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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

What if the RF field is not synchronized?

Using the swingset example: now the driving force is no longer Using the swingset example: now the driving force is no longer synchronized with the swing frequency, thus the efficiency of synchronized with the swing frequency, thus the efficiency of driving the swing is less.driving the swing is less.

In a real spin system, there is a term called “effective B1 In a real spin system, there is a term called “effective B1 field”,field”,given by given by

BB1eff1eff = = BB11 + + ∆ω∆ω //γγwhere where

∆ω∆ω = = ωω oo – – ωω e e

BB11

BB1eff1eff∆ω∆ω//γγ

Part II.1Part II.1

Image FormationImage Formation

What is image formation?

To define the spatial location of the protonTo define the spatial location of the protonpools that contribute to the MR signal afterpools that contribute to the MR signal afterspin excitation.spin excitation.

A 3-D gradient field (dB/dx, dB/dy, dB/dz) would A 3-D gradient field (dB/dx, dB/dy, dB/dz) would allow a unique correspondence between the spatial allow a unique correspondence between the spatial location and the magnetic field. Using this information,location and the magnetic field. Using this information,we will be able to generate maps that contain spatial we will be able to generate maps that contain spatial information – images.information – images.

Gradient Coils

Gradient coils generate varying magnetic field so that spins at different location precess at frequencies uniqueto their location, allowing us to reconstruct 2D or 3Dimages.

X gradient Y gradient Z gradient

x

y

z

x

z z

x

y y

Spatial Encoding – along Spatial Encoding – along xx

xx

Spatial Encoding – along Spatial Encoding – along yy

yy

Spatial Encoding of the MR Signal

0.8

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

Spatial Encoding 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

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:

♦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

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 = γγHH * G * Gslsl * d * dslsl

The bandwidth of the RF excitation pulse:The bandwidth of the RF excitation pulse: ∆ω/2π∆ω/2π

Thus the slice thickness can be derived asThus the slice thickness can be derived as ddslsl = = ∆ω∆ω / ( / (γγHH * G * Gslsl * 2 * 2π)π)

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 = γγHH (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 2⋅T2*, 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

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

PhasePhase

Phase HistoryPhase History

digitizer ondigitizer on

GradientGradient

TETE

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

readoutreadout



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(f⋅t) and cos([f+∆f]⋅t) will be more different

Calculation of the Field of View (FOV)along frequency encoding direction

γ* G* Gf f * FOV* FOVff = BW = 1/ = BW = 1/∆∆tt

Which means Which means FOVFOVff = 1/ ( = 1/ (γγ 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 φ 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

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

γ* G* Gp p * FOV* FOVpp = N = Npp / T / Tpp

Which means Which means FOVFOVpp = 1/ ( = 1/ (γγ G Gpp T Tpp/N/Npp))

= 1/ (= 1/ (γγ 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.

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

readoutreadout




ReadoutReadout

readoutreadout readoutreadout

…………

…………

…………

TRTR

Part II.2 Introduction to k-space (a space of the spatial frequency)

ImageImage k-spacek-space

Acquired MR Signal

Mathematical Representation:Mathematical Representation:

dxdyeyxIkkS ykxkiyx

yx )(2),(),( +−∫∫= π

Kx = Kx = γγ/2/2π ∫π ∫00tt Gx(t) dtGx(t) dt

Ky = Ky = γγ/2/2π ∫π ∫00tt Gy(t) dtGy(t) dt

Figure 4.7. 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. We have indicated, for four sample k-space points, which gradient vectors contribute at different image space locations to the k-space data.

..

.

..

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.+Gx-Gx 0

0

+Gy

-Gy .

Image SpaceK-Space

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Acquired MR Signal

dxdyeyxIkkS ykxkiyx

yx )(2),(),( +−∫∫= π

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.

By physically adding all the signals from each voxel up under By physically adding all the signals from each voxel up under the gradients we use.the gradients we use.

Two Spaces

FTFT

IFTIFT

k-spacek-space

kkxx

kkyy

Acquired DataAcquired Data

Image spaceImage space

xx

yy

Final ImageFinal Image

ImageImage KK

The k-space Trajectory

Kx = Kx = γγ/2/2π ∫π ∫00tt Gx(t) dtGx(t) dt

Ky = Ky = γγ/2/2π ∫π ∫00tt Gy(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

∆∆kk

∆∆k = 1 / FOVk = 1 / FOVRefer to slide Refer to slide 32, 3632, 36

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FOV: 10 cmFOV: 10 cmPixel Size: 1 cmPixel Size: 1 cm

FOV:FOV:Pixel Size:Pixel Size:

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Figure 4.16. Effects of sampling in k-space upon the resulting images. Field of view and resolution have an inverse relation between image space and k-space. Shown in (A) is a schematic representation of densely sampled k-space with a wide field of view, resulting in the high-resolution image below. If only the center of k-space is sampled (B), albeit with the same sampling density, then the resulting image below has the same field of view, but does not have as high of spatial resolution. Conversely, if k-space is sampled across a wide field of view but with limited sampling rate (C), the resulting image will have a small field of view but high resolution.

Spatial Displacement

Original image K-space trajectory Displaced image

Original image Distorted image

Figure 4.17. Spatial and intensity distortions due to magnetic field inhomogeneities during readout. If there is a systematic change in the spin frequency over time, the resulting image may be spatially displaced (B). Another possible type of distortion results from local field inhomogeneities, which in turn cause the resonant frequency to vary slightly across spatial locations, affecting the intensity of the image over space.

Figure 4.18. Image distortions caused by gradient problems during readout. Each row shows the ideal image, the problem with acquisition in k-space, and the resulting distorted image. Distortions along the x-gradient will affect the length of the trajectory in k-space, resulting in an image that appears stretched (A). Distortions along the y-gradient will affect the path taken through k-space over time, resulting in a skewed image (B). Distortions along the z-gradient will affect the match of excitation pulse and slice selection gradient, influencing the signal intensity (C).

Original image K-space trajectory Distorted Image

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