Chapter 68: magnetic resonance

68.1 NMR

68.1.1 Bloch equation

https://en.wikipedia.org/wiki/Bloch_equations

$$\dot{\boldsymbol{M}}=\boldsymbol{M}\times\gamma\boldsymbol{B}-\dfrac{\boldsymbol{M}_{{\scriptscriptstyle xy}}}{T_{{\scriptscriptstyle 2}}}-\dfrac{\boldsymbol{M}_{{\scriptscriptstyle z}}-\boldsymbol{M}_{{\scriptscriptstyle z0}}}{T_{{\scriptscriptstyle 1}}}$$

$$\begin{aligned} \dot{M}_{{\scriptscriptstyle x}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle x}}-\dfrac{M_{{\scriptscriptstyle x}}}{T_{{\scriptscriptstyle 2}}}\\ \dot{M}_{{\scriptscriptstyle y}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle y}}-\dfrac{M_{{\scriptscriptstyle y}}}{T_{{\scriptscriptstyle 2}}}\\ \dot{M}_{{\scriptscriptstyle z}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle z}}-\dfrac{M_{{\scriptscriptstyle z}}-M_{{\scriptscriptstyle z0}}}{T_{{\scriptscriptstyle 1}}} \end{aligned}$$

68.1.2 Bloch-Torrey equation

Bloch-Torrey equation = Bloch equation + diffusion term

https://journals.aps.org/pr/abstract/10.1103/PhysRev.104.563 https://journals.aps.org/pr/pdf/10.1103/PhysRev.104.563

https://en.wikipedia.org/wiki/Diffusion_MRI#Magnetization_dynamics

https://arxiv.org/abs/1608.02859 https://arxiv.org/pdf/1608.02859

$$\dot{\boldsymbol{M}}=\boldsymbol{M}\times\gamma\boldsymbol{B}-\dfrac{\boldsymbol{M}_{{\scriptscriptstyle xy}}}{T_{{\scriptscriptstyle 2}}}-\dfrac{\boldsymbol{M}_{{\scriptscriptstyle z}}-\boldsymbol{M}_{{\scriptscriptstyle z0}}}{T_{{\scriptscriptstyle 1}}}+\boldsymbol{\nabla}\cdot D\boldsymbol{\nabla}\boldsymbol{M}$$

$$\begin{aligned} \dot{M}_{{\scriptscriptstyle x}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle x}}-\dfrac{M_{{\scriptscriptstyle x}}}{T_{{\scriptscriptstyle 2}}}+\boldsymbol{\nabla}\cdot D\boldsymbol{\nabla}\left(M_{{\scriptscriptstyle x}}-M_{{\scriptscriptstyle x0}}\right)\\ \dot{M}_{{\scriptscriptstyle y}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle y}}-\dfrac{M_{{\scriptscriptstyle y}}}{T_{{\scriptscriptstyle 2}}}+\boldsymbol{\nabla}\cdot D\boldsymbol{\nabla}\left(M_{{\scriptscriptstyle y}}-M_{{\scriptscriptstyle y0}}\right)\\ \dot{M}_{{\scriptscriptstyle z}}= & \left(\boldsymbol{M}\times\gamma\boldsymbol{B}\right)_{{\scriptscriptstyle z}}-\dfrac{M_{{\scriptscriptstyle z}}-M_{{\scriptscriptstyle z0}}}{T_{{\scriptscriptstyle 1}}}+\boldsymbol{\nabla}\cdot D\boldsymbol{\nabla}\left(M_{{\scriptscriptstyle z}}-M_{{\scriptscriptstyle z0}}\right) \end{aligned}$$

68.2 MRI

68.2.1 Lin, Hsiu-Hau

Lin, Hsiu-Hau / Porcupine Lin / Hedgehog Note

https://www.youtube.com/watch?v=Y-z-1XCu7fE&list=PLS0SUwlYe8czNqxfQq2XWeAHDqT8vYjmC&index=60

68.2.2 MRI: Physics and Image Creation

https://www.youtube.com/playlist?list=PLRaiBmx1XQs5mrhyhzCmZWGHAG4bMlLXj

68.2.3 Tseng, Wen-Yih Isaac

https://www.youtube.com/playlist?list=PLTpF-A8hKVUMRaGE0Zj4WCGJX9BZraFaU