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Magnetic Resonance Imaging: Physical Principles

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Title: Magnetic Resonance Imaging: Physical Principles


1
Magnetic Resonance ImagingPhysical Principles
  • Richard Watts, D.Phil.
  • Weill Medical College of Cornell University,
  • New York, USA

2
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

3
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

4
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

5
Spin Precession
  • Magnetic Spinning Top
  • ? Gyromagnetic ratio
  • Precession frequency Larmor equation
  • For protons, ? 42.58 MHz/T

Bloch equation with no relaxation
6
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
7
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

8
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

9
RF Excitation
rotating frame
10
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

11
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

12
Slice Selective Excitation
  • Gradient field
  • Gradient strength, center frequency and bandwidth
    of rf pulse determines slice selected

13
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

14
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

15
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

16
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!
17
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(?)?

18
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

19
Fourier Transforms 3
Time/Frequency (t/?) domain Position/Spatial Frequency (x, k) domain
Fourier Transform t??, x?k
Inverse Fourier Transform ??t, k?x
20
Example Calculation, ?(x-x0)
  • Dirac delta function, spike, impulse function
  • ?(x?0) 0
  • ?(x)dx 1
  • An infinitely thin spike of unit area

21
Example Calculation, ?(x-x0)
22
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
23
Example Calculation, Boxcar
24
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

25
Continuous vs. Discrete Fourier Transform
Continuous Discrete
Fourier Transform
Inverse Fourier Transform
26
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

27
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

28
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

29
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

30
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

31
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
32
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

33
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

34
Fourier Properties
  • Linearity
  • Space scaling
  • Space shifting
  • Symmetry
  • Convolution
  • Derivative
  • Parseval

35
k-space Representation in 1D
  • Magnetic field is sum of static and time-varying
    gradient fields
  • Demodulate at ?0, accumulated phase?G

36
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

37
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

38
k-space Representation in 1D
  • s(k) and ?(z) are a Fourier pair
  • Fourier transform
  • Inverse Fourier transform

39
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

40
k-space and Image-Space
Not obvious!
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