Modelling a decay to a constant
$$ \[\begin{align} \text{FID}_{ijk} \sim& \text{gamma}\left(\alpha, \frac{\alpha}{\mu_{ijk}}\right)\\ \mu_{ijk} =& m_{jk} \times 1000 \times \left( 1 - \frac{p_{jk} \times X_i}{d_{jk} + X_i} \right) \\ \text{logit}(m_{jk}) =& \bar{m} + \beta_k^m + m_{jk}\\ \text{logit}(p_{jk}) =& \bar{p} + \beta_k^p + p_{jk}\\ \text{log}(d_{jk}) =& \bar{d} + \beta_k^d + d_{jk}\\ \begin{bmatrix} m_{k}\\ p_{k}\\ d_{k}\\ \end{bmatrix} \sim& \text{multivariate normal}\left(\begin{bmatrix}0\\0\\0\end{bmatrix}, \text{diag}(\sigma)\times \textbf{R}\times\text{diag}(\sigma)\right) \\ \bar{m}, \beta_k^m \sim& \text{normal}(.5, .5)\\ \bar{p},\beta_k^p \sim& \text{normal}(-1, .2)\\ \bar{d},\beta_k^d \sim& \text{normal}(1.5, .5)\\ \alpha \sim& \text{gamma}(6.25, .25) \\ \textbf{R} \sim&\text{LKJ}(2) \\ \sigma \sim& \text{exponential}(2) \\ \beta_k \end{align}\] $$