M R I Physics Course Jerry Allison Ph.D. Chris Wright B.S. Tom Lavin B.S. Department of Radiology Medical College of Georgia M R I Physics Course nuclear Magnetic Resonance (nMR) Nuclear Magnetism Gyromagnetic ratio Classic Mechanical Description Quantum Mechanical Description Larmor Equation RF Excitation / Detection Relaxation Bloch Equations Nuclear Magnetism • Nuclei having an odd number of protons or an odd number of neutrons or both have inherent spin (angular momentum) and a nuclear magnetic moment. • Nuclei having an even number of protons and an even number of neutrons have NO nuclear magnetic moment and cannot be observed using nMR (or MRI) techniques. Nuclear Magnetism (continued) • Important nuclei having no nuclear magnetic moment include: 16 8O ==> 8 neutrons, 8 protons, 8 electrons 12 6C ==> 6 neutrons, 6 protons, 6 electrons • Fortunately, hydrogen is abundant in the body (H2O, CH2:fat) and has a large nuclear magnetic moment (actually the largest). 1 1H ==> 1 proton, 1 electron Gyromagnetic Ratio (Magnetogyric ratio) The nMR properties of nuclei are characterized by the gyromagnetic ratio ( ). The gyromagnetic ratio is unique for each nuclide that has a nuclear magnetic moment. = nuclear magnetic moment nuclear spin angular momentum Gyromagnetic Ratio (continued) M=*S M = nuclear magnetic moment = gyromagnetic ratio S = nuclear spin angular momentum (inertia) Table 1: Nuclear Properties of Selected Nuclei Magnetogyric Resonance Natural Ratio frequency at Relative Nuclide abundance (MHz/Tesla)* 1.5T (values in Sensitivity** (%) Mhz) 1 H 42.58 63.87 99.98 1.0 2 H 6.53 9.79 0.015 9.65 x 10-3 13 C 10.71 16.07 1.11 1.59 x 10-2 14 N 3.09 4.64 99.63 1.01 x 10-3 15 N 4.34 6.51 0.37 1.04 x 10-3 17 O 5.81 8.71 0.04 2.91 x 10-2 19 F 40.05 60.08 100 0.83 23 Na 11.26 16.89 100 9.25 x 10-2 25 Mg 2.63 3.94 10.05 2.68 x 10-3 31 P 17.23 25.85 100 6.63 x 10-2 33 S 3.29 4.93 0.74 2.26 x 10-3 35 Cl 4.20 6.30 75.4 4.70 x 10-3 37 Cl 3.49 5.24 24.6 2.71 x 10-3 39 K 1.99 2.99 93.08 5.04 x 10-4 43 Ca 2.89 4.33 0.13 6.40 x 10-2 * Tesla is the magnetic field (10,000 Gauss = 1 Tesla) ** For an equal number of nuclei in an identical magnetic field Classical Mechanical Description • Some aspects of the nMR phenomenon are easiest to describe with classical mechanics, others are easiest to describe with quantum mechanics. • Classical description of a nuclear magnetic moment (spin) in an applied magnetic field: “When a force is exerted on a spinning object, the spinning object tends to move at right angles to the force”. • An example would include a spinning top in the Earth’s gravitational field. Classical Mechanical Description (continued) • Precession - continual motion of a spin at right angles to an applied force (sweeps the surface of a cone). • An example would be a gyroscope. z B0 0 M y Precession - continual motion of a spin at right angles to an applied force x (sweeps the surface of a cone). Quantum Mechanical Description Quantum mechanical description of a nuclear magnetic moment (spin) in an applied field. Spin States Spin Flip Transitions Macroscopic Magnetization Spin States • 1H (and 31P) nuclei have only two available spin states and are said to have nuclear spin of 1/2. Spin up (low energy state: parallel to applied static magnetic field) and spin down (high energy state: antiparallel to applied static magnetic field). E = 2.64 x 10-7 eV E = 8.8 x 10-8 eV E0 E0 BO = 0.5T BO = 1.5T Energy levels for hydrogen nuclei in 0.5 T (5,000 Gauss) and 1.5 T (15,000 Gauss) fields. Note the energy difference is greater at higher fields because E is directly proportional to the applied magnetic field. Espin down Energy E Espin up Magnetic Field The spin states are separated by E of energy E = h -34 h = Planck’s constant (6.62 x 10 J s) = spin frequency (cycles / s, Hertz) Spin States (continued) E is the energy difference between the spin up and spin down states. E is directly proportional to the applied magnetic field. For example: E = 2.64 x 10-7 eV @ 1.5T (63.87 MHz) E = 1.76 x 10 eV @ 1.0T (42.58 MHz) -7 E = 8.80 x 10-8 eV @ 0.5T (21.29 MHz) Spin Flip Transitions • Oscillating magnetic fields at the resonant RF frequency can cause spin flip transitions: spin up + absorbed energy ==> spin down spin down + absorbed energy ==> spin up + energy released Macroscopic Magnetization • For hydrogen nuclei (spin 1/2) at thermal equilibrium in a static magnetic field, the relative number of protons in the spin up and spin down states is given by the Boltzmann equation or Boltzmann Distribution: h B0 _______ # spin up __________ k T # spin down = e h B0 _______ k T # spin up __________ # spin down = e = gyromagnetic ratio (Hz/Tesla) h = Plank’s constant (6.626 x 10-34 J sec) B0 = magnetic field (Tesla) k = Boltzmann’s constant (1.381 x 10-34 J/K) T = temperature in Kelvin (K) where K = oC + 273.15 For example: consider hydrogen nuclei (protons) at 98.6 oF (37 oC, 310.15 K) in a 1.5 Tesla magnetic field: = 42.58 MHz/Tesla = 42.58 x 106 Hz/Tesla h = 6.626 x 10-34 J sec B0 = 1.5 Tesla k = 1.381 x 10-23 J / K T = 310.15 K (42.58MHz/Tesla)(6.626 x 10-34 J s)(1.5 Tesla) _____________________________ # spin up _________ = # spin down e (1.381 x 10-23 J / K) (310.15 K) Macroscopic Magnetization Example continued: For hydrogen nuclei (protons) at 98.6 oF in a 1.5 Tesla magnetic field: (42.58 x 106 Hz/Tesla)(6.626 x 10-34 J s)(1.5 Tesla) _____________________________ # spin up _________ = # spin down e (1.381 x 10-23 J / K) (310.15 K) # spin up protons ______________ = 1.000009882611 (4.94 ppm) # spin down protons For hydrogen nuclei (protons) at 98.6 oF in a 1.0 Tesla magnetic field: (42.58 x 106 Hz/Tesla)(6.626 x 10-34 J s)(1.0 Tesla) _____________________________ # spin up _________ = # spin down e (1.381 x 10-23 J / K) (310.15 K) # spin up protons ______________ = 1.000006588396 (3.29 ppm) # spin down protons Macroscopic Magnetization (continued) For hydrogen nuclei (protons) at 98.6 oF in a 0.5 Tesla magnetic field: # spin up protons ______________ = 1.000003294193 (1.65 ppm) # spin down protons Suppose a sample of hydrogen nuclei in a 1.5 Tesla magnetic field is heated by RF energy to 1 oC above normal body temperature: # spin up protons ______________ = 1.000009850849 (4.93 ppm) # spin down protons For physiologic temperatures, the excess of protons in the spin up state (at 1.5T) decreases about 0.01 ppm per oC. Macroscopic Magnetization (continued) • The small excess of protons in the spin up state produce a macroscopic magnetization that can be manipulated using magnetic fields oscillating at the resonant frequency. The macroscopic magnetization is also described as the thermal equilibrium magnetization or net magnetic moment. Note that the excess protons decreases as field strength decreases which results in a reduced signal-to-noise ratio at lower fields. Macroscopic Magnetization • For each gram of soft tissue, there is an excess of approximately 3 x 1016 protons in the spin up state out of 3 x 1022 protons. This excess creates the macroscopic magnetization. Larmor Equation • The Larmor equation describes the resonant precessional frequency of a nuclear magnetic moment in an applied static magnetic field. = Bo Where: = precessional frequency (resonant frequency) = gyromagnetic ratio (MHz/Tesla) Bo = magnetic field (Tesla) Larmor Equation (continued) What is the Larmor frequency of hydrogen nuclei (protons) in a 1.5 Tesla field? = Bo = (42.58 MHz / Tesla)(1.5 Tesla) = 63.87 MHz Larmor Equation (continued) What is the Larmor frequency (resonant frequency) of 23Na in a 1.5 Tesla field? = Bo = (11.26 MHz / Tesla)(1.5 Tesla) = 16.89 MHz Larmor Equation (continued) What is the Larmor frequency (resonant frequency) of 2H (deuterium) in a 1.5 Tesla field? = Bo = (6.53 MHz / Tesla)(1.5 Tesla) = 9.795 MHz Larmor Equation (continued) It should be noted that most RF coils and RF electronics used in MRI are tuned for a fairly narrow band of RF frequency. To convert from imaging 1H to 23Na would generally require having RF coils and RF electronics that can be tuned for the alternate frequency. Hydrogen is almost exclusively imaged in MRI because of it’s sensitivity and abundance. Larmor Equation (continued) Is 42.58 MHz / Tesla the for 1H in fat or water? The gyromagnetic ratio for 1H is simply 42.58 MHz / Tesla. 1H nuclei in water (H2O) and fat (~CH2) are in different molecules and experience a slightly different local magnetic field which results in slightly different resonant frequencies. These local magnetic field variations contribute to the eventual contrast between various tissues in an MRI image. RF Excitation • Spin population - Outside of the static magnetic field (Bo), the spin population can be described as a collection of randomly oriented nuclear magnetic moments (i. e. the patient): RF Excitation • Place the spin population in a static magnetic field. • Classical mechanics - individual spins precess • Quantum mechanics - energy of individual spins is quantitized (# spin up > # spin down) RF Excitation (continued) • Excess spins in the spin up state produce macroscopic magnetic moment “M” aligned with static magnetic field Bo. This condition is described as thermal equilibrium magnetization. M Bo RF Excitation is Begun • The spin population absorbs energy from magnetic fields oscillating at the resonant frequency. RF excitation can be described as a rotating magnetic field (and electric field) in the plane perpendicular to the static magnetic field. RF excitation is produced by applying an oscillating voltage waveform to an RF exciter (transmitter) coil. The magnetic field component that rotates in the transverse plane during RF excitation is termed the B1 magnetic field. RF Excitation (continued) • In quantum mechanics, RF excitation can be described as absorption of energy at the appropriate resonant RF frequency which causes spin-flip transitions. • Resonance: if the B1 frequency is at the Larmor frequency (+ a little)(i.e. 63.87 MHz for 1H at 1.5 Tesla) then: RF Excitation (continued) • Resonance (continued) 1.) Individual spins flip: spin up + absorbed energy ==> spin down spin down + absorbed energy ==> spin up + released energy 2.) Spins develop phase coherence. 3.) Macroscopic magnetization (M) is tipped away from alignment and begins to spiral at the Larmor frequency. 4.) Transverse magnetization develops. Transverse magnetization Frequently, the macroscopic magnetization is spiraled down until it precesses in the transverse plane (plane perpendicular to the static magnetic field). This is called a 90o flip. After a 90o flip, the macroscopic magnetization is precessing entirely in the transverse plane at the Larmor frequency and there are equal numbers of nuclei in the spin up and spin down states. Transverse magnetization (continued) The longitudinal component of the magnetization in the direction of the static magnetic field (Bo) is zero. The macroscopic magnetization prior to a 90o flip is entirely longitudinal and is said to point along the “Z” axis. Following a 90o flip, magnetization is entirely transverse and is said to rotate or precess in the transverse plane defined by the “X” and “Y” axes. RF Detection • The spin population “relaxes” to the thermal equilibrium magnetization • The transverse magnetization induces a voltage signal in an RF detection (receiver) coil as the spin population returns to the thermal equilibrium magnetization. The signal induced in the RF coil during the relaxation of the transverse magnetization is described as the Free Induction Decay (FID) signal. Relaxation Relaxation is the process by which a spin population returns to the thermal equilibrium distribution. Relaxation principally involves: T1 spin-lattice relaxation T2 spin-spin relaxation Homogeneity of the magnetic field, (Bo) Relaxation (continued) T1 Relaxation • Consider that the longitudinal or “Z” component of magnetization is determined by the number of spins in the spin up versus spin down energy state. The “Z” component returns exponentially to thermal equilibrium magnetization with rate constant T1 . • “... T1 is a measure of the time required to re- establish thermal equilibrium between the spins and their surroundings (lattice)…” Relaxation (continued) T1 Relaxation (continued) • Following a 90o flip, T1 is the time required for the longitudinal magnetization (“Z” component) to recover to 63% of the thermal equilibrium magnetization (MZ0). Longitudinal Magnetization MZ0 63% MZ0 T1 90o flip Time Relaxation (continued) • Thermal equilibrium: each spin (proton) is in the B0 static magnetic field of the MRI magnet and in a fluctuating magnetic field due to translation and rotation of it’s molecule and nearby molecules (4 Gauss from the adjacent proton in a water molecule). The magnetic environment of a water proton at room temperature changes with frequencies as high as 1,000,000 MHz. On average, a significant change occurs in the magnetic environment of a water proton ever 10 -12 seconds. These rapid changes can stimulate relaxation. Relaxation (continued) • Also note that proton exchange occurs in water molecules. A hydrogen nucleus (proton) in a free water molecule may exchange places with a hydrogen nucleus in a bound water molecule. Both nuclei thus experience a significant change in their magnetic environment causing relaxation to occur. Relaxation (continued) T2 Relaxation “... T2 is a measure of the time of disappearance of the transverse component of relaxation.” T2 is the time required for 63% of the transverse magnetization to decay. Transverse Magnetization 90o flip Mxy0 37% Mxy0 T2 Time Relaxation (continued) T2 Relaxation (continued) T2 has two components: Dephasing: spin-spin relaxation with no net change in energy. Spin-flip transitions (T1): spin-lattice transitions with a net increase in the number of nuclei in the spin up energy state. T2 Relaxation (continued) 1 ___ 1 ___ 1 ____ = + T2 ’ 2 T1 T2 1 ___ = dephasing component of T2 relaxation T2’ 1 ____ = T component of T relaxation 1 2 2 T1 T2 Relaxation (continued) Notice that T1 relaxation (spin flip transitions) cause dephasing and contributes to T2 relaxation. Conversely, the dephasing in T2 relaxation does not affect longitudinal magnetization and does not contribute to T1 relaxation. As a result, T2 is always smaller than T1. * T2 Relaxation The T2 relaxation observed in MRI is corrupted by inhomogeneity of the B0 magnetic field. This inhomogeneity is caused by nonuniformity in the static magnetic field and the magnetic susceptibility of patient tissue. * T2 Relaxation (continued) The observed T2 relaxation is termed T2* and has two components: 1 1 ___ ___ = B + T2* T2 B represents the effect of magnetic field inhomogeneity Relaxation (continued) • Temperature and Magnetic field dependence: T1 and T2 do not vary significantly with temperature in the physiologic temperature range. T2 does not vary significantly with magnetic field. T1 increases as the magnetic field increases (~ 200 msec / Tesla). A T1 value of 600 msec at 21 MHz (0.5T) becomes 800 msec at 63 MHz (1.5T). Relaxation (continued) • Tissue Characteristics Solids: (cortical bone protons) have extremely short T2 (microseconds) Gases: T1 = T2 Pure Water: T1 = T2 = 3 sec at 25 oC Liquids: T2 < T1 Soft Tissue: T1 800 msec, T2 80 msec Approximate T1 and T2 Values for Human Tissue (37 oC) T1 at 1.5 T T1 at 0.5 T T2 Tissue (msec) (msec) (msec) Skeletal Muscle 870 600 47 Liver 490 323 43 Kidney 650 449 58 Spleen 780 554 62 Fat 260 215 84 Gray Matter 920 656 101 White Matter 790 539 92 Cerebrospinal Fluid >4,000 >4,000 >2,000 Bloch Equations The equations describing nuclear magnetic resonance were derived by Felix Bloch in 1946. Longitudinal magnetization: - t / T1 MZ (t) = MZ0 + (MZ - MZ0) e Transverse magnetization: - t / T2 Mxy (t) = Mxy0 e B1 Maximum RF field applied: 23.5 microT (circularly polarized) For Siemens Magnetom 3T.
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