Allen W. Song, PhD Brain Imaging and Analysis Center Duke University

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

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

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

4. w q = wt q MRI Uses Frequency and Phase to Construct Image The spatial information of the proton pools contributing to MR signal is determined by the spatial frequency and phase of their magnetization.

5. Three Gradient Coils z z z y y y x x x X gradient Y gradient Z gradient 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.

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

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

8. How Do We Make a Typical MRI Image?

9. First Step in Image Formation: Slice Selection

10. z Slice Selection – along z

11. Determining Slice Thickness Resonance frequency range as the result of slice-selective gradient: Df = gH * Gsl * dsl The bandwidth of the RF excitation pulse: Dw/2p Matching the two frequency ranges, the slice thickness can be derived as dsl = Dw / (gH * Gsl * 2p)

12. Changing Slice Thickness or Selecting Difference Slices • There are two ways to do this: • Change the slope of the slice selection gradient • Change the bandwidth of the RF excitation pulse • Both are used in practice, with (a) being more popular

13. Changing Slice Thickness or Selecting Difference Slices

14. Second Step in Image Formation:Spatial encoding and resolving one dimension within a plane

15. Continuous Sampling Spatial Encoding of the MRI Signal: An Example of Two Vials Constant Magnetic Field Varying Magnetic Field w/o encoding w/ encoding

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

17. 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.

18. It turns out that all we need is just two data points: time 0 time t Element 1 Element 2 w1 = g G x, where x is determined by the voxel size Element 1 Element 2 lead B w2t B A w1t lag A G G S1 = | A*exp(-iw1t) + B*exp(-iw2t) | Time Point 2 S0 = A + B Time point 1

19. The simplest case is to wait for time t such that A and B will point along opposite direction, B w t B A -w t A time 0 time t such that S0 = A + B, S1 = A – B, resulting in A = (S0 + S1)/2, and B = (S0 – S1)/2

20. Now, let’s extrapolate to resolve 9 elements along a dimension … A9 A9 w9t9 w9t2 A9 ... A1 ... ... . . . w1t2 w1t9 A1 A1 G G G time t1=0 time t2 time t256 S2 = | A1*exp(-iw1t2) +... + A9 *exp(-iw9t2) | Time Point 2, S2 S9 = | A1*exp(-iw1t9) +... + A9 *exp(-iw9t9) | Time Point 9, S9 S1 = A1 + ... + A9 Time point 1, S1

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

22. TE Gradient Phases of spins ……… Time point #9 Time point #1 digitizer on Phase Evolution of MR Data

23. 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

24. 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 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 fvaries across subject in a known way, we can assign each component S(f) to the place it comes from

25. Third Step in Image Formation: Resolving the second in-plane dimension

26. B B A D A D C C S0 S1 Time S0 = (A + C) + (B + D) Time point 1 S1 = (A + C) - (B + D) Time point 2 Now let’s consider the simplest 2D image

27. S0 S2 S1 S3 t t G S0 = (A + C) + (B + D) Time point 1 S2 = (A + B) + (C + D) Time point 3 y S1 = (A + C) - (B + D) Time point 2 S3 = (A + B) - (C + D) Time point 4 x

28. B B A D A D C S0 S1 C Time A B B A C D C S2 D S3 Time S0 = (A + C) + (B + D) S1 = (A + C) - (B + D) S2 = (A + B) - (C + D) S3 = A – B – C + D

29. A Little More Complex Spatial Encoding x gradient

30. A Little More Complex Spatial Encoding y gradient

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

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

33. Excitation Slice Selection Frequency Encoding Phase Encoding readout ……… Readout A typical diagram for MRI phase encoding:Gradient-echo imaging Thought Question: Why can’t the phase encoding gradient be turned on at the same time with the frequency encoding gradient?

34. 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 ofS(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

35. A Brief Introduction of the Final MR Data Space (k-Space) k-space Image Motivation: direct summation is conceptually easy, but highly intensive in computation which makes it impractical for high-resolution MRI.

36. …….. Phase Encode Step 1 Time point #2 Time point #1 Time point #3 …….. Phase Encode Step 2 Time point #2 Time point #1 Time point #3 …….. Phase Encode Step 3 Time point #2 Time point #1 Time point #3 …….. Frequency Encode

37. Physical Space K-Space . . . . . . . . . . . . . . . . . . +Gy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . -Gy -Gx 0 +Gx 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.

38. Acquired MR Signal 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 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. Kx = g/2p 0tGx(t) dt Ky = g/2p 0tGy(t) dt

41. k-space Image space ky y kx x Acquired Data Final Image Two Spaces FT IFT

42. Two Spaces

43. High Signal Two Spaces K Image

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

45. Field of View, Voxel Size – a k-Space Perspective K Dk FOV FOV = 1/Dk, Dx = 1/K

46. Image Distortions: a k-Space Perspective