MRI

Vector Review

kji zyx

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0

k

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Vector Review (2)

The Dot Product

The Cross Product

)θcos(|| ||zy bababababa zyxx

(a scalar)

)(k

)(j

)(i

yxyx

zxxz

yzzy

abba

baba

bababa

zyx

zyx

bbb

aaa

kji

det (a vector)

)θsin(|| || baba

(a scalar)

MR: Classical Description: Magnetic Moments

NMR is exhibited in atoms with odd # of protons or neutrons.

Spin angular momentum creates a dipole magnetic moment

Spin angular momentum = IIntuitively current, but nuclear spin

operator in quantum mechanics

Planck’s constant / 2

μ

Model proton as a ring of current.

Which atoms have this phenomenon?1H - abundant, largest signal31P23Na

I γ μ

γ= gyromagnetic ratio : the ratio of the dipole moment to angular momentum

MR: Classical Description: Magnetic Fields

Magnetic Fields used in MR:

1) Static main field Bo

2) Radio frequency (RF) field B1

3) Gradient fields Gx, Gy, Gz

μHow do we create and detect these moments?

MR: Classical Description: Magnetic Fields: Bo

1) Static main field Bo

without Bo, spins are randomly oriented.

macroscopically, M 0μ

net magnetizationwith Bo,

a) spins align w/ Bo (polarization)

b) spins exhibit precessional behavior

- a resonance phenomena

Reference Frame

Main Magnetic FieldMain Magnetic Field

BB00z

x

y

MR: Energy of Magnetic Moment

Alignment Convention:

x

y

z

z: longitudinalx,y: transverse

Bo

At equilibrium, μM kMM o

Energy of Magnetic Moment in is equal to the dot product B

ozoz BIγBμ- B-μE

quantum mechanics - quantized states

MR: Energy states of 1H

Hydrogen has two quantized currents,

Bo field creates 2 energy states for Hydrogen where

ozoz BIγBμ- Bμ- Energy of Magnetic Moment in B

21

zI

energy separation o2

γ

21

o

B

Bγ

h

E

resonance frequency fo

MR: Nuclei spin states

There are two populations of nuclei:n+ - called paralleln- - called anti parallel

n+

n-

o2γ

BE hlower energy

higher energy

Which state will nuclei tend to go to? For B= 1.0T

Boltzman distribution: 999993.E/kT

enn

Slightly more will end up in the lower energy state. We call the net difference “aligned spins”. Only a net of 7 in 2*106 protons are aligned for H+ at 1.0 Tesla.(consider 1 million +3 in parallel and 1 million -3 anti-parallel. But...

There is a lot of a water!!!

• 18 g of water is approximately 18 ml and has approximately 2 moles of hydrogen protons

• Consider the protons in 1mm x 1 mm x 1 mm cube.• 2*6.62*1023*1/1000*1/18 = 7.73 x1019 protons/mm3

• If we have 7 excesses protons per 2 million protons, we get .25 million billion protons per cubic millimeter!!!!

Magnetic Resonance: Spins

We refer to these nuclei as spins.

At equilibrium,

- more interesting - What if was not parallel to Bo?

We return to classical physics...

- view each spin as a magnetic dipole (a tiny bar magnet)

oB || M

M

MR: Intro: Classical Physics: Top analogy

Spins in a magnetic field are analogous to a spinning top in a gravitational field.

(gravity - similar to Bo)

r

F

momentumangular dtd

Fr Torque

Top precesses about F

MR:Classical PhysicsView each spin as a magnetic dipole (a tiny bar magnet). Assume we can get dipoles away from B 0 .Classical physics describes the torque of a dipole in a B field as

Torque B

Torque is defined as

)I(

) (

dtd

dtd momentumangular

Bμ)I(

dtd

Multiply both sides by γ Bγμμ

dtd

Now sum over all

Mμ BγMM

dtd

MR: Intro: Classical Physics: Precession

rotates (precesses) about

BγMM

dtd

M

B

Solution to differential equation:

Precessional frequency: γB

is known as the Larmor frequency.Bπ2

γf

TeslaMHz 42.57

π2

γ for 1H

1 Tesla = 104 GaussUsually, Bo = .1 to 3 TeslaSo, at 1 Tesla, fo = 42.57 MHz for 1H

γB or

Other gyromagnetic ratios w/ sensitivity relative to hydrogen

• 13C 10.7MHz/ T, relative sensitivity 0.016

• 31P 17.23 MHz/ T, relative sensitivity 0.066

• 23Na 11.26 MHz/ T, relative sensitivity 0.093

MR: RF Magnetic field

Images & caption: Nishimura, Fig. 3.3

B1 induces rotation of magnetization towards the transverse plane. Strength and duration of B1 can be set for a 90 degree rotation, leaving M entirely in the xy plane.

a) Laboratory frame behavior of M b) Rotating frame behavior of M

MR: RF excitation

By design ,In the rotating frame, the frame rotates about z axis at o radians/sec

o1 B B

x

y

z

1B

M

1) B1 applies torque on M2) M rotates away from z.

(screwdriver analogy)3) Strength and duration of B1 determines torque

This process is referred to as RF excitation.Strength: B1 ~ .1 G

What happens as we leave B1 on?

Bloch Equations – Homogenous Material

It’s important to visualize the components of the vector M

at different times in the sequence.

a) Let us solve the Bloch equation for some interesting cases. In the first case, let’s use an arbitrary M vector, a homogenous material, and consider only the static magnetic field.

b) Ignoring T1 and T2 relaxation, consider the following case.

kBB ˆ0

Bγ

dt

d

MM

Solve

000

det/

B

MMM

kji

M

M

M

dtd zyx

z

y

x

The Solved Bloch Equations

kBB ˆ0

Bγdt

d

MMSolve

0

00

det/

0

0

0

z

xy

yx

zyx

z

y

x

Mdt

d

MBMdt

d

MBMdt

d

B

MMM

kji

M

M

M

dtd

The Solved Bloch Equations

A solution to the series of differential equations is:

Next we allow relaxation.

0

0

0

00

00

100

0)cos()sin(

0)sin()cos(

)(

)(

)(

)(

z

y

x

z

y

x

M

M

M

twtw

twtw

tM

tM

tM

tM

where M0 refers to the initial conditions. M0 refers to the equilibrium magnetization. This solutionshows that the vector M will precess about the B0 field.

Sample Torso Coil

z

y

x

MR: DetectionSwitch RF coil to receive mode.

x y

z

Precession of induces EMF in the RF coil. (Faraday’s Law)M

EMF time signal - Lab frame

dt

tMddt

d

))cos((-

- EMF

00

t

Voltage

(free induction decay)

M

for 90 degree excitation

Complex mm is complex.

m =Mx+iMy

Re{m} =Mx Im{m}=My

This notation is convenient:It allows us to represent a two

element vector as a scalar.

Re

Im

m

Mx

My

Transverse Magnetization ComponentThe transverse magnetization relaxes in the Bloch equation

according to

This is a decaying sinusoid. t

Transverse magnetization gives rise to the signal we “readout”.

2T

)ji(

dt

)(d yx MMtM

Solution to this equation is : tiTt eeMtM 02/0)(

MR: Detected signal and Relaxation.Rotating frame

t

Swill precess, but decays.returns to equilibriumM

M

Transverse Component

0Mxy with time constant T2

After 90º, 2T/oxy M)(M tet

MR: Intro: Relaxation: Transverse time constant T2

- spin-spin relaxation

2

xyxy T

MM

dt

d2T/

oxy M)(M tet

T2 values: < 1 ms to 250 ms

What is T2 relaxation?

- z component of field from neighboring dipoles affects the resonant frequencies.- spread in resonant frequency (dephasing) happens on the microscopic level.- low frequency fluctuations create frequency broadening.

Image Contrast:Longer T2’s are brighter in T2-weighted imaging

MR: Relaxation: Some sample tissue time constants: T2

Table: Nishimura, Table 4.2

T2 of some normal tissue types

Tissue T2 (ms)

gray matter 100

white matter 92

muscle 47

fat 85

kidney 58

liver 43

MR: RF Magnetic field

The RF Magnetic Field, also known as the B1 field

To excite equilibrium nuclei ,apply rotating field at o in x-y plane. (transverse plane)

kMM o

Image & caption: Nishimura, Fig. 3.2

B1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference

Exciting the Magnetization Vector

Bloch Equation Solution: Longitudinal Magnetization Component

1T

k)(

dt

d ozz MMM

The greater the difference from equilibrium, the faster the change

Solution:

)1()( 11 /0

/0 TtTtzz eMeMtM

Initial Mz Return to Equilibrium

pulse RFdegree 90a for 00 zM

Solution: Longitudinal Magnetization Component

initial conditions

equilibrium

Example: What happens with a 180° RF flip?

t

Mo

-Mo

Effect of T1 on relaxation- 180° flip angle

)1()( 11 /0

/0 TtTtzz eMeMtM

)21()(

)1()0()(

)0(

1

11

/0

/0

/

00

Ttz

TtTtzz

zz

eMtM

eMetMtM

MtMM

T1 Relaxation

MR Relaxation: Longitudinal time constant T1

Relaxation is complicated.T1 is known as the spin-lattice, or longitudinal time constant.

1

ozz T

)MM(M

dt

d)1(M)(M 1T/

oztet

T1 values: 100 to 2000 ms

Mechanism:- fluctuating fields with neighbors (dipole interaction)- stimulates energy exchange

n- n+

- energy exchange at resonant frequency.Image Contrast:

- Long T1’s are dark in T1-weighted images- Shorter T1’s are brighter

Is |M| constant?

MR Relaxation: More about T2 and T1

T2 is largely independent of Bo

Solids- immobile spins- low frequency interactions- rapid T2 decay: T2 < 1 ms

Distilled water- mobile spins- slow T2 decay: ~3 s- ice : T2~10 s

T1 processes contribute to T2, but not vice versa.

T1 processes need to be on the order of a period of the resonant frequency.

12 TT

MR: Relaxation: Some sample tissue time constants - T1

Image, caption: Nishimura, Fig. 4.2

fat

liver

kidney

Approximate T1 values as a function of Bo

white matter

gray matter muscle

Components of M after Excitation

Laboratory Frame

MR: Detected signal and relaxation after 90 degree RF puls.Rotating frame

t

Swill precess, but decays.returns to equilibriumM

M

Transverse Component

0Mxy with time constant T2

After 90º, 2T/oxy M)(M tet

Longitudinal Component

Mz returns to Mo with time constant T1

After 90º, )1(M)(M 1T/oz

tet

MR Contrast Mechanisms

T2-Weighted Coronal Brain T1-Weighted Coronal Brain

Putting it all together: The Bloch equation

Sums of the phenomena

12 T

k)(

T

)ji( Bγ

dt

d ozyx MMMMM

M

transverse magnetization

precession,RF excitation

longitudinal magnetization

Changes the direction of , but not the length.M

These change the length of only, not the direction.M

B

includes Bo, B1, and G

Now we will talk about affect of G

MR: Intro: So far...What we can do so far:

1) Excite spins using RF field at o

2) Record FID time signal3) Mxy decays, Mz grows4) Repeat.

More about relaxation...

Proton vs. Electron Resonance

Here g is same as

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1

B = Bohr Magneton N = Nuclear Magneton

Particle SpinLarmor/B

s-1T-1/B

Electron 1/2 1.7608 x 1011 28.025 GHz/T

Proton 1/2 2.6753 x 108 42.5781 MHz/T

Deuteron 1 0.4107 x 108 6.5357 MHz/T

Neutron 1/2 1.8326 x 108 29.1667 MHz/T23Na 3/2 0.7076 x 108 11.2618 MHz/T

31P 1/2 1.0829 x 108 17.2349 MHz/T14N 1 0.1935 x 108 3.08 MHz/T13C 1/2 0.6729 x 108 10.71 MHz/T19F 1/2 2.518 x 108 40.08 MHz/T

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1

Electron Spin Resonance – Poor RF Transmission

Graph: Medical Imaging Systems Macovski, 1983

Electron Spin Resonance

• Works on unpaired electrons– Free radicals

• Extremely short decay times– Microseconds vs milliseconds in NMR