A methodology for compartmental model indistinguishability

DISTING: A web application for fast algorithmic computation of alternative indistinguishable linear compartmental models

BACKGROUND AND OBJECTIVES

We describe and illustrate use of DISTING, a novel web application for computing alternative structurally identifiable linear compartmental models that are input-output indistinguishable from a postulated linear compartmental model. Several computer packages are available for analysing the structural identifiability of such models, but DISTING is the first to be made available for assessing indistinguishability.

METHODS

The computational algorithms embedded in DISTING are based on advanced versions of established geometric and algebraic properties of linear compartmental models, embedded in a user-friendly graphic model user interface. Novel computational tools greatly speed up the overall procedure. These include algorithms for Jacobian matrix reduction, submatrix rank reduction, and parallelization of candidate rank computations in symbolic matrix analysis.

RESULTS

The application of DISTING to three postulated models with respectively two, three and four compartments is given. The 2-compartment example is used to illustrate the indistinguishability problem; the original (unidentifiable) model is found to have two structurally identifiable models that are indistinguishable from it. The 3-compartment example has three structurally identifiable indistinguishable models. It is found from DISTING that the four-compartment example has five structurally identifiable models indistinguishable from the original postulated model. This example shows that care is needed when dealing with models that have two or more compartments which are neither perturbed nor observed, because the numbering of these compartments may be arbitrary.

CONCLUSIONS

DISTING is universally and freely available via the Internet. It is easy to use and circumvents tedious and complicated algebraic analysis previously done by hand.

An algorithm for identifiable parameters and parameter bounds for a class of cascaded mammillary models

A complex structural identifiability problem for a class of unidirectionally interconnected n-compartment linear mammillary models with multiple inputs is discussed. This class is particularly useful in the study of drug/metabolite kinetics and other interconversion kinetic processes. An explicit algorithm is developed for this model class that provides identifiable parameter combinations, parameter bounds, steady-state pool sizes, and production rates, with input forcing and output measurements in central compartments. A six-compartment model of the combined dynamics of the prohormone thyroxine (T4) and hormone triiodothyronine (T3) illustrates how physiological parameter values or their smallest ranges, such as tissue T4 to T3 conversion rates and separate T4 and T3 production rates, can be determined from stimulus-response measurements in plasma alone.

Structural identifiability and indistinguishability of certain two-compartment models incorporating nonlinear efflux from the peripheral compartment

A two-compartment model is considered where both compartments are observed and where the transfer efflux from the peripheral compartment may take three different nonlinear forms. The structural identifiability of the set of unknown parameters of each possible model is examined using the similarity transformation approach. Using the recent extension of this approach the indistinguishability of pairs of the postulated systems is also considered.

MAMCAT: an expert system for distinguishing between mammillary and catenary compartmental models

MAMCAT is a user-friendly, interactive, graphics-based PC program for addressing a common compartmental model discrimination problem in the framework of model indistinguishability theory: can mammillary and catenary models of the same order be distinguished from each other? The software is designed to teach the theory and solve specific problems. The user is guided step-by-step through definitions, examples, theory and problem solution. Topological properties are used first to screen out some distinguishable models and thereby reduce the number of candidates for indistinguishability. Transfer function analysis is then used to explore the remaining candidates. Analytical results for up to 5-compartment models are given and a more general algorithm is induced from these results.

Indistinguishability for a class of nonlinear compartmental models

Indistinguishability, as applied to nonlinear compartmental models, is analyzed by means of the local state isomorphism theorem. The method of analysis involves the determination of all local, diffeomorphic transformations connecting the state variables of two models. This is then applied to two two-compartment models, in the first instance with linear eliminations, and then with the addition of eliminations with Michaelis-Menten kinetics. In the nonlinear example, the state transformation turns out to be linear or possibly affine. It is found that the nonlinear analysis could be eased by splitting the state isomorphism equations into those of the initial linear models together with extra equations due to the nonlinearities.

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.

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.