Principles of MRI: Image Formation

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

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.

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?

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.

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

A Simple Example of Spatial Encoding

0.8

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

Spatial Decoding of the MR Signal

FrequencyDecomposition

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

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:

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

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]

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

Spatial Encoding of the MR Signal

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

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.

More Complex Spatial EncodingMore Complex Spatial Encoding

x gradientx gradient

More Complex Spatial EncodingMore Complex Spatial Encoding

y gradienty gradient

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

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

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

A typical diagram for MRI frequency encoding:Spin-echo imaging

readoutreadout



ReadoutReadout

TETE

………………

Phase HistoryPhase History

180180oo TETE

PhasePhase

GradientGradient

………………digitizer ondigitizer on

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 = 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.

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

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

A typical diagram for MRI phase encoding:Gradient-echo imaging

readoutreadout



PhasePhase EncodingEncoding

ReadoutReadout………………

A typical diagram for MRI phase encoding:Spin-echo imaging

readoutreadout




ReadoutReadout………………

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.

Part II.2 Introduction to k-space (MR data space)

ImageImage k-spacek-space

PhasePhaseEncodeEncodeStep 1Step 1



Time Time point #1point #1



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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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.+Gx-Gx 0

0

+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

Two Spaces

FTFT

IFTIFT

k-spacek-space

kkxx

kkyy

Acquired DataAcquired Data

Image spaceImage space

xx

yy

Final ImageFinal Image

ImageImage KK

HighHighSignalSignal

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

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)

A typical diagram for MRI frequency encoding:A k-space perspective

readoutreadout



ReadoutReadout

Exercise drawing its k-space representationExercise drawing its k-space representation

9090oo


A typical diagram for MRI frequency encoding:A k-space perspective

readoutreadout



ReadoutReadout


9090oo 180180oo


A typical diagram for MRI phase encoding:A k-space perspective

readoutreadout




ReadoutReadout


9090oo


A typical diagram for MRI phase encoding:A k-space perspective

readoutreadout




ReadoutReadout


9090oo 180180oo


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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



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20 cm20 cm 2 cm2 cm

Original image K-space trajectory Distorted Image

K-space can also help explain imaging distortions:K-space can also help explain imaging distortions:

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Principles of MRI: Image Formation