MRI Physics 2: Introduction to Magnetic
Resonance Imaging (MRI) Gareth J. Barker
Department of Neuroimaging
Terminology - Recap Nuclear Magnetic Resonance
– Property of (some) Nuclei – Magnetic effect – Resonance (i.e. matching) effect of some
property of nucleus to magnetic field MRS (Magnetic Resonance Spectroscopy) MRI (Magnetic Resonance Imaging)
More in later lectures … This lecture …
Terminology
sMRI • structural MRI
– Investigates physical appearance of brain
– measurement of physical properties of brain structures
• e.g. volumes, shapes, lengths = morphology
fMRI • functional-MRI:
– indirect measurement of brain activity related to function
– extremely subtle changes in tissue appearance related to changes in blood oxygenation (BOLD)
– requires detailed comparison of rapidly acquired “structural” scans
Basic principles of MRI apply to both sMRI and fMRI; application specific details in later lectures …
Magnetic Resonance Imaging
• … can create images of many parts of the body – Both structural &
functional • How?
Learning Objectives • To be aware of the basic concepts of
Magnetic Resonance Imaging (MRI), including: – Magnetic field gradients – Spin echoes and gradient echoes – “Spin Warp” imaging:
• Slice selection, frequency encoding and phase encoding
• Multi-slice, 2D and 3D sequences – K-space
GRADIENTS AND ECHOES MRI Physics
• By convention, the main magnetic field, B0, is aligned with z
Field Gradients
• A field gradient along x, for example, means that magnetic field (in z direction) varies with position along x.
-x x
y z B 0
B0 + Gxx
B0 - Gxx
Colour Gradient
From lecture 1– More Units • Magnetic field strength
is measured in Tesla
• Older unit, Gauss, is still occasionally used – 10kG=1T – 10G=1mT
• We will discuss magnetic field gradients in a later lecture
• Gradient strength is measured in mTesla per metre (mT/m) – 1mT/m = 10 G/cm
From lecture 1– More Units • Magnetic field strength
is measured in Tesla
• Older unit, Gauss, is still occasionally used – 10kG=1T – 10G=1mT
• We will discuss magnetic field gradients in a later lecture
• Gradient strength is measured in mTesla per metre (mT/m) – 10mT/m = 1G/cm
Correction!!!!
Field Gradients • What happens
when magnetic field, B, is not uniform over the object? – if field increases
linearly with position
– resonant frequency also increases linearly with position
Position
Fiel
d Si
gnal
Frequency
Projection-reconstruction imaging
• A single 90o pulse, followed by a field gradient, can give a 1 dimensional (1D) image, or projection.
Position
Sign
al Brightness
represents signal
intensity
Projection-reconstruction Imaging
• Rotating gradient can build up complete image from projections
Position
Sign
al
Position
Sign
al
Position
Sign
al
Position
Sign
al
Projection-reconstruction Imaging
• Full 2D image can be formed – similar to how a CT
scanner works – early MR scanners
used this approach
Swinburne University of Technology, Medical Imaging HET408,
David Liley ([email protected])
Projection-reconstruction Imaging
• It is difficult to measure signal immediately after 90o pulse
• We can’t control degree of T2 or T2* decay – More in next lecture
• It is difficult to
reconstruct artefact free images from projections
• Collect an echo
• Use a phase encoding technique – spin warp imaging
Echo Signals
t = 0 t = TE/2
t = TE/2
All stop, and reverse direction
t = ΤΕ
Spin & Gradient Echo Imaging
• ‘Spin Echo’ imaging uses an extra (180o) RF pulse to refocus effects of dephasing – basis for many sMRI sequences
• ‘Gradient Echo’ imaging refocuses spins without a 180o pulse – used for both sMRI and fMRI
ΤΕ/2
ΤΕ/2
180ox
Spin Echo Imaging
• 90o pulse (along x') produces transverse magnetisation along y'
• magnetisation dephases due to variations (inhomogeneities) in the local field
• 180o pulse (along x') flips spins
• spins precess in same direction and at same speed as before
• transverse magnetisation rephases along –y’
Gradient Echo Imaging
• 90o pulse (along x') produces transverse magnetisation along y'
• magnetisation dephases – applied gradient
• plus time-invariant local field inhomogeneities
• spins precess in opposite direction and at same speed as before – transverse magnetisation
rephases • effect of local inhomogeneity
is NOT refocused
ΤΕ/2
ΤΕ/2
Recap • We can use magnetic field gradients to
encode spatial positions
• For practical reasons, we often want to collect an echo signal, rather than the FID. We can do this with either: – Gradient echoes
• Uses reversal of a magnetic field gradient
– Spin echoes • Uses 180o RF pulses
SPIN WARP IMAGING MRI Physics
Spin Warp Imaging • Remember, creating an image requires:
– Slice selection – Spatial encoding
• Frequency encoding uses a gradient during data acquisition, affecting frequency of signal received
• Phase encoding uses a gradient before data acquisition, affecting phase of signal received
– All of these processes depend on the Fourier Transform
Δs
Δs = slice thickness
Frequency encoding
Phase encoding
Fourier Transformation • We can describe any
signal in two ways: – frequencies that make it
up, along with their relative proportions
– amplitude of the total signal at each instant of time
• eg: Sound – wave form (pressure wave)
• time domain – frequency (musical notes)
• frequency domain
• The Fourier Transform (FT) is a mathematical formula that translates one description into the other
• Any continuous function f(x) has a FT, F(s) – If x is time, then s is spatial
frequency – f(x) and F(s) describe same
thing in different ways
Fourier Transformation • FT of f(x)
• Inverse transform
– Note convention to use same letter for each member of a Fourier pair, with one in upper- and one in lower-case
• FT then inverse FT gets you back where you started
– there are several different conventions for positioning of the '2π's in these equations
• pick one and stick with it!
∫∞
∞−
−= dxexfsF xsi π2)()(
∫∞
∞−
= dsesfxF xsi π2)()(
∫ ∫∞
∞−
∞
∞−
−= dsedxexfxf xsixsi ππ 22 ])]([[)(
Common FT Pairs
Gaussian Gaussian ‘top hat’ sinc triangle sinc2 cosine even
impulse pair
sine odd impulse pair
constant delta function
Rep
rodu
ced
from
: Bra
cew
ell,
The
Four
ier T
rans
form
and
its
App
licat
ions
, McG
raw
Hill
, 196
5
Slice Selection • Apply a shaped
Radio Frequency (RF) pulse in presence of a slice select gradient – Only excites (affects)
spins whose resonant frequency falls within range of frequencies in pulse
⇐ FT ⇒ Bandwidth
Position R
eson
ant
Freq
uenc
y
Not excited Not excited
Exci
ted
Resonant frequency within
bandwidth of pulse
Shaped Pulses • One of the most common
RF pulse shapes is the sinc – mathematically defined as
– α is a constant which determines bandwidth
• Its FT has a 'top hat' shape – defines amplitude, and
therefore effectiveness, at various frequencies
tty
αα )sin(
= τ
Time →
2/τ = Δω Frequency →
Shaped Pulses • Another common RF
pulse shapes is a Gaussian – mathematically defined as
– α is again a constant which determines the bandwidth
• FT of a Gaussian is also a Gaussian
• Unfortunately, effect of any of these pulses at high flip angles is not exactly what would be expected from FT – Numerically optimised
pulses can be used to produce slice profiles which are closer to optimum
Time →
)exp( 2ty α−=
Slice Selection • Note that a rephase
gradient lobe is needed, to counteract dephasing during portion of pulse for which magnetisation has a transverse component – gradient area required
depends on pulse shape – typically approximately
half that of main slice select gradient
90o
Gslice
Slice Selection • As well as the
excitation pulse, we also need the spin echo refocusing pulse
• This must also be slice selective, so we apply a gradient – Note that this is
symmetrical • no rephase lobe needed
90o 180o
Frequency Encoding • Apply a gradient during
data acquisition (often called read or readout gradient) – Notice dephase lobe before
readout period. • ensures that halfway
though readout period, when spin echo occurs, spins are all in phase
• gradient area required is exactly half that of main read gradient
• can also be placed before 180o pulse, in which case amplitude is positive:
180o
Gread
Gread
Frequency Encoding • Spins at different
positions experience slightly different local fields and therefore precess at different rates – Resulting signals
thus have different frequencies
+
... etc.
Position
Local
FT
+
+
+
+
Bottles of water
fieldPosition Fi
eld
Phase Encoding
• How do we encode final direction?
• Can’t just apply another gradient at 90o to readout, as this would just give an effective gradient at 45o
– Instead, manipulate phase of the signal
Phase Encoding
-1
-0.5
0
0.5
1
low freq.high freq.
-1
-0.5
0
0.5
1phase = 0 degphase = 20 deg
Same amplitude and phase, but different frequencies
Same amplitude and frequency, but different phases
Phase Encoding • Apply a gradient before
data acquisition – usually called phase
encoding gradient • As before, spins at
different positions experience slightly different local fields and precess at different rates – By end of gradient they
have precessed through different angles, giving signals with different phases
– Can also be placed before 180o pulse:
Gread
180o
Gread
Gphase
Gphase
Phase Encoding • We can only
measure phases between 0 and 2π – can not unambiguously
distinguish the spins
• Repeat acquisition with different phase encode gradient strengths – way in which phase
changes with gradient strength is unique to spins in a particular position + ... etc.
Position
Local field
++
FT
y =
stay in phase
B0
B0
B0
B0
B0
1 2-1-2 0
y = 1 2-1-2 0
Spins at y = 0
Fiel
d
Spins at y=0 stay in phase
Position
Phase Encoding • Repetition is often
shown with dotted or dashed gradients – Sequence must be
repeated ‘matrix size’ times to collect complete information
– takes 64, 128, 256 … TR periods
Gread
Gphase
Pulse Sequence Diagrams • Putting it all together…
• Pulse sequence diagrams, show all gradients & RF pulses used – Give relative timing and
amplitudes of each pulse – not usually drawn exactly to
scale. – shading is often used to indicate
gradient areas which should be equal (or equal and opposite)
– arrows indicating exact timing of events
Data acquisition
G
G G
RF
TE TR
90 180
Acq or,
slice read
phase
Echo Time & Repetition Time
• TE and TR determine appearance of a spin echo image – different combinations will produce different
degrees of contrast (intensity difference) between tissues
• changing TR will change contrast between tissues with different T1 relaxation times
• changing TE will change contrast between tissues with different T2s
– more in next lecture
2D & 3D IMAGING
Multi Slice Imaging • TR is often much longer than
TE (to give required image contrast) – long delay between end of data
acquisition and next 90o pulse – provided we don't interfere with
recovery of longitudinal magnetisation from slice we have excited we can use this delay to measure something else
• Ideally, a slice selective pulse only affects magnetisation within the slice which it is selecting – by adjusting frequency of RF
pulse we can excite a second slice without affecting first
– can then phase and frequency encode as for first slice
Readout
Phase Encoding
Slice Selection
The magnetisation from this slice has been tilted in the xy plane and has formed the echo.
The magnetisation from anywhere outside the excited slice is still aligned along B0
and has NOT contributed to the spin echo.
Readout
Phase Encoding
Slice Selection
The magnetisation from this slice has been tilted in the xy plane and has formed the echo.
The magnetisation from anywhere outside the excited slice is still aligned along B0
and has NOT contributed to the spin echo.
Multi Slice Imaging
• If TR is long enough we can fit several slices into a single TR:
Slice 1 Slice 2 Slice 3
TR TR
Phase encode 1 Phase encode 2
Multi Slice Imaging • Pulse sequence diagrams
usually don’t show repetition of gradients needed for multiple slices
• (Also don’t typically show repetitions for averaging, multple cardiac phases, etc,)
Data acquisition
G
G G
RF
TE TR
90 180
Acq or,
slice read
phase
3D (‘Volumetric’) Imaging • Acquires data from full 3D volume
rather than single slice – typically still uses slice selective pulses, but
selects a thick (10 or 20 cm) slab • phase encode signal from this whole volume along
both y and z • phase encode gradient steps through a number of
amplitudes equal to the matrix size in that dimension, just as in 2D case
– use readout gradient to encode final dimension, as in 2D case
3D (‘Volumetric’) Imaging • Imaging time is equal to:
TR * (number of phase encoding steps in dimension 1) * (number of steps in dimension 2)
• Imaging time will be very long unless TR is short, e.g.:
2 * 256 * 192 = 98304 s = 1638.4 minutes = 27.3 hours!
• Use gradient echo sequence, with a very short TR and a low flip angle to minimize T1 weighting – more in next lecture
Recap
• To create an image we need to spatially encode the signal in 3 dimensions – For 2D imaging we use:
• Slice selection, frequency encoding in one direction & phase encoding in the other
• Can often fit multiple slices with in a TR, allowing multi-slice imaging
– For 3D imaging we use: • (Optional) slice/slab selection, frequency encoding
in one direction & phase encoding in the other two directions
K-SPACE Image Reconstruction
k-space • Remember:
– MR data points are collected as a function of: • phase encode step number (n) • time from start of readout window, (t)
• Can visualise this as a 2D plot – Turns out to be convenient to think of the axes as how
much x (or y, or both) gradient has been “seen” when data are collected
– A value ‘k’ (with components kx, ky, kz) defines where we are in ‘k-space’ ''
0))(()( dttt
t
∫= Gk γ
Spatial Frequencies • Spatial frequencies (positions in
k-space) have a useful, physical, interpretation – High spatial frequencies describe things
that change rapidly from pixel to pixel • convey detail of image
– Low spatial frequencies describe things that change slowly
• convey overall form of image
– Zero spatial frequency describes overall intensity of image
• contrast of image is largely determined by spatial frequencies close to zero
• The Fourier Transform (FT) can convert k-space data into an image
‘Rule of Thumb’
• Something large in one domain corresponds to something small in other, eg:
largest values of k smallest object resolvable
total area of k-space covered
pixel size
total area of image covered (ie field of view)
spacing between k-space points
Spin Echo • What does the k-
space sampling scheme for a simple spin echo sequence look like?
Data acquisition
Gy
Gx
Gz
RF
TE
TR
k Space Sampling Schemes a) apply a 90o pulse
• No gradients applied yet, so we must be at centre of k-space
k
kx
y
a
k Space Sampling Schemes a) apply a 90o pulse b) apply a large, negative, phase encode gradient along y
• The integral i.e. ky, is getting more negative with time
• We’re “moving” along -ky
• We’re not moving in kx
k
kx
y
a
b
∫t
y dttyG0
'' )(γ
k Space Sampling Schemes a) apply a 90o pulse b) apply a large, negative, phase encode gradient along y c) apply a positive 'dephase' gradient along x
• We’re now “moving” along kx
k
kx
y
a
b
c
k Space Sampling Schemes a) apply a 90o pulse b) apply a large, negative, phase encode gradient along y c) apply a positive 'dephase' gradient along x d) at time TE/2 after 90o pulse apply 180o pulse
• “Flips” us to the other side of k-space
k
kx
y
a
b
c d
k Space Sampling Schemes a) apply a 90o pulse b) apply a large, negative, phase encode gradient along y c) apply a positive 'dephase' gradient along x d) at time TE/2 after 90o pulse apply 180o pulse e) apply readout gradient and start measuring echo signal
k
kx
y
a
b
c d
e
k Space Sampling Schemes a) apply a 90o pulse b) apply a large, negative, phase encode gradient along y c) apply a positive 'dephase' gradient along x d) at time TE/2 after 90o pulse apply 180o pulse e) apply readout gradient and start measuring echo signal
peaks at TE/2 after 180o, then dies away again
k
kx
y
a
b
c d
e
k Space Sampling Schemes • Next phase encode
step samples different line of k space
• … process continues until all of k space is sampled
k
kx
y
k
kx
y
1234
k Space Sampling Schemes • order in which k-
space points are acquired is irrelevant, – We can come up
with complex ways of covering k-space
– may have practical or theoretical advantages
• More in later lectures
k
kx
y 12
34
Some Real Data… • Can plot out S(t,n) and
see echoes and pseudo-echoes in the frequency and phase directions:
• Can also display the intensity of raw k-space data S(kx,ky) as an image in its own right – (i.e. without Fourier
Transforming):
Fourier Transforming the Data
FT ? R
epro
duce
d fr
om: B
race
wel
l, Th
e Fo
urie
r Tra
nsfo
rm
and
its A
pplic
atio
ns, M
cGra
w H
ill, 1
965
Time
Phas
e en
code
step
Real data, as collect on the scanner:
1 25
6
Fourier Transforming the Data
FT ?
Recap • K-space is where the raw data from the
scanner “lives” – The axes of k-space are how much magnetic field
gradient has been “seen” by the object (in each direction)
– K-space “trajectories” allow us to see how the pulse sequence ensures data is sampled over the whole of k-space
• Low spatial frequencies (near the centre of k-space) encode the contrast and overall shape of an image
• High spatial frequencies (at the edges of k-space) encode all the detail
• The Fourier Transform can convert k-space data into an image
Final Recap • Creating a 2D or 3D image requires:
– Slice selection – Spatial encoding
• Frequency encoding uses a gradient during data acquisition, affecting frequency of signal received
• Phase encoding in 1 or 2 dimensions, uses gradients before data acquisition, affecting phase of signal received
– All of these processes depend on the Fourier Transform and the data collection process can best be described in k-space
Δs
Δs = slice thickness
Frequency encoding
Phase encoding
Bibliography / Suggested Reading
For a good overview: • MRI from Picture to Proton
– DW McRobbie, EA Moore, MJ Graves, MR Prince; Cambridge University Press; ISBN-10: 052168384X, 2007.
• Aimed at radiographers, but useful for everyone.
• Chapter 7 is particularly relevant
(Very) technical aspects: • Handbook of MRI Pulse
Sequences – M.A. Bernstein, K.F. King, X.J.
Zhou, Academic Press; 1 edition, ISBN-10: 0120928612, 2004
– Also available, from KCL computers, at http://www.sciencedirect.com/science/book/9780120928613
Bibliography / Suggested Reading
• Online Resources – http://www.mri-physics.com/bin/mri-physics-uk.pdf
• Freely available book on MR physics. – http://www.mritutor.org/mritutor/
• On-line resource about MR imaging.
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