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.     

真菌深层培养过程的房室结构神经网络模型

在对横纹黑蛋巢菌深层培养过程进行分析的基础上,提出一种房室结构的神经网络模型,利用ＲＢＦ网络这各房室的输入,输出关系,并进一步对整个生化过程作了建模型研究,计算结果表明,所建模型性能较佳,对真菌培养过程的观测数据拟合结果令人满意。     

Compartmental culture of embryonic stem cell-derived neurons in microfluidic devices for use in axonal biology

Axonal pathology has been clearly implicated in neurodegenerative diseases making the compartmental culture of neurons a useful research tool. Primary neurons have already been cultured in compartmental microfluidic devices but their derivation from an animal is a time-consuming and difficult work and has a limit in their sources. Embryonic stem cell (ESC)-derived neurons (ESC_Ns) overcome this limit, since ESCs can be renewed without limit and can be differentiated into ESC_Ns by robust and reproducible protocols. In this research, ESC_Ns were derived from mouse ESCs in compartmental microfluidic devices, and their axons were isolated from the somal cell bodies. Once embryoid bodies (EBs) were localized in the microfluidic culture chamber, ESC_Ns spread out from the EBs and occupied the cell culture chamber. Their axons traversed the microchannels and finally were isolated from the somata, providing an arrangement comparable to dissociated primary neurons. This ESC_N compartmental microfluidic culture system not only offers a substitute for the primary neuron counterpart system but also makes it possible to make comparisons between the two systems.     

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.     

The backward bifurcation in compartmental models for West Nile virus

In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcation. In this paper, we first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by Cruz-Pacheco et al. to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcation.     

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.     

Equilibrium points for nonlinear compartmental models.

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.     

Computational Algorithm for Least Squares Estimation of Parameters in Compartmental Analysis

This paper presents a new computational algorithm for the estimation of the parameters in multi-compartmental processes. By assuming the initial state of compartmental process known, the Newton-Raphson iterative process is vised to obtain weighted least squares estimates. The estimation procedure considers broad class of structural properties (defined in the sequel) of compartmental models, such as open-closed processes, cyclic pathways, conservative processes, connectivity etc. A computer program, which includes these structural properties is supplied. Two examples of simulated processes illustrate the estimation procedure.     

Metabolic isotopomer labeling systems. Part I: global dynamic behavior

In the last few years metabolic flux analysis (MFA) using carbon labeling experiments (CLE) has become a major diagnostic tool in metabolic engineering. The mathematical centerpiece of MFA is the solution of isotopomer labeling systems (ILS). An ILS is a high-dimensional nonlinear differential equation system that describes the distribution of isotopomers over a metabolic network during a carbon labeling experiment. This contribution presents a global analysis of the dynamic behavior of general ILSs. It is proven that an ILS is globally stable under very weak conditions that are always satisfied in practice. In particular it is shown that in some sense ILSs are a nonlinear extension to the classical theory of compartmental systems. The central stability condition for compartmental systems, i.e., the non-existence of traps in linear compartmental networks, is also the major stability condition for ILSs. As an important side result of the proof, it is shown that ILSs can be transformed to a cascade of linear systems with time-dependent inhomogeneous terms. This cascade structure has considerable consequences for the development of efficient numerical algorithms for the solution of ILSs and thus for MFA.     

Structural identifiability of compartmental systems with respect to data obtained from constant-infusion tracer experiments

The problem of structural identifiability of compartmental systems receiving constant input rates of tracer material is studied, and the relationship between this steady-state problem and that of identification using the impulse response is sought. Input connectability of the compartmental system allows exogenous inputs to produce arbitrary steady-state values anywhere in state space, resulting in sufficient conditions for the structural identifiability of the system when direct measurements can be made for every compartment. Because of the steady-state nature of the problem, the systems concept of output connectability is shown to play no role in this identification scheme. The importance of constant-infusion tracer experiments is demonstrated for a compartment model describing volatile fatty acid production and conversion in ruminants.     

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.     

Chemical mutagens: Dosimetry,Haber's rule and linear systems

We present an overall scheme for chemical mutagen risk assessment, which leads naturally to consideration of Haber's rule, a classical concept of toxicology. A rationale is given for considering compartmental models based on mammalian anatomy and physiology as the most reasonable and practical conceptual framework for risk assessment. Haber's rule is extended to the area of chemical dosimetry, defined in terms of our compartmental models. Then it is proved that Haber's rule holds for any system of linear ordinary differential equations with constant coefficients which is physically realizable. Finally, we comment on non-linearities and on the Blum-Druckey model for time-to-occurrence of tumors.     

On the identification by filtering techniques of a biologicaln-compartment model in which the transport rate parameters are assumed to be stochastic processes

In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.     

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.     

Kinetic compartmental analysis of carnitine metabolism in the dog

This study was undertaken to quantitate the dynamic parameters of carnitine metabolism in the dog. Six mongrel dogs were given intravenous injections of L-[methyl-3H]carnitine and the specific radioactivity of carnitine was followed in plasma and urine for 19-28 days. The data were analyzed by kinetic compartmental analysis. A three-compartment, open-system model [(a) extracellular fluid, (b) cardiac and skeletal muscle, (c) other tissues, particularly liver and kidney] was adopted and kinetic parameters (carnitine flux, pool sizes, kinetic constants) were derived. In four of six dogs the size of the muscle carnitine pool obtained by kinetic compartmental analysis agreed (+/- 5%) with estimates based on measurement of carnitine concentrations in different muscles. In three of six dogs carnitine excretion rates derived from kinetic compartmental analysis agreed (+/- 9%) with experimentally measured values, but in three dogs the rates by kinetic compartmental analysis were significantly higher than the corresponding rates measured directly. Appropriate chromatographic analyses revealed no radioactive metabolites in muscle or urine of any of the dogs. Turnover times for carnitine were (mean +/- SEM): 0.44 +/- 0.05 h for extracellular fluid, 232 +/- 22 h for muscle, and 7.9 +/- 1.1 h for other tissues. The estimated flux of carnitine in muscle was 210 pmol/min/g of tissue. Whole-body turnover time for carnitine was 62.9 +/- 5.6 days (mean +/- SEM). Estimated carnitine biosynthesis ranged from 2.9 to 28 mumol/kg body wt/day. Results of this study indicate that kinetic compartmental analysis may be applicable to study of human carnitine metabolism.     

On some topological properties of a strongly connected compartmental system with application to the identifiability problem

In this paper some structural properties of a strongly connected compartmental system are illustrated. In particular a suitable set of “cycles” and “paths” associated to the compartmental graph is constructed, such that an application exists between the parameter space and the space of sycles and paths, whose suitable restriction is a bijection. It is shown that this set contains the minimum number of functions necessary to uniquely identify the parametrization vector, and its relevance in identifiability analysis is illustrated.     

Mathematical modelling of immunity to malaria

A comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity boosted by exposure to infection generate qualitatively different results. Attempts to control disease by reducing transmission or increasing the recovery rate can produce an increase in prevalence in the compartmental model with discrete epidemiological states. However, the parasite density always decreases in response to disease control in the model with continuous epidemiological variables. Each model accounts for some epidemiological patterns. The increase in prevalence seen in the compartmental model is in accord with observed effects of variation in transmission. Parasite suppression in areas of antimalarial drug use is consistent with the effect of an increased recovery rate in the density model. Future work should combine the two approaches, perhaps by using the compartmental model over the low to moderate range of infection rates and switching to the density model at high infection rates. In any case, the validation of models needs to take account of the usage of antimalarial drugs as well as the intensity of transmission.     

Membrane voltage changes in a compartmental chain model of a neurone

A mathematical model of the neurone has been developed using the method of subdivision of the neurone into a number of equivalent circuit compartments. Compartmental characteristics have been investigated by calculating the shape indices of the output produced in response to a given somatic input conductance change. A generalised form of compartmental chain has been chosen to allow calculation of the shape indices produced by a variety of geometrical configurations including the straight and tapering chain forms. Equations have been deduced from the computations made on a CDC 6600 computer relating the peak amplitude of the output response to the compartmental diameter for both the straight and tapering chain forms. The effect of variation in the location of the input conductance injection site has also been related to the peak amplitude of the somatic response. The optimum characteristics of the input conductance pulse shape have been computed initially using a rectangular pulse and later the more physiologically relevant double exponential shape. The effect of alteration in the end compartmental terminal impedances over the range from open to short circuit conditions was also calculated. The establishment of optimum single compartmental chain criteria allows the future investigation of multiple chain and pyramidal cell configurations.     

Structural properties of compartmental models

This article treats structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system. Newly formulated conditions are presented for certain of these parameters to be zero or positive. Also results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem. Answers to some questions raised in the recent literature about coefficient dependence are discussed.