An integrated approach to pharmacokinetic analysis for linear mammillary systems in which input and exit may occur in/from any compartment

The general treatment of linear mammillary models employing input and disposition functions and Laplace transforms is expanded to solve concentration-time equations AUC and AUMC in any compartment without restricting sites of input or output. In this integrated approach to noncompartmental pharmacokinetic analysis, the values of AUC and AUMC can be calculated from Laplace transforms with some very simplified treatments. Tables of input functions, disposition functions, Laplace transforms, and derivatives of the Laplace transforms are presented. The relationships between the integrated parameters and various mammillary model parameters are presented using several examples.     

An integrated approach to pharmacokinetic analysis for linear mammillary systems in which input and exit may occur in/from any compartment

The general treatment of linear mammillary models employing input and disposition functions and Laplace transforms is expanded to solve concentration-time equations AUC and AUMC in any compartment without restricting sites of input or output. In this integrated approach to noncompartmental pharmacokinetic analysis, the values of AUC and AUMC can be calculated from Laplace transforms with some very simplified treatments. Tables of input functions, disposition functions, Laplace transforms, and derivatives of the Laplace transforms are presented. The relationships between the integrated parameters and various mammillary model parameters are presented using several examples.Supported in part by NIH Grants GM 26691 and HL 32243.     

Derivation of general equations for linear mammillary models when the drug is administered by different routes

A general disposition equation for a linear mammillary model consisting of ncompartments is derived. This equation is used to derive disposition equations for the central compartment when drug input occurs into the central compartment and when drug input occurs into a peripheral compartment. The derivation of equations that describe the entire time course of drug in a particular compartment after intravenous, intramuscular, oral, and rectal drug administration is also presented.     

Segmentally continuous input functions in linear multicompartment systems

The convolutions of segmentally continuous zero- and first-order input functions with the general form of pharmacokinetic multicompartmental disposition functions can be used to analyze and simulate the time course of drug invasion into the central compartment of mammillary models and certain physiologically relevant recirculating systems. The generalized model equations may be used to assess the reliability and intercorrelations of parameter estimates directly, since partial derivatives with respect to all model constants can be computed explicitly. In combination with curve-fitting algorithms, input functions identical to those of the point-area deconvolution can be obtained, although at the expense of significantly longer computing times. On the other hand, the range of potential applications goes far beyond the reconstruction of the time course of drug absorption.     

General treatment of linear pharmacokinetics

A general treatment of linear pharmacokinetics that enables equations to be obtained simply for all linear compartmental models, with input in one or more compartments, is presented. Two approaches are described: one based on a full Laplace transformation and one that avoids transformation of the input functions and the use of convolution integrals. The latter approach is of particular interest when dealing with complex input functions not having a simple Laplace transform. The concept of acceptor and donor subsystems is introduced. It is demonstrated that disposition in certain models may be simplified and analyzed in terms of disposition in subsystems of simpler composition. The treatment presented is illustrated with several examples.     

Mean residence time of drugs administered non-instantaneously and undergoing linear tissue distribution and reversible metabolism and linear or non-linear elimination from the central compartment

Based on disposition decomposition analysis (DDA), equations for the mean residence times (MRT) in the body are derived for a drug and its interconversion metabolite that undergo linear tissue distribution and linear or non-linear elimination from the central compartment after non-instantaneous administration of the drug. The MRT of the drug after non-instantaneous input can be related to the MRT of the drug after intravenous administration, the ratio of the total area under the plasma concentration—time curve of the drug after non-instantaneous administration to that after intravenous administration, the bioavailability of the drug, and the mean input time of the drug. Similar relationships also exist for the MRT of the interconversion metabolite after non-instantaneous input of the drug. The application of these equations to a non-linear reversible metabolic system is illustrated with computer simulations.     

Stepwise determination of multicompartment disposition and absorption parameters from extravascular concentration-time data. Application to mesoridazine, flurbiprofen, flunarizine, labetalol, and diazepam

When disposition is monoexponential, extravascular concentration-time (C, t) data yield both disposition and absorption parameters, the latter via the Wagner-Nelson method or deconvolution which are equivalent. Classically, when disposition is multiexponential, disposition parameters are obtained from intravenous administration and absorption data are obtained from extravascular C, t data via the Loo-Riegelman or Exact Loo-Riegelman methods or via deconvolution. Thus, in multiexponential disposition one assumes no intrasubject variation in disposition, a hypothesis that has not been proven for most drugs. Based on the classical two- and three-compartment open models with central compartment elimination, and using postabsorptive extravascular C, t data only, we have developed four equations to estimate k10 when disposition is biexponential and two other equations to estimate k10 when disposition is triexponential. The other disposition rate constants are readily obtained without intravenous data. We have analyzed extravascular data of flurbiprofen (12 sets), mesoridazine (20 sets), flunarizine (5 sets), labetalol (9 sets), and diazepam (4 sets). In the case of diazepam intravenous C, t data were also available for analysis. After disposition parameters had been estimated from the extravascular data the Exact Loo-Riegelman method with the Proost modification was applied to the absorptive extravascular data to obtain AT/VP as a function of time. These latter data for each subject and each drug studied were found to be fitted by a function indicating either simple first-order absorption, two consecutive first-order processes, or zero-order absorption. After absorption and disposition parameters had been estimated, for each set of extravascular data analyzed, a reconstruction trend line through the original C, t data was made. The new methods allow testing of the hypothesis of constancy of disposition with any given drug. There is also a need for new methods of analysis since the majority of drugs have no marketed intravenous formulation, hence the classical methods cannot be applied.     

The contribution of transplacental clearances and fetal clearance to drug disposition in the ovine maternal-fetal unit

The total clearance of drugs from the fetus is the sum of clearance by the placenta from fetus to mother and clearance by nonplacental mechanisms such as renal excretion and biotransformation. Representing the maternal-fetal unit by a general two-compartment open model, we have derived equations for the calculation of placental and nonplacental drug clearances from the maternal compartment and fetal compartment under steady-state conditions. The equations suggest that fetal drug elimination to the exterior would result in a maternal-fetal concentration gradient at steady state after maternal drug administration. These equations have been applied to the study of methadone disposition in the ovine maternal-fetal unit.     

Stepwise determination of multicompartment disposition and absorption parameters from extravascular concentration-time data. Application to mesoridazine,flurbiprofen, flunarizine,labetalol, and diazepam

When disposition is monoexponential, extravascular concentrationtime (C, t) data yield both disposition and absorption parameters, the latter via the Wagner-Nelson method or deconvolution which are equivalent. Classically, when disposition is multiexponential, disposition parameters are obtained from intravenous administration and absorption data are obtained from extravascular C, tdata via the Loo-Riegelman or Exact Loo-Riegelman methods or via deconvolution. Thus, in multiexponential disposition one assumes no intrasubject variation in disposition, a hypothesis that has not been proven for most drugs. Based on the classical two and threecompartment open models with central compartment elimination, and using postabsorptive extravascular C, tdata only, we have developed four equations to estimate k₁₀ when disposition is biexponential and two other equations to estimate k₁₀ when disposition is triexponential. The other disposition rate constants are readily obtained without intravenous data. We have analyzed extravascular data of flurbiprofen (12 sets), mesoridazine (20 sets), flunarizine (5 sets), labetalol (9 sets), and diazepam (4 sets). In the case of diazepam intravenous C, tdata were also available for analysis. After disposition parameters had been estimated from the extravascular data the Exact Loo-Riegelman method with the Proost modification was applied to the absorptive extravascular data to obtain A_T/V_p as a function of time. These latter data for each subject and each drug studied were found to befitted by a function indicating either simple firstorder absorption, two consecutive firstorder processes, or zero order absorption. After absorption and disposition parameters had been estimated, for each set of extravascular data analyzed, a reconstruction trend line through the original C, tdata was made. The new methods allow testing of the hypothesis of constancy of disposition with any given drug. There is also a need for new methods of analysis since the majority of drugs have no marketed intravenous formulation, hence the classical methods cannot be applied.     

The Cutoff Time Point of the Partial Area Method for Assessment of Rate of Absorption in Bioequivalence Studies

The partial area method has been suggested for the assessment of the absorption rate in bioequivalence studies. This paper provides a theoretical basis for the estimation of the optimal cutoff time point of the partial areas for drugs with one compartment model disposition. The analysis is performed by using the appropriate equations which relate the normalized (in terms of the extent of absorption) partial areas with time expressed in terms of multiples of half-life. Provided that the quality of experimental data ensures precise estimation of the parameters, the t_max of the formulation with the faster absorption characteristics is generally the most practical cutoff time point for calculation of the normalized partial areas, when a drug follows one compartment model disposition with linear absorption.     

Mean residence time in the body versus mean residence time in the central compartment

Mean residence time in the body may be determined by noncompartmental methods following any type of input process into the sampled compartment. Mean residence time in the central compartment can be determined following an intravenous bolus dose, as it requires calculation of the concentration at zero time. For any other input process the mean residence time in the central compartment can be calculated from the elimination rate constant from the central compartment if one accepts the same restrictions used to calculate mean residence time in the body.     

Tissue distribution of fentanyl and alfentanil in the rat cannot be described by a blood flow limited model

Traditionally, physiological pharmacokinetic models assume that arterial blood flow to tissue is the rate-limiting step in the transfer of drug into tissue parenchyma. When this assumption is made the tissue can be described as a well-stirred single compartment. This study presents the tissue washout concentration curves of the two opioid analgesics fentanyl and alfentanil after simultaneous 1-min iv infusions in the rat and explores the feasibility of characterizing their tissue pharmacokinetics, modeling each of the 12 tissues separately, by means of either a one-compartment model or a unit disposition function. The tissue and blood concentrations of the two opioids were measured by gas-liquid chromatography. The well-stirred one-compartment tissue model could reasonably predict the concentration-time course of fentanyl in the heart, pancreas, testes, muscle, and fat, and of alfentanil in the brain and heart only. In most other tissues, the initial uptake of the opioids was considerably lower than predicted by this model. The unit disposition functions of the opioids in each tissue could be estimated by nonparametric numerical deconvolution, using the arterial concentration times tissue blood flow as the input and measured tissue concentrations as the response function. The observed zero-time intercepts of the unit disposition functions were below the theoretical value of one, and were invariably lower for alfentanil than for fentanyl. These findings can be explained by the existence of diffusion barriers within the tissues and they also indicate that alfentanil is less efficiently extracted by the tissue parenchyma than the more lipophilic compound fentanyl. The individual unit disposition functions obtained for fentanyl and alfentanil in 12 rat tissues provide a starting point for the development of models of intratissue kinetics of these opioids. These submodels can then be assembled into full physiological models of drug disposition.Supported in part by the National Institute on Aging, RO1-AG-4594, the Anesthesia/ Pharmacology Research Foundation, and a travel grant from Janssen Pharma AB (Sweden).     

Mean residence time in the body versus mean residence time in the central compartment

Mean residence time in the body may be determined by noncompartmental methods following any type of input process into the sampled compartment. Mean residence time in the central compartment can be determined following an intravenous bolus dose, as it requires calculation of the concentration at zero time. For any other input process the mean residence time in the central compartment can be calculated from the elimination rate constant from the central compartment if one accepts the same restrictions used to calculate mean residence time in the body. Publication supported in part by NIH grant GM 26691.     

Mean time parameters for generalized physiological flow models (semihomogeneous linear systems)

This note gives expressions for recirculation mean time parameters of the disposition kinetics of particles in a semihomogeneous stationary linear system. In such a system each compartment may have an arbitrary single-pass disposition function, rather than a known parametric (usually monoexponential) one. Such systems provide a generalization of physiological flow models. Given observations of arterial blood concentrations and tissue amounts, and making the additional assumptions that (i) the fraction of total blood flow exiting each tissue that goes to each other tissue is constant and known, and (ii) the fraction of drug entering each tissue that is eliminated to the outside is constant and known, the input to each tissue can be known, and therefore both its total blood flow and its single-pass disposition function can be estimated. Recirculation mean time parameters can be computed from these estimates. Application to real thiopental data is presented as an example.     

Mean Residence Times and Distribution Volumes for Drugs Undergoing Linear Reversible Metabolism and Tissue Distribution and Linear or Nonlinear Elimination from the Central Compartments

Equations for the mean residence times in the body (MRT) and in the central compartment (MRTc) are derived for bolus central dosing of a drug and its metabolite which undergo linear tissue distribution and linear reversible metabolism but are eliminated either linearly or nonlinearly (Michaelis–Menten kinetics) from the central compartments. In addition, a new approach to calculate the steady-state volumes of distribution for nonlinear systems (reversible or nonreversible) is proposed based on disposition decomposition analysis. The application of these equations to a dual reversible two-compartment model is illustrated by computer simulations.     

Determination of drug absorption rate in time-variant disposition by direct deconvolution using beta clearance correction and end-constrained non-parametric regression

A novel numerical deconvolution method is presented that enables the estimation of drug absorption rates under time-variant disposition conditions. The method involves two components. (1) A disposition decomposition-recomposition (DDR) enabling exact changes in the unit impulse response (UIR) to be constructed based on centrally based clearance changes iteratively determined. (2) A non-parametric, end-constrained cubic spline (ECS) input response function estimated by cross-validation. The proposed DDR-ECS method compensates for disposition changes between the test and the reference administrations by using a "beta" clearance correction based on DDR analysis. The representation of the input response by the ECS method takes into consideration the complex absorption process and also ensures physiologically realistic approximations of the response. The stability of the new method to noisy data was evaluated by comprehensive simulations that considered different UIRs, various input functions, clearance changes and a novel scaling of the input function that includes the "flip-flop" absorption phenomena. The simulated input response was also analysed by two other methods and all three methods were compared for their relative performances. The DDR-ECS method provides better estimation of the input profile under significant clearance changes but tends to overestimate the input when there were only small changes in the clearance.     

Pulmonary absorption and excretion of compounds in the gas phase: theoretical pharmacokinetic and toxicokinetic analysis

Kinetic equations were derived that describe the plasma concentration of an inhaled compound during and following single or repeated regular and irregular pulmonary exposures. The equations are based on a diffusional type of input function and assume a linear disposition with a biexponential unit-impulse response. The use of linear system analysis avoids the complexity of modeling the disposition processes; yet, the effect of these processes is still accounted for mathematically. The approach, therefore, appears to be more general and rational than approaches based on linear compartmental modeling. The ways in which the kinetic equations can be readily applied in pharmacokinetic or toxicokinetic analyses to obtain valuable parameters that enable kinetic predictions of the cumulation during prolonged exposure are discussed. The toxicokinetic problem of comparing the effect of different work schedules in occupational environments with air contaminants is discussed. Formulas derived from considerations of the blood plasma kinetics are presented for the calculation of an adjustment factor for the adjustment of the contaminant threshold limit value for abnormal work weeks. The use of these formulas appears to be more rational than that of similar formulas that have been proposed.     

Mean time parameters for generalized physiological flow models (semihomogeneous linear systems)

This note gives expressions for recirculation mean time parameters of the disposition kinetics of particles in a semihomogeneous stationary linear system. In such a system each compartment may have an arbitrary single-pass disposition function, rather than a known parametric (usually monoexponential) one. Such systems provide a generalization of physiological flow models. Given observations of arterial blood concentrations and tissue amounts, and making the additional assumptions that (i) the fraction of total blood flow exiting each tissue that goes to each other tissue is constant and known, and (ii) the fraction of drug entering each tissue that is eliminated to the outside is constant and known, the input to each tissue can be known, and therefore both its total blood flow and its single-pass disposition function can be estimated. Recirculation mean time parameters can be computed from these estimates. Application to real thiopental data is presented as an example. Work supported in part by U.S. Department of Health, Education and Welfare grants AG03104, AG04594, GM26691.     

Monoclonal antibody disposition: a simplified PBPK model and its implications for the derivation and interpretation of classical compartment models

The structure, interpretation and parameterization of classical compartment models as well as physiologically-based pharmacokinetic (PBPK) models for monoclonal antibody (mAb) disposition are very diverse, with no apparent consensus. In addition, there is a remarkable discrepancy between the simplicity of experimental plasma and tissue profiles and the complexity of published PBPK models. We present a simplified PBPK model based on an extravasation rate-limited tissue model with elimination potentially occurring from various tissues and plasma. Based on model reduction (lumping), we derive several classical compartment model structures that are consistent with the simplified PBPK model and experimental data. We show that a common interpretation of classical two-compartment models for mAb disposition—identifying the central compartment with the total plasma volume and the peripheral compartment with the interstitial space (or part of it)—is not consistent with current knowledge. Results are illustrated for the monoclonal antibodies 7E3 and T84.66 in mice.