This document provides an introduction to nuclear magnetic resonance (NMR), including key topics such as nuclear spin and magnetism, the Larmor frequency, energy absorption and emission, NMR spectroscopy, relaxometry, and safety issues related to energy absorption in tissue. It discusses how nuclei with spin precess when placed in a magnetic field and absorb energy at the Larmor frequency. NMR spectroscopy uses the absorption spectrum to study chemical composition, while relaxometry uses relaxation properties like T1 and T2 times. Relaxation occurs through interactions between nuclei and their environments. Differences in relaxation properties between tissues enable contrast in MRI.

1. 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

2. 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

3. • M: net magnetization from collection of nuclei
• At thermal equilibrium, a sample of N protons
in a static field B0will have magnetisation
• at room temperature is very small.
Electron paramagnetism dominates nuclear
paramagnetism
kT3
h22γ
kT
IIhN
kT
o
mBh
kT
o
mBhm
hNM
3
)1(22
)/exp(
)/exp( +=
Σ
⋅Σ
= γ
λ
λ
γ Bo

4. single
voxel
net magnetization
Individual nuclear spins

5. Nuclear Magnetic Resonance
• It is very difficult to observe static
nuclear magnetism at room temperatures
• Resonance techniques can dramatically
amplify effects
• Note: uncertainty principle: Nφ~1 in
radiofrequency regime of NMR, N is very
large, s.t. a classical description is valid

6. • At thermal equilibrium, M is aligned with static
field
• application of a perturbative field nuclei
experience a torque τ
• if applied field is rotating about Bo at angular
frequency ω, (recall that torque is the angular
analogue of force: F=ma= ; τ= )
e
BM

×=
dt
dP
dt
dL
e
1 BM
dt
dM
dt
dL ×==
γ

7. Bo
B1
Be
Bo = Original static field
Bo
B1 = Applied perturbing
field
Be = Bo + B1 = resulting
“effective” field

8. • Rotating Bapp B1= B1(t)
Bloch Equation
• switching to rotating frame of M ( )
)(
e
)()( tBtM
dt
tdM ×=γ
×+→ ω
dt
d
dt
d
))(
e
()()(
γ
ωγ +×= tBtM
dt
tdM

9. • When , M is at rest in rotating frame
• there are two conditions (i.e. solutions)
M parallel to Be (only when Be = B0)
ω=-γBe=ωoLarmor frequency
ωo is the frequency at which M rotates about Be(~ B0)
0=
dt
dM

10. Resonance
• Application of perturbative field at t=0 causes
precession of M about Be (net field)
• Resonance occurs when ω1=ωo, since Β1 will
appear to be stationary in the frame of M

11. M
B1
ω=ωο
ω−ωο
M
B1
ω=ωο
Net force

12. • After a time t, the angle of M with
respect to B0 is:
α=ω1t=γB1t flip angle
M
B1
Net force
M
B1
α

13. 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

14. 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

15. • 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)
3ω∝

16. • 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.

17. 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: γ=4257 Hz/gauss
0.5T 21.28 MHz
1.0T 42.57 MHz
1.5T 63.86 MHz (Channel 3!)

18. 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)

19. 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…)

20. 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)

21. Specific Absorption Rate
ρ
ω 22
1
2
RH
SAR∝ ρ = tissue density
weightTR
2
angleflip
2
⋅
⋅
∝ oH
SAR
Practically:

22. 2.5 W/kg
1 W/kg
Specific Absorption Rate
• RF Heating occurs mostly at the surface

23. NMR Relaxation
• Energy emission occurs through interaction
with environment
– time evolution:
– Free Induction Decay solution:
)()()( toBtM
dt
tdM ×=γ
 
















−−= )
1
exp(1oMzM
T
t
−= ))cos(
2
exp(oMxyM








to
T
t ω

24. • Longitudinal relaxation
• Transverse relaxation
Mz = Mo 1−exp
−t
T
1


















1
T
2
*
=
1
T2
+
1
′T2
Mxy = Mo
2
exp
−t
T*












)cos( to
ω

25. 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 H T increases with H

26. 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





+= tenhancemen
1
1
2
1
TT

27. 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

28. x
y
x
y
x
y
x
y
Spin Dephasing after excitation
Bo

29. 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

30. 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
1
TTT

31. • Longitudinal relaxation
• Transverse relaxation
Mz = Mo 1−exp
−t
T
1


















1
T
2
*
=
1
T2
+
1
′T2
Mxy = Mo
2
exp
−t
T*












)cos( to
ω

32. T1 contrast:
Inversion-recovery
T2 contrast:
Spin Echo
T2
*
contrast:
Gradient echo
T1
FID
T2
T2
*
T2
*