Concentration of a drug after repeated injections

Question

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ is a positive constant.

a) If a dose $D$ is injected every $T$ hours, write an expresssion of the sum of the residual concentrations just before the $(n+1)$st injection.

b) Determine the limiting pre-injection concentration.

c) If the concentration of insulin must always remain at or above a critical value $C$, determine the minimal dosage $D$ in terms of $C,a$ and $T$.

What I got so far: a)

The sum can be expressed as $D\exp^{-a(t-0T)} + D\exp^{-a(t-1T)} + D\exp^{-a(t-nT)} \dots$

So it is The sum from $0$ to $\infty$ of $D\exp^{-a(t-nt)}$

I'm not sure If I am right.

b ) I think I just need to find the limit of this expression as $t$ goes to $0$? so it's simply $D$.

c) I don't know what to do here, but I tried looking up boundaries so the function is bound above some value $C$, etc. but I'm really lost here.

Answer 1

(a) We calculate the first few terms. Let $R_n$ be the concentration just before the $(n+1)$-th injection.

We have $R_1=De^{-aT}$. Thus $R_2=De^{-aT}+e^{-aT}R_1=De^{-aT}+De^{-2aT}$.

Similarly, $R_3=De^{-aT}+e^{-aT}R_2=De^{-aT}+De^{-2aT}+De^{-3aT}$.

The pattern is clear. We have $$R_n=De^{-aT}+De^{-2aT}+\cdots+De^{-naT}.$$ This is an $n$-term geometric series with first term $De^{-aT}$ and common ratio $e^{-aT}$. The sum is given by $$R_n=De^{-aT} \frac{1-e^{-naT}}{1-e^{-aT}}.\tag{1}$$

(b) Let $n\to\infty$. The $e^{-naT}$ term goes to $0$. Now from (1) we can write down the limiting pre-injection concentration.

(c) This seems too easy. The smallest concentration (after treatment has begun) is just before the second injection. So we want $De^{-aT}\ge C$. The minimal dose therefore satisfies $De^{-aT}=C$. Solve for $D$.

Answer 2

In pharmacology, a fundamental problem is how to fall concentration of a drug in the bloodstream of a patient. To keep the concentration, the dose should be repeated and for that, we have to establish what dosage of time that each application should be made. The simplest model is obtained when we assume that the rate of change in concentration is proportional to the concentration of drug in the bloodstream, ie $$ \dfrac{dC}{dt} = -kC \qquad (1) $$ where $C = C (t)$ is the concentration of $mg / ml$ and $k > 0$ is a constant experimentally found.

Suppose the patient is given an initial dose $C_0$, immediately absorbed by the blood at time $t_0$. Separating the variables in (1) and integrating, we have: $$ C (t) = C_0e ^{-kt} \qquad (2) $$ Suppose that after a while $T$ a second dose of the same amount $C_0$ is administered. then we have $$ C(T_{-}) = C_0e^{-kT} $$ is the amount of drug immediately before the second dose and $$ C_1: = C (T^{+}) = C_0e^{-kT} C_0 = C_0(1 + e^{-kT}) \qquad (3) $$ is the amount of drug in the blood immediately after the second dose, as the graph below.

The expressions (2) and (3) $$ C(t) = C_0 (1 + e ^{-kt}) e^{-k (T - T)} $$ gives the amount of drug in the blood at time $t \geq T$. Further treatment by injection amount $C_0$ at the end of each time interval equal to $T$, we obtain $$ C(2T_{-}) = C_0(1 + e^{-kT})e^{-kT} $$ and $$ C_{2}:=C(2T_{+}) = C_0(1 + e^{-kT})e^{-kT} + C_0 =C_0(1 + e^{-kT} + e^{-2kT}), \qquad t \geq 2T $$ In general, after the Nth application, the amount of drug in the blood is $$ C_{n}:=C(nT_{+}) = C_0(1 + e^{-kT} + e^{-2kT}+\ldots + e^{-nkT}) \qquad (4) $$ for $n=1,2,\ldots$. The expression (4) is the sum of a G.P. reason $e^{-kT}$ and $n +1$ terms. Where $k > 0$ and $T> 0$, then $1 - e^{-kT} > 0$. Dividing and multiplying (4) by this factor, we get: $$ C(nT_{+}) = \frac{C_0(1 + e^{-kT}+\ldots + e^{-nkT})(1- e^{-kT})}{1 - e^{-kT}} $$ from which it follows that $$ C(nT_{+}) = \frac{C_0[1 - e^{(n+1)kT}]}{1 - e^{-kT}} \qquad (5) $$ When $n$ grows, the term $e^{-(n+1)kT} \to 0$ so that $C (nT_{+})$; tends towards a saturation concentration given by $$ C_s = \frac{C_0}{1 - e ^ {-kT}} $$