Mathematical Aspects of Multiple Dosing

Assuming that a drug has been administered as a single dose, that maximum and minimum amount of the drug in the body, within a fixed dosing interval, could be easily determined irrespective of the model that could best describe the kinetics of the administered drug. For a one-compartment model drug, administered by intravenous route, the maximum quantity to be found in the body after the first dose will be the administered dose itself. Furthermore, the minimum quantity will be that which could be estimated at the end of the dosing interval as per the expression $X_{min(N=1)}=X_0 e^{-K\tau}$. Likewise, the maximum quantity that will be found upon the administration of the second dose will be this dose itself plus the minimum amount that was left over from the previous dose, i.e. $X_{nax(N=2)}=X_0+X_0 e^{-K\tau}$. This amount will be reduced as function of the dosing interval, i.e. $X_{nax(N=2)}=X_0 (1+e^{-K\tau})e^{-K\tau}$. The continuation of dosing will result in a geometric series which has been formulated by Dost as early as 1953. $$\frac{e^{-(N-1)\lambda_i \tau}-e^{-\lambda_i\tau}}{1-e^{-\lambda_i\tau}}=\left( \frac{e^{-(N-1)}-1}{1-e^{-\lambda_i\tau}} \right)e^{-\lambda_i\tau}$$ Based on the above isotherm, it could be verified that the following expression holds true, $$\left(\frac{e^{-(N-1)\lambda_i \tau-e^{-\lambda_i\tau}}}{1-e^{-\lambda_i\tau}}\right)e^{-\lambda_i t}=\left(\frac{1-e^{-(N-1)\lambda_i\tau}}{1-e^{-\lambda_i\tau}}\right)e^{-\lambda_i (t-(N-1))\tau}$$ If a PK model contains what could be regarded as discrete terms or quantities, then each such quantities has to be multiplied by Dost’ multiple dosing factor to transform the model into a multiple dosing model. $$C_N=\sum_{l=1}^nA_l\left(\frac{1-e^{-N\lambda_i\tau}}{1-e^{-\lambda_i\tau}}\right)e^{-\lambda_l t}$$ where $C_N$ is plasma levels after $N$ number of doses, $A_l$ is the zero-time intercept associated with the $\lambda_l$ rate constant, $\tau$ is the dosing interval and $t$ is the actual time.

This generalized expression could be used to quantify the plasma levels after any number of doses, during ant dosing interval. It may also be modified so that some main PK metrics be the main PK metrics ensuing from any multiple dosing situation, namely, the maximum, the minimum and the average plasma levels. The actual plasma levels during any dosing interval could also be evaluated at any dose as well as at steady state. The maximum plasma level at steady state could be determined according to the following repeated administration of intravenous bolos doses could be estimated according to the equation:

$$C_{max}^{\infty}=\sum_{l=1}^nA_l\left(\frac{1}{1-e^{-\lambda_i\tau}}\right)\text{ and }C_{min}^{\infty}=\sum_{l=1}^nA_l\left(\frac{1}{1-e^{-\lambda_i\tau}}\right)e^{-\lambda_l\tau}$$