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.

Modelling the Double Peak Phenomenon in pharmacokinetics.

Two methods of modelling the Double Peak Phenomenon in pharmacokinetics are described; both are based on compartmental models. The first method assumes that the absorption of the drug from the gut to the systemic plasma varies with the location of the drug in the gut, with negligible absorption through the jejunum. It has the advantage of clear physiological interpretation, but there are a comparatively large number of parameters to be estimated. The second method assumes simultaneous input via two parallel pathways, and has been developed with the aim of reducing the number of parameters in the model. However, this approach lacks the direct relationship to physiology. The two methods are used to model two data sets provided by AstraZeneca and a further data set from the literature, describing the pharmacokinetics of veralipride. For all three data sets, the measurement is of concentration of drug in the systemic plasma following oral administration in solution form.

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.

The tumor kinetics of multiple myeloma following autologous stem cell transplantation as assessed by measuring serum-free light chains.

In multiple myeloma, changes in serum-free immunoglobulin light chains (FLC) are a more rapid indicator of treatment response than intact immunoglobulin due to their shorter serum half-life. The present study analysed the changes in serum FLC after autologous peripheral blood stem cell transplantation (PBSCT) in 19 patients. The majority of myeloma patients (18 of 19) undergoing PBSCT had a rapid fall in FLC concentrations. In all 11 of 19 patients with raised tumor FLC, it fell within 48 h following high-dose melphalan. In patients with monoclonal intact immunoglobulin, the tumor FLC fell quicker (median half-life 4.3 days) than the monoclonal intact immunoglobulin (median half-life 14 days). FLC recovery occurred after (13 of 19) or around the time of neutrophil engraftment (6 of 19). With a median follow up of 220 days post-transplant, 16 of 19 patients have a normal FLC ratio and 3 of 19 have an elevated tumor FLC/abnormal ratio. FLC assays provided a sensitive monitor of changes in tumor and non-tumor plasma cells after PBSCT. This assay is potentially valuable as a marker of chemosensitivity, as an indicator of residual tumor and indicated time to lymphocyte engraftment. Further follow-up is required to ascertain whether differences in the kinetics of FLC responses have any prognostic clinical utility.

The structural identifiability of the susceptible infected recovered model with seasonal forcing.

In this paper, it is shown that the SIR epidemic model, with the force of infection subject to seasonal variation, and a proportion of either the prevalence or the incidence measured, is unidentifiable unless certain key system parameters are known, or measurable. This means that an uncountable number of different parameter vectors can, theoretically, give rise to the same idealised output data. Any subsequent parameter estimation from real data must be viewed with little confidence as a result. The approach adopted for the structural identifiability analysis utilises the existence of an infinitely differentiable transformation that connects the state trajectories corresponding to parameter vectors that give rise to identical output data. When this approach proves computationally intractable, it is possible to use the converse idea that the existence of a coordinate transformation between states for particular parameter vectors implies indistinguishability between these vectors from the corresponding model outputs.

A mathematical model for the in vitro kinetics of the anti-cancer agent topotecan.

In this paper a compartmental modelling approach is applied to provide a mathematical description of the activity of the anti-cancer agent topotecan, and delivery to its nuclear DNA target following administration. The activity of topotecan in defined buffers is first modelled using a linear two compartment model that then forms the basis of a cell based model for drug activity in live cell experiments. An identifiability analysis is performed before parameter estimation to ensure that the model output (i.e., continuous, perfect and noise-free data) uniquely determines the parameters. Parameter estimation is performed using experimental data which offers concentrations of active and inactive forms of topotecan from high performance liquid chromatography methods.

Structural identifiability for a class of non-linear compartmental systems using linear/non-linear splitting and symbolic computation.

Under certain controllability and observability restrictions, two different parameterisations for a non-linear compartmental model can only have the same input-output behaviour if they differ by a locally diffeomorphic change of basis for the state space. With further restrictions, it is possible to gain valuable information with respect to identifiability via a linear analysis. Examples are presented where non-linear identifiability analyses are substantially simplified by means of an initial linear analysis. For complex models, with four or more compartments, this linear analysis can prove lengthy to perform by hand and so symbolic computation has been employed to aid this procedure.