In vitro PK/PD modeling of tyrosine kinase inhibitors in non-small cell lung cancer cell lines

Tyrosine kinase inhibitors (TKIs) are routinely prescribed for the treatment of non-small cell lung cancer (NSCLC). As with all medications, patients can experience adverse events due to TKIs. Unfortunately, the relationship between many TKIs and the occurrence of certain adverse events remains unclear. There are limited in vivo studies which focus on TKIs and their effects on different regulation pathways. Many in vitro studies, however, that investigate the effects of TKIs observe additional changes, such as changes in gene activations or protein expressions. These studies could potentially help to gain greater understanding of the mechanisms for TKI induced adverse events. However, in order to utilize these pathways in a pharmacokinetic/pharmacodynamic (PK/PD) framework, an in vitro PK/PD model needs to be developed, in order to characterize the effects of TKIs in NSCLC cell lines. Through the use of ordinary differential equations, cell viability data and nonlinear mixed effects modeling, an in vitro TKI PK/PD model was developed with estimated PK and PD parameter values for the TKIs alectinib, crizotinib, erlotinib, and gefitinib. The relative standard errors for the population parameters are all less than 25%. The inclusion of random effects enabled the model to predict individual parameter values which provided a closer fit to the observed response. It is hoped that this model can be extended to include in vitro data of certain pathways that may potentially be linked with adverse events and provide a better understanding of TKI-induced adverse events.

Parameter estimation of a model describing the human fingers

The goal of this paper is twofold: firstly, to provide a novel mathematical model that describes the kinematic chain of motion of the human fingers based on Lagrangian mechanics with four degrees of freedom and secondly, to estimate the model parameters using data from able-bodied individuals. In the literature there are a variety of mathematical models that have been developed to describe the motion of the human finger. These models offer little to no information on the underlying mechanisms or corresponding equations of motion. Furthermore, these models do not provide information as to how they scale with different anthropometries. The data used here is generated using an experimental procedure that considers the free response motion of each finger segment with data captured via a motion capture system. The angular data collected are then filtered and fitted to a linear second-order differential approximation of the equations of motion. The results of the study show that the free response motion of the segments is underdamped across flexion/extension and ad/abduction.

The ingredients for an antimicrobial mathematical modelling broth

Mathematical modelling has made significant contributions to the optimisation of the use of antimicrobial treatments. In this review we discuss the key processes that such mathematical modelling should attempt to capture. In particular, we highlight that the response of the host immune system requires quantification and illustrate this with a novel model structure.

Prediction of Clinical Transporter-Mediated Drug-Drug Interactions via Comeasurement of Pitavastatin and Eltrombopag in Human Hepatocyte Models

A structurally identifiable micro‐rate constant mechanistic model was used to describe the interaction between pitavastatin and eltrombopag, with improved goodness‐of‐fit values through comeasurement of pitavastatin and eltrombopag. Transporter association and dissociation rate constants and passive rates out of the cell were similar between pitavastatin and eltrombopag. Translocation into the cell through transporter‐mediated uptake was six times greater for pitavastatin, leading to pronounced inhibition of pitavastatin uptake by eltrombopag. The passive rate into the cell was 91 times smaller for pitavastatin compared with eltrombopag. A semimechanistic physiologically‐based pharmacokinetic (PBPK) model was developed to evaluate the potential for clinical drug–drug interactions (DDIs). The PBPK model predicted a twofold increase in the pitavastatin peak blood concentration and area under the concentration‐time curve in the presence of eltrombopag in simulated healthy volunteers. The use of structural identifiability supporting experimental design combined with robust micro‐rate constant parameter estimates and a semimechanistic PBPK model gave more informed predictions of transporter‐mediated DDIs.

A mechanistic modelling approach for the determination of the mechanisms of inhibition by cyclosporine on the uptake and metabolism of atorvastatin in rat hepatocytes using a high throughput uptake method

Determine the inhibition mechanism through which cyclosporine inhibits the uptake and metabolism of atorvastatin in fresh rat hepatocytes using mechanistic models applied to data generated using a high throughput oil spin method.Atorvastatin was incubated in fresh rat hepatocytes (0.05-150 nmol/ml) with or without 20 min pre-incubation with 10 nmol/ml cyclosporine and sampled over 0.25-60 min using a high throughput oil spin method. Micro-rate constant and macro-rate constant mechanistic models were ranked based on goodness of fit values.The best fitting model to the data was a micro-rate constant mechanistic model including non-competitive inhibition of uptake and competitive inhibition of metabolism by cyclosporine (Model 2). The association rate constant for atorvastatin was 150-fold greater than the dissociation rate constant and 10-fold greater than the translocation into the cell. The association and dissociation rate constants for cyclosporine were 7-fold smaller and 10-fold greater, respectively, than atorvastatin. The simulated atorvastatin-transporter-cyclosporine complex derived using the micro-rate constant parameter estimates increased in line with the incubation concentration of atorvastatin.The increased amount of data generated with the high throughput oil spin method, combined with a micro-rate constant mechanistic model helps to explain the inhibition of uptake by cyclosporine following pre-incubation.

Parameter Identification for a Model of Neonatal Fc Receptor-Mediated Recycling of Endogenous Immunoglobulin G in Humans

Salvage of endogenous immunoglobulin G (IgG) by the neonatal Fc receptor (FcRn) is implicated in many clinical areas, including therapeutic monoclonal antibody kinetics, patient monitoring in IgG multiple myeloma, and antibody-mediated transplant rejection. There is a clear clinical need for a fully parameterized model of FcRn-mediated recycling of endogenous IgG to allow for predictive modeling, with the potential for optimizing therapeutic regimens for better patient outcomes. In this paper we study a mechanism-based model incorporating nonlinear FcRn-IgG binding kinetics. The aim of this study is to determine whether parameter values can be estimated using the limited in vivo human data, available in the literature, from studies of the kinetics of radiolabeled IgG in humans. We derive mathematical descriptions of the experimental observations-timecourse data and fractional catabolic rate (FCR) data-based on the underlying physiological model. Structural identifiability analyses are performed to determine which, if any, of the parameters are unique with respect to the observations. Structurally identifiable parameters are then estimated from the data. It is found that parameter values estimated from timecourse data are not robust, suggesting that the model complexity is not supported by the available data. Based upon the structural identifiability analyses, a new expression for the FCR is derived. This expression is fitted to the FCR data to estimate unknown parameter values. Using these parameter estimates, the plasma IgG response is simulated under clinical conditions. Finally a suggestion is made for a reduced-order model based upon the newly derived expression for the FCR. The reduced-order model is used to predict the plasma IgG response, which is compared with the original four-compartment model, showing good agreement. This paper shows how techniques for compartmental model analysis-structural identifiability analysis, linearization, and reparameterization-can be used to ensure robust parameter identification.

Three novel approaches to structural identifiability analysis in mixed-effects models

BACKGROUND AND OBJECTIVE

Structural identifiability is a concept that considers whether the structure of a model together with a set of input-output relations uniquely determines the model parameters. In the mathematical modelling of biological systems, structural identifiability is an important concept since biological interpretations are typically made from the parameter estimates. For a system defined by ordinary differential equations, several methods have been developed to analyse whether the model is structurally identifiable or otherwise. Another well-used modelling framework, which is particularly useful when the experimental data are sparsely sampled and the population variance is of interest, is mixed-effects modelling. However, established identifiability analysis techniques for ordinary differential equations are not directly applicable to such models.

METHODS

In this paper, we present and apply three different methods that can be used to study structural identifiability in mixed-effects models. The first method, called the repeated measurement approach, is based on applying a set of previously established statistical theorems. The second method, called the augmented system approach, is based on augmenting the mixed-effects model to an extended state-space form. The third method, called the Laplace transform mixed-effects extension, is based on considering the moment invariants of the systems transfer function as functions of random variables.

RESULTS

To illustrate, compare and contrast the application of the three methods, they are applied to a set of mixed-effects models.

CONCLUSIONS

Three structural identifiability analysis methods applicable to mixed-effects models have been presented in this paper. As method development of structural identifiability techniques for mixed-effects models has been given very little attention, despite mixed-effects models being widely used, the methods presented in this paper provides a way of handling structural identifiability in mixed-effects models previously not possible.

Best Practices to Maximize the Use and Reuse of Quantitative and Systems Pharmacology Models: Recommendations From the United Kingdom Quantitative and Systems Pharmacology Network

The lack of standardization in the way that quantitative and systems pharmacology (QSP) models are developed, tested, and documented hinders their reproducibility, reusability, and expansion or reduction to alternative contexts. This in turn undermines the potential impact of QSP in academic, industrial, and regulatory frameworks. This article presents a minimum set of recommendations from the UK Quantitative and Systems Pharmacology Network (UK QSP Network) to guide QSP practitioners seeking to maximize their impact, and stakeholders considering the use of QSP models in their environment.

Learning pharmacokinetic models for in vivo glucocorticoid activation

To understand trends in individual responses to medication, one can take a purely data-driven machine learning approach, or alternatively apply pharmacokinetics combined with mixed-effects statistical modelling. To take advantage of the predictive power of machine learning and the explanatory power of pharmacokinetics, we propose a latent variable mixture model for learning clusters of pharmacokinetic models demonstrated on a clinical data set investigating 11β-hydroxysteroid dehydrogenase enzymes (11β-HSD) activity in healthy adults. The proposed strategy automatically constructs different population models that are not based on prior knowledge or experimental design, but result naturally as mixture component models of the global latent variable mixture model. We study the parameter of the underlying multi-compartment ordinary differential equation model via identifiability analysis on the observable measurements, which reveals the model is structurally locally identifiable. Further approximation with a perturbation technique enables efficient training of the proposed probabilistic latent variable mixture clustering technique using Estimation Maximization. The training on the clinical data results in 4 clusters reflecting the prednisone conversion rate over a period of 4 h based on venous blood samples taken at 20-min intervals. The learned clusters differ in prednisone absorption as well as prednisone/prednisolone conversion. In the discussion section we include a detailed investigation of the relationship of the pharmacokinetic parameters of the trained cluster models for possible or plausible physiological explanation and correlations analysis using additional phenotypic participant measurements.

Predicting QRS and PR interval prolongations in humans using nonclinical data

BACKGROUND AND PURPOSE

Risk of cardiac conduction slowing (QRS/PR prolongations) is assessed prior to clinical trials using in vitro and in vivo studies. Understanding the quantitative translation of these studies to the clinical situation enables improved risk assessment in the nonclinical phase.

EXPERIMENTAL APPROACH

Four compounds that prolong QRS and/or PR (AZD1305, flecainide, quinidine and verapamil) were characterized using in vitro (sodium/calcium channels), in vivo (guinea pigs/dogs) and clinical data. Concentration-matched translational relationships were developed based on in vitro and in vivo modelling, and the in vitro to clinical translation of AZD1305 was quantified using an in vitro model.

KEY RESULTS

Meaningful (10%) human QRS/PR effects correlated with low levels of in vitro Na_v 1.5 block (3-7%) and Ca_v 1.2 binding (13-21%) for all compounds. The in vitro model developed using AZD1305 successfully predicted QRS/PR effects for the remaining drugs. Meaningful QRS/PR changes in humans correlated with small effects in guinea pigs and dogs (QRS 2.3-4.6% and PR 2.3-10%), suggesting that worst-case human effects can be predicted by assuming four times greater effects at the same concentration from dog/guinea pig data.

CONCLUSION AND IMPLICATIONS

Small changes in vitro and in vivo consistently translated to meaningful PR/QRS changes in humans across compounds. Assuming broad applicability of these approaches to assess cardiovascular safety risk for non-arrhythmic drugs, this study provides a means of predicting human QRS/PR effects of new drugs from effects observed in nonclinical studies.

Input Estimation for Extended-Release Formulations Exemplified with Exenatide

Estimating the in vivo absorption profile of a drug is essential when developing extended-release medications. Such estimates can be obtained by measuring plasma concentrations over time and inferring the absorption from a model of the drug’s pharmacokinetics. Of particular interest is to predict the bioavailability—the fraction of the drug that is absorbed and enters the systemic circulation. This paper presents a framework for addressing this class of estimation problems and gives advice on the choice of method. In parametric methods, a model is constructed for the absorption process, which can be difficult when the absorption has a complicated profile. Here, we place emphasis on non-parametric methods that avoid making strong assumptions about the absorption. A modern estimation method that can address very general input-estimation problems has previously been presented. In this method, the absorption profile is modeled as a stochastic process, which is estimated using Markov chain Monte Carlo techniques. The applicability of this method for extended-release formulation development is evaluated by analyzing a dataset of Bydureon, an injectable extended-release suspension formulation of exenatide, a GLP-1 receptor agonist for treating diabetes. This drug is known to have non-linear pharmacokinetics. Its plasma concentration profile exhibits multiple peaks, something that can make parametric modeling challenging, but poses no major difficulties for non-parametric methods. The method is also validated on synthetic data, exploring the effects of sampling and noise on the accuracy of the estimates.

Personalized Medication Response Prediction for Attention-Deficit Hyperactivity Disorder: Learning in the Model Space vs. Learning in the Data Space

Attention-Deficit Hyperactive Disorder (ADHD) is one of the most common mental health disorders amongst school-aged children with an estimated prevalence of 5% in the global population (American Psychiatric Association, 2013). Stimulants, particularly methylphenidate (MPH), are the first-line option in the treatment of ADHD (Reeves and Schweitzer, 2004; Dopheide and Pliszka, 2009) and are prescribed to an increasing number of children and adolescents in the US and the UK every year (Safer et al., 1996; McCarthy et al., 2009), though recent studies suggest that this is tailing off, e.g., Holden et al. (2013). Around 70% of children demonstrate a clinically significant treatment response to stimulant medication (Spencer et al., 1996; Schachter et al., 2001; Swanson et al., 2001; Barbaresi et al., 2006). However, it is unclear which patient characteristics may moderate treatment effectiveness. As such, most existing research has focused on investigating univariate or multivariate correlations between a set of patient characteristics and the treatment outcome, with respect to dosage of one or several types of medication. The results of such studies are often contradictory and inconclusive due to a combination of small sample sizes, low-quality data, or a lack of available information on covariates. In this paper, feature extraction techniques such as latent trait analysis were applied to reduce the dimension of on a large dataset of patient characteristics, including the responses to symptom-based questionnaires, developmental health factors, demographic variables such as age and gender, and socioeconomic factors such as parental income. We introduce a Bayesian modeling approach in a "learning in the model space" framework that combines existing knowledge in the literature on factors that may potentially affect treatment response, with constraints imposed by a treatment response model. The model is personalized such that the variability among subjects is accounted for by a set of subject-specific parameters. For remission classification, this approach compares favorably with conventional methods such as support vector machines and mixed effect models on a range of performance measures. For instance, the proposed approach achieved an area under receiver operator characteristic curve of 82-84%, compared to 75-77% obtained from conventional regression or machine learning ("learning in the data space") methods.

Analysis of a Compartmental Model of Endogenous Immunoglobulin G Metabolism with Application to Multiple Myeloma

Immunoglobulin G (IgG) metabolism has received much attention in the literature for two reasons: (i) IgG homeostasis is regulated by the neonatal Fc receptor (FcRn), by a pH-dependent and saturable recycling process, which presents an interesting biological system; (ii) the IgG-FcRn interaction may be exploitable as a means for extending the plasma half-life of therapeutic monoclonal antibodies, which are primarily IgG-based. A less-studied problem is the importance of endogenous IgG metabolism in IgG multiple myeloma. In multiple myeloma, quantification of serum monoclonal immunoglobulin plays an important role in diagnosis, monitoring and response assessment. In order to investigate the dynamics of IgG in this setting, a mathematical model characterizing the metabolism of endogenous IgG in humans is required. A number of authors have proposed a two-compartment nonlinear model of IgG metabolism in which saturable recycling is described using Michaelis–Menten kinetics; however it may be difficult to estimate the model parameters from the limited experimental data that are available. The purpose of this study is to analyse the model alongside the available data from experiments in humans and estimate the model parameters. In order to achieve this aim we linearize the model and use several methods of model and parameter validation: stability analysis, structural identifiability analysis, and sensitivity analysis based on traditional sensitivity functions and generalized sensitivity functions. We find that all model parameters are identifiable, structurally and taking into account parameter correlations, when several types of model output are used for parameter estimation. Based on these analyses we estimate parameter values from the limited available data and compare them with previously published parameter values. Finally we show how the model can be applied in future studies of treatment effectiveness in IgG multiple myeloma with simulations of serum monoclonal IgG responses during treatment.

Parameter Identifiability of Fundamental Pharmacodynamic Models

Issues of parameter identifiability of routinely used pharmacodynamics models are considered in this paper. The structural identifiability of 16 commonly applied pharmacodynamic model structures was analyzed analytically, using the input-output approach. Both fixed-effects versions (non-population, no between-subject variability) and mixed-effects versions (population, including between-subject variability) of each model structure were analyzed. All models were found to be structurally globally identifiable under conditions of fixing either one of two particular parameters. Furthermore, an example was constructed to illustrate the importance of sufficient data quality and show that structural identifiability is a prerequisite, but not a guarantee, for successful parameter estimation and practical parameter identifiability. This analysis was performed by generating artificial data of varying quality to a structurally identifiable model with known true parameter values, followed by re-estimation of the parameter values. In addition, to show the benefit of including structural identifiability as part of model development, a case study was performed applying an unidentifiable model to real experimental data. This case study shows how performing such an analysis prior to parameter estimation can improve the parameter estimation process and model performance. Finally, an unidentifiable model was fitted to simulated data using multiple initial parameter values, resulting in highly different estimated uncertainties. This example shows that although the standard errors of the parameter estimates often indicate a structural identifiability issue, reasonably “good” standard errors may sometimes mask unidentifiability issues.

Input estimation for drug discovery using optimal control and Markov chain Monte Carlo approaches

Input estimation is employed in cases where it is desirable to recover the form of an input function which cannot be directly observed and for which there is no model for the generating process. In pharmacokinetic and pharmacodynamic modelling, input estimation in linear systems (deconvolution) is well established, while the nonlinear case is largely unexplored. In this paper, a rigorous definition of the input-estimation problem is given, and the choices involved in terms of modelling assumptions and estimation algorithms are discussed. In particular, the paper covers Maximum a Posteriori estimates using techniques from optimal control theory, and full Bayesian estimation using Markov Chain Monte Carlo (MCMC) approaches. These techniques are implemented using the optimisation software CasADi, and applied to two example problems: one where the oral absorption rate and bioavailability of the drug eflornithine are estimated using pharmacokinetic data from rats, and one where energy intake is estimated from body-mass measurements of mice exposed to monoclonal antibodies targeting the fibroblast growth factor receptor (FGFR) 1c. The results from the analysis are used to highlight the strengths and weaknesses of the methods used when applied to sparsely sampled data. The presented methods for optimal control are fast and robust, and can be recommended for use in drug discovery. The MCMC-based methods can have long running times and require more expertise from the user. The rigorous definition together with the illustrative examples and suggestions for software serve as a highly promising starting point for application of input-estimation methods to problems in drug discovery.

PKPD modelling of PR and QRS intervals in conscious dogs using standard safety pharmacology data

INTRODUCTION

Pharmacokinetic-pharmacodynamic (PKPD) modelling can improve safety assessment, but few PKPD models describing drug-induced QRS and PR prolongations have been published. This investigation aims to develop and evaluate PKPD models for describing QRS and PR effects in routine safety studies.

METHODS

Exposure and telemetry data from safety pharmacology studies in conscious beagle dogs were acquired. Mixed effects baseline and PK-QRS/PR models were developed for the anti-arrhythmic compounds AZD1305, flecainide, quinidine and verapamil and the anti-muscarinic compounds AZD8683 and AZD9164. RR interval correction and circadian rhythms were investigated for predicting baseline variability. Individual PK predictions were used to drive the pharmacological effects evaluating linear and non-linear direct and effect compartment models.

RESULTS

Conduction slowing induced by the tested anti-arrhythmics was direct and proportional at low exposures, whilst time delays and non-linear effects were evident for the tested anti-muscarinics. AZD1305, flecainide and quinidine induced QRS widening with 4.2, 10 and 5.6% μM(-1) unbound drug. AZD1305 and flecainide also prolonged PR with 13.5 and 11.5% μM(-1). PR prolongations induced by the anti-muscarinics and verapamil were best described by Emax models with maximal effects ranging from 55 to 95%. RR interval correction and circadian rhythm improved PR but not QRS modelling. However, circadian rhythm had minor impact on estimated drug effects.

DISCUSSION

Baseline and drug-induced effects on QRS and PR intervals can be effectively described with PKPD models using routine data, providing quantitative safety information to support drug discovery and development.

Modeling and Simulation Approaches for Cardiovascular Function and Their Role in Safety Assessment

Systems pharmacology modeling and pharmacokinetic-pharmacodynamic (PK/PD) analysis of drug-induced effects on cardiovascular (CV) function plays a crucial role in understanding the safety risk of new drugs. The aim of this review is to outline the current modeling and simulation (M&S) approaches to describe and translate drug-induced CV effects, with an emphasis on how this impacts drug safety assessment. Current limitations are highlighted and recommendations are made for future effort in this vital area of drug research.

Observation of the time-course for peptidoglycan lipid intermediate II polymerization by Staphylococcus aureus monofunctional transglycosylase

The polymerization of lipid intermediate II by the transglycosylase activity of penicillin-binding proteins (PBPs) represents an important target for antibacterial action, but limited methods are available for quantitative assay of this reaction, or screening potential inhibitors. A new labelling method for lipid II polymerization products using Sanger's reagent (fluoro-2,4-dinitrobenzene), followed by gel permeation HPLC analysis, has permitted the observation of intermediate polymerization products for Staphylococcus aureus monofunctional transglycosylase MGT. Peak formation is inhibited by 6 µM ramoplanin or enduracidin. Characterization by mass spectrometry indicates the formation of tetrasaccharide and octasaccharide intermediates, but not a hexasaccharide intermediate, suggesting a dimerization of a lipid-linked tetrasaccharide. Numerical modelling of the time-course data supports a kinetic model involving addition to lipid-linked tetrasaccharide of either lipid II or lipid-linked tetrasaccharide. Observation of free octasaccharide suggests that hydrolysis of the undecaprenyl diphosphate lipid carrier occurs at this stage in peptidoglycan transglycosylation.

Reprint of "Pharmacokinetic modelling of the anti-malarial drug artesunate and its active metabolite dihydroartemisinin"

A four compartment mechanistic mathematical model is developed for the pharmacokinetics of the commonly used anti-malarial drug artesunate and its principle metabolite dihydroartemisinin following oral administration of artesunate. The model is structurally unidentifiable unless additional constraints are imposed. Combinations of mechanistically derived constraints are considered to assess their effects on structural identifiability and on model fits. Certain combinations of the constraints give rise to locally or globally identifiable model structures. Initial validation of the model under various combinations of the constraints leading to identifiable model structures was performed against a dataset of artesunate and dihydroartemisinin concentration-time profiles of 19 malaria patients. When all the discussed constraints were imposed on the model, the resulting globally identifiable model structure was found to fit reasonably well to those patients with normal drug absorption profiles. However, there is wide variability in the fitted parameters and further investigation is warranted.

Kinetic modelling of haemodialysis removal of myoglobin in rhabdomyolysis patients

An extended two compartment model is proposed to describe the dynamics of myoglobin in rhabdomyolysis patients undergoing dialysis. Before using clinical data to estimate the model's unknown parameters, structural identifiability analysis was performed to determine the parameters uniqueness given certain clinical observations. A Taylor series expansion method was implemented which found that the model was structurally globally/uniquely identifiable for both on- and off-dialysis phases. The fitted model was then used in a predictive capacity showing that the use of Theralite high cut-off (HCO) or HCO 1100 dialyser gave a significant reduction in myoglobin renal exposure compared to standard haemodialysis (HD).

Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake

In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available.

Pharmacokinetic modelling of the anti-malarial drug artesunate and its active metabolite dihydroartemisinin

A four compartment mechanistic mathematical model is developed for the pharmacokinetics of the commonly used anti-malarial drug artesunate and its principle metabolite dihydroartemisinin following oral administration of artesunate. The model is structurally unidentifiable unless additional constraints are imposed. Combinations of mechanistically derived constraints are considered to assess their effects on structural identifiability and on model fits. Certain combinations of the constraints give rise to locally or globally identifiable model structures. Initial validation of the model under various combinations of the constraints leading to identifiable model structures was performed against a dataset of artesunate and dihydroartemisinin concentration-time profiles of 19 malaria patients. When all the discussed constraints were imposed on the model, the resulting globally identifiable model structure was found to fit reasonably well to those patients with normal drug absorption profiles. However, there is wide variability in the fitted parameters and further investigation is warranted.

The input-output relationship approach to structural identifiability analysis

Analysis of the identifiability of a given model system is an essential prerequisite to the determination of model parameters from physical data. However, the tools available for the analysis of non-linear systems can be limited both in applicability and by computational intractability for any but the simplest of models. The input-output relation of a model summarises the input-output structure of the whole system and as such provides the potential for an alternative approach to this analysis. However for this approach to be valid it is necessary to determine whether the monomials of a differential polynomial are linearly independent. A simple test for this property is presented in this work. The derivation and analysis of this relation can be implemented symbolically within Maple. These techniques are applied to analyse classical models from biomedical systems modelling and those of enzyme catalysed reaction schemes.

Describing the effectiveness of immunosuppression drugs and apheresis in the treatment of transplant patients

When any foreign object is found in the human body antibodies are generated that mark it for removal by the immune system. In most cases these are natural and healthy responses; however, when considering organ transplants the immune response to the implanted organ must be kept to a minimum to avoid host rejection. To reduce the host's immune response to the implant, clinicians are able to manipulate the antibody dynamics through drug therapy, to minimise the antibody synthesis (immunosuppression), and by the removal of antibodies directly from the patients' blood, a process known as apheresis. In this paper models are presented that describe the in vivo kinetics of three immune complexes which are routinely measured pre- and post-operatively in implant patients, namely IgA, IgG and IgM. These models are then used to analyse the effective clearance rates of different apheresis methods (plasmapheresis, plasma absorption or plasma exchange) and to quantify the impact immune-suppression drugs have on the underlying antibody synthesis. It is hoped that the simplicity of the mathematical models, and associated implementation, will allow the translation of knowledge gained of the process dynamics to positively impact future patient diagnosis and treatment.

Structural identifiability and indistinguishability analyses of the minimal model and a euglycemic hyperinsulinemic clamp model for glucose-insulin dynamics

Many mathematical models have been developed to describe glucose-insulin kinetics as a means of analysing the effective control of diabetes. This paper concentrates on the structural identifiability analysis of certain well-established mathematical models that have been developed to characterise glucose-insulin kinetics under different experimental scenarios. Such analysis is a pre-requisite to experiment design and parameter estimation and is applied for the first time to these models with the specific structures considered. The analysis is applied to a basic (original) form of the Minimal Model (MM) using the Taylor Series approach and a now well-accepted extended form of the MM by application of the Taylor Series approach and a form of the Similarity Transformation approach. Due to the established inappropriate nature of the MM with regard to glucose clamping experiments an alternative model describing the glucose-insulin dynamics during a Euglycemic Hyperinsulinemic Clamp (EIC) experiment was considered. Structural identifiability analysis of the EIC model is also performed using the Taylor Series approach and shows that, with glucose infusion as input alone, the model is structurally globally identifiable. Additional analysis demonstrates that the two different model forms are structurally distinguishable for observation of both glucose and insulin.

The use of a formal sensitivity analysis on epidemic models with immune protection from maternally acquired antibodies

This paper considers the outcome of a formal sensitivity analysis on a series of epidemic model structures developed to study the population level effects of maternal antibodies. The analysis is used to compare the potential influence of maternally acquired immunity on various age and time domain observations of infection and serology, with and without seasonality. The results of the analysis indicate that time series observations are largely insensitive to variations in the average duration of this protection, and that age related empirical data are likely to be most appropriate for estimating these characteristics.

Indistinguishability and identifiability of kinetic models for the MurC reaction in peptidoglycan biosynthesis

An important question in Systems Biology is the design of experiments that enable discrimination between two (or more) competing chemical pathway models or biological mechanisms. In this paper analysis is performed between two different models describing the kinetic mechanism of a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the three products; the competing model also assumes ordered substrate binding, but with fast release of the three products. The two versions are shown to be distinguishable; however, if standard quasi-steady-state assumptions are made distinguishability cannot be determined. Once model structure uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate constant values given the experiment observations, a process known as structural identifiability. Structural identifiability analysis is carried out for both models to determine which of the unknown reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs. This structural analysis forms an integrated step towards the modelling of the full pathway of the cytoplasmic phase of peptidoglycan biosynthesis.

Kinetic modelling of the role of the aldehyde dehydrogenase enzyme and the breast cancer resistance protein in drug resistance and transport

A compartmental model for the in vitro uptake kinetics of the anti-cancer agent topotecan (TPT) has been extended from a previously published model. The extended model describes the drug activity and delivery of the pharmacologically active form to the DNA target as well as the catalysis of the aldehyde dehydrogenase (ALDH) enzyme and the elimination of drug from the cytoplasm via the efflux pump. Verification of the proposed model is achieved using scanning-laser microscopy data from live human breast cancer cells. Before estimating the unknown model parameters from the experimental in vitro data it is essential to determine parameter uniqueness (or otherwise) from this imposed output structure. This is formally performed as a structural identifiability analysis, which demonstrates that all of the unknown model parameters are uniquely determined by the output structure corresponding to the experiment.

A new general glucose homeostatic model using a proportional-integral-derivative controller

The glucose-insulin system is a challenging process to model due to the feedback mechanisms present, hence the implementation of a model-based approach to the system is an on-going and challenging research area. A new approach is proposed here which provides an effective way of characterising glycaemic regulation. The resulting model is built on the premise that there are three phases of insulin secretion, similar to those seen in a proportional-integral-derivative (PID) type controller used in engineering control problems. The model relates these three phases to a biological understanding of the system, as well as the logical premise that the homeostatic mechanisms will maintain very tight control of the system. It includes states for insulin, glucose, insulin action and a state to simulate an integral function of glucose. Structural identifiability analysis was performed on the model to determine whether a unique set of parameter values could be identified from the available observations, which should permit meaningful conclusions to be drawn from parameter estimation. Although two parameters--glucose production rate and the proportional control coefficient--were found to be unidentifiable, the former is not a concern as this is known to be impossible to measure without a tracer experiment, and the latter can be easily estimated from other means. Subsequent parameter estimation using Intravenous Glucose Tolerance Test (IVGTT) and hyperglycaemic clamp data was performed and subsequent model simulations have shown good agreement with respect to these real data.

The use of deconvolution techniques to identify the fundamental mixing characteristics of urban drainage structures

Mixing and dispersion processes affect the timing and concentration of contaminants transported within urban drainage systems. Hence, methods of characterising the mixing effects of specific hydraulic structures are of interest to drainage network modellers. Previous research, focusing on surcharged manholes, utilised the first-order Advection-Dispersion Equation (ADE) and Aggregated Dead Zone (ADZ) models to characterise dispersion. However, although systematic variations in travel time as a function of discharge and surcharge depth have been identified, the first order ADE and ADZ models do not provide particularly good fits to observed manhole data, which means that the derived parameter values are not independent of the upstream temporal concentration profile. An alternative, more robust, approach utilises the system's Cumulative Residence Time Distribution (CRTD), and the solute transport characteristics of a surcharged manhole have been shown to be characterised by just two dimensionless CRTDs, one for pre- and the other for post-threshold surcharge depths. Although CRTDs corresponding to instantaneous upstream injections can easily be generated using Computational Fluid Dynamics (CFD) models, the identification of CRTD characteristics from non-instantaneous and noisy laboratory data sets has been hampered by practical difficulties. This paper shows how a deconvolution approach derived from systems theory may be applied to identify the CRTDs associated with urban drainage structures.

A coupled drug kinetics-cell cycle model to analyse the response of human cells to intervention by topotecan

A model describing the response of the growth of single human cells in the absence and presence of the anti-cancer agent topotecan (TPT) is presented. The model includes a novel coupling of both the kinetics of TPT and cell cycle responses to the agent. By linking the models in this way, rather than using separate (disjoint) approaches, it is possible to illustrate how the drug perturbs the cell cycle. The model is compared to experimental in vitro cell cycle response data (comprising single cell descriptors for molecular and behavioural events), showing good qualitative agreement for a range of TPT dose levels.

Novel phenomena-based dynamic model of carbon black/composite vapour sensors

A novel physically based mathematical model of carbon black/polymer vapour sensors is described, which incorporates parameters that have physical meaning. This model has an analytical solution and so requires negligible computational power to analyse a sensor's response to a particular form of input. Another advantage of this modelling approach is that the environmental dependencies of sensor responses may be compensated for and so help in the design of better pattern-recognition algorithms for electronic nose systems. This also means that the underlying chemistry of the sensors may be decoupled from their physical non-analyte specific properties. Experimentally, three different conducting nanocomposite polymers, poly(styrene- co -butadine), poly(ethyl- co -vinyl acetate) and poly(caprolactone), were tested. Each experiment consisted of separate exposures of the sensors to acetone and ethanol vapour in ambient air. A total of 336 such experiments were performed over a two-week period. The model was validated with respect to these data and was then fitted to the two vapour responses simultaneously, demonstrating its applicability to ‘real world’ systems. The temperature dependence of the model parameters was judged to be the most important factor and it needs to be compensated for when applying this type of sensor in practice.

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 transmission and control of mastitis in dairy cows: A theoretical approach

A multi-species model that incorporates the transmission of both major and minor mastitis pathogens as well as the interaction between them via coinfection of a quarter is fitted to data from seven dairy herds. The results suggest that major and minor pathogens can interact, on occasion, in a counter-intuitive way with implications for the control of clinical mastitis. The key finding is that delaying culling of cows with major pathogen infections for more than 100 days post infection could result in a higher prevalence of major pathogen infections, whereas early culling would reduce the levels. A theoretical exploration of current and proposed control strategies is carried out, informed by parameters estimated from the model and data. The results at each stage suggest of areas of further research such as: field-testing of the hypotheses presented; the exploration of a stochastic formulation of the model; analysis of the raw repeated measures data; application of control theory to determine the most effective combination of control strategies; inclusion of economic factors into the modelling framework.

The mechanism distinguishability problem in biochemical kinetics: The single-enzyme, single-substrate reaction as a case study

A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models. Evans et al. [N.D. Evans, M.J. Chappell, M.J. Chapman, K.R. Godfrey, Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems, Automatica 40 (2004) 1947-1953] have shown that if, in addition, either of the mechanisms satisfies a particular criterion then such a transformation always exists when the models are indistinguishable from their experimentally observable outputs. The approach is applied to the single enzyme-single substrate reaction mechanism. In principle, mechanisms can be distinguished using this analysis, but we show that our ability to distinguish mechanistic models depends both on the precise measurements made, and on our knowledge of the system prior to performing the kinetics experiments.

Data reduction in headspace analysis of blood and urine samples for robust bacterial identification

This paper demonstrates the application of chemical headspace analysis to the problem of classifying the presence of bacteria in biomedical samples by using computational tools. Blood and urine samples of disparate forms were analysed using a Cyrano Sciences C320 electronic nose together with an Agilent 4440 Chemosensor. The high dimensional data sets resulting from these devices present computational problems for parameter estimation of discriminant models. A variety of data reduction and pattern recognition techniques were employed in an attempt to optimise the classification process. A 100% successful classification rate for the blood data from the Agilent 4440 was achieved by combining a Sammon mapping with a radial basis function neural network. In comparison a successful classification rate of 80% was achieved for the urine data from the C320 which were analysed using a novel nonlinear time series model.

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.

Combined index of heart rate variability and oximetry in screening for the sleep apnoea/hypopnoea syndrome

Many sleep centres employ a preliminary screening test in order to reduce the number of polysomnographies required in the routine diagnosis of the sleep apnoea/hypopnoea syndrome (SAHS). We investigated the combination of heart rate and oximetry information as a means of performing this test. A retrospective study of 100 patients with suspected SAHS was made. All patients had in-hospital polysomnography on one night. We estimated the number of respiratory event-related arousals by counting the number of autonomic arousals (assessed on the basis of changes in the heart interbeat interval) that were coincident with a rise in oximetry. The hourly index of such events was denoted the "cardiac-oximetry disturbance index" (CODI). The median apnoea/hypopnoea index (AHI) was 16.5 (range 1.0-93.6) h-1. The CODI correlated significantly with the AHI (Spearman correlation coefficient rs = 0.88, P < 0.01), and the area (+/- standard error) under the receiver operating characteristic (ROC) was 0.94 +/- 0.05. Oximetry alone (based on 4% dips) was a less effective screening test (rs = 0.80, P < 0.01; area under ROC 0.83 +/- 0.06). Using 2% dips in oximetry offered comparable performance with the CODI (rs = 0.91, P < 0.01; area under ROC 0.93 +/- 0.04). The CODI was better correlated with the electroencephalograph arousal index (rs = 0.84, P < 0.01) than was oximetry (2% dips, rs = 0.57, P < 0.01). The CODI algorithm also offers an informal measure of self-validation: a large discrepancy between the number of autonomic arousals and the number of rises in oximetry indicates the presence of autonomic arousals without changes in oximetry (or vice versa). This self-validation mechanism identified several patients in this study, and may be useful in identifying sleep disruption due to chronic pain or other causes.

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows

A structural identifiability analysis is performed on a mathematical model for the coupled transmission of two classes of pathogen. The pathogens, classified as major and minor, are aetiological agents of mastitis in dairy cows that interact directly and via the immunological reaction in their hosts. Parameter estimates are available from experimental data for all but four of the parameters in the model. Data from a longitudinal study of infection are used to estimate these unknown parameters. A novel approach and application of structural identifiability analysis is combined in this paper with the estimation of cross-protection parameters using epidemiological data.

A multispecies model for the transmission and control of mastitis in dairy cows

Mastitis in dairy cows is a significant economic and animal welfare issue in the dairy industry. The bacterial pathogens responsible for infection of the mammary gland may be split into two main categories: major and minor pathogens. Infection with major pathogens generally results in clinical illness or strong inflammatory responses and reduced milk yields, whereas minor pathogen infection is usually subclinical. Previous investigations have considered the transmission of these pathogens independently. Experimental evidence has shown cross-protection between species of pathogens. In this study a mathematical model for the coupled transmission of major and minor pathogens along with their interaction via the host was developed in order to consider various methods for controlling the incidence of major pathogen infection. A stability analysis of the model equilibria provides explanations for observed phenomena and previous decoupled modelling results. This multispecies model structure has provided a basis for quantifying the extent of cross-protection between species and assessing possible control strategies against the disease.

Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences

In this paper methods from differential algebra are used to study the structural identifiability of biological and pharmacokinetics models expressed in state-space form and with a structure given by rational functions. The focus is on the examples presented and on the application of efficient, automatic methods to test for structural identifiability for various input-output experiments. Differential algebra methods are coupled with Gröbner bases, Lie derivatives and the Taylor series expansion in order to obtain efficient algorithms. In particular, an upper bound on the number of derivatives needed for the Taylor series approach for a structural identifiability analysis of rational function models is given.

The sleep apnoea/hypopnoea syndrome depresses waking vagal tone independent of sympathetic activation

The modest daytime hypertension and sympathetic upregulation associated with the sleep apnoea/hypopnoea syndrome (SAHS), does not explain the relatively large increased risk of cardiac morbidity and mortality in the SAHS patients population. Therefore, efferent vagal and sympathetic activity was evaluated during wakefulness in SAHS subjects and matched healthy controls, in order to determine if vagal downregulation may play a role in the aetiology of cardiac disease in the SAHS. The awake autonomic nervous system function of 15 male subjects, with mild-to-moderate SAHS was compared to that of 14 healthy controls matched for age, body mass index, gender and blood pressure. All subjects were free from comorbidity. Vagal activity was estimated from measurements of heart rate variability high frequency power (HF) and sympathetic activity was measured from urine catecholamine excretion. The %HF power was significantly (p < 0.03) reduced in SAHS patients (10+/-1.6 (mean+/-SEM)) as compared to controls (17 +/- 3). In addition, HF power correlated with the apnoea/hypopnoea index in the SAHS subjects (R = -0.592, p = 0.02). There was no statistically significant difference in the daytime excretion of nonadrenaline between control (242 +/- 30 nmol x collection(-1)) and SAHS (316 +/- 46 nmol x collection(-1)) subjects (p = 0.38). In these sleep apnoea/hypopnoea syndrome patients there was limited evidence of increased waking levels of urine catecholamines. The principal component altering waking autonomic nervous system function, in the sleep apnoea/hypopnoea syndrome subjects, was a reduced daytime efferent vagal tone.