Question

A patient is continually receiving a drug. If the drug is eliminated from the body over time so that the fraction that remains after t hours is $e^{-0.4 t},$ at what constant rate should the drug be administered to maintain a steady level of the drug in the bloodstream?

Solution

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Use the formula about survival and renewal at the beginning of this section.

\begin{align*} P(T) &= S(T) \cdot P_0 + \int_0^T S(T-t) \ R(t) \ dt \end{align*}

Given $S(t) = e^{-0.4 t}$ we want $P(T) = P_0$. Also note that they say the rate of administration, $R(t)$, is constant so let's just call it $R$ and we can move it outside the integral.

\begin{align*} P_0 &= e^{-0.4T} P_0 + R \int_0^T e^{-0.4(T-t) } \ dt \\\\ P_0 &= e^{-0.4T} P_0 + R \int_0^T e^{-0.4T} e^{0.4t } \ dt \\\\ P_0 &= e^{-0.4T} P_0 + Re^{-0.4T} \bigg[ \dfrac{ 1}{0.4 } e^{0.4t} \bigg]_0^T \\\\ P_0 &= e^{-0.4T} P_0 + Re^{-0.4T} \bigg[ \dfrac{ 1}{0.4 } e^{0.4T} - \dfrac{ 1}{0.4 } \bigg] \end{align*}

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