MRI

Vector Review x y z

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

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

MR: Classical Description: Magnetic Fields Magnetic Fields used in MR: 1) Static main field B o 2) Radio frequency (RF) field B 1 3) Gradient fields G x, G y, G z How do we create and detect these moments?

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

Reference Frame z x y

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

MR: Energy states of 1 H Hydrogen has two quantized currents, B o field creates 2 energy states for Hydrogen where Energy of Magnetic Moment in energy separation resonance frequency f o

MR: Nuclei spin states There are two populations of nuclei: n + - called parallel n - - called anti parallel n+n+ n-n- lower energy higher 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*10 6 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*10 23 *1/1000*1/18 = 7.73 x10 19 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 B o ? 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 B o ) 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 Torque is defined as Multiply both sides by Now sum over all

MR: Intro: Classical Physics: Precession rotates (precesses) about Solution to differential equation: Precessional frequency: is known as the Larmor frequency. for 1 H 1 Tesla = 10 4 Gauss Usually, B o =.1 to 3 Tesla So, at 1 Tesla, f o = MHz for 1 H or

Other gyromagnetic ratios w/ sensitivity relative to hydrogen 13 C 10.7MHz/ T, relative sensitivity P MHz/ T, relative sensitivity Na MHz/ T, relative sensitivity 0.093

MR: RF Magnetic field Images & caption: Nishimura, Fig. 3.3 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. a) Laboratory frame behavior of M b) Rotating frame behavior of M

MR: RF excitation By design, In the rotating frame, the frame rotates about z axis at  o radians/sec x y z 1) B 1 applies torque on M 2) M rotates away from z. (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)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. b)Ignoring T1 and T2 relaxation, consider the following case. Solve

The Solved Bloch Equations Solve

The Solved Bloch Equations A solution to the series of differential equations is: Next we allow relaxation. 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 B0 field.

Sample Torso Coil z y x

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

Complex m m is complex. m =M x +iM y Re{m} =M x Im{m}=M y This notation is convenient: It allows us to represent a two element vector as a scalar. Re Im m MxMx MyMy

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

MR: Detected signal and Relaxation. Rotating frame t S will precess, but decays. returns to equilibrium 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 T2’s are brighter in T2-weighted imaging

MR: Relaxation: Some sample tissue time constants: T 2 Table: Nishimura, Table 4.2 T 2 of some normal tissue types TissueT 2 (ms) gray matter100 white matter92 muscle47 fat85 kidney58 liver43

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) Image & caption: Nishimura, Fig. 3.2 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

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? t MoMo -M o Effect of T 1 on relaxation - 180° flip angle

T1 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 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 T 2 and T 1 T 2 is largely independent of B o 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 Image, caption: Nishimura, Fig. 4.2 fat liver kidney Approximate T 1 values as a function of B o white matter gray matter muscle

Components of M after Excitation Laboratory Frame

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

MR Contrast Mechanisms T2-Weighted Coronal BrainT1-Weighted Coronal Brain

Putting it all together: The Bloch equation Sums of the phenomena transverse magnetization precession, RF excitation longitudinal magnetization Changes the direction of, but not the length. These change the length of only, not the direction. includes B o, 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) M xy decays, M z grows 4) Repeat. More about relaxation...

Proton vs. Electron Resonance Here g is same as   B = Bohr Magneton  N = Nuclear Magneton

ParticleSpin  Larmor /B s -1 T -1 /B Electron1/ x GHz/T Proton1/ x MHz/T Deuteron x MHz/T Neutron1/ x MHz/T 23 Na3/ x MHz/T 31 P1/ x MHz/T 14 N x MHz/T 13 C1/ x MHz/T 19 F1/ x MHz/T

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