2.4 Establishment of the PK–PD model
A PK–PD model was established by combining a separately constructed PK model and a PD model. The PK model is a mathematical model showing time-dependent changes in concentrations of drug and biological matter, whereas the PD model shows the relationship between drug efficacy and drug concentration. The PK–PD model is used to predict the drug efficacy on the target site using the drug dosage. However, alternative indices of drug concentration are needed in models involving drug concentration changes over time because the PD model expresses drug efficacy as a function of the drug concentration. Normally, the area under the concentration–time curve (AUC) is used as an alternative index to drug concentration and is an integral value of a drug concentration–time function as it represents the full drug dosein vivo in pharmacokinetics study.
We established a PK–PD model that predicts the drug efficacy on the lung cancer part from values of the AUC obtained by the PK model. The microchannel volume of the MO–MPS and changes in the drug concentration in the organ parts must be considered because pharmacokinetics depends on the drug concentration in each organ part and the metabolism of compounds by these organs (Fig. 1E). Only the metabolism of the liver part was considered in the proposed model because the number of metabolic enzymes in the lung cancer part, the drug target part, was considerably lower than that of the liver. The gradient of the prodrug concentration is expressed as follows:
\(\frac{d\left[C_{P}\right]}{\text{dt}}={-k}_{P}\left[C_{P}\right]\)(1)
where kP is the elimination rate constant of the prodrug. The change in prodrug concentration is as follows:
\(C_{P}\left[t\right]=X_{0}\times e^{-k_{P}\times t}\)(2)
where X0 is the initial concentration of the prodrug.
The concentrations of the metabolites simultaneously increase because of prodrug metabolism and decrease because of metabolic enzyme reactions. The metabolites are different due to species of enzymes that metabolize prodrugs. Therefore, the rate at which the metabolic enzyme contributes to the overall metabolism must considered. The gradient of each metabolite concentration can be expressed by:
\(\frac{d\left[C_{M}\right]}{\text{dt}}={\text{fm}\times k}_{P}\left[C_{P}\right]-k_{M}\left[C_{M}\right]\)(3)
where km is the elimination rate constant of the metabolite; andfm is the fraction metabolized defined as the metabolized rate by each metabolic enzyme. The change in metabolite concentration then becomes as follows:
\(C_{M}\left[t\right]=\frac{k_{P}\times fm\times X_{0}}{k_{P}-k_{M}}\times\left(e^{-k_{M}\times t}-e^{{-k}_{P}\times t}\right)\)(4)
The elimination rate constants can be expressed as:
\(k=\frac{\text{CL}}{\text{Vd}}=\frac{Q\times E}{\text{Vd}}\) (5)
where CL is the clearance in the liver part [µL/h]; Vdis the distribution volume [µL]; Q is the flow rate in the liver part; and E is the extraction ratio in the liver part. The change in metabolite concentration can be expressed as:
\(C_{M}\left[t\right]=\frac{E_{P}\times fm\times X_{0}}{E_{P}-E_{M}}\times\left(e^{-\frac{Q\times E_{M}}{\text{Vd}}\times t}-e^{-\frac{Q\times E_{P}}{\text{Vd}}\times t}\right)\)(6)
A PD model was developed based on the drug efficacy of SN-38, whose concetration was evaluated previously, on A549 cells.(Mijatovic et al., (2006)) SN-38, metabolite of CPT-11 used in this study, has strong anticancer effect. The drug efficacy is expressed as:
\(\frac{\text{Cell\ density}}{\text{Control\ cell\ density}}=-0.086\times\ln\left(\text{Auc}\right)+0.512\)(7)
AUC [h∙ng/µL] is an integral part of the concentration change function. Therefore, the drug efficacy after t hours from the administration of drug dose can be expressed as:
\(\frac{\text{Cell\ density}}{\text{Control\ cell\ density}}=-0.086\times\ln\left(\int_{t}^{0}{C_{M}\left[t\right]}\right)+0.512\)(8)