M R I Physics Course

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M R I Physics Course

• Jerry Allison Ph.D.
• Chris Wright B.S.
• Tom Lavin B.S.
• Department of Radiology
• Medical College of Georgia

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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 Bloc
h Equations
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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.

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Nuclear Magnetism (continued)

• Important nuclei having no nuclear magnetic
  moment include
• 168O gt 8 neutrons, 8 protons, 8 electrons
• 126C gt 6 neutrons, 6 protons, 6 electrons
• Fortunately, hydrogen is abundant in the body
  (H2O, CH2fat) and has a large nuclear magnetic
  moment (actually the largest).
• 11H gt 1 proton, 1 electron

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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

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Gyromagnetic Ratio (continued)
M ? S
M nuclear magnetic moment ? gyromagnetic
ratio S nuclear spin angular momentum
(inertia)
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Table 1 Nuclear Properties of Selected Nuclei
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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
  Earths gravitational field.

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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.

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z
B0
? 0
M
y
Precession - continual motion of a spin at right
angles to an applied force (sweeps the surface
of a cone).
x
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Quantum Mechanical Description

• Quantum mechanical description of a nuclear
  magnetic moment (spin) in an applied field.
• Spin States
• Spin Flip Transitions
• Macroscopic Magnetization

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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).

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?E 2.64 x 10-7 eV
?E 8.8 x 10-8 eV
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.
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Espin down
?E
Energy
Espin up
Magnetic Field
The spin states are separated by ?E of energy
?E h? h Plancks constant (6.62 x 10-34 J
s) ? spin frequency (cycles / s, Hertz)
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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 _at_ 1.5T (63.87 MHz) ?E
1.76 x 10-7 eV _at_ 1.0T (42.58 MHz) ?E 8.80 x
10-8 eV _at_ 0.5T (21.29 MHz)
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Spin Flip Transitions

• Oscillating magnetic fields at the resonant RF
  frequency can cause spin flip transitions
  
  spin up absorbed energy gt
  spin down spin down absorbed
  energy gt spin up energy released

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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
  

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? gyromagnetic ratio (Hz/Tesla) h Planks
constant (6.626 x 10-34 J sec) B0 magnetic
field (Tesla) k Boltzmanns constant (1.381 x
10-34 J/K) T temperature in Kelvin
(K) where K oC 273.15
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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)
_____________________________
e
(1.381 x 10-23 J / K) (310.15 K)

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Macroscopic Magnetization
Example continued For hydrogen nuclei
(protons) at 98.6 oF in a 1.5 Tesla magnetic
field
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For hydrogen nuclei (protons) at 98.6 oF in a
1.0 Tesla magnetic field

1.000006588396 (3.29 ppm)
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Macroscopic Magnetization (continued)
For hydrogen nuclei (protons) at 98.6 oF in a
0.5 Tesla magnetic field
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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
For physiologic temperatures, the excess of
protons in the spin up state (at 1.5T)
decreases about 0.01 ppm per oC.
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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.

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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.

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Larmor Equation

• The Larmor equation describes the resonant
  precessional frequency of a nuclear magnetic
  moment in an applied static magnetic field.

w g Bo
Where w precessional frequency
(resonant frequency) g gyromagnetic
ratio (MHz/Tesla) Bo magnetic field
(Tesla)
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Larmor Equation (continued)
What is the Larmor frequency of hydrogen nuclei
(protons) in a 1.5 Tesla field? w g Bo w
(42.58 MHz / Tesla)(1.5 Tesla) w 63.87 MHz
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Larmor Equation (continued)
What is the Larmor frequency (resonant
frequency) of 23Na in a 1.5 Tesla field? w
g Bo w (11.26 MHz / Tesla)(1.5 Tesla) w
16.89 MHz
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Larmor Equation (continued)
What is the Larmor frequency (resonant
frequency) of 2H (deuterium) in a 1.5 Tesla
field? w g Bo w (6.53 MHz / Tesla)(1.5
Tesla) w 9.795 MHz
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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 its
sensitivity and abundance.
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Larmor Equation (continued)
Is 42.58 MHz / Tesla the g 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.
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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)

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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 gt spin down)

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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
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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.

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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

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RF Excitation (continued)

• Resonance (continued)
• 1.) Individual spins flip
• spin up absorbed energy gt spin down
• spin down absorbed energy gt 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.

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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.

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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.

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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.

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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)

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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)

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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).

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MZ0
Longitudinal Magnetization
63 MZ0
Time
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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 its 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.

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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.

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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.

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90o flip
Mxy0
Transverse Magnetization
37 Mxy0
T2
Time
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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.

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T2 Relaxation (continued)
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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.
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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.
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T2 Relaxation (continued)
The observed T2 relaxation is termed T2 and
has two components
g D B


g D B represents the effect of magnetic
field inhomogeneity
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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).

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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 lt T1 Soft Tissue
  T1 ? 800 msec, T2 ? 80 msec

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Bloch Equations
The equations describing nuclear magnetic
resonance were derived by Felix Bloch in 1946.
Longitudinal magnetization
Transverse magnetization
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B1
Maximum RF field applied 23.5 microT
(circularly polarized) For Siemens Magnetom 3T.