Statistical Analysis of Two-Compartment Pharmacokinetic Models with Drug Non-adherence

Abstract

Poor drug adherence is considered one of major barriers to achieving the clinical and public health benefits of many pharmacotherapies. In the current paper, we aim to investigate the impact of dose omission on the plasma concentrations of two-compartment pharmacokinetic models with two typical routes of drug administration, namely the intravenous bolus and extravascular first-order absorption. First, we reformulate the classical two-compartment pharmacokinetic models with a new stochastic feature, where a binomial random model of dose intake is integrated. Then, we formalize the explicit expressions of expectation and variance for trough concentrations and limit concentrations, with the latter proved of the existence and uniqueness for steady-state distribution. Moreover, we mathematically demonstrate the strict stationarity and ergodicity of trough concentrations as a Markov chain. In addition, we numerically simulate the impact of drug non-adherence to different extents on the variability and regularity of drug concentration and compare the drug pharmacokinetic preference between one and two compartment pharmacokinetic models. The results of sensitivity analysis also suggest the drug non-adherence as one of the most sensitive model parameters to the expectation of limit concentration. Our modelling and analytical approach can be integrated into the chronic disease models to estimate or quantitatively predict the therapy efficacy with drug pharmacokinetics presumably affected by random dose omissions.

Appendix A: Derivation of Eqs. (59)

We claim the relationships (Eqs. (59)) in the case of IV bolus administrations, the similar procedure can be used for the case of first-order absorption. The one-compartment IV bolus model is

$$\begin{aligned} {\widetilde{C}}'(t)=-{\widetilde{k}}_{10}{\widetilde{C}}(t),\,\, {\widetilde{C}}(0^+)=D/\widetilde{V_1}. \end{aligned}$$

(A.1)

In order to obtain indistinguishable models, we assume that ${\widetilde{C}}(t)=C_1(t)$ for all time $t\ge 0$. Firstly, the equation ${\widetilde{C}}(0^+)=C_1(0^+)$ directly implies $D/\widetilde{V_1}=D/V_1$, which means

$$\begin{aligned}{\widetilde{V}}_1=V_1.\end{aligned}$$

Since the functions of ${\widetilde{C}}(t)$ and $C_1(t)$ are infinite differentiable at the right hand side of time zero, then we have ${\widetilde{C}}'(0^+)=C'_1(0^+)$ which is equivalent to

$$\begin{aligned}-{\widetilde{k}}_{10}{\widetilde{C}}(0^+)=k_{21}\frac{V_2}{V_1}C_2(0^+)-(k_{10}+k_{12})C_1(0^+)=-(k_{10}+k_{12})C_1(0^+).\end{aligned}$$

This implies

$$\begin{aligned}{\widetilde{k}}_{10}=k_{12}+k_{10}\end{aligned}$$

because of $C_2(0^+)=0$.

Taking derivatives of ${\widetilde{C}}(t)$ and $C'_1(t)$ with respect to time t give rise to

$$\begin{aligned} {\widetilde{C}}''(t)={\widetilde{k}}^2_{10}{\widetilde{C}}(t) \end{aligned}$$

and

$$\begin{aligned} C''_1(t)=k_{12}k_{21}C_1(t)-k^2_{21}/V_1 A_2(t)-(k_{12}+k_{10})k_{21}/V_1 A_2(t)+(k_{12}+k_{10})^2 C_1(t). \end{aligned}$$

Substituting $t=0^+$ into above two equations and using the identity ${\widetilde{C}}''(0^+)=C''_1(0^+)$ yields

$$\begin{aligned}k_{21}=0.\end{aligned}$$