1. Principles of MRI Physics and Engineering Allen W. Song Brain Imaging and Analysis Center Duke University

2. What if the RF field is not synchronized? Using the swingset example: now the driving force is no longer synchronized with the swing frequency, thus the efficiency of driving the swing is less. In a real spin system, there is a term called “effective B1 field”, given by B 1eff = B 1 +  /  B 1eff = B 1 +  / where  =  o –  e  =  o –  e B1B1B1B1 B 1eff  / 

4. Part II.1 Image Formation

5. What is image formation? To define the spatial location of the proton pools that contribute to the MR signal after spin excitation.

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

7. Gradient Coils Gradient coils generate 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

9. Spatial Encoding – along y y

10. Spatial Encoding of the MR Signal 0.8 w/o encoding w/ encoding Constant Magnetic Field Varying Magnetic Field

11. Spatial Encoding of the MR Signal Frequency Decomposition

12. Steps in 3D Localization  Can only detect total RF signal from inside the “RF coil” (the detecting antenna)  Excite and receive M xy 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

13. RF Field: Excitation Pulse 0t Fo Fo Fo+1/ t TimeFrequency t Fo Fo  F= 1/ t FT FT

14. 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 f 60 KHz Left = –7 cmRight = +7 cm G x = 1 Gauss/cm = 10 mTesla/m = strength of gradient field Center frequency [63 MHz at 1.5 T]

15.  Exciting and Receiving M xy in a Thin Slice of Tissue Source of RF frequency on resonance Addition of small frequency variation Amplitude modulation with “sinc” function RF power amplifier RF coil Excite:

16.  Exciting and Receiving M xy in a Thin Slice of Tissue RF coil RF preamplifier Filters Analog-to-Digital Converter Computer memory Receive:

17. Slice Selection

18. Slice Selection – along z z

19. 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:   Thus the slice thickness can be derived as d sl =  / (  H * G sl * 2  d sl =  / (  H * G sl * 2 

20. Changing slice thickness 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

21. Changing slice thickness new slice thickness

22. Selecting different slices In theory, there are two ways to select different slices: (a)Change the position of the zero point of the slice selection gradient with respect to isocenter selection gradient with respect to isocenter (b) Change the center frequency of the RF to correspond to a resonance frequency at the desired slice to a resonance frequency at the desired slice F =  H (Bo + G sl * L sl ) Option (b) is usually used as it is not easy to change the isocenter of a given gradient coil.

23. Selecting different slices new slice location

24.  Readout Localization (frequency encoding)  After RF pulse (B 1 ) 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

25. Readout of the MR Signal w/o encoding w/ encoding Constant Magnetic Field Varying Magnetic Field

26. Readout of the MR Signal Fourier Transform

27. 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

28. Phase Phase History digitizer on Gradient TE

29. A typical diagram for MRI frequency encoding: Spin-echo imaging readout Excitation Slice SliceSelection Frequency Encoding Encoding Readout TE

30. 180 o TE Phase Phase History Gradient

31. 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

32. Calculation of the Field of View (FOV) along frequency encoding direction  * G f * FOV f = BW = 1/  t Which means FOV f = 1/ (  G f  t) where BW is the bandwidth for the receiver digitizer.

33.  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 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 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

34. A typical diagram for MRI phase encoding: Gradient-echo imaging readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout

35. A typical diagram for MRI phase encoding: Spin-echo imaging readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout

36. Calculation of the Field of View (FOV) along phase encoding direction  * G p * FOV p = N p / T p Which means FOV p = 1/ (  G p T p /N p ) = 1/ (  G p  t) = 1/ (  G p  t) where T p is the duration and N p the number of the phase encoding gradients, Gp is the maximum amplitude of the phase encoding gradient.

37. Multi-slice acquisition Total acquisition time = Number of views * Number of excitations * TR Number of views * Number of excitations * TR Is this the best we can do? Interleaved excitation method

38. readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout readoutreadout …… …… …… TR

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

40. Acquired MR Signal Mathematical Representation: Kx =  /2  0 t  Gx(t) dt Ky =  /2  0 t  Gy(t) dt

41. 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......... +Gx-Gx0 0 +Gy -Gy. Image Space K-Space........................................................................

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

43. Two Spaces FT IFTk-space kxkxkxkx kykykyky Acquired Data Image space x y Final Image

44. Image K

45. The k-space Trajectory Kx =  /2  0 t  Gx(t) dt Ky =  /2  0 t  Gy(t) dt Equations that govern k-space trajectory: time 0t Gx (amplitude) Kx (area)

46. A typical diagram for MRI frequency encoding: A k-space perspective readout Excitation Slice SliceSelection Frequency Encoding Encoding Readout Exercise drawing its k-space representation 90 o

47. The k-space Trajectory

48. A typical diagram for MRI frequency encoding: A k-space perspective readout Excitation Slice SliceSelection Frequency Encoding Encoding Readout Exercise drawing its k-space representation 90 o 180 o

49. The k-space Trajectory

50. A typical diagram for MRI phase encoding: A k-space perspective readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout Exercise drawing its k-space representation 90 o

51. The k-space Trajectory

52. A typical diagram for MRI phase encoding: A k-space perspective readout Excitation Slice SliceSelection Frequency Encoding Encoding Phase Phase Encoding Encoding Readout Exercise drawing its k-space representation 90 o 180 o

53. The k-space Trajectory

54. .......... Sampling in k-space k max kkkk  k = 1 / FOV Refer to slide 32, 36

55. ............... A B FOV: 10 cm Pixel Size: 1 cm FOV: Pixel Size:

56. .......... A B FOV: 10 cm Pixel Size: 1 cm FOV: Pixel Size:

57. .......... A B FOV: 10 cm Pixel Size: 1 cm FOV: Pixel Size:

58. 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.

59. Spatial Displacement Original imageK-space trajectoryDisplaced image Original imageDistorted 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.

60. 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 imageK-space trajectoryDistorted Image