On the stochastic theory of compartments: Solution forn-compartment systems with irreversible,time-dependent transition probabilities

The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.     

A stochstic model for a two-compartment reversible system

This paper discusses a general stochastic model for a two-compartment reversible system with non-homogeneous Poisson inputs, arbitrary residence times at each of the compartments and time-dependent transition probabilities. The probability distributions of the number of particles in each compartment and in the system are obtained together with the number of particles which depart from the system. In addition, various covariance functions with a time lag are obtained. Some of the above obtained results are deduced for time-independent arrivals, exponential residence times and time-independent transition probabilities. Fluctuations of the particles present in the system are also analysed. Similar analysis is provided for the model into which some particles are initially introduced at the system. Some possible applications are discussed at the end.     

On the distribution of the general irreversiblen-compartmental model having time-dependent transition probabilities

The cumulant generating function and first two moments are derived for the stochastic distribution of units in a general irreversiblen-compartment model with time-dependent transition probabilities. In this model, a unit in the first compartment can transfer to any one of the remainingn−1 compartments and a unit in the second compartment can transfer to any of the remainingn−2 compartments and so on. In addition, a unit can enter or leave the system through any compartment. The work is related to previous research and a numerical example is given.     

An analytical solution for a class of delay-differential equations in pharmacokinetic compartment modeling

The authors obtain the analytic expression for the solution of a differential system with time lags for any n-compartment linear model with a single input, and, by convolution, for all intakes. The theoretical result is applied to the case of one-, two-, and three-compartment models and gives insight into the pharmacokinetics of drug undergoing enterohepatic circulation: the amount of drug in any compartment is expressed for all time. Statistical results, such as the mean residence time of drug, are obtained by the same calculation.     

On the time-dependent reversible stochastic compartmental model—II. A class ofn-compartment systems

This paper discusses the solution of a generaln-compartment system with time dependent transition probabilities utilizing the technique described by Cardenas and Matis (1975) (hereafter abbreviated (CM)). In addition, the cumulant generating function is derived for a special class of reversiblen-compartment systems where the time-dependent intensity coefficients corresponding to the migration and death rates are some multiple of each other. The immigration rates can be any integrable function of time. The moments are also obtained and the solution to the two-compartment system is presented explicitly. The solution is illustrated with a linear and a periodic function which forms have been widely reported in the literature.     

Rate Constants and the Variation of Random Concentration Curves in a 2-Compartment System

In biology and medicine many substances and drugs enter the system not at regular time intervals but rather according to a random process. In the present article a situation is investigated where input enters a 2-compartment system according to a Poisson process. The arising two random concentration curves y(t), one for the central and one for the peripheral compartment are discussed (shot noise). The equations for E[y(t)] and Var [y(t)] are derived. The dependence of E[y(t)] and Var [y(t)] and of the index of dispersion ID[y(t)] on the rate parameters is analysed and discussed in both compartments. The arising calculations were considerably simplified by means of “Mathematica”, a computer program which allows to perform symbolic calculations.     

Stochastic theory of compartments: An open,two compartment,reversible system with independent poisson arrivals

This paper discusses two compartment models with interaction allowed between the compartments. The total number of particles in the system at any time is discussed along with the number to the found in each separate compartment. An interesting result is that the number of particles in each of the two compartments areindependent random variables. Some asymptotic results are also given. The paper is a continuation of some earlier work by the author.     

Numerical simulations of stochastic circulatory models

Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1–22 were investigated. The models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. The reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. The Kolmogorov-Smirnov test revealed that even for 1000 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions.     

Compartmental mean residence time in a mammillary model with an effect compartment after intravenous or oral administration

A general estimation of mean residence time (MRT) in an effect compartment E, associated with a linear mammillary n-compartment model is presented: elimination takes place from the central and the effect compartments. Even though no sample is available from E, the MRT of the drug in this compartment can be estimated after intravenous or oral administation. Furthermore, the effect of MRT is independent of the route of administration. Also, with no new calculation, the method provides the area under the amount-time curve in compartment E.     

Stochastic compartmental modeling and parameter estimation with application to cancer treatment follow-up studies

In this paper we use marginal probabilities to derive expressions for the means, variances and covariances ofm-compartment systems. We also present an efficient algorithm for the estimation of the parameters of the system using time series data when measurements are available fromk of them compartments. An application of the analysis and parameter estimation procedure for a model representing the results of a cancer treatment follow-up study is given. Supported in part by NSF Grant Number DCR74-17282.     

Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation

A complete analysis has been performed of the mean residence times in linear compartmental systems, closed or open, with or without traps and with zero input. This analysis allows the derivation of explicit and simple general symbolic formulae to obtain the mean residence time in any compartment of any linear compartmental system, closed or open, with or without traps, as well as formulae to evaluate the mean residence time in the entire system like the above situations. The formulae are given as functions of the fractional transfer coefficients between the compartments and, in the case of open systems, they also include the excretion coefficients to the environment from the different compartments. The relationship between the formulae derived and the particular connection properties of the compartments is discussed. Finally, some examples have been solved.     

Mean residence time in multicompartmental models with time delays

Calculation of the mean residence time (MRT) of a drug in a stationary compartmental model is classically carried out from several expressions. Nevertheless, one or more time delays between compartments modify the mean residence times. It is the aim of this paper to propose a general method for MRT calculations, in any n-compartmental models which may include time delays. As examples, catenary and mammillary models are considered.     

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.     

Comparison of mathematical models for the maternal age dependence of Down's syndrome rates

Summary The maternal age dependence of Down's syndrome rates was analyzed by two mathematical models, a discontinuous (DS) slope model which fits different exponential equations to different parts of the 20–49 age interval and a CPE model which fits a function that is the sum of a constant and exponential term over this whole 20–49 range. The CPE model had been considered but rejected by Penrose, who preferred models postulating changes with age assuming either a power function X¹⁰, where X is age or a Poisson model in which accumulation of 17 events was the assumed threshold for the occurrence of Down's syndrome. However, subsequent analyses indicated that the two models preferred by Penrose did not fit recent data sets as well as the DS or CPE model. Here we report analyses of broadened power and Poisson models in which n (the postulated number of independent events) can vary. Five data sets are analyzed. For the power models the range of the optimal n is 11 to 13; for the Poisson it is 17 to 25. The DS, Poisson, and power models each give the best fit to one data set; the CPE, to two sets. No particular model is clearly preferable. It appears unlikely that, with a data set from any single available source, a specific etiologic hypothesis for the maternal age dependence of Down's syndrome can be clearly inferred by the use of these or similar regression models.     

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.     

A stochastic compartmental model with continuous infusion

This paper deals with stochasticm-compartmental systems with continuous time-dependent infusions into all compartments and reversible time-independent flows between any two compartments. A methodology for the first two moments of the distribution of the number of units in the different compartments at any point in time is outlined without resorting to the usual techniques of generating functions and inverse Laplace transforms. A possible application to a systems analysis of the kidney transplant system is discussed.     

A Statistical Mechanical Deconvolution of the Differential Scanning Calorimetric Profiles of the Thermal Denaturation of Cyanomethemoglobin

A differential scanning calorimetric study of the thermal unfolding of horse cyanomethemoglobin (as an irreversible protein system) was carried out in phosphate-EDTA buffer (20 mM phosphate, 1 mM EDTA) pH 7.2. The calorimetric rescanning of the protein solution was found to be irreversible and the process unfolded state → final state appears to follow first order kinetic. Assuming the system to be comprised of n reversible states and one irreversible final state, the number of particles participating in the reversible states changes with time because they ultimately transit to the final irreversible denatured state. Hence, we carried out the deconvolution analysis using the grand canonical ensembles instead of just the canonical ensembles. This change was effected by introducing a correction term into the related equations which determines the outlet share of those particles exiting from the reversible states and converting into the final irreversible state. This approach provided an improved interpretation of the experimental data, which supports the following two-step process for the thermal denaturation of cyanomethemoglobin: α₂β₂ → (α + αβ + β)^excited → α^melt + (αβ)^melt + (β^melt.     

The budget of dissolved trace metals in Lake Biwa,Japan

A comprehensive study on the dynamics of dissolved elements (Mg, Al, Si, P, Ca, V, Cr, Mn, Fe, Ni, Zn, As, Sr, Y, W, and U) in Lake Biwa was carried out using a clean technique. Lake water samples (n = 523) were collected from six stations in the North Basin and three stations in the South Basin. River water samples (n = 178) were collected from 14 major rivers flowing into the North Basin. Rainwater samples (n = 89) were collected at Otsu. The river water was enriched with Mn, Al, Fe, P, and Zn and the rainwater was enriched with Zn, Al, Fe, and Mn compared to North Basin water during winter mixing. The residence times of dissolved species were estimated on the basis of input through the rivers and rain. The residence times for Ca, Mg, and Sr were about 8 years, the same as that for water. Mn, Al, Fe, and Zn showed the shortest residence times (0.05–0.19 year). A budget calculation suggested that more than 60% of the input of dissolved Si, P, V, Cr, Mn, Fe, Ni, and Zn was scavenged and retained in the lake sediments and/or discharged as suspended particles.     

Some conditions for sustained oscillations in biochemical chains with feedback inhibition

A chain ofn reactions is considered in which the last substance inhibits the production of the first with degreep, p being the order of the inhibition. Maintained oscillations are possible for certain values of the parameters under the following conditions: (1) If there is no time delay, then there must be at least three compartments (n=3) and either the degree of inhibition is sufficiently large (p>8 forn=3) or there must be enzymatic removal from the first compartment, in which casep≥1. (2) If there are time delays, but there is no enzymatic removal, the degree of the inhibition must be greater than or equal to 2 for any value ofn. (3) If there is a time delay in addition to enzymatic removal, one compartment with simple first order inhibition is sufficient. Conditions on the parameters necessary for maintained oscillations are given for many of the cases discussed.     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.