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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
           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.