=
z
zzrect
otherwise
zzzzps
0
0
0
2/1)(
⇒ Necessary slice selection gradient is,
),0,0( zss GG =r
In the presence of this gradient, the total field is
kzGB
krGBB
z
SSr
rrrr
)(
)(
0
0
+=
⋅+=
z
z0z∆
)(zps
The Fourier Transform Approach
• The Larmor frequency at position z is
given by
• Thus, given a slice selection function,
��, it is possible to find the
corresponding excitation frequencies.
• The required RF pulse must be
composed of frequencies ranging as
given in the following figure.
EE 519 Advanced Topics in
Medical Imaging14
zGz zγωω += 0)(
z∆
Gz
z
ω(z)
ω0
ω1
ω2
z0
ωcω∆
00 zG
zG
zc
z
γωωγω
+=∆=∆
(Gençer, 2009)
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EE 519 Advanced Topics in Medical Imaging 8
The desired frequency
selection function
EE 519 Advanced Topics in
Medical Imaging15
f
fcf∆
)( fps
( ){ }tBF
f
ffrect
otherwise
ffffp
c
cs
1
0
2/1)(
=
∆−
=
∆<−
=
)(sin)(
2
)(sin)(
1
00
2
1
ftcAtB
zGf
etfcftB
e
zcrf
tfi c
∆=
+==∆∆∝ −
πγωπω
π π
where A is a constant to be determined by the desired flip angle:
∫=p
dttB e
τ
γα0
1 )(
Physically realizable pulses
EE 519 Advanced Topics in
Medical Imaging16
pp
e ttfcAtB ττπ ≤≤−∆= 0))2/((sin)(1
pτ2/pτ
(Gençer, 2009)
not realizable
realizable
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EE 519 Advanced Topics in Medical Imaging 9
Post-excitation rephasing
• The resulting slice selection profile of
the above shifted and truncated pulse
(temporarily ignoring the pulse
truncation effect)
EE 519 Advanced Topics in
Medical Imaging17
2/)(2
2/)(2
0
2/)(
pc
pc
ffic
c
ffi
s
ef
ffrect
otherwise
fffefp
τπ
τπ
−
−
∆−=
∆<−
=
)( 0
00
0 zzGffzGff
zGffzc
zc
z −=−
+=+=
γγγ
2/)(2/)(2 0 pzpc zzGiffi ee τγτπ −− =
Post excitation rephasing
• Thus, a linear phase shift is introduced
accross the slice thickness by the slice
selective gradient.
• If not corrected, this phase shift can
lead to undesirable signal loss.
• Since the phase shift is a linear
function of z, it can be removed by
applying a refocusing z-gradient after
the RF pulse.
• This procedure is called post excitation
rephasing.
EE 519 Advanced Topics in
Medical Imaging18
Rephasing gradient
Rephasing period
Phase angle during
the rephasing period),(
,
tz
G
r
zr
φτ
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EE 519 Advanced Topics in Medical Imaging 10
How to find rephasing parameters?
EE 519 Advanced Topics in
Medical Imaging19
ppzr
p
z ttzzGzzGtz ττγτ
γφ ≥−−+−= ))((2
)(),( 0,0
0)(2
)(),( 0,0 =−+−=+ rzr
p
zrp zzGzzGz τγτ
γττφ
rzr
p
z GG ττ
,2−=
Therefore, fixing τr, one can determine Gr,z, or vice versa.
Let τr = τp/2 then Gr,z =-Gz
In general, if Gss=(Gx,Gy,Gz) is used for slice selection, the refocusing gradient Gr=(Grx,Gry,Grz) satisfies the following equation:
pzrzr
pyryr
pxrxr
GG
GG
GG
ττ
ττ
ττ
2
12
12
1
,
,
,
−=
−=
−=
Pulse truncation effects
• To have a perfect rectangular
excitation profile in the frequency
domain, we need a pulse of infinite
length in the time domain.
• In practice, the pulse has to be
truncated to finite duration to be
useful.
• Pulse truncation can be characterized
mathematically by the multiplication
of an infinite pulse with a rectangular
function.
• The truncated sinc pulse can be
expressed as
EE 519 Advanced Topics in
Medical Imaging20
−
−∆=p
ppe trecttfcAtB
τττ
π2/
)2
(sin)(ˆ1
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EE 519 Advanced Topics in Medical Imaging 11
Pulse truncation results
• A non-uniform excitation profile
accross the slice thickness.
• Excitation of spins in the neighboring
slices (cross-talk artifact)
• To minimize the pulse truncation
effects, it is necessary to pack as many
sidelobes into the pulse as possible.
EE 519 Advanced Topics in
Medical Imaging21
Number of side lobes
versus gradient field strength
EE 519 Advanced Topics in
Medical Imaging22
pτ�
ωπ /
2
zGz
∆= γω
ω t
If one assumes n side lobes on each side of the sinc function,
( )
p
z
p
zG
n
n
τγ
π
τωπ
=
∆
=
2
2
2
zG
n
z
p ∆=
γπτ 4
For a given pulse length , if the number of side lobes in the pulse is increased to reduce the cross-talk,
must be increased to maintain the same slice thickness .
pτ
zGz∆
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EE 519 Advanced Topics in Medical Imaging 12
Spatial Information Encoding
• After a signal has been activated by a
selective or nonselective pulse, spatial
information can be encoded into the
signal during free precession period.
• Since an activated MR signal is in the
form of a complex exponential, we
have essentially two ways to encode
spatial information
1. frequency encoding, and
2. phase encoding.
EE 519 Advanced Topics in
Medical Imaging23
Frequency Encoding
• Frequency encoding makes the
oscillation frequency of an activated
MR signal linearly dependent on its
spatial origin.
• Consider a one-dimensional object
with spin density function ρ(x)
(number of spins per unit length).
• The object experiences (after an
excitation) a homogeneous B0 field
plus a gradient field Gxx, i.e., the
Larmor frequency at position x is
EE 519 Advanced Topics in
Medical Imaging24
xGx xγωω += 0)(
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EE 519 Advanced Topics in Medical Imaging 13
Frequency Encoding
• Omitting the transverse relaxation
effect, the signal received from the
entire object is
• After demodulation, i.e., removal of
the carrier signal, we have
• The signal is frequency encoded
because its oscillation frequency is
linearly related to the spatial location.
For the same reason Gx is called a
frequency encoding gradient.
EE 519 Advanced Topics in
Medical Imaging25
∫∞
∞−
+−= dxextS txGBi x )( 0)()( γρ
∫∞
∞−
−= dxextS xtGi xγρ )()(
26
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EE 519 Advanced Topics in Medical Imaging 14
Generalized case
• The frequency-encoded FID signal
after demodulation is, in general,
given by
where Gfe is the frequency-encoding
gradient defined by
• A general frequency-encoded echo
signal can be expressed as
EE 519 Advanced Topics in
Medical Imaging27
rdertSobject
trGi ferr rr
∫⋅−= γρ )()(
),,( zyxfe GGGG =
rdertSobject
TtrGi Eferr rr
∫−⋅−= )()()( γρ
Iso-frequency lines established by
the frequency-encoding gradient
EE 519 Advanced Topics in
Medical Imaging28
x
y
feGr
(Gençer, 2009)
For a fixed freqeuency-encoded gradient vector Gfe,spatial information is frequency encoded along thedirection of Gfe. In other words, only one-dimensionalspatial localization along the gradient direction is achieved.
For multi-dimensional localization, multiple encodedsignals are necessary.
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EE 519 Advanced Topics in Medical Imaging 15
NMR signal characteristics in the presence
of a frequency-encoding gradient
EE 519 Advanced Topics in
Medical Imaging29
(Bernstein, 2004)
Phase encoding
• Assume we turn on a gradient Gx for a
short interval Tpe, and then turn it off.
• The local signal under the influence of
this gradient is
• During the interval 0 < t < Tpe, the local
signal is frequency encoded. As a
result of this frequency encoding,
signals from different x positions
accumulate different phase angles
after a time interval Tpe:
EE 519 Advanced Topics in
Medical Imaging30
≥≤≤
= +−
+−
peTxGBi
petxGBi
Ttedxx
TtedxxtxdS
pex
x
)(
)(
0
0
)(
0)(),( γ
γ
ρρ
pex xTGx γφ −=)(
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EE 519 Advanced Topics in Medical Imaging 16
31
Phase encoding terms
• Since φ(x) is linearly related to the
signal we shall call
– the signal phase encoded,
– the gradient Gx phase encoding gradient,
– Tpe as the phase encoding interval.
• Phase encoding along an arbitrary
direction can also be done for a
multidimensional object by turning on
Gx, Gy, and Gz simultaneously during
0< t <Tpe.
• The initial phase angle is now given by
EE 519 Advanced Topics in
Medical Imaging32
pepe TrGrrrr ⋅−= γφ )(
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EE 519 Advanced Topics in Medical Imaging 17
Phase encoding along
an arbitrary direction
EE 519 Advanced Topics in
Medical Imaging33
pepe TrGrrrr ⋅−= γφ )(Initial phase:
ti
object
TrGierdertS pepe 0)()( ωγρ −⋅−
= ∫
rr rr
The received signal:
pezyxpe TtGGGG ≤≤= 0),,(r
The gradient vector:
To gain a better understanding of both phase and frequency encoding schemes, we next look at them from a k-space perspective.
A k-space interpretation
• This section establishes an important
connection between spatial encoding
(phase encoding and frequency
encoding) and the Fourier Transform.
Frequency Encoded Signals
Consider the frequency encoded signal
and Fourier Transform relationship
below:
EE 519 Advanced Topics in
Medical Imaging34
∫
∫∞
∞−
−
∞
∞−
−
=
=
dxexkS
dxextS
xkix
xtGi
x
x
π
γ
ρ
ρ
2)()(
)()(
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EE 519 Advanced Topics in Medical Imaging 18
Frequency encoded signals
• In the above equation we have the
following relations:
• The role of the frequency-encoding
gradient Gx is to map a time signal to a
k-space signal.
• When multiple gradients are used for
frequency encoding, the mapping
relationship between t and k is given
by
EE 519 Advanced Topics in
Medical Imaging35
−=
signalsechoTtG
signalsFIDtGk
Ex
xx )()2/(
)2/(
πγπγ
−=
signalsechoTtG
signalsFIDtGk
Efe
fe
)()2/(
)2/(r
rr
πγπγ
Frequency-encoded signals
• When multiple gradients are used for
frequency encoding, the corresponding
k-space signal is
• Note that the k-space sampling
trajectory of a frequency-encoded signal
is a straight line only if constant
gradients are used. If the gradients
are time-varying, then
EE 519 Advanced Topics in
Medical Imaging36
∫⋅−=
object
rki rderkSrrr rr
πρ 2)()(
∫=t
fe dGtk0
)()2/()( ττπγr
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EE 519 Advanced Topics in Medical Imaging 19
37
xk
yk
φ
Examples
EE 519 Advanced Topics in
Medical Imaging38
=
−+=−=
−=−=
=
−
x
y
EyxEfe
Eyy
Exx
G
G
TtGGTtGk
TtGk
TtGkk
1
22
tan
)()2/()()2/(
)()2/(
)()2/(
φ
πγπγ
πγπγ
r
r
xk
yk
φ
=
+==
==
=
−
x
y
yxfe
yy
xx
G
G
tGGtGk
tGk
tGkk
1
22
tan
)2/()2/(
)2/(
)2/(
φ
πγπγ
πγπγ
r
r
xk
yk
φ
FID signal
Echo signal
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EE 519 Advanced Topics in Medical Imaging 20
39
Example:
40
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EE 519 Advanced Topics in Medical Imaging 21
41 42
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EE 519 Advanced Topics in Medical Imaging 22
43
Example
ky
kx
44
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EE 519 Advanced Topics in Medical Imaging 23
45
Phase encoded signals:
46
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EE 519 Advanced Topics in Medical Imaging 24
47 48
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EE 519 Advanced Topics in Medical Imaging 25
49
Basic Imaging Methods:
50
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EE 519 Advanced Topics in Medical Imaging 26
51
x
x
52
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EE 519 Advanced Topics in Medical Imaging 27
53 54
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EE 519 Advanced Topics in Medical Imaging 28
55 56
Two dimensional imaging:
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EE 519 Advanced Topics in Medical Imaging 29
57 58
• A basic task of planar imaging is to generate
sufficient data to cover k-space. Following
imaging pulse sequence excites an object
periodically with a pair of slice-selective 90o
and 180o pulses, which generate a set of
spin-echo signals. Spatial information is then
encoded in the spin-echo signals by two-
dimensional frequency encoding.
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EE 519 Advanced Topics in Medical Imaging 30
59 60
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61 62
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63 64
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65 66
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EE 519 Advanced Topics in Medical Imaging 34
67