mri is a fourier transform magnetic resonance imaging · •this is not a physics lecture … but,...
TRANSCRIPT
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Magnetic Resonance Imaging
6.003
Elfar Adalsteinssonwith materials from LL Wald, MGH Martinos Center
MRI is a Fourier Transform
• This is not a physics lecture … but, we’ll make plausibility arguments for the signal source and signal encoding in MRI that lead to this Fourier interpretation.
!(#) = &',)
* +, , -./01(23 4 '526 4 ))7+7,
MRI = Magnetic Resonance Imaging
formerly known as
Nuclear Magnetic Resonance ImagingDerives from
(nuclear) spin angular momentumand associated magnetic dipole moment, m
Proper treatment of spins: Quantum Mechanics
Here:Classical picture
“charged, spinning sphere”
gives rise to current loopthat creates a magnetic dipole moment, m
Hydrogen 1HOther nuclei, 23Na, 31P, …much lower concentrations
Abundance~ 80 M
… MRI in medicine is imaging
of water
Signal Source in MRI
Spins in a strong magnetic field, B0
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No net alignment:!M =
!m∑ = 0
Spins in earth’s magnetic field Spins in a strong, external field B0
B0 ~0.2T
Net alignment of spins in the presence of B0
yields signal source in MRI:!M =
!m∑
B0 ~0.2T
Spins in a strong, external field B0
Spins after RF excitation, B1
Direction of precession
Direction of B0
Spinning nucleus
Precession of Magnetization
xy
z
!Mxy
Mxy: Precesses at ω = γ "#
$ = "#'
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Signal Detection
Faraday detection: a generator
( ) ( )0 cost M twF µ
0
( )( )
( ) sin( )
d tV tdt
V t M tw w
F= -
µ -
V(t)
Imaging
Image Encoding
What is imaging?
We want to estimate Mxy as a function of x, y, z …
How do we do this?
Image encoding embeds spatial information in FIDs
Image reconstruction infers spatial mapping from encoded FIDs
Can we do imaging with precessing spins in a B0 field?
Gradient Fields for
Frequency Encoding of Spatial Information
We introduce intentional spatial variation in the precession frequency of spins
Frequency and spatial location map 1-to-1 in the presence of a constant gradient field
Thus, piano comparison to frequency encoding
X-gradient coil produces a linearly varying z-directed field that is characterized by its slope, !", as a function of x,
!B(x, y, z) = B0 z Gx ⋅ x z
!B(x, y, z) = (B0 +Gx ⋅ x)z
Bz
x
Bz
x
Bz
x
Gx =∂Bz∂x
Uniform magnet Field from x-gradient coil
Total field, sumof#$ and !" % &directed along (
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Uniform magnet Field from x-gradient coil
Gx =∂Bz∂x
Bz
x
Bz
x
Bz
x
X-gradient coil produces a linearly varying z-directed field that is characterized by its slope, !", as a function of x,
!B(x, y, z) = B0 z Gx ⋅ x z
!B(x, y, z) = (B0 +Gx ⋅ x)z
Total field, sumof#$ and !" % &directed along (
Nobel Prize
MR Nobel prizes:1943 physics, Stern1944, physics, Rabi1952, physics, Bloch, Purcell1991, Chemistry, Ernst2002, Chemistry, Wurthrich2003, Medicine, Lauterbur, Mansfield
Precession: ! = #(%& +∆%), or * = +,-(%& +∆%)
A gradient field, ./, maps space to frequency
After demodulation
i.e. ! =!& +#∆%, or * = *& + +,-∆%
∆% from ./: ! = !& +#./1, or * = *& + +,-./1
! = #./1, or * = +,-./1
Precession: ! = #(%& +∆%), or * = +,-(%& +∆%)
A gradient field, ./, maps space to frequency
After demodulation
i.e. ! =!& +#∆%, or * = *& + +,-∆%
∆% from ./: ! = !& +#./1, or * = *& + +,-./1
! = #./1, or * = +,-./1
!"($) = '(,*
+ ,, - ./01((,*,2)3,3-
The MRI Signal Equation
What’s is the phase term,4(,, -, $)?
By definition of phase, and by Larmor relation:
5152 = 6 = 789 ,
!"($) = '(,*
+ ,, - ./01((,*,2)3,3-
The MRI Signal Equation
What’s is the phase term, 4(,, -, $)?
By definition of phase, and by Larmor relation:
51((,*,2)52 = 6(,, -, $) = 789(,, -, $),
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!"($) = '(,*
+ ,, - ./01((,*,2)3,3-
The MRI Signal Equation
What’s is the phase term, 4(,, -, $)?
i.e.
4(,, -, $) = ∫627 ,, -, 8 38 = 9∫6
2:; ,, -, 8 38
!"($) = '(,*
+ ,, - ./01((,*,2)3,3-
The MRI Signal Equation
What’s is the phase term, 4(,, -, $)?
If 56 ,, -, $ = 57 +9( $ , +9* $ -, then
4 ,,-, $ = :57$ +:;
7
29( < ,3< +
:;7
29* < -3<
!(#) = &',)
* +, , -./01(23 4 '526 4 ))7+7,
The MRI Signal Equation
where
8' # = 901∫;
4<' = 7=
8) # = 901∫;
4<) = 7=
After demodulation,
!
"
#$
#%&(#$, #%)
Fourier0(!, ")
⋯
Lab: Given raw data in k-space,reconstruct an image