Resources: - [thePIRL YouTube](https://www.youtube.com/@thepirl903) - [How MRI Works - Part 1 - NMR Basics](https://www.youtube.com/watch?v=TQegSF4ZiIQ) - [How MRI Works - Part 2 - The Spin Echo](https://www.youtube.com/watch?v=M7yh0To6Wbs) - [How MRI Works - Part 3 - Fourier Transform and K-Space](https://www.youtube.com/watch?v=R_4GuyJTzMo) - [How MRI Works - Part 4 - The Gradient Recalled Echo (GRE)](https://www.youtube.com/watch?v=vapJRr6gAds) - [MRI4ALL Paper](https://onlinelibrary.wiley.com/doi/10.1002/jmri.29771) - [MRI4ALL Github](https://github.com/mri4all) - [50 mT Halbach MRI paper](https://pubmed.ncbi.nlm.nih.gov/32627235/) - [Ultra low field MRI study review paper](https://www.sciencedirect.com/science/article/pii/S2666956026000073) - [Tabletop MRI paper](https://www.sciencedirect.com/science/article/abs/pii/S1090780719302642) - [Laron Principles of MRI](https://larsonlab.github.io/MRI-education-resources/Introduction.html) ### How an MRI Works 1. $\vec{B_{0}}$ Magnetic field aligns quantum spin of hydrogen atoms (protons) 2. RF coils send an RF pulse $B_{1}$ at Larmor Frequency $\omega$ to resonate with the nuclear precession and tip their spin to the desired flip angle $\theta$ 3. Nuclei precess giving off an RF signal for time $T_{2}$ 1. The magnetic field follows a gradient so the signal frequency variesx with position for imaging 4. Nuclei realigns with the field over time $T_{1}$ The **Larmor Equation** describes the magnetic precession frequency of a proton which is placed in a time-varying magnetic field at resonance. $ \omega = \gamma \vec{B} $ $\omega =$ precession frequency (~100 MHz) $\gamma =$ gyromagnetic ratio of (hydrogen) nucleus $=42.6 \mathrm{~MHz/T}$ $\vec{B} =$ magnetic field Magnetic resonance can be visualized with [CompassMR](https://www.drcmr.dk/CompassMR/) The sum of precessions of the $N$ particles creates a changing magnetic field which creates a measurable voltage in the RF coils accord to Lenz's Law $\epsilon = -\frac{ \partial \Phi }{ \partial t }$. This gives us an equation for the signal voltage: $ S(t) = N \sin(\theta)\gamma \vec{B}\cos(\omega t)e^{-t/T_{2}} $ $N =$ number of particles with spin $\theta =$ flip angle (angle of spin with respect to magnetic field) $\omega =$ precession frequency $\gamma =$ gryomagnetic ratio $\vec{B} =$ magnetic field Tissue Signal Also known as Total magnetization Vector $S(\text{TR},\text{TE}) =M_{0}\underbrace{ (1-e^{-\text{TR}/T_{1}}) }_{ \text{longitudinal} }\underbrace{ e^{-\text{TE}/T_{2}} }_{ \text{transverse} }$ $M_{0} =$ Boltzman Magnetization $T_{1} =$ Time constant of realignment with magnetic field $T_{2} =$ Time constant of Free Induction Decay (FID), i.e. duration of nuclei's RF signal $\text{TR}=$ Repetition time $TE =$ Echo time If you account for dephasing mechanisms in the sample such as field inhomogeneity, instead of $T_{2}$ we observe $\frac{1}{T_{2}^{\star}}=\frac{1}{T_{2}}+\gamma\Delta B$ so $T_{2}^{\star}< \frac{1}{\gamma\Delta B}$ If you follow the 90° pulse with a 180° pulse after time $\tau$, the dephased vectors are rotated 180 degrees about the pulse vector such that time $\tau$ after the second pulse, the vectors will realign and the signal peaks at the $T_{2}$ decay curve. ![[Pasted image 20261016171952.png|453]] The Boltzmann Magnetization $M_{0}$ is the sum of magnetic moments accounting for the density and polarization (% of unpaired spins) of the protons in the sample. This relation is described by the Curie Law: ${} M_{0} = \frac{N\gamma^{2}\hbar^{2}B_{0}}{4kT}=\chi_{0}B_{0} {}$ $\chi_{0} =$ Static Nuclear Susceptability ## Contrast Methods - **$T_{2}$-Weighted Image**: Let $\omega =0$ and choose an Echo Time (TE) at which to compare tissues with maximal contrast from $T_{2}$ relaxation - **Spin Density Image**: Let $\omega=0$ and ${} \text{TE}=0 {}$ to contrast spin counts $N$ - Not as useful in the brain where proton density doesn't vary too much - **$T_{1}$-Weighted Image**: Choose a Repetition Time (TR) at which to compare magnetization $M(t)$ to maximize contrast from $T_{1}$ relaxation - Longer TR results in more signal but a longer scan time ($T_{1}\gg T_{2}$) | | $\text{TR} \approx T_{1}$ | ${} \text{TR}\gg T_{1} {}$ | | ------------------------------ | ------------------------- | -------------------------- | | ${} \text{TE}\approx T_{2} {}$ | Not useful | $T_{2}$-Weighted Image | | ${} \text{TE}\ll T_{1} {}$ | $T_{1}$-Weighted Image | $M_{0}$ Spin Density Image | $S = \frac{N_{\text{tot}}\gamma^{3} \hbar^{2} B_{0}^{2}}{4kT}$ $\gamma = 267\times \times 10^{6}~\mathrm{T^{-1}s ^{-1}}=$ $\hbar =$ $B_{0} =$ magnetic field strength $k =$ Boltzman's Constant $T =$ Temperature Polarization $P = \frac{\gamma \hbar B_{0}}{2kT}$ is the percent of orbitals with unpaired spins. At 1T and body temperature, we get 3.4 ppm polarized spins, but there are enough atoms in the body for this to still result in a measurable signal. Some gases such as HP$^{129}$Xe can be polarized 50% (100,000x higher than hydrogen), allowing you to see a patient's lungs very clearly by having them inhale it. ### Bloch Equation Describes precession around $\vec{B}$ axis, $\hat{z}$ $ \frac{d\vec{M}}{dt} = \vec{M} \times \gamma \vec{B} - \frac{M_{x}\hat{x}+M_{y}\hat{y}}{T_{2}}-\frac{(M_{z}-M_{0})\hat{z}}{T_{1}} $ ${} \vec{B} =\vec{B}_{0}+\vec{B}_{1}= {}$ Total Magnetic Field, $\hat{z}$ $\vec{M} =$ Magnetization vector | Tissue | $T_{1}$ \[ms] | $T_{2}$ \[ms] | | ------------ | ------------- | ------------- | | White Matter | 800 | 110 | | Gray Matter | 1300 | 80 | | CSF | 3700 | 1700 | | Muscle | 1000 | 30 | | Fat | 400 | 100 | | Liver | 800 | 40 | | Cartilage | 1200 | 40 | | Tissue | $T_{2}$ | | ------------ | -------- | | Bone | 1 ms | | Lung | 0.1-2 ms | | Spinal fluid | 160 ms | | Water | 1000 ms |