Magnetic Resonance Imaging: Physical Principles

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Magnetic Resonance ImagingPhysical Principles

• Richard Watts, D.Phil.
• Weill Medical College of Cornell University,
• New York, USA

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Physics of MRI, Lecture 1

• Nuclear Magnetic Resonance
• Nuclear spins
• Spin precession and the Larmor equation
• Static B0
• RF excitation
• RF detection
• Spatial Encoding
• Slice selective excitation
• Frequency encoding
• Phase encoding
• Image reconstruction

• Fourier Transforms
• Continuous Fourier Transform
• Discrete Fourier Transform
• Fourier properties
• k-space representation in MRI

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Bibliography

• Magnetic Resonance Imaging Physical Principles
  and Sequence Design. Haacke, Brown, Thompson,
  Venkatesan. Wiley 1999.
• The Fourier Transform and Its Applications.
  Bracewell. McGraw-Hill 2000.
• Medical books simple, but little mathematical
  depth
• Physics books more depth, but more complicated
• Web http//mri.med.cornell.edu/links.html

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Nuclear Spins

• Rotating charges correspond to an electrical
  current
• Current gives a magnetic moment, ?
• Nuclear spin, electron spin and electron orbit
• Moments align with external magnetic field

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Spin Precession

• Magnetic Spinning Top
• ? Gyromagnetic ratio
• Precession frequency Larmor equation
• For protons, ? 42.58 MHz/T

Bloch equation with no relaxation
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NMR Nuclei
Nuclei must have an unpaired proton to give a net
magnetic moment
Nucleus g (MHz/T) r (M)
1H(1/2) 42.58 88
23Na(3/2) 11.27 0.08
31P(1/2) 17.25 0.075
17O(5/2) -5.77 0.016
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B0 Field

• Higher static field gives greater polarization of
  the spins at a given temperature ? more signal
• B0 lt 0.5T can be achieved using permanent magnets
  (large, heavy, but cheap to maintain)
• B0 gt 0.5T requires superconducting magnet
  (expensive, requires liquid nitrogen and helium
  cooling). Quenching possible!
• B0 must be uniform to high accuracy (1 in 106)
  over the volume to be imaged
• Typical clinical scanner uses 1.5T
  superconducting magnet. Larmor frequency 64MHz

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RF Excitation

• Energy is put into the spins by a small
  alternating magnetic field, b1, at the resonant
  frequency of the spin precession
• For typical DC field strengths 0.5-7T the
  resonant frequency is in the radio frequency (RF)
  part of the electromagnetic spectrum
• Interference to/from other RF sources can be a
  problem requires shielding

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RF Excitation
rotating frame
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RF Detection

• Faraday induction
• The rate of change of flux multiplied by the
  number of coils
• Coil design for maximum sensitivity in the region
  of interest

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Spatial Encoding

• Three gradient coils allow the magnetic field to
  vary in any direction (linear combination)
• Spatial information comes from the variation in
  Larmor frequency due to the field gradient
• No moving parts to acquire different
  views/acquisition parameters
• Gradient coils require switching current rapidly
  in a high magnetic field ? vibration ? noise

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Slice Selective Excitation

• Gradient field
• Gradient strength, center frequency and bandwidth
  of rf pulse determines slice selected

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Gradient Coils

• Field always along z
• Gradient along x, y, or z
• Usually, only one gradient active at a time, but
  can combine to give any arbitrary direction

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Frequency Encoding

• Apply magnetic field gradient along x-axis during
  data read-out
• Frequency of signal depends on the x-position of
  the spins from which it is generated
• Fourier transform from (spatial) frequency space
  to image space

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Phase Encoding

• Apply magnetic field gradient along y-axis prior
  to read-out for a time ?t
• Phase of spins depends on y-position
• Repeat acquisition for different phase encoding
  amplitudes

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Image Reconstruction

• Spins from all positions (voxels) contribute
  signals to each measurement (sample)
• The frequency and phase of the signal from each
  voxel is determined by its spatial position
• How do we form an image?

Fourier Transforms!
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Fourier Transforms 1

• Example Sound
• What is sound? An oscillating pressure wave in
  the air. Pressure is a function of time f(t)
• How do we hear? Our ears contain hair-like
  structures that resonate at different
  frequencies. What we detect is the amplitude of
  each frequency, F(?)
• What is the mathematical relation between f(t)
  and F(?)?

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Fourier Transforms 2

• To measure the signal at a given frequency,
  multiply by a sine/cosine wave at that frequency
  and integrate all other frequencies ? 0
• This is the continuous Fourier transform
• Note that

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Fourier Transforms 3
Time/Frequency (t/?) domain Position/Spatial Frequency (x, k) domain
Fourier Transform t??, x?k
Inverse Fourier Transform ??t, k?x
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Example Calculation, ?(x-x0)

• Dirac delta function, spike, impulse function
• ?(x?0) 0
• ?(x)dx 1
• An infinitely thin spike of unit area

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Example Calculation, ?(x-x0)
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Example Calculation, Boxcar
Boxcar (Rect) function of width a at the
origin f(-a/2 ltxlt a/2) 1, otherwise f(x)0
Note f(x) is even so F(k) must be real
Useful result relating to the point-spread
function for finite sampling - later
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Example Calculation, Boxcar
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Discrete Fourier Transform (DFT)

• Signal in MRI is continuous over all spatial
  frequencies, but only a set of uniformly spaced
  measurements are made.
• Discrete sampling at k p?k and x q?x
• (p,q integers). Replace integral with finite
  summation

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Continuous vs. Discrete Fourier Transform
Continuous Discrete
Fourier Transform
Inverse Fourier Transform
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Sampling Effects

• ?k Spatial frequency step
• Due to the cyclical nature of the sine wave,
• Hence, a signal at xx0 cannot be distinguished
  from one at xx01/?k
• 1/?k Field of view, FOV

• p?k Highest spatial frequency
• This determines the resolution of the image
  obtained
• The point-spread function can be calculated
  intrinsic blurring of the reconstructed image

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Nyquist Sampling Criterion

• The sampling field of view, 1/?k must be greater
  than the object dimension, A
• ie.
• If this is not fulfilled, wrap-around artifacts
  are produced

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

f(x) F(k) F(k)
f(x) Real Part Imaginary Part
Real Even Odd
Imaginary Odd Even
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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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Fourier Properties

• Linearity
• Space scaling
• Space shifting
• Symmetry
• Convolution
• Derivative
• Parseval

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k-space Representation in 1D

• Magnetic field is sum of static and time-varying
  gradient fields
• Demodulate at ?0, accumulated phase?G

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Aside - Signal Demodulation

• Signal measured has a frequency of 64 MHz at
  1.5T. We are only interested in frequency or
  phase changes relative to ?0.
• Signal is demodulated by multiplying by a sine
  wave at ?0
• Effectively we move into the frame of reference
  of the center frequency. Analogue band-pass
  filters get rid of higher frequencies

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k-space Representation in 1D

• Signal is the integration of the spin density
  ?(z) over z, accounting for phase variations
• Define kk(t) as
• Hence,
• The signal measured s(k) is the Fourier transform
  of the spin density

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k-space Representation in 1D

• s(k) and ?(z) are a Fourier pair
• Fourier transform
• Inverse Fourier transform

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k-space Representation in 3D

• s(k)?s(k), ?(z) ? ?(r)
• s(k) and ?(r) are a Fourier pair
• Fourier transform
• Inverse Fourier transform
• Integration is now over a volume

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k-space and Image-Space
Not obvious!