allen w. song, phd brain imaging and analysis center duke university

Allen W. Song, PhDBrain Imaging and Analysis CenterDuke University

MRI: Image Formation

What is image formation?

To define the spatial location of the sourcesthat contribute to the detected signal.

But MRI does not use projection, reflection, or refractionmechanisms commonly used in optical imaging methodsto form image …

A Simple Example

MRI Uses Frequency and Phase to Construct Image

= = tt

The spatial information of the proton pools contributing to MR signal is determined by the spatial frequency and phase of their magnetization.

Three 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

The Use of Gradient Coils for Spatial Encoding

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

MR Signal

Spatial Decoding of the MR Signal

FrequencyDecomposition

a 1-D Image !

How Do We Make a Typical MRI Image?

First Step in Image Formation: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: /2/2

Matching the two frequency ranges, the slice Matching the two frequency ranges, the slice thickness can be derived asthickness can be derived as ddslsl = = / ( / (HH * G * Gslsl * 2 * 2))

Changing Slice Thickness or Selecting Difference Slices

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 or Selecting Difference Slices

Second Step in Image Formation:

Spatial encoding and resolving one dimension within a plane

Spatial Encoding of the MRI Signal:An Example of Two Vials

w/o encoding w/ encoding

ConstantMagnetic Field

VaryingMagnetic Field

Continuous Sampling

Spatial Decoding of the MR Signal

FrequencyDecomposition

a 1-D Image !

It’d be inefficient to collect data points continuously over time, actually, if all weneed to resolve are just two elements in space.

There is a better way to resolve these twoelements discretely.

t

2t

G

Element 1 Element 2

G

Element 1 Element 2

A B

A

B

time 0

S0 = A + B

Time point 1S1 = | A*exp(-i1t) + B*exp(-i2t) |

Time Point 2

lag

lead

It turns out that all we need is just two data points:

1 = G x, where x is determined by the voxel size

A B

time t

The simplest case is to wait for time t such that A and B will point along opposite direction,

A B

A

B

such that S0 = A + B, S1 = A – B,

resulting in A = (S0 + S1)/2, and B = (S0 – S1)/2

t

t

time 0 time t

GG

A1

time t1=0 time t256

S1 = A1 + ... + A9

Time point 1, S1

S9 = | A1*exp(-i1t9) +... + A9 *exp(-i9t9) |

Time Point 9, S9

1t2

9t2

GA1

A9

time t2

S2 = | A1*exp(-i1t2) +... + A9 *exp(-i9t2) |

Time Point 2, S2

A9... ...

1t9A1

A9

...9t9

. . .

Now, let’s extrapolate to resolve 9 elements along a dimension …Now, let’s extrapolate to resolve 9 elements along a dimension …

A1 A2 A3 A4 A5 A6 A7 A8 A9

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

Image Resolution (in Plane)

Spatial resolution depends on how well we can separate frequencies in the data S(t) 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 S(t) because phases of cos(ft) and cos([f+f]t) will be more different

Summary: Second Step in Image FormationFrequency Encoding

After slice selection, in-plane spatial encoding 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 S(t) into frequency components

S(f ) — using the Fourier transform Since frequency f varies across subject in a known way, we can

assign each component S(f ) to the place it comes from

Third Step in Image Formation:

Resolving the second in-plane dimension

Now let’s consider the simplest 2D image

A B

C D

A B

C DA

B

C

D

S0 = (A + C) + (B + D)

Time point 1

S1 = (A + C) - (B + D)

Time point 2

S0 S1

TimeGx

x

Gx

x

A B

C D

G

xS0 = (A + C) + (B + D)

Time point 1

S1 = (A + C) - (B + D)

Time point 2

S0 S1

t

S2 = (A + B) + (C + D)

Time point 3

S3 = (A + B) - (C + D)

Time point 4

S2 S3

t

y

A B

C DA

B

C

D

S0 S1

TimeGx

x

Gx

xA

B

CD

A B

C D

S2 S3

y

Gy

y

Gy

TimeGx

x

Gx

x

S0 = (A + C) + (B + D)

S1 = (A + C) - (B + D)

S2 = (A + B) - (C + D)

S3 = A – B – C + D

A Little More Complex Spatial EncodingA Little More Complex Spatial Encoding

x gradientx gradient

y gradienty gradient

A Little More Complex Spatial EncodingA Little More Complex Spatial Encoding

Physical SpaceA 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 1 data point1 data point another data pointanother data point

Physical SpaceA 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 1 data point1 data point 1 more data point1 more data point another pointanother point

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

readoutreadout

ExcitationExcitation

SliceSliceSelectioSelectionnFrequencyFrequency EncodingEncoding

PhasePhase EncodingEncoding

ReadoutReadout………………

Thought Question: Why can’t the phase encoding gradient be

Thought Question: Why can’t the phase encoding gradient be

turned on at the same time with the frequency encoding gradient?

turned on at the same time with the frequency encoding gradient?

Summary: Third Step in Image Formation Phase Encoding

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 S(f ) component of

S(t), as well as the frequency f By collecting data with many different amounts of phase encoding

strength, can break each S(f ) into phase components, and so assign them to spatial locations in 3D

A Brief Introduction of the Final MR Data Space (k-Space)

ImageImage k-spacek-space

Motivation: direct summation is conceptually easy,but highly intensive in computation which makes it impractical for high-resolution MRI.

PhasePhaseEncodeEncodeStep 1Step 1



Time Time point #1point #1



…………....




…………....




…………....

……

……

....

Frequency Encode

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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..

.

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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, ky) is the 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 physical space, sum of all the little signal from each voxel I(x,y) in the physical space, under the gradient field at that particular momentunder 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

IFTIFT

FTFT

k-spacek-space

kkxx

kkyy

Acquired DataAcquired Data

Image spaceImage space

xx

yy

Final ImageFinal Image

Two Spaces

ImageImage KK

Two Spaces

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

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FOV = 1/k, x = 1/K

FOV

k

Field of View, Voxel Size – a k-Space Perspective

K

Image Distortions: a k-Space Perspective

allen w. song, phd brain imaging and analysis center duke university

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