Pharmacodynamic system analysis of the biophase level predictor and the transduction function

This work deals with the pharmacodynamic problem of relating a drug effect E(t) to an observable pharmacokinetic (PK) predictor variable r(t), which may be a venous and/or arterial drug level, some other PK variable, or a drug infusion scheme. It is proposed that the relationship between E(t) and r(t) may, in many cases, be appropriately modeled as E(t) = N(F(r(t))) [for practical analysis reasons, F(r(t)) is denoted the biophase level, cb(t), the operator F() is accordingly denoted the biophase level predictor (BLP), and N() is denoted the transduction function (TF)]. This work proposes a method for determining the two fundamental components, BLP and TF, that define the E(t)-r(t) relationship. The BLP is determined by a hysteresis minimization (HM) technique with the following features: (1) the method considers errors in both HM variables; (2) the method is suitable for dealing with the important case in which the predictor variable is a drug infusion scheme; (3) the approach is noncompartmental in contrast to the effect-compartment approaches and it does not require a specific structured modeling of the cb(t)-r(t) relationship when dealing with drugs with linear PKs in a general operational sense; and (4) the method makes use of a dimensionless transformation of the hysteresis variable that eliminates numerical scaling problems so that a complex penalty function optimization approach can be avoided. The TF is determined by a cross validation procedure in conjunction with the BLP determined by HM. The method is demonstrated using pharmacodynamic data for several drugs, considering both concentration-based and drug input-based r(t) values. The significance of the information obtained from the determined BLP and TF is discussed, including the concepts of equilibration dynamics and overloading. The limitations and potential problems of the methodology are discussed.