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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

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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

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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

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Medical Imaging6

z=z0

transverseslice

y=y0coronalslice

x=x0sagittalslice

(Gençer, 2009)

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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.

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Medical Imaging8

sµr

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Slice selection gradients

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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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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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The Fourier Transform Approach

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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.

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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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The desired frequency

selection function

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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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Post-excitation rephasing

• The resulting slice selection profile of

the above shifted and truncated pulse

(temporarily ignoring the pulse

truncation effect)

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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.

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Medical Imaging18

Rephasing gradient

Rephasing period

Phase angle during

the rephasing period),(

,

tz

G

r

zr

φτ

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How to find rephasing parameters?

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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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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.

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Medical Imaging21

Number of side lobes

versus gradient field strength

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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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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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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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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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NMR signal characteristics in the presence

of a frequency-encoding gradient

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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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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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Phase encoding along

an arbitrary direction

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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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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

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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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39

Example:

40

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41 42

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43

Example

ky

kx

44

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45

Phase encoded signals:

46

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47 48

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49

Basic Imaging Methods:

50

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51

x

x

52

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53 54

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55 56

Two dimensional imaging:

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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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59 60

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61 62

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63 64

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65 66

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67