Compartmental models with Erlang distributed residence times and random rate coefficients

In this paper a stochastic model for a two-compartment system which incorporates Erlang residence time distributions (i.e. the residence times have the gamma distribution where the shape parameters assume integer values only) into each compartment is generalized to include random rate coefficients. Analytical forms of the model are derived for the case where the rate coefficients have gamma densities. A relationship is established between the new models and existing models that are in current practical usage.     

Moments of physiological transit time distributions and the time course of drug disposition in the body

Characterizing the tissue distribution kinetics of drugs by physiological and physico-chemical parameters and using a circulatory model the time course of blood concentration after intravenous injection is predicted for linear pharmacokinetic systems. The interrelationships between the first three (zero to second) moments of the distribution functions of organ transfer times, circulation times and residence times of drug molecules in the body are described. Utilizing literature data the model is applied to the analysis of lidocain kinetics in humans.     

Analysis of growth kinetics by division tracking

Cell division tracking using fluorescent dyes, such as carboxyfluorescein diacetate succinimidyl ester, provides a unique opportunity for analysis of cell growth kinetics. The present review article presents new methods for enhancing resolution of division tracking data as well as derivation of quantities that characterize growth from time-series data. These include the average time between successive divisions, the proportion of cells that survive and the proliferation per division. The physical significance of these measured quantities is interpreted by formulation of a two-compartment model of cell cycle transit characterized by stochastic and deterministic cell residence times, respectively. The model confirmed that survival is directly related to the proportion of cells that enter the next cell generation. The proportion of time that cells reside in the stochastic compartment is directly related to the proliferation per generation. This form of analysis provides a starting point for more sophisticated physical and biochemical models of cell cycle regulation.     

On the stochastic theory of compartments: III. General time-dependent reversible systems

General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Semiclassical approximations of stochastic epidemiological processes towards parameter estimation using as prime example the SIS system with import

In this paper we investigate several schemes to approximate the stationary distribution of the stochastic SIS system with import. We begin by presenting the model and analytically computing its stationary distribution. We then approximate this distribution using Kramers–Moyal approximation, van Kampen's system size expansion, and a semiclassical scheme, also called WKB or eikonal approximation depending on its different applications in physics. For the semiclassical scheme, done in the context of the Hamilton–Jacobi formalism, two approaches are taken. In the first approach we assume a semiclassical ansatz for the generating function, while in the second the solution of the master equation is approximated directly. The different schemes are compared and the semiclassical approximation, which performs better, is then used to analyse the time dependent solution of stochastic systems for which no analytical expression is known. Stochastic epidemiological models are studied in order to investigate how far such semiclassical approximations can be used for parameter estimation.     

Bayesian flux balance analysis applied to a skeletal muscle metabolic model

In this article, the steady state condition for the multi-compartment models for cellular metabolism is considered. The problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. The problem has been addressed previously by a number of authors, and optimization-based approaches as well as extreme pathway analysis have been proposed. These approaches are briefly discussed here. The main emphasis of this work is a Bayesian statistical approach to the flux balance analysis (FBA). We show how the bound constraints and optimality conditions such as maximizing the oxidative phosphorylation flux can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. We propose an effective Markov chain Monte Carlo (MCMC) scheme to explore the posterior densities, and compare the results with those obtained via the previously studied linear programming (LP) approach. The proposed methodology, which is applied here to a two-compartment model for skeletal muscle metabolism, can be extended to more complex models.     

On dependency properties of the ISIs generated by a two-compartmental neuronal model

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.     

Quantifying the hydraulic performance of treatment wetlands using the moment index

A new hydraulic index was derived according to residence time distribution theory. The approach quantifies hydraulic inefficiencies according to the juxtaposition of the hold back parameter relative to the residence time distribution. The index was evaluated for its ability to detect variation, for conformity with qualitative assessments, and for correlation to effluent pollutant fractions in order to assess its suitability as a predictor of treatment.The moment index overcomes many of the weaknesses inherent in existing indices. The index can be computed from a dataset considering just one volume exchange so arbitrary truncation of data due to the finite nature of data collection has no impact on the moment index. The moment index appears to be more sensitive than existing indices in detecting attenuation of a residence time distribution as well. The new index demonstrated excellent correlation to the effluent pollutant fraction predicted by a first-order reduction implying the index could be the good predictor of treatment. In addition to correlation with treatment, the moment index matched qualitative assessment precisely for eight specific cases considered.The moment index could substantially aid in the design and management of treatment wetlands for balancing cost and efficacy by resolving some of the uncertainty associated with residence time. The index could be used to help identify the optimal wetland configuration for maximizing residence time. Not only would it be useful in quantifying the effects of vegetation, bathymetry, and wetland shape on residence time; it could have utility in supplying the bounds for pollutant reduction.     

Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

Spinal motor neurons have voltage gated ion channels localized in their dendrites that generate plateau potentials. The physical separation of ion channels for spiking from plateau generating channels can result in nonlinear bistable firing patterns. The physical separation and geometry of the dendrites results in asymmetric coupling between dendrites and soma that has not been addressed in reduced models of nonlinear phenomena in motor neurons. We measured voltage attenuation properties of six anatomically reconstructed and type-identified cat spinal motor neurons to characterize asymmetric coupling between the dendrites and soma. We showed that the voltage attenuation at any distance from the soma was direction-dependent and could be described as a function of the input resistance at the soma. An analytical solution for the lumped cable parameters in a two-compartment model was derived based on this finding. This is the first two-compartment modeling approach that directly derived lumped cable parameters from the geometrical and passive electrical properties of anatomically reconstructed neurons.     

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.     

A CSTR-hollow fiber system for continuous hydrolysis of proteins. Performance and kinetics

The effect of four operating variables (enzyme concentration, substrate concentration, flow rate, and reaction volume) on the performance of CSTR-hollow fiber membrane reactor was studied for the continuous hydrolysis of a soy protein isolate using Pronase. Based on a residence time distribution study, the reactor system was modeled as an ideal CSTR in combination with the Michaelis-Menten equation of enzyme kinetics. This kinetic model correlated conversion with a space-time parameter modified to include all four independent variables. An empirical model based on curvilinear regression analysis was also developed. Both models predicted conversion fairly well, although the kinetic model slightly underpredicts at high conversion.     

Modelling molecular interaction pathways using a two-stage identification algorithm

In systems biology, molecular interactions are typically modelled using white-box methods, usually based on mass action kinetics. Unfortunately, problems with dimensionality can arise when the number of molecular species in the system is very large, which makes the system modelling and behavior simulation extremely difficult or computationally too expensive. As an alternative, this paper investigates the identification of two molecular interaction pathways using a black-box approach. This type of method creates a simple linear-in-the-parameters model using regression of data, where the output of the model at any time is a function of previous system states of interest. One of the main objectives in building black-box models is to produce an optimal sparse nonlinear one to effectively represent the system behavior. In this paper, it is achieved by applying an efficient iterative approach, where the terms in the regression model are selected and refined using a forward and backward subset selection algorithm. The method is applied to model identification for the MAPK signal transduction pathway and the Brusselator using noisy data of different sizes. Simulation results confirm the efficacy of the black-box modelling method which offers an alternative to the computationally expensive conventional approach.     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

Simple Epidemic Models with Positive Latent Period

This paper deals with two types of simple epidemic models, namely, deterministic and stochastic wherein the latent period is assumed to be positive. In the deterministic epidemic model, the distributions of susceptibles, inactive infectives, active infectives and that of epidemic curve which gives the rate at which new infections take place have been obtained. The expression for the expected time of the entire epidemic has been derived. Also the partial differential equation for the moment generating function of the proportion of susceptibles in the population is established. In the end, we have studied a stochastic approach of the system.     

On the stochastic theory of compartments: The leaving process of the two-compartment systems

In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.     

Analysis and modeling of suspended solids from high-frequency monitoring in a stormwater treatment wetland

Tools for modeling pulsed flows and constituent fluxes in wetlands, although well developed in theory, have not been well tested in practice. High-frequency monitoring of suspended solids and flows in a stormwater treatment wetland enabled application and analysis of these tools. A dynamic flow- and volume-weighted time variable, analogous to the retention time in steady-flow systems, is one important tool tested in this study. Cross-correlations with time lags demonstrated that the dynamic time variable was a better predictive variable of pulsed events than was the standard, static time variable. Although plug-flow models are typically used for steady-flow wetlands, residence time distribution (RTD) models are indispensable for describing pulsed flows and constituent fluxes in wetlands. This study demonstrated that RTD modeling with reaction kinetics of suspended solids during storm events produces a better explanation of outflow data than possible with steady, plug-flow models. Using only input and output data, an RTD model explained sedimentation rates with less unexplained variance than the standard, plug-flow model. The results of this study underscore the importance and utility of RTD modeling for complex flows.     

The effect of temporal variability on persistence conditions in rivers

There has been great interest in the invasion and persistence of algal and insect populations in rivers. Recent modeling approaches assume that the flow speed of the river is constant. In reality, however, flow speeds in rivers change significantly on various temporal scales due to seasonality, weather conditions, or many human activities such as hydroelectric dams. In this paper, we study persistence conditions by deriving the upstream invasion speed in simple reaction-advection-diffusion equations with coefficients chosen to be periodic step functions. The key methodological idea to determine the spreading speed is to use the exponential transform in order to obtain a moment generating function. In a temporally periodic environment, the averages of each coefficient function determine the minimal upstream and downstream propagation speeds for a single-compartment model. For a two-compartment model, the temporal variation can enhance population persistence.