Counterexamples of a sufficient condition for local identifiability in a linear compartment system

The problem of the structural identifiability of biological compartment systems, i.e., the question of the possibility of estimating all their unknown parameters by an input-output experiment, is very important. Delforge studied a structural condition for local identifiability in a linear compartment system. Unfortunately, he made a mistake in the principal result of his paper: the proposed condition is not sufficient, but only necessary. This paper shows some counterexamples to the condition.     

Contributions to the treatment of tracer distribution equations in multicompartmental systems

Summary A tracer kinetic method is outlined for the treatment of multicompartmental non steady-state systems. Metabolic rates are considered as consisting of two components: one of them (exchange) is balanced by a reverse process of the same rate, and the other (transport) results in concentration change. Assuming the transport rates to be known, the method allows of excluding a lot of logically possible compartmental models as inconsistent with the experimental data. Thus the real compartment structure of the system can be approached.     

Development of a compartmental model of human zinc metabolism: identifiability and multiple studies analyses

A compartmental model of zinc metabolism has been developed from stable isotope tracer studies of five healthy adults. Multiple isotope tracers were administered orally and intravenously, and the resulting enrichment was measured in plasma, erythrocytes, urine, and feces for as long as 3 wk. Data from total zinc measurements and model-independent calculations of various steady-state parameters were also modeled with the kinetic data. A structure comprised of 14 compartments and as many as 25 unknown kinetic parameters was developed to adequately model the data from each of the individual studies. The structural identifiability of the model was established using the GLOBI2 identifiability analysis software. Numerical identifiability of parameter estimates was evaluated using statistical data provided by SAAM. A majority of the model parameters was estimated with sufficient statistical certainty to be considered well determined. After the fitting of the model and data from the individual studies using SAAM/CONSAM, results were submitted to SAAM extended multiple studies analysis for aggregation into a single set of population parameters and statistics. The model was judged to be valid based on criteria described elsewhere.     

Analysis of unique structural identifiability via submodels

The paper presents a sufficient and necessary condition for unique structural identifiability of linear compartmental models. By virtue of this result unique identifiability can be tested via the analysis of some submodels of the original model. Thus, the identifiability problem is reduced step by step to simpler and, finally, to rather trivial problems. In addition to the knowledge of the symbolic expression of the transfer-function matrix, the proposed method of full-rank submodels requires only some numerical rank determinations, and hence allows for a quick and interactive test for unique structural identifiability. The procedure also gives a lower bound on the number of different solutions.     

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.     

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.     

The relationship of intercompartmental excitation transfer rate constants to those of an underlying physical model

In studies on photosynthetic systems it is common practice to interpret the results of time-resolved fluorescence experiments on the basis of compartmental, or target, models. Each compartment represents a group of molecules with similar fluorescence characteristics. In cases of practical interest, the members of each compartment are spatially contiguous and make up part of an overall energy-transferring system. Since a rate constant describing the overall transfer between compartments is not that of any pair of molecules in the system, this question naturally rises: what do we learn about the microscopic structure from these data? In this note we introduce ‘compartment melting’, a smooth mathematical connection between the compartmental and microscopic levels. We then show, on the basis of model calculations on finite lattices in one, two, and three dimensions, that average microscopic rates at the interfaces between compartments may be estimated from observed intercompartmental rates. The estimate involves a modest number of structural assumptions about the system. As examples of the method, which is applicable mainly to systems containing homogeneous pigment pools, some recent chlorophyll-protein antenna studies are analyzed.     

Kinetics of glucose transport and sequestration in lactating bovine mammary glands measured in vivo with a paired indicator/nutrient dilution technique.

To quantify kinetics of mammary glucose utilization in vivo, 24 paired glucose and extracellular indicator (p-aminohippuric acid) dilution curves across intact bovine mammary glands were obtained after bolus injections into the external iliac artery. Dilution curves were analyzed using a compartmental capillary, convolution integration model. Four candidate submodels of glucose transport and metabolism in capillary supply zones were fit to the glucose dilution curves and evaluated. Model I, with one extracellular compartment for glucose and first-order unidirectional uptake, failed, indicating that efflux of glucose from the intracellular space could not be ignored. Model II, with first-order exchanges between extracellular and intracellular compartments and sequestration from the latter, was overdefined because unidirectional clearance of glucose was at least five times the blood flow rate and 20 times the net clearance rate. Model III, combining extracellular and intracellular space into one compartment, was superior in its goodness-of-fit to curves and identifiability of parameters. Michaelis-Menten parameters of sequestration were not identifiable. Parameters of the optimal compartmental capillary, convolution integration model were applicable to both the dynamics of injected glucose dilution and the steady-state background arteriovenous difference of glucose. Glucose sequestration followed first-order kinetics between 0 and 7 mM extracellular glucose with an average rate constant of 0.006 s(-1) or a clearance of 44 ml/s. The ratio of intracellular to extracellular glucose distribution space was 0.34, which is considerably lower than the expected intracellular volume and suggests an intracellular occlusion compartment with which extracellular glucose rapidly exchanges.     

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.     

Decomposition-based qualitative experiment design algorithms for a class of compartmental models.

Qualitative experiment design, to determine experimental input/output configurations that provide identifiability for specific parameters of interest, can be extremely difficult if the number of unknown parameters and the number of compartments are relatively large. However, the problem can be considerably simplified if the parameters can be divided into several groups for separate identification and the model can be decomposed into smaller submodels for separate experiment design. Model decomposition-based experiment design algorithms are proposed for a practical class of large-scale compartmental models representative of biosystems characterized by multiple input sources and unidirectional interconnectivity among subsystems. The model parameters are divided into three types, each of which is identified consecutively, in three stages, using simpler submodel experiment designs. Several practical examples are presented. Necessary and sufficient conditions for identifiability using the algorithm are also discussed.     

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.     

Steady state analysis of influx and transbilayer distribution of ergosterol in the yeast plasma membrane

Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.     

A class of general stochastic compartmental systems

In this paper a general class of semi-Markov compartmental systems is studied. Two models for different input processes are analysed. Attention has been paid to the recurrence times associated with each compartment and to the distribution of the number of particles in each compartment. As an example, a three-compartment system is discussed to study the movement between three health states of patients with chronic diseases.     

Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.     

Analysis of tracer experiments in non-conservative steady-state systems

A method is developed for finding the transfer and localization rates and the volumes ofN compartment steady-state biological systems from experimental results. It is shown that a complete solution for certain systems in which the rates and volumes remain constant and in which there is access to all compartments can be obtained by using a single radioactive tracer. The information obtainable from experiments wherein some compartments are not accessible is analyzed for mammillary and catenary systems. Conservative systems are handled as special cases in which the localization is zero while anisotropic membranes separating compartments are shown to introduce no additional mathematical difficulty whenever all compartments are accessible. The limitations on the use of this method of multi-compartment tracer analysis are briefly discussed. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

A priori analysis of metabolic flux identifiability from (13)C-labeling data

The (13)C-labeling technique was introduced in the field of metabolic engineering as a tool for determining fluxes that could not be found using the 'classical' method of flux balancing. An a priori flux identifiability analysis is required in order to determine whether a (13)C-labeling experiment allows the identification of all the fluxes. In this article, we propose a method for identifiability analysis that is based on the recently introduced 'cumomer' concept. The method improves upon previous identifiability methods in that it provides a way of systematically reducing the metabolic network on the basis of structural elements that constitute a network and to use the implicit function theorem to analytically determine whether the fluxes in the reduced network are theoretically identifiable for various types of real measurement data. Application of the method to a realistic flux identification problem shows both the potential of the method in yielding new, interesting conclusions regarding the identifiability and its practical limitations that are caused by the fact that symbolic calculations grow fast with the dimension of the studied system.     

Modeling and identification of endocrine-metabolic systems. theoretical aspects and their importance in practice

Some guidelines for modeling, identification, and validation in endocrinology and metabolism are presented. Fundamentals based on recent theoretical developments are given, and their essential role for solving physiological and clinical problems is shown. Concepts and issues discussed include a priori and a posteriori identifiability; optimal experiment design; assessment of model validity; simple and complex models; compartmental and noncompartmental models. Recent studies on the glucose-insulin system and on ketone body metabolism are used to illustrate the methodological line of thought discussed in the paper.     

An integral equation approach to measuring turnover in nonsteady compartmental and distributed systems

Physiological systems are often modelled by a set of compartments. Alternatively they can be described by the diffusion-convection-reaction equations governing distributed systems. The problem considered here is that of identifying a continuously changing input of some metabolite )tracee), endogenous to the system and hence inaccessible, when a nonlinear or time-varying component is also introduced into the loss parameter, as for example through feedback mechanisms. A tracer is used to determine the steady-state impulse response under time-invariant, linear conditions. A known input of tracer is also administered when the system is driven out of steady state. The integral equations developed utilize the predetermined impulse response, the measured concentrations of both tracer and tracee (output) in some region of the system to estimate the changing loss parameter and the unknown input in a continuous fashion.     

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.