Kinetics of non-equilibrium metabolism-coupled passive transport in biosystems

Expressions for time course of solute concentration in an arbitrary compartment of a biosystem were derived using simplifying assumptions of unidirectional transport and first order metabolism kinetics. The coefficients of the resulting exponential-summation function comprise, in addition to the volumes and the connecting areas of individual compartments, the rate parameters of the processes mentioned. The equations presented were verified using results obtained in drug potency testing.     

Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve

Resolution of kinetic equations and parameter identification are discussed for n-compartment linear catenary models with elimination allowed from any compartment. For a given input, general formulas are derived to describe the tracer amount in any compartment as a function of the model parameters. Conversely, explicit procedures are given to identify the model parameters when the concentration-time curve is known in one arbitrary compartment, the tracer being injected into the same compartment. In this inverse problem, the solution is not unique: the model transfer rate constants can only be localized in a finite set of intervals.     

Stochastic modeling of controlled-drug release

A drug release process by the oral route is random in nature and thus is subject to constant fluctuations. Moreover, individuals have varied tolerances to such fluctuations. The objective of this work is to characterize these fluctuations by a stochastic formalism. The system under consideration, i.e., the gastrointestinal tract consists of four consecutive compartments, i.e., stomach, duodenum, jejunum, and ileum. The master equation of the system as well as the governing equations for the means, variances, and covariances of the random variables, each representing the number of microspheres in an individual compartment, have been derived through the probabilistic population balance. These equations have been numerically solved to predict the total release fraction of drug and its internal fluctuations, and the dynamic statistics (means, variances, and covariances) of the amount of drug in each compartment at any time after administration. The dissolution-intensity functions in the model have been recovered from the available in vitro dissolution data from controlled-release pellets of isosorbide-5-nitrate (IS-5-N) by assuming that the rate of release is of the first order. The residence times and transition-intensity functions of drug in the individual compartments have been estimated from the available data generated by the gamma scintigraphies of IS-5-N pellets labeled by ¹¹¹In. Based on these parameters, the total numbers of dissolved drug microspheres and their fluctuations at any instance have been calculated. The model is in accord with the existing in vivo dissolution data of the same drug independently obtained through plasma analysis. More important, the model predicts that fluctuations in terms of the standard deviations of the numbers of particles in the duodenum, jejunum, and ileum can be of the same orders of magnitude as the corresponding mean numbers when 100 microspheres are simultaneously administered orally; in practice, such fluctuations characterized by these deviations could result in an undesirable release profile. Discussion is given of the potential direct clinical application of the results obtained as well as the plausible indirect application of these results and the model derived to the analyses of chemical and biochemical reactors.     

BrdU—Hoechst flow cytometry: A unique tool for quantitative cell cycle analysis

Unlike other techniques, flow cytometric analysis of BrdU-quenched 33258 Hoechst fluorescence may be used to measure cell activation and the G₁, S, and G₂/M compartment distributions in each of three successive cell cycles after growth stimulation of human peripheral blood lymphocytes. Cell cycle kinetic curves can be constructed from the BrdU—Hoechst flow data which allow the simultaneous assessment of growth fraction, lagtime, compartment exit rate, compartment duration, and compartment arrest. Applications of this new versatile technique include the evaluation of drug and growth factor effets, cell aging, and diagnosis in medicine and immunology.     

Biological half-time and body burden of cadmium in dogs after a long-term oral administration of cadmium

To investigate the kinetic behavior of cadmium, we conducted a long-term oral administration experiment, using beagle dogs. The experimental animals were given a commercial diet and pelleted food containing 1, 3, 10, 50, and 100 mg of cadmium per day in the form of cadmium chloride for 8 yr. A single injection of cadmium (as CdCl2) into dogs was also performed in order to obtain fundamental kinetic information for a dog. The kinetic behavior of cadmium in chronic experiment is described theoretically, using a two-compartment model. The model was selected based on the elimination pattern of cadmium from the blood in the single injection experiment. The parameters of the model were estimated from the acute and chronic experimental data. The theoretical value of the cumulative amount of cadmium excreted in urine agreed with the experimental one. This result suggests that the two-compartment model used in this study is useful to elucidate the kinetic behavior of cadmium after a long-term exposure to cadmium. The terminal biological half-time in the two-compartment model was estimated at about 1 to 2 yr for both male and female dogs given 1, 3, 10, and 50 mg of cadmium, and for the male dog given 100 mg of cadmium, but only 0.3 to 0.5 yr for the female dog given 100 mg of cadmium. The amount of cadmium in the central compartment and tissue compartment increased continuously and then gradually reached a steady state. The amount of tissue compartment was much higher than that of the central compartment for each beagle dog.     

Kinetic Model Development for Accelerated Stability Studies

Accelerated stability coupled with modeling to predict the stability of compounds, blends, and products at long-term storage conditions provides significant benefits in science-based decision-making throughout drug substance and drug product development. The study can often be completed, including data analysis in the space of three working weeks, and the information gathered and learning made in this time period can rival years of traditional analysis. The speed of the studies allows an earlier assessment of risk to quality enabling appropriate risk mitigation strategies to be implemented in a timely manner. The scientific foundation is based upon Arrhenius kinetic equations that can be linear or nonlinear in time, and can be based upon water vapor pressure or liquid water activity (relative humidity). A variety of kinetic models are evaluated, and the best model is chosen based upon both Bayesian information criteria and an automated assessment of kinetic model parameters fitting within acceptable ranges. Confidence intervals are estimated based upon a bootstrapping approach. Moisture vapor transmission rate models are applied on top of the resulting kinetic models in order to simulate different packaging types and the use of desiccant. The kinetic models are integrated with the prediction of packaging humidity over time to create a long-term prediction of impurities and other phenomena. The resulting models have been shown to be useful for not only the prediction of drug product impurities in long-term storage but other physical phenomena as well such as hydrate development and solvate loss.     

Mathematical models of hierarchically structured cell populations under equilibrium with application to the epidermis

Abstract. There are three categories of keratinocytes in the germinative compartment of the epidermis – stem, transit-amplifying and post-mitotic. Their population structure is hierarchical. This means that stem cells differentiate into transit-amplifying cells which, after a few rounds of division, become post-mitotic cells. The cell processes of birth, differentiation, death and migration affect the composition and proliferation rate of the germinative compartment. These phenomena are quantified by various cell kinetic parameters. In this paper we derive equations that relate these parameters for different models of hierarchically structured cell populations in equilibrium. We include in the models asymmetric and symmetric division, variations in cell-cycle times, apoptosis and variation in the number of transit generations. We conclude that variation in cell-cycle times need only be considered if apoptosis is not negligible. If it is negligible, then only average cell-cycle times are needed. Unfortunately, it is impossible to predict the importance of apoptosis from the available experimental data. However, the strength of its effect is determined by the other parameters, especially the fraction of cycling stem cells. We show that variation in the number of transit generations can have a potentially large effect on cell birth rate. We also show that cell birth rate does not directly depend on the mean transit-amplifying cell-cycle time, only on the mean stem cell-cycle time. We argue that 'homogeneous cell population' equations should not be used to study hierarchical cell populations as has been done in the past. Finally we argue that stem cell parameters and transit-amplifying cell parameters should not be lumped together.     

Effects of hematocrit on thixotropic properties of human blood

The rheological properties of whole human blood exhibit thixotropic behavior at low shear rates up to about ten reciprocal seconds (1). The accepted cause of this shear rate-dependent and time-dependent behavior is the progressive breakdown of rouleaux into individual red cells. Huang developed a rheological equation which incorporates the kinetics of rouleau breakdown in his models (2). This five-parameter equation was used successfully to represent the hysteresis loop and the torque-decay curve of whole human blood. Numerical values of these five thixotropic parameters, which characterize the rheological behavior of the blood from apparently healthy human subjects, were established (3). In this communication, we examined the effect of hematocrit on each of the above mentioned parameters. The results show that the following parameters will increase their values with an increase in hematocrit: the yield stress, Newtonian contribution of viscosity, non-Newtonian contribution of viscosity, apparent viscosity and the equilibrium value of the structural parameter which indicates the relative amount of rouleaux in blood. Mathematical equations were developed to give the relationship between parameters and hematocrit. Two other thixotropic parameters, viz. the kinetic rate constant of rouleaux breakdown into individual red cells and the order of the breakdown reaction, were found to be independent of the hematocrit. It is consistent with reaction kinetic theory that the rate constant and the order of reaction are independent of the concentration of reactants.     

Theoretical considerations on potentiation in drug interaction

To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.     

Simple kinetic model for the study of lactate metabolic adaptation to exercise in sportsmen routine evaluation

The interindividual specific lactate metabolic adaptation to exercise has been studied. A simple kinetic model was used which did not require labelled molecules. An one open compartment model with a first-order release rate constant described the release of lactate from the muscle. Six volunteers performed five times the same session: pedaling as long as possible at 400 W and 110 rpm. The lactate concentration was measured along the 60 min of recovery. The theoretical curve corresponding to the model was fitted to the experimental data using a non-linear regression method. The values of the following kinetic parameters were obtained: elimination rate constant (ke), release rate constant (ka), apparent amount released into the compartment divided by the volume of distribution (FQo/Vd) and area under the lactate concentration-time curve (AUC). Two way-ANOVA, Scheffé test and discriminant analysis were used to test the statistical significance of these parameters. No significant intra-individual variations were shown. Significant differences were observed between subjects (ke: P = 0.0020; ka: P less than 0.0001; FQo/Vd: P = 0.0002; AUC: P = 0.0395). A correlation was also found between FQo/Vd and ke (r = 0.72; P less than 0.001). In conclusion, the computed parameters provided by the model are sufficient to discriminate and characterize the metabolic response of each subject after short and intensive exercises.     

Metabolism of apolipoproteins A-I and A-II in human high-density lipoprotein: a mathematical approach for analysis of their specific activity decay curves

The differential rate equations describing the compartmental model of human high-density lipoprotein (HDL) were integrated by means of Laplace transforms and an exponential equation was obtained for each of the three compartments. These equations were used to fit the observed plasma decay data and give estimates for the rate constants of the system by means of a written computer program. Furthermore, these estimates were used to calculate the exponential constants of the integrated equations. Consequently, the amount of label in any of the intravascular, extravascular, and urine compartments can be calculated as a fraction of the original dose of label at any time point. This method was tested using data for the (AI)HDL subclass because it contains only apolipoprotein A-I as the major apolipoprotein and does not contain apolipoprotein A-II. The calculated plasma and urine radioactivity data were compared with the experimentally obtained data from two normolipoproteinemic subjects and found to be in good agreement. The significance of this method is its application to the analysis of the decay data of the individual apolipoproteins of (AI + AII) HDL subclass where the urinary radioactivity data resulting from the individual apolipoprotein breakdown on the native particle cannot be measured experimentally at present. Such data are essential for the detailed calculation of the kinetic parameters of these apolipoproteins.     

Kinetic analysis of cell size and DNA content distributions during tumor cell proliferation: Ehrlich ascites tumor study.

In order to study the growth dynamics of proliferating and non-proliferating cells utilizing discrete-time state equations, the cell cycle was divided into a finite number of age compartments. In analysing tumor growth, the kinetic parameters associated with a retardation in the growth rate of tumors were characterized by computer simulation in which the simulated results of the growth curve, the growth fraction, and the mean generation time were adjusted to fit the experimental data. The cell age distibution during the period of growth was obtained and by a linear transformation of the state transition matrices, was employed to specify the cell size and DNA content distributions. In an application of the model, the time-course behavior of cell cycle parameters of Ehrlich ascites tumor is illustrated, and the parameters important for the transition of cells in the proliferating compartment to the non-proliferating compartment are discussed, particularly in relation to the G1-G0 and G2-G0 transitions of non-cycling cells as revealed by the variation of cell size distribution.     

Formulation Variables Influencing Drug Release from Layered Matrix System Comprising Chitosan and Xanthan Gum

The purpose of this study was to investigate the formulation variables influencing the drug release from the layered tablets containing chitosan and xanthan gum as matrix component. Increasing the amount of lactose could diminish pH sensitive release behavior of these matrix tablets. Effect of formulation variables on drug release from the prepared three-layered matrix tablets was investigated. The amount of drug loading did not affect the drug release which was influenced by the hydrodynamic force and the matrix composition. An increase in stirring rate correspondingly increased the release rate. Moreover, incorporation of soluble diluents in core or barrier could enhance the drug release. Least square fitting the experimental dissolution data to the mathematical expressions (power law, first order, Higuchi’s and zero order) was carried out to study the drug release mechanism. Most dissolution profiles of the prepared three-layered tablets provided a better fit to zero order kinetic than to first order kinetic and Higuchi’s equation.     

Nonsteady-State Three Compartment Tracer Kinetics: I. Theory

A set of differential equations is derived which describes the four unidirectional fluxes of a substance across the boundaries of the central compartment of a serially arranged three compartment system, and the amount of this substance present in the central compartment. An analytic solution is obtained which yields all of these quantities as functions of time. The analysis is associated with a defined set of repetitive experiments from which the necessary data are obtained and during which the two outer compartments must be subject to experimental control. The solution is applicable to both the initial steady state and a transient, time-dependent state created by making a step change in the initial conditions. It describes the fluxes and compartment size without assuming that constant kinetic coefficients relate the fluxes to compartmental quantities but is limited by the requirement that the response of the system be repeatable in time.     

Relationship between loading dose and infusion rate to achieve any fraction of the steady-state level in any compartment model

When a drug is infused at a constant rate K₀, the time necessary for the concentration to reach a satisfying threshold of effectiveness may be too long. To achieve this level faster, it is useful to give simultaneously a dose D, of the same drug by intravenous injection. This paper proposes the calculation, as a function of K₀ and model parameters, of the loading dose D necessary to reach, in a time T, any fraction of the asymptotic value of the amount of drug in a compartment receiving a constant rate infusion, for any n-compartment model. As an example, the expression of D for mammillary and catenary pharmacokinetic models is derived.     

THE DISCRETE–TIME KINETIC MODEL ANALYSIS OF DNA CONTENT DISTRIBUTIONS IN EXPERIMENTAL TUMOUR CELLS

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMF measurements of DNA content distributions and to analyse their relationship to the cell kinetic parameters, namely cell loss rate, cell cycle times and growth fraction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coefficients of variation, of the L1210 ascites tumour during the growth period.     

The discrete-time kinetic model analysis of DNA content distributions in experimental tumour cells.

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMf measurements of DNA content distributions and to analyse their relationship tothe cell kinetic parameters, namely cell loss rate, cell cycle times and grwoth graction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coeffficients of variation, of the L1210 ascites tumour during the growth period.     

Parameter identification in N-compartment mamillary models

We consider a general mamillary model with a central compartment (compartment 1) and n?1 peripheral compartments, each bidirectionally connected to the first. Elimination is allowed from any compartment and effectively occurs from the system. With input introduced into an arbitrary compartment and measurement performed in an arbitrary compartment, explicit equations are given to derive the parameters of the model from the input-output procedure. The calculations include essentially the determination of the roots of a polynomial plus some elementary algebra. If input and measurement are performed in the same compartment, then a set of 2n elementary combinations of the model parameters can be uniquely determined. However, the model parameters themselves can only be localized, each within an interval. These intervals are explicitly calculated and their width discussed.     

KINETIC ANALYSIS OF CELL SIZE AND DNA CONTENT DISTRIBUTIONS DURING TUMOR CELL PROLIFERATION: EHRLICH ASCITES TUMOR STUDY

In order to study the growth dynamics of proliferating and non-proliferating cells utilizing discrete-time state equations, the cell cycle was divided into a finite number of age compartments. In analysing tumor growth, the kinetic parameters associated with a retardation in the growth rate of tumors were characterized by computer simulation in which the simulated results of the growth curve, the growth fraction, and the mean generation time were adjusted to fit the experimental data. The cell age distribution during the period of growth was obtained and by a linear transformation of the state transition matrices, was employed to specify the cell size and DNA content distributions. In an application of the model, the time-course behavior of cell cycle parameters of Ehrlich ascites tumor is illustrated, and the parameters important for the transition of cells in the proliferating compartment to the non-proliferating compartment are discussed, particularly in relation to the G₁-G₀ and G₂-G₀ transitions of non-cycling cells as revealed by the variation of cell size distribution.     

Understanding glucose transport by the bacterial phosphoenolpyruvate:glycose phosphotransferase system on the basis of kinetic measurements in vitro

The kinetic parameters in vitro of the components of the phosphoenolpyruvate:glycose phosphotransferase system (PTS) in enteric bacteria were collected. To address the issue of whether the behavior in vivo of the PTS can be understood in terms of these enzyme kinetics, a detailed kinetic model was constructed. Each overall phosphotransfer reaction was separated into two elementary reactions, the first entailing association of the phosphoryl donor and acceptor into a complex and the second entailing dissociation of the complex into dephosphorylated donor and phosphorylated acceptor. Literature data on the K(m) values and association constants of PTS proteins for their substrates, as well as equilibrium and rate constants for the overall phosphotransfer reactions, were related to the rate constants of the elementary steps in a set of equations; the rate constants could be calculated by solving these equations simultaneously. No kinetic parameters were fitted. As calculated by the model, the kinetic parameter values in vitro could describe experimental results in vivo when varying each of the PTS protein concentrations individually while keeping the other protein concentrations constant. Using the same kinetic constants, but adjusting the protein concentrations in the model to those present in cell-free extracts, the model could reproduce experiments in vitro analyzing the dependence of the flux on the total PTS protein concentration. For modeling conditions in vivo it was crucial that the PTS protein concentrations be implemented at their high in vivo values. The model suggests a new interpretation of results hitherto not understood; in vivo, the major fraction of the PTS proteins may exist as complexes with other PTS proteins or boundary metabolites, whereas in vitro, the fraction of complexed proteins is much smaller.