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.     

Properties of the transfer functions of compartmental models

This paper is concerned with the nonlinear system of algebraic equations relating the positive parameters of a linear time-invariant compartmental model to its transfer function coefficients. The general form that these equations must take is shown, and simple necessary conditions for the existence of positive solutions are given. An immediate use of these conditions is the development of necessary conditions for a polynomial with positive coefficients to have negative roots. A method is then outlined which triangularizes the system and reduces the complete solution problem to one of finding and counting roots of a polynomial. Sufficient conditions for the existence of real and positive solutions are demonstrated.     

A methodology for compartmental model indistinguishability

The problem of constructing all minimal compartmental models that are indistinguishable through input-output knowledge alone from some given model is examined. The main tool in this analysis is a set of geometric properties that can be deduced from input-output knowledge and hence must be equally true in any two indistinguishable models. These properties, together with preservation of the form of the model's transfer function(s), provide an effective means for producing a set of candidate models for indistinguishability.     

A Stochastic Compartmental Model with Long Lasting Infusion

Most of the compartmental models in current use to model pharmacokinetic systems are deterministic. Stochastic formulations of pharmacokinetic compartmental models introduce stochasticity through either a probabilistic transfer mechanism or the randomization of the transfer rate constants. In this paper we consider a linear stochastic differential equation (LSDE) which represents a stochastic version of a one‐compartment linear model when input function undergoes random fluctuations. The solution of the LSDE, its mean value and covariance structure are derived. An explicit likelihood function is obtained either when the process is observed continuously over a period of time or when sampled data are available, as it is generally feasible. We discuss some asymptotic properties of the maximum likelihood estimators for the model parameters. Furthermore we develop expressions for two random variables of interest in pharmacokinetics: the area under the time‐concentration curve, M₀(T), and the plateau concentration, x_ss. Finally the estimation procedure is illustrated by an application to real data.     

Indistinguishability and identifiability analysis of linear compartmental models.

Two compartmental model structures are said to be indistinguishable if they have the same input-output properties. In cases in which available a priori information is not sufficient to specify a unique compartmental model structure, indistinguishable model structures may have to be generated and their attributes examined for relevance. An algorithm is developed that, for a given compartmental model, investigates the complete set of models with the same number of compartments and the same input-output structure as the original model, applies geometrical rules necessary for indistinguishable models, and test models meeting the geometrical criteria for equality of transfer functions. Identifiability is also checked in the algorithm. The software consists of three programs. Program 1 determines the number of locally identifiable parameters. Program 2 applies several geometrical rules that eliminate many (generally most) of the candidate models. Program 3 checks the equality between system transfer functions of the original model and models being tested. Ranks of Jacobian matrices and submatrices and other criteria are used to check patterns of moment invariants and local identifiability. Structural controllability and structural observability are checked throughout the programs. The approach was successfully used to corroborate results from examples investigated by others.     

Use of transfer function and compartmental analysis to quantify 11C-labelled photoassimilate export from wheat leaves

Compartmental analysis of tracer loss from a leaf after pulse-labellingwith carbon isotopes has often been used to infer the flow ofphotosynthate through the leaf. Recently, a more general approachhas been suggested based upon estimation of the transfer functionusing data from pulse-labelling as well as continuouslabellingexperiments. A comparison of these two approaches shows thatwith the same data set they give equivalent physiological interpretations.The measured decline of ¹¹C activity from a wheat leaf after¹¹CO₂ pulse-labelling was extrapolated by compartmental as wellas transfer function analysis. Both methods estimated a 66.4%loss of the initially fixed ¹¹C due to export and respiration.The advantage of transfer function analysis, however, is itsapplicability to continuous-labelling experiments. The modelallows the use of the net photosynthetic rate as the reference(100%) value. Data from continuous-labelling experiments withwheat plants indicate diurnal variations in the export of freshlylabelled assimilate of between 32.7% and 43.6% of net photosynthesis. Key words: Triticum aestivum L, ¹¹CO₂, carbon partitioning, transfer function analysis, compartmental analysis     

Compartmental models with multiple sources of stochastic variability: The one-compartment,time invariant hazard rate case

Previous stochastic compartmental models have introduced the primary source of stochasticity through either a probabilistic transfer mechanism or a random rate coefficient. This paper combines these primary sources into a unified stochastic compartmental model. Twelve different stochastic models are produced by combining various sources of stochasticity and the mean value and the covariance for each of the twelve models is derived. The covariance of each model has a different form whereby the individual sources of stochasticity are identificable from data. The various stochastic models are illustrated for certain specified distributions of the rate coefficient and of the initial count. Several properties of the models are derived and discussed. Among these is the fact that the expected count of a model with a random rate coefficient will always exceed the expected count of a model with a fixed coefficient evaluated at the mean rate. A general modeling strategy for the onecompartment, time invariant hazard rate is also proposed.     

Qualitative theory of compartmental systems with lags

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.     

Compartmental systems and generating functions

Compartmental systems can be represented by direct graphs in which each node corresponds to a generating function and each arm to a transfer generating function. A homomorphism is established between a compartmental system and this representation, in analogy with that obtained through the use of the Laplace transformation. From the values obtained experimentally in a given compartment, through the solution of a difference equation, the generating function for the corresponding node can be calculated and the graph of the system can be built up within the degrees of freedom of the model. From the graph it is possible to calculate the transfer generating function between any two connected nodes, the mean permanence time in a given node, the mean transit time between two nodes, and their precursor-successor order.     

Density functions of residence times for deterministic and stochastic compartmental systems

A significant consideration in modeling systems with stages is to obtain models for the individual stages that have probability density functions (pdfs) of residence times that are close to those of the real system. Consequently, the theory of residence time distributions is important for modeling. Here I show first that linear deterministic compartmental systems with constant coefficients and their corresponding stochastic analogs (stochastic compartmental systems with linear rate laws) have the same pdfs of residence times for the same initial distributions of inputs. Furthermore, these are independent of inflows. Then I show that does not hold for non-linear deterministic systems and their stochastic analogs (stochastic compartmental systems with non-linear rate laws). In fact, for given initial distributions of inputs, the pdfs of non-linear determistic systems without inflows and of their stochastic analogs, are functions of the initial amounts injected. For systems with inflows, the pdfs change as the inflows influence the occupancies of the compartments of the system; they are state-dependent pdfs.     

On the identifiability of linear compartmental systems: a revisited transfer function approach based on topological properties

This work deals with the problem of the a priori identifiability of compartmental systems from input-output experiments. A new approach is presented, in which, having associated a directed graph with the matrix to be identified, a set of “forms” is defined which are functions of the elements of matrix itself. It is shown how, by exploiting the topological properties of the graph and its subgraphs, the problem can be simplified into one of smaller dimensions. Examples are provided to illustrate this new approach.     

Multi-compartmental modelling and experimental design for glucose transport studies in insulin-resistant rats

Many pathologies are associated with abnormalities of glucose metabolism or with perturbations of its transport (type 2 diabetes or insulin-resistance). The pre-diabetic state is characterised by a state of insulin-resistance, in others words a defect of glucose transport in insulin-sensible tissues, such as muscles and adipose tissues. The mathematical modelling of experimental data can be an excellent method to explore the mechanisms implied in the studied biological phenomenon. Thus, starting from a symbolic formulation like the compartmental modelling, it can be possible to develop a theoretical basis for the observation and to consider the best-adapted experiments for the study. We showed with mathematical models that [123I]-6-deoxy-6-iodo-D-glucose (6-DIG), shown as a tracer of glucose transport in vitro, could point out this transport abnormality. To quantify the insulin resistance, we estimated the fractional transfer coefficients of 6-DIG from the blood to the organs. We realised many studies to lead to a satisfying model; special attention has been paid to the precision of the parameter to select the best model. The results showed that by associating experimental data obtained with 6-DIG activities and an adapted mathematical model, discriminating parameters (in and out fractional transfer coefficients) between the two groups (control and insulin-resistant rats) could be pointed out.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Cupredoxins--a study of how proteins may evolve to use metals for bioenergetic processes

Cupredoxins are small proteins that contain type I copper centers, which are ubiquitous in nature. They function as electron transfer shuttles between proteins. This review of the structure and properties of native cupredoxins, and those modified by site-directed mutagenesis, illustrates how these proteins may have evolved to specifically bind copper, develop recognition sites for specific redox partners, tune redox potential for a particular function, and allow for efficient electron transfer through the protein matrix. This is relevant to the general understanding of the roles of metals in energy metabolism, respiration and photosynthesis.     

On stochastic compartmental modeling

This communication contains a proof of the fact that the coefficient of variation of the contents of a compartment of a stochastic compartmental model with deterministic rate parameters is small for large populations. We can therefore conclude that the use of stochastic compartmental models is not of great consequence in the case of systems involving large populations when only the randomness of the transfer mechanism is considered.     

Petri Net and Probabilistic Model Checking Based Approach for the Modelling,Simulation and Verification of Internet Worm Propagation

Internet worms are analogous to biological viruses since they can infect a host and have the ability to propagate through a chosen medium. To prevent the spread of a worm or to grasp how to regulate a prevailing worm, compartmental models are commonly used as a means to examine and understand the patterns and mechanisms of a worm spread. However, one of the greatest challenge is to produce methods to verify and validate the behavioural properties of a compartmental model. This is why in this study we suggest a framework based on Petri Nets and Model Checking through which we can meticulously examine and validate these models. We investigate Susceptible-Exposed-Infectious-Recovered (SEIR) model and propose a new model Susceptible-Exposed-Infectious-Recovered-Delayed-Quarantined (Susceptible/Recovered) (SEIDQR(S/I)) along with hybrid quarantine strategy, which is then constructed and analysed using Stochastic Petri Nets and Continuous Time Markov Chain. The analysis shows that the hybrid quarantine strategy is extremely effective in reducing the risk of propagating the worm. Through Model Checking, we gained insight into the functionality of compartmental models. Model Checking results validate simulation ones well, which fully support the proposed framework.     

Peak drug levels in linear pharmacokinetic systems—III. Multimodal impulse responses in multicompartment systems

This paper deals with the unimodality of the ‘impulse response’ in compartmental systems, where the ‘impulse response’ in any given compartment is the time course of the amount of diffusing substance in that compartment after an initial instantaneous injection of the substance into that or some other compartment. It is shown that in certain compartmental structures, with injection in certain compartments, the impulse response is always unimodal or monotonic in all compartments, regardless of the numerical values of the various transfer rate coefficients. Structures with this property are here named ‘UM structures’, and they include the familiar mammillary and catenary structures. In this paper, the set of all UM structures is described. Structures which are not UM (NUM structures) are identified by showing that, by removal of certain connections and compartments according to certain rules, they can be reduced to small structures which can be shown to be NUM by numerical computation. Computations on two systems with bimodal impulse responses show that with constant infusions of a fixed amount of substance the peak level may increase paradoxically with decrease in the infusion rate over a certain range. This effect is extremely small, however. Supported in part by U.S. Public Health Service Research Grant GM 21269 from the National Institute of General Medical Sciences, and in part by Biomedical Research Support Grant S07 RR 05392 from the National Institutes of Health.     

Compartmental-model response function for dendritic trees

The membrane-potential response function of a dendritic tree is constructed using a compartmental model. It is shown that if certain conditions are imposed on the compartments at branching nodes and terminals of an arbitrary tree, then the response function has a simple matrix solution that may be evaluated explicitly. Moreover, it reduces to the corresponding path-integral expression of cable theory in the continuum limit. The response function in the presence of synaptic inputs with shunting is also constructed. The compartmental approach presented here provides a particularly simple method for exploring the effects of complex geometries on spatio-temporal pattern processing in neurons.