M Physics Course by sanmelody

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```									M R I Physics Course
Jerry Allison Ph.D.
Chris Wright B.S.
Tom Lavin B.S.
Medical College of Georgia
M R I Physics Course

nuclear Magnetic Resonance
(nMR)
Nuclear Magnetism
Gyromagnetic ratio
Classic Mechanical Description
Quantum Mechanical Description
Larmor Equation
RF Excitation / Detection
Relaxation
Bloch Equations
Nuclear Magnetism
• Nuclei having an odd number of protons or
an odd number of neutrons or both have
inherent spin (angular momentum) and a
nuclear magnetic moment.
• Nuclei having an even number of protons
and an even number of neutrons have NO
nuclear magnetic moment and cannot be
observed using nMR (or MRI) techniques.
Nuclear Magnetism (continued)
• Important nuclei having no nuclear magnetic
moment include:
16
8O ==> 8 neutrons, 8 protons, 8 electrons
12
6C ==> 6 neutrons, 6 protons, 6 electrons

• Fortunately, hydrogen is abundant in the body
(H2O, CH2:fat) and has a large nuclear magnetic
moment (actually the largest).
1
1H ==> 1 proton, 1 electron
Gyromagnetic Ratio
(Magnetogyric ratio)
The nMR properties of nuclei are characterized by


the gyromagnetic ratio ( ). The
gyromagnetic ratio is unique for each nuclide
that has a nuclear magnetic moment.

=        nuclear magnetic moment
nuclear spin angular momentum
Gyromagnetic Ratio (continued)

M=*S
M = nuclear magnetic moment
 = gyromagnetic ratio
S = nuclear spin angular
momentum (inertia)
Table 1: Nuclear Properties of Selected Nuclei
Magnetogyric     Resonance
Natural
Ratio        frequency at                   Relative
Nuclide                                    abundance
(MHz/Tesla)*   1.5T (values in               Sensitivity**
(%)
Mhz)
1
H       42.58          63.87         99.98            1.0
2
H        6.53           9.79         0.015        9.65 x 10-3
13
C      10.71          16.07          1.11        1.59 x 10-2
14
N       3.09           4.64         99.63        1.01 x 10-3
15
N       4.34           6.51          0.37        1.04 x 10-3
17
O       5.81           8.71          0.04        2.91 x 10-2
19
F      40.05          60.08          100            0.83
23
Na      11.26          16.89          100         9.25 x 10-2
25
Mg       2.63           3.94         10.05        2.68 x 10-3
31
P      17.23          25.85          100         6.63 x 10-2
33
S      3.29           4.93          0.74        2.26 x 10-3
35
Cl       4.20           6.30          75.4        4.70 x 10-3
37
Cl       3.49           5.24          24.6        2.71 x 10-3
39
K       1.99           2.99         93.08        5.04 x 10-4
43
Ca       2.89           4.33          0.13        6.40 x 10-2
* Tesla is the magnetic field (10,000 Gauss = 1 Tesla)
** For an equal number of nuclei in an identical magnetic field
Classical Mechanical Description
• Some aspects of the nMR phenomenon are
easiest to describe with classical mechanics,
others are easiest to describe with quantum
mechanics.
• Classical description of a nuclear magnetic
moment (spin) in an applied magnetic field:
“When a force is exerted on a spinning object, the
spinning object tends to move at right angles to
the force”.
• An example would include a spinning top in the
Earth’s gravitational field.
Classical Mechanical Description
(continued)
• Precession - continual motion of a spin at
right angles to an applied force (sweeps the
surface of a cone).
• An example would be a gyroscope.
z

B0

0
M
y

Precession - continual motion of a spin
at right angles to an applied force
x       (sweeps the surface of a cone).
Quantum Mechanical Description
Quantum mechanical description of a nuclear
magnetic moment (spin) in an applied field.
Spin States
Spin Flip Transitions
Macroscopic Magnetization
Spin States
• 1H (and 31P) nuclei have only two available
spin states and are said to have nuclear spin
of 1/2. Spin up (low energy state: parallel
to applied static magnetic field) and spin
down (high energy state: antiparallel to
applied static magnetic field).
E = 2.64 x 10-7 eV

E = 8.8 x 10-8 eV
E0                               E0
BO = 0.5T
BO = 1.5T

Energy levels for hydrogen nuclei in 0.5 T (5,000 Gauss) and 1.5 T
(15,000 Gauss) fields. Note the energy difference is greater at
higher fields because E is directly proportional to the applied
magnetic field.
Espin down

Energy                         E

Espin up

Magnetic Field

The spin states are separated by E of energy
E = h
-34
h = Planck’s constant (6.62 x 10 J s)
 = spin frequency (cycles / s, Hertz)
Spin States (continued)
E is the energy difference between the
spin up and spin down states.
E is directly proportional to the applied
magnetic field.

For example:
E = 2.64 x 10-7 eV @ 1.5T (63.87 MHz)
E = 1.76 x 10 eV @ 1.0T (42.58 MHz)
-7

E = 8.80 x 10-8 eV @ 0.5T (21.29 MHz)
Spin Flip Transitions
• Oscillating magnetic fields at the resonant RF
frequency can cause spin flip transitions:

spin up + absorbed energy ==> spin down

spin down + absorbed energy ==> spin up +
energy released
Macroscopic Magnetization
• For hydrogen nuclei (spin 1/2) at thermal
equilibrium in a static magnetic field, the
relative number of protons in the spin up
and spin down states is given by the
Boltzmann equation or Boltzmann
Distribution:
 h B0
_______
# spin up
__________                k T
# spin down
=
e
 h B0
_______
k T
# spin up
__________
# spin down
=   e
   = gyromagnetic ratio (Hz/Tesla)
h = Plank’s constant (6.626 x 10-34 J sec)
B0 = magnetic field (Tesla)
k = Boltzmann’s constant (1.381 x 10-34 J/K)
T = temperature in Kelvin (K)
where K = oC + 273.15
For example: consider hydrogen nuclei (protons)
at 98.6 oF (37 oC, 310.15 K) in a 1.5 Tesla
magnetic field:
 =     42.58 MHz/Tesla = 42.58 x 106 Hz/Tesla
h =     6.626 x 10-34 J sec
B0 =    1.5 Tesla
k =     1.381 x 10-23 J / K
T =     310.15 K
(42.58MHz/Tesla)(6.626 x 10-34 J s)(1.5 Tesla)
_____________________________
# spin up
_________ =
# spin down    e     (1.381 x 10-23 J / K) (310.15 K)
Macroscopic Magnetization
Example continued: For hydrogen nuclei (protons)
at 98.6 oF in a 1.5 Tesla magnetic field:

(42.58 x 106 Hz/Tesla)(6.626 x 10-34 J s)(1.5 Tesla)
_____________________________
# spin up
_________ =
# spin down     e     (1.381 x 10-23 J / K) (310.15 K)

# spin up protons
______________ = 1.000009882611 (4.94 ppm)
# spin down protons
For hydrogen nuclei (protons) at 98.6 oF in a
1.0 Tesla magnetic field:

(42.58 x 106 Hz/Tesla)(6.626 x 10-34 J s)(1.0 Tesla)
_____________________________
# spin up
_________ =
# spin down     e     (1.381 x 10-23 J / K) (310.15 K)

# spin up protons
______________ = 1.000006588396 (3.29 ppm)
# spin down protons
Macroscopic Magnetization
(continued)

For hydrogen nuclei (protons) at 98.6 oF in a
0.5 Tesla magnetic field:

# spin up protons
______________ = 1.000003294193 (1.65 ppm)
# spin down protons
Suppose a sample of hydrogen nuclei in a 1.5 Tesla
magnetic field is heated by RF energy to 1 oC
above normal body temperature:

# spin up protons
______________ = 1.000009850849 (4.93 ppm)
# spin down protons

For physiologic temperatures, the excess of protons
in the spin up state (at 1.5T) decreases
about 0.01 ppm per oC.
Macroscopic Magnetization
(continued)

• The small excess of protons in the spin up
state produce a macroscopic magnetization
that can be manipulated using magnetic
fields oscillating at the resonant frequency.
The macroscopic magnetization is also
described as the thermal equilibrium
magnetization or net magnetic moment.
Note that the excess protons decreases as
field strength decreases which results in a
reduced signal-to-noise ratio at lower fields.
Macroscopic Magnetization
• For each gram of soft tissue, there is an
excess of approximately 3 x 1016
protons in the spin up state out of
3 x 1022 protons. This excess creates
the macroscopic magnetization.
Larmor Equation
• The Larmor equation describes the resonant
precessional frequency of a nuclear magnetic
moment in an applied static magnetic field.
 =  Bo
Where:     = precessional frequency
(resonant frequency)
    = gyromagnetic ratio (MHz/Tesla)
Bo   = magnetic field (Tesla)
Larmor Equation (continued)

What is the Larmor frequency of hydrogen nuclei
(protons) in a 1.5 Tesla field?
 =  Bo
 = (42.58 MHz / Tesla)(1.5 Tesla)
 = 63.87 MHz
Larmor Equation (continued)

What is the Larmor frequency (resonant frequency)
of 23Na in a 1.5 Tesla field?
 =  Bo
 = (11.26 MHz / Tesla)(1.5 Tesla)
 = 16.89 MHz
Larmor Equation (continued)

What is the Larmor frequency (resonant frequency)
of 2H (deuterium) in a 1.5 Tesla field?
 =  Bo
 = (6.53 MHz / Tesla)(1.5 Tesla)
 = 9.795 MHz
Larmor Equation (continued)

It should be noted that most RF coils and RF
electronics used in MRI are tuned for a fairly
narrow band of RF frequency. To convert from
imaging 1H to 23Na would generally require having
RF coils and RF electronics that can be tuned for the
alternate frequency. Hydrogen is almost exclusively
imaged in MRI because of it’s sensitivity and
abundance.
Larmor Equation (continued)

Is 42.58 MHz / Tesla the  for 1H in fat or water?
The gyromagnetic ratio for 1H is simply
42.58 MHz / Tesla. 1H nuclei in water (H2O) and
fat (~CH2) are in different molecules and experience
a slightly different local magnetic field which results
in slightly different resonant frequencies. These
local magnetic field variations contribute to the
eventual contrast between various tissues in an
MRI image.
RF Excitation
• Spin population - Outside of the static
magnetic field (Bo), the spin population can
be described as a collection of randomly
oriented nuclear magnetic moments
(i. e. the patient):
RF Excitation
• Place the spin population in a static magnetic
field.

• Classical mechanics - individual spins precess

• Quantum mechanics - energy of individual spins
is quantitized (# spin up > # spin down)
RF Excitation (continued)
• Excess spins in the spin up state produce
macroscopic magnetic moment “M” aligned with
static magnetic field Bo. This condition is
described as thermal equilibrium magnetization.
M

Bo
RF Excitation is Begun
• The spin population absorbs energy from
magnetic fields oscillating at the resonant
frequency. RF excitation can be described as a
rotating magnetic field (and electric field) in the
plane perpendicular to the static magnetic field.
RF excitation is produced by applying an
oscillating voltage waveform to an RF exciter
(transmitter) coil. The magnetic field component
that rotates in the transverse plane during RF
excitation is termed the B1 magnetic field.
RF Excitation (continued)
• In quantum mechanics, RF excitation can be
described as absorption of energy at the
appropriate resonant RF frequency which causes
spin-flip transitions.

• Resonance: if the B1 frequency is at the Larmor
frequency (+ a little)(i.e. 63.87 MHz for 1H at 1.5
Tesla) then:
RF Excitation (continued)
• Resonance (continued)
1.) Individual spins flip:
spin up + absorbed energy ==> spin down
spin down + absorbed energy ==> spin up + released energy
2.) Spins develop phase coherence.
3.) Macroscopic magnetization (M) is tipped away
from alignment and begins to spiral at the Larmor
frequency.
4.) Transverse magnetization develops.
Transverse magnetization
Frequently, the macroscopic magnetization is
spiraled down until it precesses in the transverse
plane (plane perpendicular to the static magnetic
field). This is called a 90o flip. After a 90o flip,
the macroscopic magnetization is precessing
entirely in the transverse plane at the Larmor
frequency and there are equal numbers of nuclei
in the spin up and spin down states.
Transverse magnetization              (continued)

The longitudinal component of the
magnetization in the direction of the static
magnetic field (Bo) is zero. The macroscopic
magnetization prior to a 90o flip is entirely
longitudinal and is said to point along the “Z”
axis. Following a 90o flip, magnetization is
entirely transverse and is said to rotate or precess
in the transverse plane defined by the “X” and
“Y” axes.
RF Detection
• The spin population “relaxes” to the thermal
equilibrium magnetization
• The transverse magnetization induces a voltage
signal in an RF detection (receiver) coil as the
spin population returns to the thermal
equilibrium magnetization. The signal induced in
the RF coil during the relaxation of the transverse
magnetization is described as the Free Induction
Decay (FID) signal.
Relaxation
Relaxation is the process by which a spin
population returns to the thermal equilibrium
distribution.

Relaxation principally involves:
T1 spin-lattice relaxation
T2 spin-spin relaxation
Homogeneity of the magnetic field, (Bo)
Relaxation (continued)
T1 Relaxation
• Consider that the longitudinal or “Z” component of
magnetization is determined by the number of
spins in the spin up versus spin down energy state.
The “Z” component returns exponentially to
thermal equilibrium magnetization with rate
constant T1 .
• “... T1 is a measure of the time required to re-
establish thermal equilibrium between the spins
and their surroundings (lattice)…”
Relaxation (continued)

T1 Relaxation      (continued)
• Following a 90o flip, T1 is the time required for the
longitudinal magnetization (“Z” component) to
recover to 63% of the thermal equilibrium
magnetization (MZ0).
Longitudinal Magnetization

MZ0

63% MZ0

T1
90o flip

Time
Relaxation (continued)
• Thermal equilibrium: each spin (proton) is in the B0
static magnetic field of the MRI magnet and in a
fluctuating magnetic field due to translation and rotation
of it’s molecule and nearby molecules (4 Gauss from the
adjacent proton in a water molecule). The magnetic
environment of a water proton at room temperature
changes with frequencies as high as 1,000,000 MHz. On
average, a significant change occurs in the magnetic
environment of a water proton ever 10 -12 seconds.
These rapid changes can stimulate relaxation.
Relaxation (continued)
• Also note that proton exchange occurs in water
molecules. A hydrogen nucleus (proton) in a free
water molecule may exchange places with a
hydrogen nucleus in a bound water molecule.
Both nuclei thus experience a significant change
in their magnetic environment causing relaxation
to occur.
Relaxation (continued)

T2 Relaxation

“... T2 is a measure of the time of disappearance
of the transverse component of relaxation.”

T2 is the time required for 63% of the transverse
magnetization to decay.
Transverse Magnetization              90o flip

Mxy0

37% Mxy0

T2
Time
Relaxation (continued)

T2 Relaxation (continued)
T2 has two components:
Dephasing: spin-spin relaxation with no net
change in energy.
Spin-flip transitions (T1): spin-lattice
transitions with a net increase in the
number of nuclei in the spin up energy
state.
T2 Relaxation (continued)

1
___          1
___       1
____
=          +
T2 ’    2 T1
T2
1
___    = dephasing component of T2 relaxation
T2’
1
____ = T component of T relaxation
1              2
2 T1
T2 Relaxation (continued)

Notice that T1 relaxation (spin flip transitions)
cause dephasing and contributes to
T2 relaxation. Conversely, the dephasing
in T2 relaxation does not affect longitudinal
magnetization and does not contribute to
T1 relaxation. As a result, T2 is always
smaller than T1.
*
T2 Relaxation

The T2 relaxation observed in MRI is corrupted
by inhomogeneity of the B0 magnetic
field. This inhomogeneity is caused by
nonuniformity in the static magnetic
field and the magnetic susceptibility of
patient tissue.
*
T2 Relaxation (continued)

The observed T2 relaxation is termed T2* and
has two components:
1                1
___
___      = B +
T2*              T2

  B represents the effect of magnetic field
inhomogeneity
Relaxation (continued)
• Temperature and Magnetic field dependence:
T1 and T2 do not vary significantly with
temperature in the physiologic
temperature range.
T2 does not vary significantly with magnetic
field.
T1 increases as the magnetic field increases
(~ 200 msec / Tesla).
A T1 value of 600 msec at 21 MHz (0.5T)
becomes 800 msec at 63 MHz (1.5T).
Relaxation (continued)
• Tissue Characteristics
Solids: (cortical bone protons) have extremely
short T2 (microseconds)
Gases:           T1 = T2
Pure Water:       T1 = T2 = 3 sec at 25 oC
Liquids:          T2 < T1
Soft Tissue:      T1  800 msec, T2  80 msec
Approximate T1 and T2 Values for Human Tissue
(37 oC)
T1 at 1.5 T   T1 at 0.5 T     T2
Tissue
(msec)        (msec)      (msec)
Skeletal Muscle          870           600          47
Liver                    490           323          43
Kidney                   650           449          58
Spleen                   780           554          62
Fat                      260           215          84
Gray Matter              920           656         101
White Matter             790           539          92
Cerebrospinal Fluid     >4,000        >4,000      >2,000
Bloch Equations
The equations describing nuclear magnetic resonance
were derived by Felix Bloch in 1946.
Longitudinal magnetization:
- t / T1
MZ (t) = MZ0 + (MZ - MZ0)        e
Transverse magnetization:
- t / T2
Mxy (t) = Mxy0   e
B1

Maximum RF field applied: 23.5 microT (circularly
polarized) For Siemens Magnetom 3T.

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