Nonlinear Pharmacokinetics: Drug Elimination for IV Bolus Injection

In pharmacokinetics, the elimination rate of a drug following a capacity-limited model is primarily controlled by two parameters: V_max and K_M. These parameters are crucial in how the drug behaves inside the body after administration.

Following the administration of a single intravenous (IV) bolus injection, we can determine the concentration of the drug in the plasma at any given time. This calculation is achieved using a specific equation that integrates the values of V_max and K_M.

We can also calculate the total amount of drug present in the body post-injection. This calculation utilizes D₀, which represents the initial drug amount in the body at time zero (t = 0).

One of the critical aspects of drug administration is understanding how long it takes for the drug dose to decline to a certain amount within the body. For instance, consider a drug administered as a single 400-mg dose with a K_M value of 38 mg/L. If the V_max varies from 200 to 100 mg/h, it will take around 2.46 hours for the drug concentration to decline to 20 mg at a V_max of 200 mg/h. In contrast, at a V_max of 100 mg/h, it would take approximately 4.93 hours — almost twice as long — to reach the same concentration.

Interestingly, an inverse relationship exists between the time the dose takes to decline and the V_max. For example, when V_max is held constant at 200 mg/h, the time required for the drug to decline to 20 mg is 2.46 hours when K_M is 38 mg/L. However, if K_M is increased to 76 mg/L, this time increases to 3.03 hours, indicating that an increase in K_M will subsequently increase the time taken for the drug to be eliminated from the body.