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Principles of MRI Physics and Engineering. Allen W. Song Brain Imaging and Analysis Center Duke University. Part II.1 Image Formation. What is image formation?. Define the spatial location of the proton pools that contribute to the MR signal. Steps in 3D Localization. - PowerPoint PPT Presentation
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Principles of MRI Principles of MRI Physics and EngineeringPhysics and Engineering
Principles of MRI Principles of MRI Physics and EngineeringPhysics and Engineering
Allen W. Song Allen W. Song
Brain Imaging and Analysis CenterBrain Imaging and Analysis Center
Duke UniversityDuke University
Part II.1Part II.1
Image FormationImage Formation
What is image formation?
Define the spatial location of the protonDefine the spatial location of the protonpools that contribute to the MR signal.pools that contribute to the MR signal.
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
RF Field: Excitation Pulse
00 tt
FoFo
FoFo Fo+1/ tFo+1/ t
TimeTime FrequencyFrequency
tt
FoFo FoFo
F= 1/ tF= 1/ t
FTFT
FTFT
Gradient Fields: Spatially Nonuniform B: Extra static magnetic fields (in addition to B0) that
vary their intensity in a linear way across the subject
Precession frequency of M varies across subject This is called frequency encoding — using a
deliberately applied nonuniform field to make the precession frequency depend on location
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]
Spin phase with x gradient onSpin phase with x gradient on
xx
Spin phase with y gradient onSpin phase with y gradient on
yy
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:
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:
Slice Selection
Slice Selection – along Slice Selection – along zz
zz
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
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
Changing slice thickness
new slicenew slicethicknessthickness
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.
Selecting different slices
new slicenew slicelocationlocation
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
Readout of the MR Signal
w/o encoding w/ encoding
ConstantMagnetic Field
VaryingMagnetic Field
Readout of the MR Signal
Fourier Transform
A typical diagram for MRI frequency encoding:Gradient-echo imaging
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
TETE
PhasePhase
Phase HistoryPhase History
digitizer ondigitizer on
GradientGradient
TETE
A typical diagram for MRI frequency encoding:Spin-echo imaging
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
TETE
180180oo TETE
PhasePhase
Phase HistoryPhase History
GradientGradient
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
Calculation of the Field of View (FOV)along frequency encoding direction
* G* Gf f * FOV* FOVff = BW, = BW,
where BW is the bandwidth for thewhere BW is the bandwidth for thereceiver digitizer.receiver digitizer.
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
A typical diagram for MRI phase encoding:Gradient-echo imaging
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
A typical diagram for MRI phase encoding:Spin-echo imaging
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
Calculation of the Field of View (FOV)along phase encoding direction
* G* Gp p * FOV* FOVpp = N = Npp / T / Tpp
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.
Multi-slice acquisition
Total acquisition time =Total acquisition time = Number of views * Number of excitations * TRNumber of views * Number of excitations * TR
Is this the best we can do?Is this the best we can do?
Interleaved excitation methodInterleaved excitation method
Part II.2 Introduction to k-space (a space of the spatial frequency)
Acquired MR Signal
Mathematical Representation:Mathematical Representation:
dxdyeyxIkkS ykxkiyx
yx )(2),(),(
This equation is obtained by physically adding all the signalsThis equation is obtained by physically adding all the signalsfrom each voxel up under the gradients we use.from each voxel up under the gradients we use.
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.
Two Spaces
FTFT
IFTIFT
k-spacek-space
kkxx
kkyy
Acquired DataAcquired Data
Image spaceImage space
xx
yy
Final ImageFinal Image
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:
A typical diagram for MRI frequency encoding:A k-space perspective
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI frequency encoding:A k-space perspective
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI phase encoding:A k-space perspective
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI phase encoding:A k-space perspective
digitizer ondigitizer on
ExcitationExcitation
SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding
PhasePhase EncodingEncoding
ReadoutReadout
Exercise drawing its k-space representationExercise drawing its k-space representation
The k-space Trajectory
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Sampling in k-spaceSampling in k-space
kkmaxmax
kk
k = 1 / FOVk = 1 / FOV2k2kmaxmax = 1 / = 1 / xx
Link back to slides 26 and 30Link back to slides 26 and 30
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AA
BB
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
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AA BB
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
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AA
BB. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .
FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm
FOV:FOV:Pixel Size:Pixel Size:
Examples of images and their k-space map