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 e0.4t,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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Answered 11 months ago
Answered 11 months ago
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Use the formula about survival and renewal at the beginning of this section.

P(T)=S(T)P0+0TS(Tt) R(t) dt\begin{align*} P(T) &= S(T) \cdot P_0 + \int_0^T S(T-t) \ R(t) \ dt \end{align*}

Given S(t)=e0.4tS(t) = e^{-0.4 t} we want P(T)=P0P(T) = P_0. Also note that they say the rate of administration, R(t)R(t), is constant so let's just call it RR and we can move it outside the integral.

P0=e0.4TP0+R0Te0.4(Tt) dtP0=e0.4TP0+R0Te0.4Te0.4t dtP0=e0.4TP0+Re0.4T[10.4e0.4t]0TP0=e0.4TP0+Re0.4T[10.4e0.4T10.4]\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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