Pharmacokinetics of Missed Doses

Although missed doses represents a concern for patients and clinical practitioners alike, its PK considerations have not been fully dealt with in the entirety of PK literature. Interestingly, both regulatory authorities and drug manufacturers provide the same recommendation if a single dose has been missed irrespective of the very characteristics of the drug in question. Such recommendation is not based on any PK or scientific grounds. Needless to say that no answer is provided where more than one dose went missing!

Lack of compliance may be encountered in some occasions. This is manifested by patients not taking the doses at the recommended times or failing to take some doses altogether. The former case is not expected to have drastic impact on plasma levels, whereas the latter may disrupt such levels significantly, depending on the PK nature of the drug in question. To examine the impact of lack of compliance, the missing quantities must be estimated in mathematical terms. In the first place, one has to revert to the mathematical model that is suited to describe the drug’s PK properties. It is appreciated that such model must contain a multiple dosing function of the sort of $(1-e^{-nK\tau})/(1-e^{-K\tau})$ that will determine the magnitude of accumulation of the drug in the body. This function is basically determined by some constant elements, such as the number of doses and the rate constants, and a time element which represents a continuous quantity. These are in turn multiplied by similar quantities representing the exponential decay terms, like $e^{-K\tau}$, in the model. According, assessment of the impact of missing doses should be interpreted in terms of the effect of such event (a missed dose) on these quantities. The mathematical treatment for this situation should focus on the impact of the missed dose on whatever Pk metric of interest consequent to such missing dose. The two models shown hereunder are used for the assessment of missing a dose in the case of a one-compartment intravenous bolus model drug, $$C_n=\frac{X_0}{V_d}\left[ \frac{1-e^{-NK\tau}}{1-e^{-K\tau}}e^{-Kt}-e^{-Kt_{miss}}\right]$$ $$C_{\infty}=\frac{X_0}{V_d}\left[ \frac{e^{-K\tau}}{1-e^{-K\tau}}e^{-Kt}-e^{-Kt_{miss}}\right]$$ These two equations have been transformed into the right equation to determine the impact of a missing dose on any consequent dose by subtracting the value of $e^{-Kt_{miss}}$ from the multiple dosing factor at any time during a dosing interval. The time element in this exponential term $t_{miss}$ is equated with the number of elapsed doses prior to a particular dose of interest plus at any time point with its interval. Similar treatment could be contemplated for orally administered drugs that confer upon the body the characteristics of a one compartment model. $$C_n=\frac{K_\alpha FX_0}{V_d(K_\alpha - K)}\left[ \frac{1-e^{-nK\tau}}{1-e^{-K\tau}}e^{-Kt}-e^{-Kt_{miss}} - \frac{1-e^{-nK_\alpha \tau}}{1-e^{-K_\alpha \tau}}e^{-K_\alpha t}-e^{-K_\alpha t_{miss}}\right]\label{ref5.1}\tag{5.1}$$ $$C_\infty=\frac{K_\alpha FX_0}{V_d(K_\alpha - K)}\left[ \frac{e^{-Kt}}{1-e^{-K\tau}}-e^{-Kt_{miss}} - \frac{e^{-K_\alpha t}}{1-e^{-K_\alpha \tau}}e^{-K_\alpha t}-e^{-K_\alpha t_{miss}}\right]\label{ref5.2}\tag{5.2}$$ The impact is not significant for drug with relatively short half-lives and requires no intervention by a booster dose. By contrast, for drugs with relatively long half-lives, the impact is significant which may necessitate the administration of booster dose to restore the disrupted steady state condition.

PK Generalization of Missed Doses

PK models may be generally described as consisting of constant quantities that are associated with some driving force that causes their accumulation or regression. Accordingly, the temporal element in such model assumes critical significance since it will drive these constant quantities in up- or down-ward direction. Close examination of Equation (\ref{ref5.2}) reveals this basic characteristic of any PK model that tends to account for missed doses. This model consists of the accumulation function $(1-e^{-NK\tau})/(1-e^{-K\tau})$ as well as the decay functions $e^{-Kt_\tau}$ and $e^{-Kt_{miss}}$. With the exception of intravenous infusion; the presence of exponential terms in these models transforms them into typical geometric series which allows the prediction of the time dependent variable they estimate, i.e., the plasma levels. Whereas the exponential term $e^{-Kt_\tau}$ accounts for the decay of plasma level during any dosing interval of a multiple dosing regimen, the other term $e^{-Kt_{miss}}$ determines the amount of the drug that could have been contributed to the plasma level if the missed dose were not lost. This term may be considered as an intruder on the well-established PK models that are routinely used to predict plasma levels. Hence, it will be the only element to be subject to generalization within the context of devising PK model that account for missed doses.

For all purposes, questions regarding missed doses are usually posed in conjunction with a certain point in time where the effect of such missing is to be determined. This time point may be regarded as the point where the patient has informed the healthcare profession or has asked for advice on the matter. The current doses number $N_C$ may be considered as the point of time to be used for the estimation of the elapsed time for one or more than one dose(s) that might have been missed. Thus a generalized expression for this elapsed time may be provided as follows: $$t_M=\tau (N_C-N_M)+t_\tau$$ where $t_M$ is the elapsed time since a dose(s) was missed, $N_C$ and $N_M$ are the respective numbers of the current and missed dose(s) and $t_\tau$ is the time within any dosing interval.

According to this definition of the elapsed time, Equation (\ref{ref5.1}) may be re-written according to this definition of the elapsed time for a multiple dosing condition, where a single dose was missed, as follows: $$C_P^{N_C}=\frac{X_0}{V_d}\left[ \frac{1-e^{-N_C K\tau}}{1-e^{-K\tau}}e^{-Kt_\tau}-e^{-K\tau (N_C-N_M)+t_\tau}\right]\label{ref5.3}\tag{5.3}$$ In cases where more than one doses has been missed, their contribution to the decay in any plasma level will correspond to the sum of the individual exponential decay terms associated with the respective elapsed times of these doses. Hence, a generalized expression for summing up the values of all exponential terms of missed dose(s) may be provided as: $$\sum_{l=1}^{n} e^{-\lambda_l[\tau(N_C-N_M)+t_\tau]}$$

Intravenous Bolus

A model could be suggested to determine the impact of missed doses for drugs administered by intravenous bolus mode as follows: $$C_P^{N_C}=\frac{X_0}{V_d}\left[ \frac{1-e^{-N_C K\tau}}{1-e^{-K\tau}}e^{-Kt_\tau}-\sum_{\text{All }N_M}e^{-K\tau (N_C-N_M)+t_\tau}\right]$$ This equation could be used to determine the plasma level(s) at any time prior to the administration of the current dose after the most recent missed dose. It could be also used for predicting the plasma level for any consequently administered doses. In the latter case the term $N_C$ will assume the number of the future dose(s) at which plasma levels need to be predicted.

First-order absorption

The conventional 1-compartment oral model could be readily transformed into a model that accounts for missed doses as follows: $$C_n=\frac{K_\alpha F X_0}{V_d(K_\alpha-K)}\left[ \left( \frac{1-e^{-NK\tau}}{1-e^{-K\tau}}e^{-Kt_\tau}-\sum_{\text{All }N_M}e^{-K[\tau (N_C-N_M)+t_\tau]} \right) - \left( \frac{1-e^{-NK_\alpha \tau}}{1-e^{-K_\alpha \tau}}e^{-K_\alpha t_\tau}-\sum_{\text{All }N_M}e^{-K_\alpha [\tau (N_C-N_M)+t_\tau]} \right) \right]$$

Intravenous infusion

Unlike most other convoluted PK models with continuous time flow, the PK models used to predict drug levels consequent to intravenous infusion consist of discrete terms that account for infusion and post-infusion phases. This implies that for any such model two mathematical expressions will be needed to describe the entire course of plasma levels during both phases. Hence, the following expression may be used to determine the impact of missed doses(s) during the infusion phase: $$C_{p(Inf)}^N=\frac{k_0}{KV_d}\left[ \frac{1}{1-e^{-K\tau}} (1-e^{-Kt_i}\{1-e^{-KT_{pi}}+e^{-NK\tau}(e^{KT_i-1})\}-e^{-K\tau}) -(1-e^{-Kt_{iN_M}})-\sum_{\text{All }N_M}(1-e^{-KT_i})e^{-K\tau[(N_C-N_M)-(T_i+t_i)]}\right]$$ Similarly, a plasma levels during the post-infusion phase could be estimated, in the presence of missed doses according to the following expression: $$C_{p(P.Inf)}^N=\frac{k_0}{KV_d}\left[ \frac{1-e^{-NK\tau}}{1-e^{-K\tau}}(1-e^{-KT_i})e^{-Kt_{pi}}-\sum_{\text{All }N_M}(1-e^{-KT_i})e^{-K[\tau(N_C-N_M)+t_{pi}]}\right]$$