Chapter 3.P2, Problem 3P

Assuming that $x_1(0)=0$ and $x_2(0)=0$, use the parameter values listed in the table below to perform numerical simulations of the system (iii) with the goal of recommending two different encapsulated dosage strengths $A=R{T}_b$ for distribution.

$\begin{array}{cccccc}{k}_{21}& k_{12}& K& V_1& V_2& T_b\\ \text{0}\text{.29/h}& \text{0}\text{.31/h}& \text{0}\text{.16/h}& \text{6}\text{L}& \text{0}\text{.25}\text{L}& \text{1}\text{h}\end{array}$

Use the following guidelines to arrive at your recommendations:

➤ It is desirable to keep the target concentration levels in the brain as close as possible to constant levels between $10\text{mg/L}$ and $30\text{mg/L}$, depending on the individual patient. The therapeutic range must be above the minimum effective concentration and below the minimum toxic concentration. For the purpose of this project, we will specify that concentration fluctuations should not exceed $25\%$ of the average of the steady-state response.

➤ As a matter of convenience, a lower frequency of administration is better that a higher frequency of administration; once every $24$ hours or once every $12$ hours is best. Once every $9.5$ hours is unacceptable and more that $4$ times per day is unacceptable. Multiple doses are acceptable, that is, “take two capsules every $12$ hours.”

$\frac{d{c}_1}{dt}=-\left(K+k_{21}\right)c_1+\frac{k_{12}{V}_2}{V_1}{c}_2+\frac{1}{V_1}d(t)$

$\frac{d{c}_2}{dt}=\frac{V_1{k}_{21}}{V_2}{c}_1-k_{12}{c}_2$ (iii)

$d\left(t\right)=\{\begin{array}{l}R,0\le t\le T_b\\ 0,T_b\le t\le T_p\end{array}$