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In this paper, the authors examine a common differential equation model for the concentration of a drug in the blood.

In the paper, they begin with the following expression

$$ \mu(t)=\frac{D k_{a}}{V\left(k_{a}-k_{e}\right)}\left\{\exp \left(-k_{e} t\right)-\exp \left(k_{a} t\right)\right\} $$

They then collect some terms

$$\mu(t)=D \exp \left(\beta_{0}+\beta_{1} t\right)\left[1-\exp \left\{-\left(k_{a}-k_{e}\right) t\right\}\right]$$

With $\beta_{0}=\log \left\{k_{a} /\left[V\left(k_{a}-k_{e}\right)\right]\right\} \text { and } \beta_{1}=-k_{e}$

And finally make the leap saying that this can be modeled using

$$\mu_{i j}=D_{i} \exp \left(\beta_{0}+\beta_{1} t_{i j}+\beta_{2} / t_{i j}\right)$$

This last part is where they lose me. I'm certain they make some approximation, but it isn't clear what approximation is made.

Can anyone shed some light onto how that last jump was made?

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With this kind of thing, a good place to start is to expand things out so you can do a clear comparison of them. On the one hand we have: $$\mu(t) = D\exp (\beta_0 +\beta_1 t)[1 - \exp \{-(k_a-k_e)t\}] $$ which expands to $$\mu(t) = D\exp (\beta_0 +\beta_1 t) - D\exp (\beta_0 +\beta_1 t)\exp \{-(k_a -k_e)t\} \tag{1}\label{eqn1}$$ and on the other we have $$\mu_{ij} = D_i \exp(\beta_0 +\beta_1 t_{ij} +\beta_2/t_{ij}) \tag{2} \label{eqn2} $$ Comparing (\ref{eqn1}) and (\ref{eqn2}) we see that it's likely that $\beta_2 = (k_a-k_e)$ and the minus sign has been associated with the $t$ to obtain $\beta_2/t_{ij}$ in (\ref{eqn2}).

Without knowing more about the problem it's impossible to say why the first term, $D\exp(\beta_0 +\beta_1 t)$ is being ignored. I followed your link, but all the details are paid-for and I'm not interested enough in this question to spend money to be able to answer it :) It seems likely that it is either

  • small relative to the second term, and so can be neglected
  • being incorporated into the $D_i$ term after time discretization

with the second option being most likely since $D_i$ wouldn't need a subscript if $t$ weren't involved somehow.

Since this is a question about mathematical modelling, it might be worth me noting that a model is of less use if it's not well-explained, so your question would benefit from indicating what $k_a$ and $k_e$ are and explicitly talking about the time discretization in order to help answerers. $t_{ij}$ suggests a two-dimensional mesh to me, but I'm guessing. If I knew for certain, then I might be able to explain the $D_i$ term better.

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