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Classical Description of MR and Magnetic Resonance Imaging Techniques

Dive deep into the classical description of MRI and magnetic resonance, covering topics such as magnetic moments, energy states, nuclear spin, and more. Understand the fundamentals of creating, detecting, and utilizing magnetic fields in MR. Explore the behavior of spins, nuclei, and magnetic dipoles within different contexts, unraveling the complexities of MR technology.

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Classical Description of MR and Magnetic Resonance Imaging Techniques

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  1. MRI

  2. Vector Review z x y

  3. Vector Review (2) (a scalar) The Dot Product The Cross Product (a vector) (a scalar)

  4. MR: Classical Description: Magnetic Moments Intuitively current, but nuclear spin operator in quantum mechanics Spin angular momentum = Planck’s constant / 2 NMR is exhibited in atoms with odd # of protons or neutrons. Spin angular momentum creates a dipole magnetic moment = gyromagnetic ratio : the ratio of the dipole moment to angular momentum Which atoms have this phenomenon? 1H - abundant, largest signal 31P 23Na Model proton as a ring of current.

  5. MR: Classical Description: Magnetic Fields How do we create and detect these moments? Magnetic Fields used in MR: 1) Static main fieldBo 2) Radio frequency (RF) field B1 3) Gradient fieldsGx, Gy, Gz

  6. MR: Classical Description: Magnetic Fields: Bo 1) Static main field Bo without Bo, spins are randomly oriented. macroscopically, net magnetization with Bo, a) spins align w/ Bo (polarization) b) spins exhibit precessional behavior - a resonance phenomena

  7. Reference Frame z y x

  8. MR: Energy of Magnetic Moment Bo x z: longitudinal x,y: transverse Alignment Convention: z y At equilibrium, Energy of Magnetic Moment in is equal to the dot product quantum mechanics - quantized states

  9. MR: Energy states of 1H Energy of Magnetic Moment in Hydrogen has two quantized currents, Bo field creates 2 energy states for Hydrogen where energy separation resonance frequency fo

  10. MR: Nuclei spin states There are two populations of nuclei: n+ - called parallel n- - called anti parallel higher energy n- n+ lower energy Which state will nuclei tend to go to? For B= 1.0T Boltzman distribution: Slightly more will end up in the lower energy state. We call the net difference “aligned spins”. Only a net of 7 in 2*106 protons are aligned for H+ at 1.0 Tesla. (consider 1 million +3 in parallel and 1 million -3 anti-parallel. But...

  11. There is a lot of a water!!! • 18 g of water is approximately 18 ml and has approximately 2 moles of hydrogen protons • Consider the protons in 1mm x 1 mm x 1 mm cube. • 2*6.62*1023*1/1000*1/18 = 7.73 x1019 protons/mm3 • If we have 7 excesses protons per 2 million protons, we get .25 million billion protons per cubic millimeter!!!!

  12. Magnetic Resonance: Spins We refer to these nuclei as spins. At equilibrium, - more interesting - What if was not parallel to Bo? We return to classical physics... - view each spin as a magnetic dipole (a tiny bar magnet)

  13. MR: Intro: Classical Physics: Top analogy Spins in a magnetic field are analogous to a spinning top in a gravitational field. (gravity - similar to Bo) Top precesses about

  14. MR:Classical Physics Torque View each spin as a magnetic dipole (a tiny bar magnet). Assume we can get dipoles away from B0 .Classical physics describes the torque of a dipole in a B field as Torque is defined as Multiply both sides by Now sum over all

  15. MR: Intro: Classical Physics: Precession Solution to differential equation: rotates (precesses) about Precessional frequency: or is known as the Larmor frequency. for 1H Usually, Bo = .1 to 3 Tesla So, at 1 Tesla, fo = 42.57 MHz for 1H 1 Tesla = 104 Gauss

  16. Other gyromagnetic ratios w/ sensitivity relative to hydrogen • 13C 10.7MHz/ T, relative sensitivity 0.016 • 31P 17.23 MHz/ T, relative sensitivity 0.066 • 23Na 11.26 MHz/ T, relative sensitivity 0.093

  17. MR: RF Magnetic field B1 induces rotation of magnetization towards the transverse plane. Strength and duration of B1 can be set for a 90 degree rotation, leaving M entirely in the xy plane. a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig. 3.3

  18. MR: RF excitation z By design , In the rotating frame, the frame rotates about z axis at o radians/sec 1) B1 applies torque on M 2) M rotates away from z. (screwdriver analogy) 3) Strength and duration of B1 determines torque y x This process is referred to as RF excitation. Strength: B1 ~ .1 G What happens as we leave B1 on?

  19. Bloch Equations – Homogenous Material • Let us solve the Bloch equation for some interesting cases. In the first case, let’s use an arbitrary M vector, a homogenous material, and consider only the static magnetic field. • Ignoring T1 and T2 relaxation, consider the following case. It’s important to visualize the components of the vector M at different times in the sequence. Solve

  20. The Solved Bloch Equations Solve

  21. The Solved Bloch Equations A solution to the series of differential equations is: where M0 refers to the initial conditions. M0 refers to the equilibrium magnetization. This solution shows that the vector M will precess about the B0 field. Next we allow relaxation.

  22. Sample Torso Coil y z x

  23. Precession of induces EMF in the RF coil. (Faraday’s Law) MR: Detection z Switch RF coil to receive mode. y x M EMF time signal - Lab frame Voltage t for 90 degree excitation (free induction decay)

  24. Complex m m is complex. m =Mx+iMy Re{m} =Mx Im{m}=My This notation is convenient: It allows us to represent a two element vector as a scalar. Im m My Re Mx

  25. Transverse Magnetization Component The transverse magnetization relaxes in the Bloch equation according to Solution to this equation is : This is a decaying sinusoid. t Transverse magnetization gives rise to the signal we “readout”.

  26. will precess, but decays. returns to equilibrium MR: Detected signal and Relaxation. S t Rotating frame Transverse Component with time constant T2 After 90º,

  27. MR: Intro: Relaxation: Transverse time constant T2 T2 values: < 1 ms to 250 ms What is T2 relaxation? - z component of field from neighboring dipoles affects the resonant frequencies. - spread in resonant frequency (dephasing) happens on the microscopic level. - low frequency fluctuations create frequency broadening. Image Contrast: Longer T2’s are brighter in T2-weighted imaging - spin-spin relaxation

  28. MR: Relaxation: Some sample tissue time constants: T2 T2 of some normal tissue types Table: Nishimura, Table 4.2

  29. MR: RF Magnetic field The RF Magnetic Field, also known as the B1 field To excite equilibrium nuclei , apply rotating field at o in x-y plane. (transverse plane) B1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference Image & caption: Nishimura, Fig. 3.2

  30. Exciting the Magnetization Vector

  31. Bloch Equation Solution: Longitudinal Magnetization Component The greater the difference from equilibrium, the faster the change Solution: Return to Equilibrium Initial Mz

  32. Solution: Longitudinal Magnetization Component equilibrium initial conditions Example: What happens with a 180° RF flip? Effect of T1 on relaxation - 180° flip angle Mo t -Mo

  33. T1 Relaxation

  34. MR Relaxation: Longitudinal time constant T1 Relaxation is complicated. T1 is known as the spin-lattice, or longitudinal time constant. T1 values: 100 to 2000 ms Mechanism: - fluctuating fields with neighbors (dipole interaction) - stimulates energy exchange n- n+ - energy exchange at resonant frequency. Image Contrast: - Long T1’s are dark in T1-weighted images - Shorter T1’s are brighter Is |M| constant?

  35. MR Relaxation: More about T2 and T1 T2 is largely independent of Bo Solids - immobile spins - low frequency interactions - rapid T2 decay: T2 < 1 ms Distilled water - mobile spins - slow T2 decay: ~3 s - ice : T2~10 s T1 processes contribute to T2, but not vice versa. T1 processes need to be on the order of a period of the resonant frequency.

  36. Approximate T1 values as a function of Bo MR: Relaxation: Some sample tissue time constants - T1 gray matter muscle white matter Image, caption: Nishimura, Fig. 4.2 kidney liver fat

  37. Components of M after Excitation Laboratory Frame

  38. will precess, but decays. returns to equilibrium MR: Detected signal and relaxation after 90 degree RF puls. S t Rotating frame Transverse Component with time constant T2 After 90º, Longitudinal Component Mz returns to Mo with time constant T1 After 90º,

  39. MR Contrast Mechanisms T2-Weighted Coronal Brain T1-Weighted Coronal Brain

  40. Putting it all together: The Bloch equation Sums of the phenomena precession, RF excitation transverse magnetization longitudinal magnetization Changes the direction of , but not the length. These change the length of only, not the direction. includes Bo, B1, and Now we will talk about affect of

  41. MR: Intro: So far... What we can do so far: 1) Excite spins using RF field at o 2) Record FID time signal 3) Mxy decays, Mz grows 4) Repeat. More about relaxation...

  42. Proton vs. Electron Resonance Here g is same as g mB = Bohr Magneton mN = Nuclear Magneton http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1

  43. Particle Spin wLarmor/Bs-1T-1 n/B Electron 1/2 1.7608 x 1011 28.025 GHz/T Proton 1/2 2.6753 x 108 42.5781 MHz/T Deuteron 1 0.4107 x 108 6.5357 MHz/T Neutron 1/2 1.8326 x 108 29.1667 MHz/T 23Na 3/2 0.7076 x 108 11.2618 MHz/T 31P 1/2 1.0829 x 108 17.2349 MHz/T 14N 1 0.1935 x 108 3.08 MHz/T 13C 1/2 0.6729 x 108 10.71 MHz/T 19F 1/2 2.518 x 108 40.08 MHz/T http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nmr.html#c1

  44. Electron Spin Resonance – Poor RF Transmission Graph: Medical Imaging Systems Macovski, 1983

  45. Electron Spin Resonance • Works on unpaired electrons • Free radicals • Extremely short decay times • Microseconds vs milliseconds in NMR

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