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

Physics of Magnetic ResonanceSummary

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

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.

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

Net Magnetization Vector

zMtMtM

tzyxtM

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)()(

This is bad news since )(tM xy

is where are signal comes from!

Somehow we need to get these guys spinning together!

RF Excitationtime

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

dttBe )(0 1

)(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()(

tMdtdtV

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

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

T2* Relaxation

T2* Decay: Dephasing due to field inhomogeneity

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

90 degreeRF pulses

180 degreeRF pulses

90 degreeRF pulses

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

What do we measure?

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?

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

90 degreeRF pulses

180 degreeRF pulses

CASE II: TR>>T1 , TE~T2

90 degreeRF pulses

180 degreeRF pulses

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?

90 degreeRF pulses

180 degreeRF pulses

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!

90 degreeRF pulses

180 degreeRF pulses

CASE III: TR~T1 , TE<T2

What do we measure?

90 degreeRF pulses

180 degreeRF pulses

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?

90 degreeRF pulses

180 degreeRF pulses

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

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

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

γ)()()(

tBttBttBtBttBtBt

tMtMtM

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

γ)()()(

tBttBttBtBttBtBt

tMtMtM

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

0)(M-)(M

γ)()()(

tMtMtM

0)(M-)(M

γ)()()(

tMtMtM

Initial conditions for 90 degree pulse:

)0(0)0(

0)(M-)(M

γ)()()(

tMtMtM

)1()( 1Tt

oz eMtM

)cos()(

)sin()(

Solutions

0)(M-)(M

γ)()()(

tMtMtM

)1()( 1Tt

oz eMtM

)()()(

tjMtMtM

Solutions

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

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

γ)()()(

tBttBttBtBttBtBt

tMtMtM

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

0)(M-)(M

γ)()()(

tMtMtM

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

0)(M-)(M

γ)()()(

tMtMtM

Initial conditions for 90 degree pulse:

)cos()0(

)sin()0(0)0(

0)(M-)(M

γ)()()(

tMtMtM

)cos()1(

)0()1()(

TttBjxyxy

eMeMtM

Solutions

Solve full Bloch equation with only B=Bo

Solution for transverse components Mx and My

)sin()0()0(

)0()(2

Tttjxy

TttBjxyxy

eeMtMo

Where a is the flip angle after RF excitation

11 )cos()1()( Tt

oz eMeMtM

Signal Detection

Signal Detection via RF coil

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

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.

Signal Detection via RF coil

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

)0,,,()(

zyxTtxy

zyxTttjxy

dxdydzezyxMAets

dxdydzeezyxMAts

After demodulation:

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

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

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

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

zzyyxx

ˆˆˆ

Larmor frequency within a gradient field

rGBzyxB

azayaxr

aGaGaGG

zzyyxx

ˆˆˆ

Slice Selection

Slice Selection Gradient

Coil 1

Coil 2

Helmholtz Coils

zGzzGBz

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?

zGBrGBzyxB

509144.154.21210000

490856.154.21210000

1000035.1

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

26.64509144.158.4248.63490856.158.42

/58.42),()(

509144.154.21210000

490856.154.21210000

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

Gradient in Z

Slice Selection

64 MHz

65 MHz

66 MHz

63 MHz

62 MHz

Slice SelectionHow do we determine the slice width and center?

After z selectiongradient and excitationz-gradient

zDz (slice width)(slice center)

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

GBBzzz

minmaxminmax

maxmax

minmin ,

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

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

6.85110516.810000

158.422.0

Selecting different slices

minmax

minmaxminmax

maxmax

minmin

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.

RF Excitation (RF Pulse)

Fo Fo+1/ t

Time Frequency

DF= 1/ t

RF Excitation (RF Pulse)

tjettAts

2)sin()(DD

12 vvv

vrectAS

RF Excitation: Flip angle

tjettAts

2)sin()(DD

12 vvv

vrectAS

e dttBa

Envelope of the pulse

tjettAts

2)sin()(DD

12 vvv

vrectAS

21 )()(

e dtetsdttB a

tjettAts

2)sin()(DD

12 vvv

vrectAS

)sin()(max

zzzrectAvrectA

dtttAdttB

RF Excitation: Flip angle (truncated sinc)

)()sin()(~ 2

tj trectettAts

))((sin)( zzGczzzrectA zpp *

)sin(p

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)

After z selectiongradient and excitation

z-gradient

Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis

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

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

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

dxexMAetsts xTtxy

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)

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

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

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

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

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

dydxeyxfts txGjo

x2),()(

dydxeeyxMAts yxTtxtGjxyo

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

Effective proton density

dydxeyxfts txGjo

x2),()(

dydxeeyxMAts yxTtxtGjxyo

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

Let tGu x

dydxeyxfGutsvuF uxj

2),()()0,(

dydxeyxfts txGjo

x2),()(

Let tGu x

dydxeyxfGutsvuF uxj

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

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

dydxeeyxfts tyGjtxGjo

yx 22),()(

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.

TttxGxdtGtxp

After negative x-gradient

TttxGxdtGtxp

After negative x-gradient

Now impose positive x-gradient for Ts

222),(2

sppxsx

TtTtxGxTGtx

TtxGTxGxdtGTxGtxs

Now impose positive x-gradient for Ts

pspxsxsp

sppxsx

TxGxTGTtx

TtTtxGxTGtx

Phase Encoding

Pulse Repetition

Image Contrast

Image Quality

Vcoverage

Ucoverage

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

zfz max,

max, 2121

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

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

ofxGffxf

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

To avoid aliasing only frequency given below are measured

fxGffxf

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

uTGFOV

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

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

Fourier space coverage in MRI

PEyyyerage

xxxerage

TGNvNV

TGNuNU

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

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

VvNNFOV

UuMMFOVx

Vcoverage

Ucoverage

Fourier Plane Spatial Domain

Field of View and Resolution in MRI

eragey

eragex

ELEG 479 Lecture #12 Magnetic Resonance (MR) Imaging

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