Magnetization Transfer Imaging

\[ \frac{d M_{x,f}}{dt}= \underbrace{-\frac{M_{x,f}}{T_{2,f}}}_{\color{blue}{\text{relaxation}}} +\underbrace{\Delta M_{y,f}}_{\color{orange}{\text{precession}}} -\underbrace{\text{Im}\left( \omega_{1} \right)M_{z,f}}_{\color{purple}{\text{nutation}}} \]

\[ \frac{d M_{y,f}}{dt}= \underbrace{-\frac{M_{y,f}}{T_{2,f}}}_{\color{blue}{\text{relaxation}}} +\underbrace{\Delta M_{x,f}}_{\color{orange}{\text{precession}}} -\underbrace{\text{Re}\left( \omega_{1} \right)M_{z,f}}_{\color{purple}{\text{nutation}}} \]

\[ \frac{d M_{z,f}}{dt}= \underbrace{-\frac{\left( M_{0,f}-M_{z,f} \right)}{T_{1,f}}}_{\color{blue}{\text{relaxation}}} +\underbrace{k_{r}M_{z,r}}_{\color{green}{\text{exchange}}} +\underbrace{\left( \text{Im}\left( \omega_{1} \right)M_{x,f}-\text{Re}\left( \omega_{1} \right)M_{y,f} \right)}_{\color{purple}{\text{nutation}}} \]

\[ \frac{d M_{z,r}}{dt}= \underbrace{-\frac{\left( M_{0,r}-M_{z,r} \right)}{T_{1,r}}}_{\color{blue}{\text{relaxation}}} +\underbrace{\left( -k_{r}M_{z,r}+k_{f}M_{z,f} \right)}_{\color{green}{\text{exchange}}} +\underbrace{W\left(\omega_1,\Delta,T_{2,r} \right)M_{z,r}}_{\color{red}{\text{RF pumping}}} \]

\[ MTsat=\left( A_{app}\cdot \frac{\alpha_{MT}}{S_{MT}}-1 \right)\cdot R_{1}\text{TR}_{\text{MT}}-\frac{\alpha_{MT}^{2}}{2} \]

\[ A_{app}=S_{PD}S_{T_{1}}\frac{\text{TR}_{\text{PD}}\frac{\alpha_{T_{1}}}{\alpha_{PD}}-\text{TR}_{T_{1}}\frac{\alpha_{PD}}{\alpha_{T_{1}}}}{\text{TR}_{\text{PD}}S_{T_{1}}\alpha_{T_{1}}-\text{TR}_{T_{1}}S_{PD}\alpha_{PD}} \]

\[ R_{1}=\frac{1}{2}\cdot \frac{\frac{S_{T_{1}}\alpha_{T_{1}}}{\text{TR}_{T_{1}}}-\frac{S_{PD}\alpha_{PD}}{\text{TR}_{PD}}}{\frac{S_{PD}}{\alpha_{PD}}-\frac{S_{T_{1}}}{\alpha_{T_{1}}}} \]