Limited exponential growth: Introduction
Administration of a drug via a constant rate intravenous infusion
Pharmacokinetic differential equation for a constant rate intravenous infusion The second example of limited exponential growth comes from pharmacokinetics: the administration of a drug via a constant rate intravenous infusion. With such intravenous infusion, an intravenous bolus injection is repeated several times, with an infinitely short dose interval, but with a low dosage. This represents a continuous administration of a drug at a constant speed, the so-called infusion rate \(R_\mathrm{inf}\). Initially, the plasma concentration \(C\) of the drug in the body increases rapidly: in the beginning, the amount of drug that enters the body with each drip is larger than the amount that is cleared. We assume elimination via a first-order process with elimination rate constant \(k\). Then have the following formula for the change of the amount of drug in the body by intravenous infusion, under the assumptions that the administered drug is rapidly distributed throughout the body and that \(D(t)\) represents the amount of drug in the body at time \(t\): \[\frac{\dd D}{\dd t}=R_\text{inf}-\mathrm{Cl}\cdot C\] where \(\mathrm{Cl}\) is the total body clearance of the pharmacon. We denote the volume of distribution as a \(V_d\). Then we can write \[D(t)=V_d\cdot C(t)\quad\text{and}\quad k=\frac{\mathrm{Cl}}{V_d}\] Thus, the following initial value problem is the mathematical description of this model: \[\frac{\dd C}{\dd t}=\frac{R_\mathrm{inf}}{V_d}-k\cdot C, \quad C(0)=C_0.\]
Calculation without a loading dose If no a\dd Ditional intravenous loading dose of the drug is administered at the beginning, then \(C_0\) equals zero. In this case, as the plasma concentration increases, per unit of time more pharmacon will be cleared. Ultimately, the plasma concentration approximates an equilibrium value, the so-called steady-state concentration \(C_\mathrm{ss}\). In this steady state, the plasma concentration does not change anymore. So it follows: \[C_\mathrm{ss}=\frac{R_\mathrm{inf}}{k\cdot V_d}=\frac{R_\mathrm{inf}}{\mathrm{Cl}}.\] Thus, the steady state concentration depends only on the infusion rate \(R_\mathrm{inf}\) and clearance \(\mathrm{Cl}\). In terms of the half-life \(t_{1/2}\) of the drug, for which \(t_{1/2}=\ln(2)/k\) holds, and the volume of distribution \(V_d\), the formula for the steady-state concentration\(C_\mathrm{ss}\) can also be written as \[C_\mathrm{ss}=\frac{R_\mathrm{inf}\cdot t_{1/2}}{\ln(2)\cdot V_d}\tiny.\] A doctor or nurse can, of course, only adjust the rate of infusion for a given drug. The height of the plateau \(C_\mathrm{ss}\) is inversely proportional to the clearance \(\mathrm{Cl}\) and the volume of distribution \(V_d\), and proportional to the half-life \(t_{1/2}\). The time profile of \(C\) is given by the following formula, during the infusion without a loading dose: \[C(t)=C_\mathrm{ss}\cdot\left(1-e^{-k\cdot t}\right)\] In theory, it never reaches the equilibrium, but a physician will be satisfied when a concentration has reached the 'therapeutic level'. Suppose that this level is equivalent to 97% of the steady-state concentration, then this level is only reached after 5 half-lives (after the first time 50% is achieved, after 2 times 75%, and after \(n\) times \(100\cdot (1-(\tfrac{1}{2})^n)\) % is reached). For a drug with a long half-life, it takes proportionately long before it has therapeutic effect.
Calculation with a loading dose If rapid treatment is required, it is wise to administer at the start of the infusion a loading dose via intravenous bolus injection so that the therapeutic effect occurs earlier. The concentration profile is then given by \[C(t)=C_\mathrm{ss}\cdot\left(1-e^{-k\cdot t}\right)+\frac{D_0}{V_d}\cdot e^{-k\cdot t}\] The a\dd Ditional term corresponds with an exponential decay of the loading dose. Ideally, however, a loading dose \(D_0\) smaller than \(V_d\cdot C_\mathrm{ss}\) is chosen in order to prevent the occurrence of a too high, harmful plasma concentration. If this accidentally happens then it follows from the initial value problem that the plasma concentration decreases and will approach the steady-state concentration. But prevention is better than cure: The above relationships between the steady-state concentration and the pharmacokinetic characteristic values of a pharmacon may help in the preparation of dosing schedule, taking into account uncertainties in the values of several quantities (such as, for example, the volume of distribution \(V_d\) or the clearance \(\mathrm{Cl}\) in the case of impaired renal function due to illness or old age).
