MRI Vector Review z x y Vector Review

MRI

Vector Review z x y

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

MR: Classical Description: Magnetic Moments NMR is exhibited in atoms with odd # of protons or neutrons. Spin angular momentum = Intuitively current, but nuclear spin operator in quantum mechanics Planck’s constant / 2 Spin angular momentum creates a dipole magnetic moment = gyromagnetic ratio : the ratio of the dipole moment to angular momentum Which atoms have this phenomenon? 1 H - abundant, largest signal 31 P 23 Na 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 field Bo 2) Radio frequency (RF) field B 1 3) Gradient fields Gx, Gy, Gz

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

Reference Frame y x z

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

MR: Energy states of 1 H 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- lower energy n+ Which state will nuclei tend to go to? For B= 1. 0 T 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 1 mm x 1 mm cube. • 2*6. 62*1023*1/1000*1/18 = 7. 73 x 1019 protons/mm 3 • 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 View each spin as a magnetic dipole (a tiny bar magnet). Assume we can get dipoles away from B 0. 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 1 H 1 Tesla = 104 Gauss Usually, Bo =. 1 to 3 Tesla So, at 1 Tesla, fo = 42. 57 MHz for 1 H

Other gyromagnetic ratios w/ sensitivity relative to hydrogen • 13 C • • 10. 7 MHz/ T, relative sensitivity 0. 016 31 P 17. 23 MHz/ T, relative sensitivity 0. 066 23 Na 11. 26 MHz/ T, relative sensitivity 0. 093

MR: RF Magnetic field a) Laboratory frame behavior of M b) Rotating frame behavior of M B 1 induces rotation of magnetization towards the transverse plane. Strength and duration of B 1 can be set for a 90 degree rotation, leaving M entirely in the xy plane. mages & caption: Nishimura, Fig. 3. 3

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

Bloch Equations – Homogenous Material It’s important to visualize the components of the vector M at different times in the sequence. a) b) 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 T 1 and T 2 relaxation, consider the following case. Solve

The Solved Bloch Equations Solve

The Solved Bloch Equations A solution to the series of differential equations is: where M 0 refers to the initial conditions. M 0 refers to the equilibrium magnetization. This solution shows that the vector M will precess about the B 0 field. Next we allow relaxation.

Sample Torso Coil y z x

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

Complex m m is complex. m =Mx+i. My 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 Mx Re

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

MR: Detected signal and Relaxation. Rotating frame will precess, but decays. returns to equilibrium S t Transverse Component with time constant T 2 After 90º,

MR: Intro: Relaxation: Transverse time constant T 2 - spin-spin relaxation T 2 values: < 1 ms to 250 ms What is T 2 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 T 2’s are brighter in T 2 -weighted imaging

MR: Relaxation: Some sample tissue time constants: T 2 of some normal tissue types Tissue T 2 (ms) gray matter 100 white matter muscle 92 47 fat 85 kidney liver 58 43 Table: Nishimura, Table 4. 2

MR: RF Magnetic field The RF Magnetic Field, also known as the B 1 field To excite equilibrium nuclei , apply rotating field at o in x-y plane. (transverse plane) B 1 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

Exciting the Magnetization Vector

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

Solution: Longitudinal Magnetization Component initial conditions equilibrium Example: What happens with a 180° RF flip? Effect of T 1 on relaxation - 180° flip angle Mo t -Mo

T 1 Relaxation

MR Relaxation: Longitudinal time constant T 1 Relaxation is complicated. T 1 is known as the spin-lattice, or longitudinal time constant. T 1 values: 100 to 2000 ms Mechanism: - fluctuating fields with neighbors (dipole interaction) - stimulates energy exchange nn+ - energy exchange at resonant frequency. Image Contrast: - Long T 1’s are dark in T 1 -weighted images - Shorter T 1’s are brighter Is |M| constant?

MR Relaxation: More about T 2 and T 1 T 2 is largely independent of Bo Solids - immobile spins - low frequency interactions - rapid T 2 decay: T 2 < 1 ms Distilled water - mobile spins - slow T 2 decay: ~3 s - ice : T 2~10 s T 1 processes contribute to T 2, but not vice versa. T 1 processes need to be on the order of a period of the resonant frequency.

MR: Relaxation: Some sample tissue time constants - T 1 Approximate T 1 values as a function of Bo gray matter muscle white matter kidney live r fa t Image, caption: Nishimura,

Components of M after Excitation Laboratory Frame

MR: Detected signal and relaxation after 90 degree RF puls. Rotating frame will precess, but decays. returns to equilibrium S t Transverse Component with time constant T 2 After 90º, Longitudinal Component Mz returns to Mo with time constant T 1 After 90º,

MR Contrast Mechanisms T 2 -Weighted Coronal Brain T 1 -Weighted Coronal Brain

Putting it all together: The Bloch equation Sums of the phenomena precession, transverse RF excitation magnetization Changes the direction of , but not the length. longitudinal magnetization These change the length of only, not the direction. includes Bo, B 1, 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 B = Bohr Magneton N = Nuclear Magneton http: //hyperphysics. phy-astr. gsu. edu/hbase/nuclear/nmr. html#c 1

Larmor/B n/B -1 -1 s T 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 Particle Spin Neutron 23 Na 1/2 3/2 1. 8326 x 108 29. 1667 MHz/T 0. 7076 x 108 11. 2618 MHz/T 1/2 1 1. 0829 x 108 17. 2349 MHz/T 0. 1935 x 108 3. 08 MHz/T 1/2 0. 6729 x 108 2. 518 x 108 31 P 14 N 13 C 19 F 10. 71 MHz/T 40. 08 MHz/T http: //hyperphysics. phy-astr. gsu. edu/hbase/nuclear/nmr. html#c 1

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