Mammillary Compartmental Models Research Articles

Distribution Clearance: Significance and Underlying Mechanisms

PurposeEvaluation of distribution kinetics is a neglected aspect of pharmacokinetics. This study examines the utility of the model-independent parameter whole body distribution clearance (CLD) in this respect.MethodsSince mammillary compartmental models are widely used, CLD was calculated in terms of parameters of this model for 15 drugs. The underlying distribution processes were explored by assessment of relationships to pharmacokinetic parameters and covariates.ResultsThe model-independence of the definition of the parameter CLD allowed a comparison of distributional properties of different drugs and provided physiological insight. Significant changes in CLD were observed as a result of drug-drug interactions, transporter polymorphisms and a diseased state.ConclusionTotal distribution clearance CLD is a useful parameter to evaluate distribution kinetics of drugs. Its estimation as an adjunct to the model-independent parameters clearance and steady-state volume of distribution is advocated.

The Performance of an Artificial Neural Network Model in Predicting the Early Distribution Kinetics of Propofol in Morbidly Obese and Lean Subjects.

Induction of anesthesia is a phase characterized by rapid changes in both drug concentration and drug effect. Conventional mammillary compartmental models are limited in their ability to accurately describe the early drug distribution kinetics. Recirculatory models have been used to account for intravascular mixing after drug administration. However, these models themselves may be prone to misspecification. Artificial neural networks offer an advantage in that they are flexible and not limited to a specific structure and, therefore, may be superior in modeling complex nonlinear systems. They have been used successfully in the past to model steady-state or near steady-state kinetics, but never have they been used to model induction-phase kinetics using a high-resolution pharmacokinetic dataset. This study is the first to use an artificial neural network to model early- and late-phase kinetics of a drug. Twenty morbidly obese and 10 lean subjects were each administered propofol for induction of anesthesia at a rate of 100 mg/kg/h based on lean body weight and total body weight for obese and lean subjects, respectively. High-resolution plasma samples were collected during the induction phase of anesthesia, with the last sample taken at 16 hours after propofol administration for a total of 47 samples per subject. Traditional mammillary compartment models, recirculatory models, and a gated recurrent unit neural network were constructed to model the propofol pharmacokinetics. Model performance was compared. A 4-compartment model, a recirculatory model, and a gated recurrent unit neural network were assessed. The final recirculatory model (mean prediction error: 0.348; mean square error: 23.92) and gated recurrent unit neural network that incorporated ensemble learning (mean prediction error: 0.161; mean square error: 20.83) had similar performance. Each of these models overpredicted propofol concentrations during the induction and elimination phases. Both models had superior performance compared to the 4-compartment model (mean prediction error: 0.108; mean square error: 31.61), which suffered from overprediction bias during the first 5 minutes followed by under-prediction bias after 5 minutes. A recirculatory model and gated recurrent unit artificial neural network that incorporated ensemble learning both had similar performance and were both superior to a compartmental model in describing our high-resolution pharmacokinetic data of propofol. The potential of neural networks in pharmacokinetic modeling is encouraging but may be limited by the amount of training data available for these models.

Effect of non‐instantaneous disposition on the general measures of distribution kinetics in a modified mammillary compartmental model

A theoretical investigation on the evaluation of the general measures of distribution kinetics is proposed by taking into account the non‐instantaneous drug disposition in the central compartment after a bolus intravenous injection. The analytical expressions for the drug concentration, the area under curve (AUC), the mean equilibrium time (MEQT), and the clearance (CL) are derived. Comparing the present model with the corresponding instantaneous distribution model (IDM), we found that the latter underestimates the AUC value and overestimates the MEQT and CL values. The following situations lead to a greater discrepancy in the AUC and MEQT values between the present model and the IDM in general: (1) a greater number of compartments; (2) a slower blood‐circulation flow rate; (3) a smaller steady‐state volume of distribution, and (4) a larger intercompartmental clearance. For a large steady‐state volume of distribution, the deviation in the MEQT value from the present model due to the assumption of instantaneous distribution for a large intercompartmental clearance is more serious than that for a small intercompartmental clearance. For a small steady‐state volume of distribution, the reverse is true. The present model can describe the distribution of anesthetics such as isoflurane, enflurane, halothane and methoxyflurane in humans. An increase in mass‐transfer rate constant from central compartment (CC) to peripheral compartment (PC) has the effect of a decrease in AUC of CC, and the reverse is true for mass‐transfer rate constant from PC to CC. Drug concentration in PC (Ci ) increases with time and at a fixed time, C 1>C 2>C 3>C 4. AUC of PC decreases as volume of PC increases.

Interpretation of General Measures of Distribution Kinetics in Terms of a Mammillary Compartmental Model

Interpretation of General Measures of Distribution Kinetics in Terms of a Mammillary Compartmental Model

Hidden oscillations in generalized linear mammillary compartmental models

Oscillations due to complex eigenvalues, known to exist but difficult to detect, are sometimes totally hidden in the output of compartmental models, i.e., none of their modes appear in the output. An example is constructed of a class of linear compartmental models with complex eigenvalues, which have oscillating modes appearing in the output for some single-pool-input/single-pool-output (SpISpO) configurations, while for other such configurations all oscillations are totally hidden in the output. To generate the example, generalized mammillary compartmental models are defined in which a central pool exchanges with peripheral submodels called clusters, through individual connector pools, and their transfer functions are calculated corresponding to all SpISpO configurations. When such a model is repetitive, i.e., when it has identical peripheral clusters, and the input or the output is in the central pool, then it is zero-state equivalent, up to a multiplicative constant, with a reduced model having one peripheral cluster only. We analyze the visibility of an eigenvalue, i.e., whether or not the modes associated with it appear in the output, for repetitive generalized mammillary models. Sufficient conditions are given for such models to have oscillating modes appearing in the impulse response for some input/output configurations, while for other such configurations all oscillations are totally hidden, i.e., none appear in the output. A particularly interesting example is presented of a class of linear models with complex eigenvalues satisfying these conditions. This class has the structure of nonlinear models used to describe the process of protein synthesis and turnover.

An algorithm for the identifiable parameter combinations of the general mammillary compartmental model

Mammillary compartmental models are not uniquely identifiable except in special cases where parameter subsets of rate constants have known values. However, even for the most general n-pool mammillary model, an intrinsic set of parameter combinations is identifiable. An explicit algorithm for computing the uniquely identifiable sums and products of unidentifiable parameters for this class of models is developed for the most practical experimental conditions, input into and output from the central pool. Also, all steady state unidirectional mass fluxes from noncentral pools to the central pool (products of pool masses Q i and fractional transport rates k 1 i ) are shown to be always uniquely identifiable, even when the Q i and k 1 i are not, and the lower bounds are the same and the upper bounds are the same for all mass fluxes to the invironment.

Applications of a recirculatory stochastic pharmacokinetic model: Limitations of compartmental models

General equations for the time integral on [0, infinity) of the venous drug concentration-time function after intravenous and oral drug administration are derived. A physiologically realistic stochastic recirculating model is applied in the derivations. The quotient of the intravenous drug dose and the integral on [0, infinity) of the resulting venous blood drug concentration function is equivalent to a summation of organ clearances only provided that drug elimination does not occur in the pulmonary system, and in general it is not equivalent to "total body clearance." In general, mammillary compartmental models are not isomorphic with recirculating models. A necessary condition for isomorphism is that the pulmonary system be conservative toward the drug. Equations for the pulmonary "first-pass" effect derived via the compartmental analysis are invalid. A valid expression for the pulmonary "first-pass" effect is derived. General equations derived via compartmental analysis for the extent of hepatic metabolism and the hepatic first-pass" effect are shown to be valid. A generally applicable expression for the advantage of close intraarterial drug administration is derived. The limitations of compartmental models for representing drug distribution and elimination are discussed, and the advantages of recirculating models are emphasized.

A mathematical model of cadmium transport in a biological system

The research was undertaken to analyze the uptake and distribution of 109Cd in the mouse following repeated administrations and to develop a mathematical model that would describe the transport of cadmium in an animal organism. The model used was a mammillary compartmental model with the compartments corresponding to selected organs and tissues analyzed for 109Cd content. The nine-compartment model was described by first-order kinetics. From the research it was observed that no administration of cadmium was influenced by previous administrations, and this allowed the use of the same rate constants in the model for each of the multiple dosings. The linear mammillary compartmental model was able to approximate the observed laboratory values in the present investigation. The derived model was also shown to be valuable for predicting the retention of cadmium in other species without adjusting any of the rate constants.

A note on the use of a stochastic mammillary compartmental model as an environmental safety model.

Environmental safety testing typically requires procedures for extrapolating from the relatively high experimental to the very low use doses of potentially harmful substances. In the present paper, a stochastic mammillary compartmental model for environmental safety testing is proposed and extrapolation procedures based on its dose-response relationship are developed. The proposed model is a direct generalization of one of the basic safety models, the one-hit model, in that a harmful reaction is assumed to occur if at any time any of the peripheral compartments attains a specified threshold of particles. Consideration of a closed model yields an upper bound on the probability of attaining a certain threshold level, thus providing a conservative procedure for extrapolating to a low dose, while a lower bound obtained from a related open model provides a useful monitoring device as to the sharpness of the upper, bound. The extrapolation procedure is illustrated with simulated data and approximations for initial values are developed.

A note on the use of a stochastic mammillary compartmental model as an environmental safety model

Environmental safety testing typically requires procedures for extrapolating from the relatively high experimental to the very low use doses of potentially harmful substances. In the present paper, a stochastic mammillary compartmental model for environmental safety testing is proposed and extrapolation procedures based on its dose-response relationship are developed. The proposed model is a direct generalization of one of the basic safety models, the one-hit model, in that a harmful reaction is assumed to occur if at any time any of the peripheral compartments attains a specified threshold of particles. Consideration of a closed model yields an upper bound on the probability of attaining a certain threshold level, thus providing a conservative procedure for extrapolating to a low dose, while a lower bound obtained from a related open model provides a useful monitoring device as to the sharpness of the upper bound. The extrapolation procedure is illustrated with simulated data and approximations for initial values are developed.

On the probability of reaching a threshold in a stochastic mammillary system

The multivariate distribution over time of a particular stochastic mammillary compartmental model is obtained for any point in time. The maximum expectation of the peripheral compartments is then derived and used to determine lower bounds on the probability that the maximum of the peripheral compartments reaches any arbitrary threshold level. A bound on the probability is illustrated by an example and some of its implications are explored.