Introduction to Nuclear Magnetic Resonance

• Topics– Nuclear spin and magnetism– Resonance behavior and the Larmor Frequency

• Larmor frequency• flip angle

– Energy Absorption and Emission• NMR spectroscopy • Energy absorption in tissue (safety issues)• Relaxometry

– T1,T2,T2* relaxation

Nuclear Magnetism• Nucleons (protons, neutrons) have a quantum property

known as spin.• Nucleons have been shown to obey Fermi statistics, and thus

have a maximum spin magnitude of 1/2 Bohr magneton. (spin=1/2)

• In the absence of a magnetic field, nuclear spin is not an observable

• In the presence of a homogeneous magnetic field, the energy of the nucleus depends on the relative orientation of the magnetic field and the nuclear spin vector

• M: net magnetization from collection of nuclei• At thermal equilibrium, a sample of N protons in a static

field B0 will have magnetisation

• at room temperature is very small.

Electron paramagnetism dominates nuclear paramagnetismkT3h22

kTIIhN

kTomBh

kTomBhmhNM

3)1(22

)/exp(

)/exp(

Bo

single voxel

net magnetization

Individual nuclear spins

Nuclear Magnetic Resonance

• It is very difficult to observe static nuclear magnetism at room temperatures

• Resonance techniques can dramatically amplify effects

• Note: uncertainty principle: N~1 in radiofrequency regime of NMR, N is very large, s.t. a classical description is valid

• At thermal equilibrium, M is aligned with static field• application of a perturbative field nuclei

experience a torque

• if applied field is rotating about Bo at angular

frequency , (recall that torque is the angular analogue of force: F=ma= ; = )

eBM

dtdP

dtdL

e1 BM

dtdM

dtdL

Bo

B1

Be

Bo = Original static field

BoB1 = Applied perturbing field

Be = Bo + B1 = resulting “effective” field

• Rotating Bapp B1= B1(t)

Bloch Equation

• switching to rotating frame of M ( )

)(e)()( tBtMdt

tdM

dtd

dtd

))(e()()( tBtM

dttdM

• When , M is at rest in rotating frame

• there are two conditions (i.e. solutions) M parallel to Be (only when Be = B0)

=-Be=o Larmor frequency

o is the frequency at which M rotates about Be (~ B0)

0dt

dM

Resonance

• Application of perturbative field at t=0 causes precession of M about Be (net field)

• Resonance occurs when 1=o, since 1 will appear to be stationary in the frame of M

M

B1

M

B1

Net force

• After a time t, the angle of M with respect to B0 is:

t=B1t flip angle

M

B1

Net force M

B1

Nuclear Magnetic Resonance:Properties in Matter

• Energy Absorption– In matter, resonance frequency depends on

magnetic field at the nucleus• in complex molecules, electron moments will alter the

field seen by the nucleus (chemical shift)Absorption spectrum is a reflection of the chemical

composition

Nuclear Magnetic Resonance:Properties in Matter

• Relaxation– After we have delivered energy to the nuclei

in our sample at the Larmor frequency, there are two possible ways for the sample to lose this energy (back to lowest energy state):

• spontaneous emission • induced emission

• Spontaneous emission:– negligible effect at RF frequencies (dominant at visible

frequencies)

• Induced emission– Energy emission requires interaction of the nucleus with

its external environment

The nature of energy emission depends strongly on the environment of the excited nucleus (Relaxation)

3

• NMR Spectroscopy is the study of the chemistry of matter using the NMR absorption spectrum

• Relaxometry is the study of the chemistry of matter using the NMR relaxation properties.

MRI generates tissue contrast based (mostly) on NMR relaxation differences.

NMR in tissue• Protons in water molecules are the dominant nuclear species in

the human body• At 1.5T, 10-6 more protons are aligned with the static field than

anti-aligned at room temperature very small magnetic moment.

• Proton Resonance frequencies: =4257 Hz/gauss0.5T 21.28 MHz1.0T 42.57 MHz1.5T 63.86 MHz (Channel 3!)

NMR Absorption in Tissue

• RF energy at the Larmor frequency will be absorbed by water protons in tissue

• MRI scanner: 16 Kilowatt RF transmitter• Dosage: “Specific Absorption Rate (SAR)”

– mass normalized rate of RF energy coupling to biologic tissue (watts/kg)

Specific Absorption Rate

• Depends on:– frequency– pulse sequence (shape of RF pulse,repetition

time, pulse width)– RF coil– Volume of tissue in coil (i.e. exposed)– resistivity of tissue– geometry (spherical vs. cylindrical volume…)

Specific Absorption Rate

• Regulated by the FDA– 0.4 W/kg averaged over the whole body, or 8.0

W/kg peak SAR in any 1g of tissue, and 3.2 W/kg averaged over the head

– RF energy insufficient to produce a 1o C rise in core temp. and localized heating less than 38o C in the head, 39o C in the trunk, and 40o C in the extremities (except pts. with impaired circulation)

Specific Absorption Rate

221

2 RHSAR tissue density

weightTR

2angle flip2

oH

SAR

Practically:

2.5 W/kg

1 W/kg

Specific Absorption Rate

• RF Heating occurs mostly at the surface

NMR Relaxation• Energy emission occurs through interaction

with environment– time evolution:

– Free Induction Decay solution:

)()()( toBtMdt

tdM

)1

exp(1oMzMTt

))cos(2

exp(oMxyM

toTt

• Longitudinal relaxation

• Transverse relaxation

MzMo 1 exp tT1

1

T2* 1

T2 1

T 2

MxyMo

2

exp t

T*

)cos( to

NMR Relaxation• T1 relaxation

– time constant of recovery of longitudinal component of magnetization

– physics• reflection of spin thermal interactions with the environment

(i.e. the lattice)• induced emission: molecules moving near the Larmor

frequency will induce relaxation– pure water: molecular motion too fast long T1

– solids: molecular motion too slow long T1

– tissue: molecular motion near Larmor freq short T1

• Field strength: fraction of protons moving near Larmor frequency decreases with Ho T1 increases with Ho

NMR Relaxation• T2 Relaxation

– Time constant of disappearance of transverse magnetization

– Geometry dictates that T1 is a part of T2 (as longitudinal component grows, transverse component decays)

T2 is always greater or equal to T1

tenhancemen1

1

2

1TT

NMR Relaxation• T2 Relaxation (cont’d)

– physics:• Induced emission from interactions with immediate surroundings (spin-

spin interactions)• Each nucleus experiences slight, temporary changes in local field due to

slow interactions with other nuclei. This causes temporary changes in Larmor frequency leading to permanent phase dispersion

• Field strength: change in Larmor frequency doesn’t affect much

• T1 versus T2 in tissue– T1 and T2 roughly correlate (e.g. low T1 implies low T2)– T1 = ~5 T2

x

y

x

y

x

y

x

y

Spin Dephasing after excitation

Bo

NMR Relaxation

• Relaxation times (msec)

0.5T 1.0T 1.5T T1 T2 T1 T2 T1 T2

Gray Matter 650 100 800 100 900 100

Muscle 550 50 700 50 880 50

Fat 200 80 250 80 270 80

NMR Relaxation

• T2 versus T2*

– True T2: decay of transverse magnetization due to “natural” processes at the molecular level

– T2*: the observed or effective decay of transverse

magnetization due to magnetic field inhomogeneity and susceptibility effects

2

1

2

1*2

1TTT

• Longitudinal relaxation

• Transverse relaxation

MzMo 1 exp tT1

1

T2* 1

T2 1

T 2

MxyMo

2

exp t

T*

)cos( to

T1 contrast:Inversion-recovery

T2 contrast:Spin Echo

T2* contrast:

Gradient echo

T1

FID

T2

T2*

T2*