eleg 479 lecture #12 magnetic resonance (mr) imaging

ELEG 479Lecture #12

Magnetic Resonance (MR) Imaging

Mark Mirotznik, Ph.D.Associate Professor

The University of Delaware

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Physics of Magnetic ResonanceSummary

Protons and electrons have a property called spin that results in them looking like tiny magnets.

S

NN

S

In the absence of an external magnetic field all the magnetic dipole are oriented randomly so we get zero net magnetic field when we add them all up.

RandomOrientation

=No Net

Magnetization

Physics of Magnetic ResonanceSummary

When we add a large external magnetic field we can get the protons to line up in 1 of 2 orientations (spin up or spin down) with a few more per million in one of the orientations than the other. This produces a net magnetization along the axis of the applied magnetic field.

When we add a large external magnetic field we cause a torque on the already spinning proton that causes it to precess like a top around the applied magnetic field. The frequency is precesses , called its Larmor frequency., is determined from Larmor’s equation.

Do

o PkT

BM4

22

z

x

y

z

xy

The net magnetization vector is the sum of all of these little magnetic moments added together. This is what we measure.

x

y

z

z

xy

x

y

z

z

xy

z

x

y

z

xy

x

y

z

z

xy

Net Magnetization Vector

zMtMtM

tzyxtM

zxy

N

nnnnn

ˆ)()(

),,,()(1

Physics of Magnetic ResonanceSummary from Last Lecture


However since all the spinning protons are precessing out of phase with each other this results in zero net magnetization in the transverse plane.

zMtMtM zxy ˆ)()(

= 0

This is bad news since )(tM xy

is where are signal comes from!

Somehow we need to get these guys spinning together!

RF Excitationtime

B1

B1

To get them to all spin together we add a RF field whose frequency is the same as the Larmor resonant frequency of the proton and is oriented in the xy or transverse plane.


a B1Dt

Tip Angle Amplitude of RFPulse

Time of Applicationof RF Pulse


x

y

z

M

xyM

a

B1


x

y

z

M

xyM

a

dttBe )(0 1

a

)(1 tBe

)(1 tBe= envelope of the RF signalIn general


To get the signal out we place a coil near the sample. A time-varying transverse magnetic field will produce a voltage on the coil that can be digitized and stored for processing.

)sin()(

)()(

a

oo

xy

MKtV

tMdtdtV

Do

o PkT

BM4

22

recall

oo Band

Relaxation Processes

After the RF field is removed over time the spin system will return back to it’s equilibrium state due to several relaxation processes.

Physics of Magnetic ResonanceRelaxation

Physics of Magnetic ResonanceRelaxation

After the RF field is removed over time the spin system will return back to it’s equilibrium state due to several relaxation processes. These are:

(1) Spin-Spin relaxation (also called the T2 relaxation): Due to random processes in which neighboring proton spins effect each other spin system will lose coherence and Mxy will decay. This is an irreversible process.

(2) Spin-Lattice relaxation (also called T1 relaxation): Due to another random process the Mz will begin to recover back to it’s original equilibrium state. Also irreversible.

(3) T2* relaxation: Due to inhomogenities in the external Bo

field Mxy will decay much faster than T2. This is a reversible process.

T1 Relaxation

T2 Relaxation (FID)

T2 Decay

RF

RF

T2* Relaxation

T2* Decay: Dephasing due to field inhomogeneity

x'

y'

z'

Mxy = 0

T2* relaxation is dephasing of transverse magnetization too

but it turns out to be reversible

Animation of T2* Dephasing

Spin- Echo

Spin Echo

Summary of Relaxation Processes

MRI Image An MRI image is determined by two things

three intrinsic properties of the tissue. These are: T1, T2 and Pd. (two relaxation time constants and the density of protons)

the details of the external magnetic fields (Bo, B1 and the gradient magnetics (have not talked about these yet)). How they are configured and how we turn them on and off (pulse sequence) effects what the image looks like.

By varying the pulse sequence we can control which of the intrinsic properties to emphasize in the image.

Tissue Contrast

White matterT1=813 msT2=101 ms

TR

TE

90 degreeRF pulses

180 degreeRF pulses

0.5TE

0.5TE

TE

0.5TE

0.5TE


TR

TE

90 degreeRF pulses

180 degreeRF pulses CASE I: TR>>T1 , TE<T2

What do we measure?

TE


TR

TE

90 degreeRF pulses

180 degreeRF pulses

CASE I: TR>>T1 , TE<T2

Short TE means that the signal has not decayed much due to T2relaxation. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?

TE


TR

TE

90 degreeRF pulses

180 degreeRF pulses

CASE I: TR>>T1 , TE<T2

Short TE means that the signal has not decayed much due to T2relaxation. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?

PD Weighted imaging! TE


TR

TE

90 degreeRF pulses

180 degreeRF pulses

TE

CASE II: TR>>T1 , TE~T2


TR

TE

90 degreeRF pulses

180 degreeRF pulses

TE


TE on the order of T2means that the signal is proportional to the T2relaxation constant. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?


TR

TE

90 degreeRF pulses

180 degreeRF pulses

TE


TE on the order of T2means that the signal is proportional to the T2relaxation constant. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?

T2 Weighted imaging!

TR

90 degreeRF pulses

180 degreeRF pulses

TE

CASE III: TR~T1 , TE<T2

What do we measure?

TR

90 degreeRF pulses

180 degreeRF pulses

TE


TE is shorter than T2means that the signal is not heavily weighted on the T2relaxation constant. TR on the order of T1 means that by the next pulse the system is not back at equilibrium (Mz due to T1 relaxation has not fully recovered) So what are we measuring?

TR

90 degreeRF pulses

180 degreeRF pulses

TE


TE is shorter than T2means that the signal is not heavily weighted on the T2relaxation constant. TR on the order of T1 means that by the next pulse the system is not back at equilibrium (Mz due to T1 relaxation has not fully recovered) So what are we measuring?

T1 Weighted imaging!

Tissue ContrastSummary

TE TR

PD weighted TE<T2 (short TE) TR>>T1 (long TR)

T1weighted TE<T2 (short TE) TR~T1 (short TR)

T2 weighted TE~T2 (long TE) TR>>T1 (long TR)

Bloch Equations

Full Bloch equation including relaxation

12 Tz))((

T)y(t)x(t)(

(t)Bγ)(dt

d ozyx MtMMMtMM

transverse magnetization

precession,RF excitation longitudinal

magnetization

)(B(t) B 1o tB

includes Bo and B1

Example: Solve for the transverse components of M after a 90 degree pulse.

12 Tz))((

T)y(t)x(t)(

(t)Bγ)(dt

d ozyx MtMMMtMM

-(t)1

(t)1

(t)1

)()(M)()(M)()(M)(M-)()(M)(M

γ)()()(

dtd

1

2

2

yx

zx

zy

oz

y

x

xy

xo

yo

z

y

x

MMT

MT

MT

tBttBttBtBttBtBt

tMtMtM


-(t)1

(t)1

(t)1

)()(M)()(M)()(M)(M-)()(M)(M

γ)()()(

dtd

1

2

2

yx

zx

zy

oz

y

x

xy

xo

yo

z

y

x

MMT

MT

MT

tBttBttBtBttBtBt

tMtMtM

After 90 degree pulse the RF field is shut down and only Bo is non-zero

-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM



-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM

Initial conditions for 90 degree pulse:

0)0(

)0(0)0(

z

oy

x

M

MMM



-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM

)1()( 1Tt

oz eMtM

2

2

)cos()(

)sin()(

Tt

ooy

Tt

oox

etMtM

etMtM

Solutions



-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM

)1()( 1Tt

oz eMtM

2)(

)()()(

Tt

tjoxy

yxxy

eeMtM

tjMtMtM

o

Solutions

Example: Solve for the transverse components of M after an arbitrary flip angle (a)

-(t)1

(t)1

(t)1

)()(M)()(M)()(M)(M-)()(M)(M

γ)()()(

dtd

1

2

2

yx

zx

zy

oz

y

x

xy

xz

yz

z

y

x

MMT

MT

MT

tBttBttBtBttBtBt

tMtMtM

After an arbitrary RF pulse the RF field is shut down and only Bo is non-zero

-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM


After an arbitrary RF pulse the RF field is shut down and only Bo is non-zero

-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM

Initial conditions for 90 degree pulse:

)cos()0(

)sin()0(0)0(

a

a

oz

oy

x

MM

MMM


-(t)1

(t)1

(t)1

0)(M-)(M

γ)()()(

dtd

1

2

2

0x

0y

oz

y

x

z

y

x

MMT

MT

MT

BtBt

tMtMtM

11

11

2

2

)cos()1(

)0()1()(

)sin(

)0()(

Tt

oTt

o

Tt

zTt

oz

Tttjo

TttBjxyxy

yxxy

eMeM

eMeMtM

eeM

eeMtM

jMMM

o

o

a

a

Solutions

Solve full Bloch equation with only B=Bo

Solution for transverse components Mx and My

)sin()0()0(

)0(

)0()(2

2

a

zxy

Tttjxy

TttBjxyxy

MM

eeM

eeMtMo

o

Where a is the flip angle after RF excitation

11 )cos()1()( Tt

oTt

oz eMeMtM

a

Signal Detection

Signal Detection via RF coil


max

min

max

min

max

min

2 ),,(/)0,,,()(z

z

y

y

x

x

zyxTttBjxy dxdydzeezyxMAts o

Coils oriented as shown above will only respond to changes in the transverse magnetic field (this is what we want)

Assuming the magnetic fields are homogenous the signal will be a weighted integration of all the protons within the coil.

The waiting will be based on the total magnetization at location x,y,z at the start of the pulse (Mxy(x,y,z,0)) and the tissue decay time T2(x,y,z)

This is not an image!!

Transverse magnetization at t=0.


max

min

max

min

max

min

2 ),,(/)0,,,()(z

z

y

y

x

x


max

min

max

min

max

min

2

max

min

max

min

max

min

2

),,(/

),,(/

)0,,,()(

)0,,,()(

z

z

y

y

x

x

zyxTtxy

tj

z

z

y

y

x

x

zyxTttjxy

dxdydzezyxMAets

dxdydzeezyxMAts

o

o

After demodulation:

max

min

max

min

max

min

2 ),,(/)0,,,()(z

z

y

y

x

x

zyxTtxyo dxdydzezyxMAts

Creating an Image

Creating an ImageTo create an image using NMR we need to figure out a way to encode the proton spins spatially in three dimensions.

But how?

Frequency and Phase Are Our Friends in MR Imaging

q

q = t

The spatial information of the proton pools contributingMR signal is determined by the spatial frequency andphase of their magnetization.

Gradient Coils

Gradient coils generate spatially varying magnetic field so that spins at different location precess at frequencies unique to their location, allowing us to reconstruct 2D or 3D images.

X gradient Y gradient Z gradient

x

y

z

x

z z

x

y y

Gradient Coils

Sounds generated during imaging due to mechanical stress within gradient coils.

Purpose: Spatially alter magnitude of B0 (not direction)

Vector Notation

rGBzyxB

azayaxr

aGaGaGG

oz

zyx

zzyyxx

),,(

ˆˆˆ

ˆˆˆ

Larmor frequency within a gradient field

rGBr

rGBzyxB

azayaxr

aGaGaGG

o

oz

zyx

zzyyxx

)(

),,(

ˆˆˆ

ˆˆˆ

Slice Selection

Slice Selection Gradient

BG

Coil 1

Coil 2

Helmholtz Coils

zGzzGBz

z

zo

0)()(

Z-Gradient Fields

By adding a z-gradient field we cause a variation in theresonant frequency from head to toe.

A sample is put inside a 1.5T magnet. A z-gradient of 3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?

Example

A sample is put inside a 1.5T magnet. A z-gradient of3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?

TB

TB

z

zGBrGBzyxB

z

z

zooz

509144.154.21210000

35.1

490856.154.21210000

35.1

1000035.1

),,(

max

min

Example

A sample is put inside a 1.5T magnet. A z-gradient of3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?

MHzMHzteslaMHzzBz

TB

TB

z

z

26.64509144.158.4248.63490856.158.42

/58.42),()(

509144.154.21210000

35.1

490856.154.21210000

35.1

max

min

max

min

Example

A Field Gradient Makes the Larmor Frequency Depend upon Position

B(Z) B G Zo Z *

(z) B(z)

64,260,000 Hz63,480,000 Hz

1.500 T 1.501 T

Z

B0

Gradient in Z

Slice Selection

64 MHz

65 MHz

66 MHz

63 MHz

62 MHz

G

(-)

(+)

Slice SelectionHow do we determine the slice width and center?

z

x

Bo

After z selectiongradient and excitationz-gradient

zDz (slice width)(slice center)

Determining slice thicknessResonance frequency range as the result of slice-selective gradient:

z

oo

z

o

z

o

zo

GBBzzz

GBz

GBz

zGBz

D

minmaxminmax

maxmax

minmin ,

)(

zGz

D

D

Changing slice thickness

There are two ways to do this:

(a) Change the slope of the slice selection gradient

(b) Change the bandwidth of the RF excitation pulse

Both are used in practice, with (a) being more popular

zGz

D

D

Suppose we wish to have a slice thickness of 2 mm and we are using a z-gradient of 1.0 G/cm ? What range of RF frequencies should we use?

Example

Suppose we wish to have a slice thickness of 2 mm and we are using a z-gradient of 1 G/cm ? What range of RF frequencies should we use?

Example

HzMHz

cm

Gz

z

6.85110516.810000

158.422.0

4 D

D

DD

Selecting different slices

z

o

z

o

z

o

z

o

z

o

zo

GGBz

GBzzz

GBz

GBz

zGBz

2/2

22

,

)(

minmax

minmaxminmax

maxmax

minmin

z

o

Gz

Selecting different slicesIn theory, there are two ways to select different slices:

(a) Change the position of the zero point of the slice selection gradient with respect to isocenter

(b) Change the center frequency of the RF to correspond to a resonance frequency at the desired slice

Option (b) is usually used as it is not easy to change theisocenter of a given gradient coil.

z

o

Gz

RF Excitation (RF Pulse)

0 t

Fo

Fo Fo+1/ t

Time Frequency

t

Fo Fo

DF= 1/ t

FT

FT

RF Excitation (RF Pulse)

t

Fo

D

FT

tjettAts

2)sin()(DD

D

2

)()(

12 vvv

vrectAS

D

A

21

RF Excitation: Flip angle

t

Fo

D

FT

tjettAts

2)sin()(DD

D

2

)()(

12 vvv

vrectAS

D

A

21

max

min

)(1

t

t

e dttBa

Envelope of the pulse


t

Fo

D

FT

tjettAts

2)sin()(DD

D

2

)()(

12 vvv

vrectAS

D

A

21

max

min

max

min

21 )()(

t

t

tjt

t

e dtetsdttB a


t

Fo

D

FT

tjettAts

2)sin()(DD

D

2

)()(

12 vvv

vrectAS

D

A

21

)()(

)sin()(max

min

1

zzzrectAvrectA

dtttAdttB

t

t

e

D

D

DD

D

a

a

RF Excitation: Flip angle (truncated sinc)

)()sin()(~ 2

p

tj trectettAts

DD

D

))((sin)( zzGczzzrectA zpp *

D

a

DD

D2

2

)sin(p

p

dtttA

a

2/p 2/p

A potential problemWait a minute! Dr. M I remember you telling us that if the magnetic field varied from place to place (inhomogeneous) then we would get rapid dephasing of spins. Since some spins are spinning faster than others they quickly get out of phase. That was the whole reason behind the spin echo stuff!

Won’t that happen again?

Wait a minute! Dr. M I remember you telling us that if the magnetic field varied from place to place (inhomogeneous) then we would get rapid dephasing of spins. Since some spins are spinning faster than others they quickly get out of phase. That was the whole reason behind the spin echo stuff! Won’t that happen again?

Spinning fast

Spinning slow

Why yes it will! It is called gradient dephasingGood question!

Spinning fast

Spinning slow

Why yes it will! It is called gradient dephasing. It will quickly kill our signal much faster than T2 or even T2*

Any ideas on how to get around this?

Localization in xy plane

Lets Start with a Simple Flat Person(only xz plane)

z

x

Bo

Lets Start with a Simple Flat Person(only xz plane)

z

x

Bo

After z selectiongradient and excitation

z-gradient

Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis

max

min

max

min

max

min

2 ),,(/)0,,,()(z

z

y

y

x

x


Recall that the signal we measure is given by:

Now we have selected only a single slice in z (z=zo) and we have no y dependence (flat person)

dxexMAets xTtxy

tj o )(/2 2)0,()(

After demodulation (envelope detection)

dxexMAetsts xTtxy

tjo

o )(/2 2)0,()()(



dxexMAetsts xTtxy

tjo

o )(/2 2)0,()()(

Let )(/ 2)0,()( xTtxy exMAxf

dxxftso )()(

This is what we want to image (called the effective proton density)

Lets Start with a Simple Flat Person(frequency encoding using x-gradient)

z

x

Bo

After z selectiongradient and excitation


Now lets apply a gradient in the x direction (Gx)

)(/ 2)0,()( xTtxy exMAxf

dxeexMAts xTttxGjxy

xo )(/2 2)0,()(

dxeexMAets xTttxGjxy

tj xo )(/22 2)0,()(


dxeexMAts xTttxGjxyo

x )(/2 2)0,()(

dxexfts txGjo

x2)()( What does this look like?



dxexfts txGjo

x2)()(

Let tGu x

dxexfGusuF uxj

xo

2)()()(

The received signal is related to the Fourier transform (THIS IS THE KEY!)


)(/ 2)0,()( xTtxy exMAxf


dxexfts txGjo

x2)()(

Let tGu x

dxexfGusuF uxj

xo

2)()()(

We can now find our image as a function of x by taking an inverse Fourier Transform

A Simple Example of Spatial Encoding with Frequency Encoding

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

FrequencyDecomposition

A Simple Example of Spatial Encoding with Frequency Encoding

Decays faster than T2*

Extend this to a full 3D person

x

y

Extend this to a full 3D personAfter slice selection we need to image in xy plane

Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis

Now lets apply a gradient in the x direction (Gx)

),(/ 2)0,,(),( yxTtxy eyxMAyxf

dydxeeyxMAts yxTttxGjxy

xo

),(/2 2)0,,()(


dydxeyxfts txGjo

x2),()(

dydxeeyxMAts yxTtxtGjxyo

x

),(/2 2)0,,()(

Effective proton density




dydxeyxfts txGjo

x2),()(

dydxeeyxMAts yxTtxtGjxyo

x

),(/2 2)0,,()(

Let tGu x

dydxeyxfGutsvuF uxj

xo

2),()()0,(


dydxeyxfts txGjo

x2),()(

Let tGu x

dydxeyxfGutsvuF uxj

xo

2),()()0,(

Corresponds to a single line or trajectory in the uv plane


Now lets apply gradients in both the x direction (Gx) and y direction (Gy)


dydxeeeyxMAts yxTtytGjxtGjxyo

yx

),(/22 2)0,,()(


dydxeeyxfts tyGjtxGjo

yx 22),()(

Let

Effective proton density


Now lets apply gradients in both the x direction (Gx) and y direction (Gy)

dydxeeyxfts tyGjtxGjo

yx 22),()(

Let tGvtGu yx ,

dydxeeyxfvuF vxjuxj 22),(),(


Let tGvtGu yx ,

dydxeeyxfvuF vxjuxj 22),(),(

Polar Scanning

Gradient Echo(A brief detour)

Spatial Encoding in xy planeGradient Echo Mathematical Analysis

Spins will dephase very quickly (quicker than T2*) due to the

gradient fields.

2)2,(

2,),(

sx

sp

spppx

t

x

TxGTx

TttxGxdtGtxp

After negative x-gradient


2)2,(

2,),(

sx

sp

spppx

t

x

TxGTx

TttxGxdtGtxp

After negative x-gradient

Now impose positive x-gradient for Ts

23

2,),(

222),(2

sp

sppxsx

t

T

spx

sxx

sx

TtTtxGxTGtx

TtxGTxGxdtGTxGtxs

p


Now impose positive x-gradient for Ts

0),(

23

2,),(

pspxsxsp

sp

sppxsx

TxGxTGTtx

TtTtxGxTGtx

Phase Encoding

Pulse Repetition

Image Contrast

Image Quality

Dv

Du

Vcoverage

Ucoverage

Dx

Dy

FOVx

FOVy

Fourier PlaneSpatial Domain

Field of View and Resolution in MRI

Nyquist Sampling Theorem: Review

• Assume we have a continuous signal with maximum frequency of fmax

• To avoid aliasing we must sample the signal at a sampling frequency of fs>=2 fmax

• The sampling interval T=1/ fs

• fmax<=1/(2T)

Sampling in MRI

• Slice selection direction: sampling in z-directionSlice thickness (Dz) controlled by RF excitation

bandwidth (D)To avoid aliasing

Where fmax,z is the highest spatial frequency in along the z-axis

zz f

zfz max,

max, 2121

DD

Sampling in MRI

• Within each slice: sampling in xy plan We sample in the Fourier domain (u,v)

(called k-space in MRI literature, kx=u, ky=v) Rectilinear Scan

Du depends on sampling interval T during readout (ADC)

Du depends on sampling interval during this time

Sampling in MRI• Within each slice: sampling in xy plan

We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Rectilinear Scan

Dv depends on spacing between phase encoding

Dv depends on the integrated phase shift here


We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Polar Scan

Angle scan depends on steps in Gy/Gx

Angle scan depends on steps in Gy/Gx


We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Polar Scan

Rho spacing depends on sampling interval T during readout

r spacing depends on sampling interval during this time

X-gradient relates dimension x with Larmor freq byxGx xo )(

To avoid aliasing only frequency given below are measured

222)(

2s

xss

os

ofxGffxf

X-gradient relates dimension x with Larmor freq byxGx xo )(

To avoid aliasing only frequency given below are measured

x

s

x

s

sx

sso

so

Gfx

Gf

fxGffxf

22

222)(

2

Field of view in the x-direction (FOVx) is thus given by

uTGFOV

TGGf

Gf

GfFOV

xxFOV

xx

xx

s

x

s

x

sx

x

D

11

1)2

(2

minmax

Dependant on the phase encoding gradient Gy. The amount of phase change is given by

PEyTGv DDD

Field of view in y

vTGyyFOV

PEyy D

11

minmax

While FOV is limited by the sampling interval in the UV plane (Fourier plane) the resolution is limited by the total extent of the UV plane being sampled. If we ignore high spatial frequency content we will have lower resolution (blur our image). Since MRI scans cover only a finite area of the Fourier space we can expect a finite resolution.

Fourier space coverage in MRI

PEyyyerage

xxxerage

TGNvNV

TGNuNU

DD

D

cov

cov

Fourier space coverage in MRI

PEyyyerage

xxxerage

TGNvNV

TGNuNU

DD

D

cov

cov

Outside of this range we assume the contributions to be zero. This is equivalent to passing the actual image through a low-pass filter in the uv plane whose transfer function is given by

)()(),(covcov erageerage Vvrect

UurectvuH

In the spatial domain this is then given by the point spread function (PSF)

)(sin)(sin),( covcovcovcov yVcxUcVUyxh erageerageerageerage

In the spatial domain this is then given by the PSF

)(sin)(sin),( covcovcovcov yVcxUcVUyxh erageerageerageerage

The full width half max (FWHM) resolution is given by the width of the sinc function’s main lobe

PEyyyeragey

xxxeragex

TGNvNVFWHM

TGNuNUFWHM

D

D

D

111

111

cov

cov

Increasing the U,V (coverage area in Fourier space) reduces blurring.

erage

y

erage

x

VvNNFOV

y

UuMMFOVx

cov

cov

11

11

D

D

D

D

Dv

Du

Vcoverage

Ucoverage

Dx

Dy

FOVx

FOVy

Fourier Plane Spatial Domain

Field of View and Resolution in MRI

eragey

eragex

VFWHM

UFWHM

cov

cov

1

1

vFOV

uFOV

y

x

D

D

1

1

eleg 479 lecture #12 magnetic resonance (mr) imaging

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