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![Bloch Equation – Gradient Fields
• Gradient coils create changes in the magnetic field versus
position
• Changes precession frequency to be a function of position
(enabling image formation):
G(t) = [GX (t), GY (t), GZ(t)]
~B(~r, t) =
2
4
0
0
B0 + ~G(t) · ~r
3
5
!0 = (B0 + ~G(t) · ~r)
d ~M(t)
dt
= ~M(t) ⇥ ~B(t) +
2
4
1/T2 0 0
0 1/T2 0
0 0 1/T1
3
5 ~M(t) +
2
4
0
0
M0/T1
3
5](https://image.slidesharecdn.com/principlesofnmrimaging-pederlarsonenc2017-200204185843/85/Principles-of-N-MR-Imaging-9-320.jpg)
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Principles of (N)MR Imaging
- 1. Principles of (N)MR Imaging Peder Larson, Ph.D. University of California – San Francisco, Department of Radiology and Biomedical Imaging Experimental NMR Conference, Educational Presentation, Asilomar, Pacific Grove, CA March 28, 2017 https://radiology.ucsf.edu/research/labs/larson/educational-materials (Google: PederLarson Lab, Educational Materials link on sidebar)
- 2. Overview 1. MRI Systems – Magnetic field gradients 2.Imaging Principles – Slice-selective RF Pulses – Image Formation 3. MR spectroscopic imaging (MRSI) – Spectroscopic image formation – Spectral-Spatial RF pulses
- 3. From NMR to MRI • 1946: NMR phenomenon first discovered by Felix Bloch and Edward Purcell (Physics Nobel Prize in 1952) • 1973: First description of NMR Imaging (Medicine Nobel Prize to Paul Lauterburand Peter Mansfield in 2003) • 1982: Clinical magnetic resonance imaging (MRI) systems (“Nuclear was associated with bombs and wars and God knows what”– John Mallard)
- 4. MRI has come a long way… MR: What’s theattraction? Figure 1.3 First ever human head image using MRI at 0.1 T from EMI Central Research Laboratories. For this image CT type “back projection” was used. Courtesy of Ian Young. The early history of NMR ‘Nuclear induction’, as it was first described, was dis- covered in 1945, soon after the close of World War II, by Bloch and independently by Purcell and Pound. It is said that the development of radio communica- tions in the war effort, to which Purcell had con- tributed scientifically, was one of the factors underpinning this important scientific discovery. Another important factor, as in the development of atomic physics, was the expulsion or fleeing of European physicists from the Nazi regime, an exodus that included Bloch and Bloembergen. What did these MR pioneers discover? That you can detect a signal (a voltage in a coil) when you place a sample in a magnetic field and irradiate it with radiofrequency (RF) energy of a certain frequency, the resonant or Larmor frequency. The signal is produced by the interaction of the sample nuclei with the magnetic field. The spin echo was ‘stumbled upon’ by Hahn in 1949. He discovered that you could get a repeat of the NMR signal at a delayed time by adding a second burst of RF energy. That’s all you need to know for now. So what were NMR researchers doing between the forties and the seventies – that’s a long time in cultural and scientific terms. The answer: they were doing chemistry, including Lauterbur, a professor of chemistry at the same institution as Damadian, albeit on different campuses. NMR developed into a laboratory spectroscopic technique capable of examining the molecular structure of compounds, until Damadian’s ground-breaking discovery in 1971. First human head MRI (published 1978) 2010
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- 7. Gradient Encoding Position (z) Magnetic Field/Frequency Magnetic field gradient B0 ω0 GZ •Applied magnetic field gradients (G) add or subtract to the main magnetic field (B0) • This changes the resonance frequency as a function of position: ω = γB = γ(B0 + GZz) x y Mxy Net Magnetization Vectors (Rotating Frame at ω0) B0
- 8. Gradient Encoding Position (x) Magnetic field gradient B0 GX Magnetic Field/Frequency • Gradients included for all three axes (x,y,z) , and can be modulated independently •Create image by separating signals at different frequencies ω0B0 Mxy Net Magnetization Vectors (Rotating Frame at ω0) x y
- 9. Bloch Equation – Gradient Fields • Gradient coils create changes in the magnetic field versus position •Changes precession frequency to be a function of position (enabling image formation): G(t) = [GX (t), GY (t), GZ(t)] ~B(~r, t) = 2 4 0 0 B0 + ~G(t) · ~r 3 5 !0 = (B0 + ~G(t) · ~r) d ~M(t) dt = ~M(t) ⇥ ~B(t) + 2 4 1/T2 0 0 0 1/T2 0 0 0 1/T1 3 5 ~M(t) + 2 4 0 0 M0/T1 3 5
- 10. Magnetic Fields in MRI z x y Field component NotationDirection Approximate Strength Main field ± inhomogeneity B0 ± ΔB0 z 104 ± 10-2 G Chemical shift σ z 10-2 - 10-1 G Magnetic Susceptibility χ z 10-2 - 10-1 G Gradients GX, GY, GZ z 5 G/cm è101 - 102 G Radiofrequency (RF) B1 x,y 10-1 G B0 • Chemical shift and magnetic susceptibility are inherent in the body and are sources of “off-resonance” • Gradients and RF are controlled fields and manipulated to create images
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- 12. Frequency Encoding – 1D imaging • Gradient field applied •Different positions distinguishable in MR signal based upon frequency Position Frequency G > 0 Fourier Transform f/x ω0 t Reference (ω0) 0 s(t)
- 13. Typical 2D MRI Pulse Sequence 1. RF Excitation 2. Spatial Encoding 3.Data Acquisition TE = Echo Time ”Frequency Encoding” ”Phase Encoding”
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- 18. Decoding Position • General behavior of Mxy in the presence of gradients •Defining “k-space” as • And neglecting relaxation • Results in General case: time varying gradients on x,y,z Proportional to phase of Mxy ( ) ( ) tt p g dGtk t ò= 0 2 Demodulate received signal at Larmor (rotating frame) frequency for s(t) Looking like a Fourier Transform...
- 19. Fourier Transform Signal Relationship • MR Signal is the Fourier Transform of the object net magnetization • k-space location (defined by gradients) determines where in FT space the signal is coming from AssumeT2 large relative to t, then Received signal is the spatial Fourier Transform of the transverse (xy) net magnetization! Evaluated at k-space location that depends on gradients Example : consider signal from three locations - x= 0, x1, x2 - with magnetic field gradient on ( ) ( ) tt p g dGtk t ò= 0 2
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- 21. Figure 7.10 Imagesand their 2D spectra (k-space) showing: (a) reconstruction from all spatial frequencies, (b) low spatial (a) (b) (c) k-space (frequency domain)Image-space (real domain) Fourier Transform Low-frequency only High-frequency only McRobbie et al. MRI: From Picture to Proton
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- 24. Encoding Gradients kx ky t GX tGY ( ) () tt p g dGtk t ò= 0 2 s(t) = M(kx(t),ky(t)) M(kx,ky)
- 25. Encoding Gradients kx ky t GX tGY ( ) () tt p g dGtk t ò= 0 2 s(t) = M(kx(t),ky(t)) M(kx,ky)
- 26. Encoding Gradients kx ky t GX tGY ( ) () tt p g dGtk t ò= 0 2 s(t) = M(kx(t),ky(t)) M(kx,ky)
- 27. Encoding Gradients kx ky t GX tGY ( ) () tt p g dGtk t ò= 0 2 s(t) = M(kx(t),ky(t)) M(kx,ky)
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- 30. More Trajectories GX DAQ GY a b kx ky kx ky kx ky kx ky kx ky c de 2D FT Echo-planar Imaging (EPI) Radial or Projection reconstruction PROPELLER (for motion correction) Spiral
- 31. Cartesian Encoding: FOV and resolution kx ky 1 FOVy k-space (sampling pattern) Image space (pointspread function)Fourier Transform x y FOVy 1/resx resx resy 1/resy
- 32. Selective Excitation • Every RF pulse is selective in frequency • (Approximate Fourier Transform relationship between RF pulse shape and Magnetization profile) •Profile characterized by the “Bandwidth” γΔBZ (resonance frequency) |MXY| Fourier Transform 0 (γB0) f f
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- 35. Spatially Selective RF Excitation Position Flip Angle Fourier Transform Slice thickness • RF pulse with applied Gradient pulse excites only a limited region of spins •Received RF signal will only come from this region • (Approximate Fourier Transform relationship, valid for “small” tip angles, < 60°)
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- 39. MRSI Sampling Requirements kz kf kz kf 1/resz 1/FOVf = 1/bandwidth 1/resz 1/FOVf = 1/bandwidth •Fast Acquisition • Reduced SNR efficiency • Tradeoff between spatial resolution and bandwidth Phase Encoding Echo-planar spectroscopic imaging (EPSI)
- 40. 0 2 4 6 8 10 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Ky(cm-1 ) time (ms) Kx (cm-1 ) EPSI (Echo Planar Spectroscopic Imaging) -1-0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Kx (cm-1 ) Ky(cm-1 )
- 41. Spiral Spectroscopic Imaging 0 10 2030 40 50 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 time(ms) Kx(cm-1 ) Ky(cm-1 ) -1 -0.5 0 0.5 1 -1 -0.8 -0.4 0 0.4 0.8 1 Kx(cm-1 ) Ky(cm-1 )
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- 43. Tradeoffs • Speed • SNR efficiency •Robustness to hardware imperfections • Bandwidth • Resolution Comparison of Accelerated MRSI Strategies 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 3000 Spectral Bandwidth Resolution (cm) SBW(Hz) Concentric Rings Symmetric EPSI Flyback EPSI Spiral 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 Acquisition Time Resolution (cm) AcquisitonTime(s) Concentric Rings Symmetric EPSI Flyback EPSI Spiral 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 3000 Spectral Bandwidth with Interleaves Resolution (cm) SBW(Hz) Concentric Rings Symmetric EPSI Flyback EPSI Spiral 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 SNR Efficiency Resolution (cm) SNREfficiency Concentric Rings Symmetric EPSI Flyback EPSI Spiral Flyback EPSI Symmetric EPSI Concentric Rings Spiral Speed - - + ++ SNR - ++ + ++ Robustness ++ - ++ -- Jiang W, et al, MRM 2014.
- 44. Spectral-spatial RF pulses • Designed with spectral k-space (kf = t) in mind •Spectrally and spatially selective • Typically use echo-planar gradient during RF pulse Meyer et al. MRM 15:287-304 (1990)
- 45. Excitation Spectral k-space Gf t RF t GZ t kZ = 0 kf= t Frequency shifts (e.g. chemical shift) is identical to a constant gradient kz kf
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- 48. Spectral-Spatial Design 1. Design spectral pulse 2. Design spatial pulse and slice select gradient 3.Use multiple spatial pulses and gradients – Weight spatial pulses with spectral pulse envelope – Alternate sign of gradient (EP) or add rewinder (flyback) z f z f z f
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- 50. Spectral-Spatial RF Example A. Schricker et al. MRM 46:1079-1087 2001 • Replacespin echo 180 pulses with spectral spatial pulses in PRESS • Design such that NAA/Cr/Cho only within passband and water/fat are in stopband • No need for CHESS (Spectrally-selective water suppression pulses)
- 51. Recommended MRI Resources • Nishimura. Principles of Magnetic Resonance Imaging. Available from lulu.com: Paperback or Hardcover –Complete and coherent description of MRI, targeted towards engineers and physicists • McRobbie, Moore, Graves, and Prince. MRI From Picture to Proton (2nd edition). Cambridge University Press. – Comprehensive description of MRI, targeted towards a less technical audience – Many useful imaging examples and practical tips • Schröder, Faber. In Vivo NMR Imaging. Spinger. http://link.springer.com/book/10.1007/978-1- 61779-219-9/page/1 – Useful chapters on image formation, special contrast in MRI, and applications – Very detailed descriptions • Bernstein, King, Zhou. Handbook of MRI Pulse Sequences. Academic Press. http://www.sciencedirect.com/science/book/9780120928613 – Detailed descriptions of specialized MRI topics – Assumes introductory knowledge of MRI – Essential for MRI scientists • Danish Research Centre for Magnetic Resonance. Educational Materials: http://www.drcmr.dk/educations/education-material – Introduction to MRI notes, targeted towards physicists and engineers – Bloch equation and “compassMR” simulations for teaching – Videos explaining simulations