Structural and practical identifiability analysis in bioengineering: a beginner's guide.

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

Experimental and mathematical analysis of in vitro Pitavastatin hepatic uptake across species.

1. To investigate the non-linear kinetics of in vitro hepatocyte uptake across species, the OATP substrate Pitavastatin was used as a probe. 2. Experiments were conducted at AstraZeneca (Alderley Park, Macclesfield) using freshly isolated rat, dog and human hepatocytes, utilising the "oil spin" methodology described by Hassen et al. (1996). Very few mechanistic models have previously been used to characterise the uptake process. 3. Here two candidate pharmacokinetic non-linear compartmental models are proposed. Both models have been shown to be structurally identifiable and distinghishable previously, which establishes that all unknown parameters could be identified from the experimental observations available and that input/output relationships for both the candidate models were structurally different. 4. A kinetic modelling software package, FACSIMILE (MCPA Software, Faringdon, UK), was used to obtain numerical solutions for the system equations and for parameter estimation. Model fits gave good agreement with the in vitro data and suggest the current widely accepted assumption that the rate of diffusion across the hepatocyte cell membrane is the same at both 4 °C and 37 °C is not valid, at least for Pitavastatin. Although this finding has already been proposed, this is the first time it is comprehensively debunked using statistical testing.

Compartmental modelling of the pharmacokinetics of a breast cancer resistance protein.

A mathematical model for the pharmacokinetics of Hoechst 33342 following administration into a culture medium containing a population of transfected cells (HEK293 hBCRP) with a potent breast cancer resistance protein inhibitor, Fumitremorgin C (FTC), present is described. FTC is reported to almost completely annul resistance mediated by BCRP in vitro. This non-linear compartmental model has seven macroscopic sub-units, with 14 rate parameters. It describes the relationship between the concentration of Hoechst 33342 and FTC, initially spiked in the medium, and the observed change in fluorescence due to Hoechst 33342 binding to DNA. Structural identifiability analysis has been performed using two methods, one based on the similarity transformation/exhaustive modelling approach and the other based on the differential algebra approach. The analyses demonstrated that all models derived are uniquely identifiable for the experiments/observations available. A kinetic modelling software package, namely FACSIMILE (MPCA Software, UK), was used for parameter fitting and to obtain numerical solutions for the system equations. Model fits gave very good agreement with in vitro data provided by AstraZeneca across a variety of experimental scenarios.

Modelling the dynamics of intramammary E. coli infections in dairy cows: understanding mechanisms that distinguish transient from persistent infections.

The majority of intramammary infections with Escherichia coli in dairy cows result in transient infections with duration of about 10 days or less, although more persistent infections (2 months or longer) have been identified. We apply a mathematical model to explore the role of an intracellular mammary epithelial cell reservoir in the dynamics of infection. We included biological knowledge of the bovine immune response and known characteristics of the bacterial population in both transient and persistent infections. The results indicate that varying the survival duration of the intracellular reservoir reproduces the data for both transient and persistent infections. Survival in an intracellular reservoir is the most likely mechanism that ensures persistence of E. coli infections in mammary glands. Knowledge of the pathogenesis of persistent infections is essential to develop preventive and treatment programmes for these important infections in dairy cows.

Efficient removal of immunoglobulin free light chains by hemodialysis for multiple myeloma: in vitro and in vivo studies.

Of patients with newly diagnosed multiple myeloma, approximately 10% have dialysis-dependent acute renal failure, with cast nephropathy, caused by monoclonal free light chains (FLC). Of these, 80 to 90% require long-term renal replacement therapy. Early treatment by plasma exchange reduces serum FLC concentrations, but randomized, controlled trials have shown no evidence of renal recovery. This outcome can be explained by the low efficiency of the procedure. A model of FLC production, distribution, and metabolism in patients with myeloma indicated that plasma exchange might remove only 25% of the total amount during a 3-wk period. For increasing FLC removal, extended hemodialysis with a protein-leaking dialyzer was used. In vitro studies indicated that the Gambro HCO 1100 dialyzer was the most efficient of seven tested. Model calculations suggested that it might remove 90% of FLC during 3 wk. This dialyzer then was evaluated in eight patients with myeloma and renal failure. Serum FLC reduced by 35 to 70% within 2 hr, but reduction rates slowed as extravascular re-equilibration occurred. FLC concentrations rebounded on successive days unless chemotherapy was effective. Five additional patients with acute renal failure that was caused by cast nephropathy then were treated aggressively, and three became dialysis independent. A total of 1.7 kg of FLC was removed from one patient during 6 wk. Extended hemodialysis with the Gambro HCO 1100 dialyzer allowed continuous, safe removal of FLC in large amounts. Proof of clinical value now will require larger studies.