principles of mri: image formation allen w. song brain imaging and analysis center duke university
TRANSCRIPT
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Principles of MRI:Principles of MRI:Image Formation Image Formation Principles of MRI:Principles of MRI:Image Formation Image Formation
Allen W. Song Allen W. Song
Brain Imaging and Analysis CenterBrain Imaging and Analysis Center
Duke UniversityDuke University
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What is image formation?
To define the spatial location of the sourcesTo define the spatial location of the sourcesthat contribute to the detected signal.that contribute to the detected signal.
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But MRI does not use projection, reflection, or refractionBut MRI does not use projection, reflection, or refractionmechanisms commonly used in optical imaging methodsmechanisms commonly used in optical imaging methodsto form image. So how are the MR images formed?to form image. So how are the MR images formed?
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Frequency and Phase Are Our Friends in MR ImagingFrequency and Phase Are Our Friends in MR Imaging
= = tt
The spatial information of the proton pools contributingThe spatial information of the proton pools contributingMR signal is determined by the spatial frequency andMR signal is determined by the spatial frequency andphase of their magnetization.phase of their magnetization.
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Gradient Coils
Gradient coils generate spatially varying magnetic field so that spins at different location precess at frequencies unique to their location, allowing us to reconstruct 2D or 3D images.
X gradient Y gradient Z gradient
x
y
z
x
z z
x
y y
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A Simple Example of Spatial Encoding
0.8
w/o encoding w/ encoding
ConstantMagnetic Field
VaryingMagnetic Field
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Spatial Decoding of the MR Signal
FrequencyDecomposition
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Steps in 3D Localization Can only detect total RF signal from inside the “RF
coil” (the detecting antenna) Excite and receive Mxy in a thin (2D) slice of the
subject The RF signal we detect must come from this slice Reduce dimension from 3D down to 2D
Deliberately make magnetic field strength B depend on location within slice Frequency of RF signal will depend on where it comes from Breaking total signal into frequency components will
provide more localization information
Make RF signal phase depend on location within slice
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Exciting and Receiving Mxy in a Thin Slice of Tissue
Source of RF frequency on resonanceSource of RF frequency on resonance
Addition of small frequency variationAddition of small frequency variation
Amplitude modulation with “sinc” functionAmplitude modulation with “sinc” function
RF power amplifierRF power amplifier
RF coilRF coil
Excite:Excite:
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Electromagnetic Excitation Pulse (RF Pulse)
00 tt
FoFo
FoFo Fo+1/ tFo+1/ t
TimeTime FrequencyFrequency
tt
FoFo FoFo
F= 1/ tF= 1/ t
FTFT
FTFT
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Gradient Fields: Spatially Nonuniform B:
During readout (image acquisition) period, turning on gradient field is called frequency encoding --- using a deliberately applied nonuniform field to make the precession frequency depend on location
Before readout (image acquisition) period, turning on gradient field is called phase encoding --- during the readout (image acquisition) period, the effect of gradient field is no longer time-varying, rather it is a fixed phase accumulation determined by the amplitude and duration of the phase encoding gradient.
x-axis
f60 KHz
Left = –7 cm Right = +7 cm
Gx = 1 Gauss/cm = 10 mTesla/m = strength of gradient field
Centerfrequency
[63 MHz at 1.5 T]
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Exciting and Receiving Mxy in a Thin Slice of Tissue
RF coilRF coil
RF preamplifierRF preamplifier
FiltersFilters
Analog-to-Digital ConverterAnalog-to-Digital Converter
Computer memoryComputer memory
Receive:Receive:
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Slice Selection
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Slice Selection – along Slice Selection – along zz
zz
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Determining slice thickness
Resonance frequency range as the resultResonance frequency range as the resultof slice-selective gradient:of slice-selective gradient: F = F = HH * G * Gslsl * d * dslsl
The bandwidth of the RF excitation pulse:The bandwidth of the RF excitation pulse:
Thus the slice thickness can be derived asThus the slice thickness can be derived as ddslsl = = / ( / (HH * G * Gslsl * 2 * 2
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Changing slice thickness
There are two ways to do this:There are two ways to do this:
(a)(a) Change the slope of the slice selection gradientChange the slope of the slice selection gradient
(b)(b) Change the bandwidth of the RF excitation pulseChange the bandwidth of the RF excitation pulse
Both are used in practice, with (a) being more popularBoth are used in practice, with (a) being more popular
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Changing slice thickness
new slicenew slicethicknessthickness
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Selecting different slices
In theory, there are two ways to select different slices:In theory, there are two ways to select different slices:(a)(a) Change the position of the zero point of the sliceChange the position of the zero point of the slice selection gradient with respect to isocenterselection gradient with respect to isocenter
(b) Change the center frequency of the RF to correspond(b) Change the center frequency of the RF to correspond to a resonance frequency at the desired sliceto a resonance frequency at the desired slice
F = F = HH (Bo + G (Bo + Gslsl * L * Lsl sl ))
Option (b) is usually used as it is not easy to change theOption (b) is usually used as it is not easy to change theisocenter of a given gradient coil.isocenter of a given gradient coil.
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Selecting different slices
new slicenew slicelocationlocation
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Readout Localization (frequency encoding)
After RF pulse (B1) ends, acquisition (readout) of NMR RF signal begins During readout, gradient field perpendicular to slice
selection gradient is turned on Signal is sampled about once every few microseconds,
digitized, and stored in a computer• Readout window ranges from 5–100 milliseconds (can’t be longer
than about 2T2*, since signal dies away after that)
Computer breaks measured signal V(t) into frequency components v(f ) — using the Fourier transform
Since frequency f varies across subject in a known way, we can assign each component v(f ) to the place it comes from
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Spatial Encoding of the MR Signal
w/o encoding w/ encoding
ConstantMagnetic Field
VaryingMagnetic Field
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It’d be easy if we image with only 2 voxels …It’d be easy if we image with only 2 voxels …
But often times we have imaging matrix at 256 or higher.But often times we have imaging matrix at 256 or higher.
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More Complex Spatial EncodingMore Complex Spatial Encoding
x gradientx gradient
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More Complex Spatial EncodingMore Complex Spatial Encoding
y gradienty gradient
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Physical Space
A 9A 9××9 case9 case
Before EncodingBefore Encoding After Frequency Encoding After Frequency Encoding (x gradient)(x gradient)
So each data point contains information from all the voxelsSo each data point contains information from all the voxels
MR data spaceMR data space
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A typical diagram for MRI frequency encoding:Gradient-echo imaging
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
TETE
Data points collected during thisData points collected during thisperiod corrspond to one-line in k-spaceperiod corrspond to one-line in k-space
………………Time point #1Time point #1 Time point #9Time point #9
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Phase Evolution of MR DataPhase Evolution of MR Data
digitizer ondigitizer on
Phases of spinsPhases of spins
GradientGradient
TETE
………………Time point #1Time point #1 Time point #9Time point #9
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A typical diagram for MRI frequency encoding:Spin-echo imaging
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
TETE
………………
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Phase HistoryPhase History
180180oo TETE
PhasePhase
GradientGradient
………………digitizer ondigitizer on
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Image Resolution (in Plane)
Spatial resolution depends on how well we can separate frequencies in the data V(t) Resolution is proportional to f = frequency accuracy Stronger gradients nearby positions are better separated
in frequencies resolution can be higher for fixed f Longer readout times can separate nearby frequencies
better in V(t) because phases of cos(ft) and cos([f+f]t) will be more different
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Calculation of the Field of View (FOV)along frequency encoding direction
* G* Gf f * FOV* FOVff = BW = 1/ = BW = 1/tt
Which means Which means FOVFOVff = 1/ ( = 1/ ( G Gff t)t)
where BW is the bandwidth for thewhere BW is the bandwidth for thereceiver digitizer.receiver digitizer.
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The Second Dimension: Phase Encoding Slice excitation provides one localization dimension Frequency encoding provides second dimension The third dimension is provided by phase encoding:
We make the phase of Mxy (its angle in the xy-plane) signal depend on location in the third direction
This is done by applying a gradient field in the third direction ( to both slice select and frequency encode)
Fourier transform measures phase of each v(f ) component of V(t), as well as the frequency f
By collecting data with many different amounts of phase encoding strength, can break each v(f ) into phase components, and so assign them to spatial locations in 3D
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Physical Space
A 9A 9××9 case9 case
Before EncodingBefore Encoding After Frequency EncodingAfter Frequency Encodingx gradientx gradient
After Phase EncodingAfter Phase Encodingy gradienty gradient
So each point contains information from all the voxelsSo each point contains information from all the voxels
MR data spaceMR data space
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A typical diagram for MRI phase encoding:Gradient-echo imaging
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout………………
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A typical diagram for MRI phase encoding:Spin-echo imaging
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout………………
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Calculation of the Field of View (FOV)along phase encoding direction
* G* Gp p * FOV* FOVpp = N = Npp / T / Tpp
Which means Which means FOVFOVpp = 1/ ( = 1/ ( G Gpp T Tpp/N/Npp))
= 1/ (= 1/ ( G Gpp t)t)
where Twhere Tpp is the duration and N is the duration and Npp the number the number
of the phase encoding gradients, Gp is theof the phase encoding gradients, Gp is themaximum amplitude of the phase encodingmaximum amplitude of the phase encodinggradient.gradient.
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Part II.2 Introduction to k-space (MR data space)
ImageImage k-spacek-space
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PhasePhaseEncodeEncodeStep 1Step 1
PhasePhaseEncodeEncodeStep 2Step 2
PhasePhaseEncodeEncodeStep 3Step 3
Time Time point #1point #1
Time Time point #2point #2
Time Time point #3point #3
…………....
Time Time point #1point #1
Time Time point #2point #2
Time Time point #3point #3
…………....
Time Time point #1point #1
Time Time point #2point #2
Time Time point #3point #3
…………....
……
……
....
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Contributions of different image locations to the raw k-space data. Each data point in k-space (shown in yellow) consists of the summation of MR signal from all voxels in image space under corresponding gradient fields.
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+Gy
-Gy .
Physical SpaceK-Space
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Acquired MR Signal
dxdyeyxIkkS ykxkiyx
yx )(2),(),(
From this equation, it can be seen that the acquired MR signal,From this equation, it can be seen that the acquired MR signal,which is also in a 2-D space (with kx, ky coordinates), is the which is also in a 2-D space (with kx, ky coordinates), is the Fourier Transform of the imaged object.Fourier Transform of the imaged object.
For a given data point in k-space, say (kx, ky), its signal S(kx, For a given data point in k-space, say (kx, ky), its signal S(kx, ky) is the sum of all the little signal from each voxel I(x,y) in the ky) is the sum of all the little signal from each voxel I(x,y) in the physical space, under the gradient field at that particular momentphysical space, under the gradient field at that particular moment
Kx = Kx = /2/200ttGx(t) dtGx(t) dt
Ky = Ky = /2/200ttGy(t) dtGy(t) dt
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Two Spaces
FTFT
IFTIFT
k-spacek-space
kkxx
kkyy
Acquired DataAcquired Data
Image spaceImage space
xx
yy
Final ImageFinal Image
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ImageImage KK
HighHighSignalSignal
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Full k-spaceFull k-space Lower k-spaceLower k-space Higher k-spaceHigher k-space
Full ImageFull Image Intensity-Heavy ImageIntensity-Heavy Image Detail-Heavy ImageDetail-Heavy Image
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The k-space Trajectory
Kx = Kx = /2/200ttGx(t) dtGx(t) dt
Ky = Ky = /2/200ttGy(t) dtGy(t) dt
Equations that govern k-space trajectory:Equations that govern k-space trajectory:
time0 t
Gx (amplitude)
Kx (area)
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A typical diagram for MRI frequency encoding:A k-space perspective
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
9090oo
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The k-space Trajectory
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A typical diagram for MRI frequency encoding:A k-space perspective
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
9090oo 180180oo
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The k-space Trajectory
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A typical diagram for MRI phase encoding:A k-space perspective
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
9090oo
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The k-space Trajectory
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A typical diagram for MRI phase encoding:A k-space perspective
readoutreadout
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
9090oo 180180oo
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The k-space Trajectory
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Sampling in k-spaceSampling in k-space
kkmaxmax
k = k = GGtt
k = 1 / FOVk = 1 / FOV
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AA
BB
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
10 cm10 cm 2 cm2 cm
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AA BB
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
5 cm5 cm1 cm1 cm
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AA
BB. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
20 cm20 cm 2 cm2 cm
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Original image K-space trajectory Distorted Image
K-space can also help explain imaging distortions:K-space can also help explain imaging distortions: