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MRI. Vector Review. z. x. y. Vector Review (2). (a scalar). The Dot Product. The Cross Product. (a vector). (a scalar). MR: Classical Description: Magnetic Moments. Intuitively current, but nuclear spin operator in quantum mechanics. Spin angular momentum =.
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Vector Review z x y
Vector Review (2) (a scalar) The Dot Product The Cross Product (a vector) (a scalar)
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.
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
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
Reference Frame z y x
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
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
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...
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!!!!
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)
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
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
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
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
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
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?
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
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.
Sample Torso Coil y z x
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)
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
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”.
will precess, but decays. returns to equilibrium MR: Detected signal and Relaxation. S t Rotating frame Transverse Component with time constant T2 After 90º,
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
MR: Relaxation: Some sample tissue time constants: T2 T2 of some normal tissue types Table: Nishimura, Table 4.2
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
Bloch Equation Solution: Longitudinal Magnetization Component The greater the difference from equilibrium, the faster the change Solution: Return to Equilibrium Initial Mz
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
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?
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.
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
Components of M after Excitation Laboratory Frame
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,
MR Contrast Mechanisms T2-Weighted Coronal Brain T1-Weighted Coronal Brain
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
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...
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
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
Electron Spin Resonance – Poor RF Transmission Graph: Medical Imaging Systems Macovski, 1983
Electron Spin Resonance • Works on unpaired electrons • Free radicals • Extremely short decay times • Microseconds vs milliseconds in NMR