principles of mri physics and engineering

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


GradientGradient

TETE

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




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




PhasePhase EncodingEncoding

ReadoutReadout

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





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




ReadoutReadout

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

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




ReadoutReadout


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





ReadoutReadout


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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



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Examples of images and their k-space map

principles of mri physics and engineering

Documents

rf frequency

slice frequency of rf

slice rf field

rf signal phase

slice of tissuesource

total rf signal

frequency f

center frequency