Allen W. Song, PhD Brain Imaging and Analysis Center Duke University MRI: Image Formation

What is image formation? To define the spatial location of the sources that contribute to the detected signal.

But MRI does not use projection, reflection, or refraction mechanisms commonly used in optical imaging methods to form image … A Simple Example

MRI Uses Frequency and Phase to Construct Image   =  t 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 zz x yy

The Use of Gradient Coils for Spatial Encoding w/o encoding w/ encoding Constant Magnetic Field Varying Magnetic Field MR Signal

Spatial Decoding of the MR Signal Frequency Decomposition a 1-D Image !

How Do We Make a Typical MRI Image?

First Step in Image Formation: Slice Selection

Slice Selection – along z z

Determining Slice Thickness Resonance frequency range as the result of slice-selective gradient:  f =  H * G sl * d sl  f =  H * G sl * d sl The bandwidth of the RF excitation pulse:  /2   /2  Matching the two frequency ranges, the slice thickness can be derived as d sl =  / (  H * G sl * 2  ) d sl =  / (  H * G sl * 2  )

Changing Slice Thickness or Selecting Difference Slices There are two ways to do this: (a)Change the slope of the slice selection gradient (b)Change the bandwidth of the RF excitation pulse Both 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 Constant Magnetic Field Varying Magnetic Field Continuous Sampling

Spatial Decoding of the MR Signal Frequency Decomposition a 1-D Image !

It’d be inefficient to collect data points continuously over time, actually, if all we need to resolve are just two elements in space. There is a better way to resolve these two elements discretely.

tt 2t2t G Element 1 Element 2 G Element 1 Element 2 A B A B time 0 S 0 = A + B Time point 1 S 1 = | A*exp(-i  1 t) + B*exp(-i  2 t) | 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 S 0 = A + B, S 1 = A – B, resulting in A = (S 0 + S 1 )/2, and B = (S 0 – S 1 )/2  t  t time 0time t

G G A1A1 time t1=0 time t256 S 1 = A A 9 Time point 1, S 1 S 9 = | A 1 *exp(-i  1 t 9 ) A 9 *exp(-i  9 t 9 ) | Time Point 9, S 9 1t21t2 9t29t2 G A1A1 A9A9 time t2 S 2 = | A 1 *exp(-i  1 t 2 ) A 9 *exp(-i  9 t 2 ) | Time Point 2, S 2 A9A9... 1t91t9 A1A1 A9A9 9t99t9 Now, let’s extrapolate to resolve 9 elements along a dimension … A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 A7A7 A8A8 A9A9

A typical diagram for MRI frequency encoding: Gradient-echo imaging readout Excitation Slice SliceSelection Frequency Encoding Encoding Readout TE Data points collected during this period corrspond to one-line in k-space ……… Time point #1 Time point #9

Phase Evolution of MR Data digitizer on Phases of spins GradientTE……… Time point #1 Time 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(f  t) and cos([f+  f]  t) will be more different

Summary: Second Step in Image Formation Frequency 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 2  T2*, 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 AB CD A B C D A B C D S 0 = (A + C) + (B + D) Time point 1 S 1 = (A + C) - (B + D) Time point 2 S0S0 S1S1 Time

AB CD G x S 0 = (A + C) + (B + D) Time point 1 S 1 = (A + C) - (B + D) Time point 2 S0S0 S1S1 t S 2 = (A + B) + (C + D) Time point 3 S 3 = (A + B) - (C + D) Time point 4 S2S2 S3S3 t y

A B C D A B C D S0S0 S1S1 Time A B C D A B C D S2S2 S3S3 S 0 = (A + C) + (B + D) S 1 = (A + C) - (B + D) S 2 = (A + B) - (C + D) S 3 = A – B – C + D

A Little More Complex Spatial Encoding x gradient

y gradient A Little More Complex Spatial Encoding

Physical Space A 9×9 case Before Encoding After Frequency Encoding (x gradient) So each data point contains information from all the voxels MR data space 1 data point another data point

Physical Space A 9×9 case Before Encoding After Frequency Encoding x gradient After Phase Encoding y gradient So each point contains information from all the voxels MR data space 1 data point 1 more data point another point

A typical diagram for MRI phase encoding: Gradient-echo imaging readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout ……… Thought Question: Why can’t the phase encoding gradient be 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 M xy (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) Image k-space Motivation: direct summation is conceptually easy, but highly intensive in computation which makes it impractical for high-resolution MRI.

PhaseEncode Step 1 PhaseEncode Step 2 PhaseEncode Step 3 Time point #1 Time point #2 Time point #3 …….. Time point #1 Time point #2 Time point #3 …….. Time point #1 Time point #2 Time point #3 …….. …….. 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.

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 Fourier Transform of the imaged object. 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, under the gradient field at that particular moment Kx =  /2  0 t  Gx(t) dt Ky =  /2  0 t  Gy(t) dt

Two Spaces IFT FTk-space kxkxkxkx kykykyky Acquired Data Image space x y Final Image

Two Spaces

Image K HighSignal

Full k-space Lower k-space Higher k-space Full Image Intensity-Heavy Image Detail-Heavy Image

FOV = 1/  k,  x = 1/K FOV kk Field of View, Voxel Size – a k-Space Perspective K

Image Distortions: a k-Space Perspective