Mathematical Modelling of Alternative Pathway of Complement System.

The complement system (CS) is an integral part of innate immunity and can be activated via three different pathways. The alternative pathway (AP) has a central role in the function of the CS. The AP of complement system is implicated in several human disease pathologies. In the absence of triggers, the AP exists in a time-invariant resting state (physiological steady state). It is capable of rapid, potent and transient activation response upon challenge with a trigger. Previous models of AP have focused on the activation response. In order to understand the molecular machinery necessary for AP activation and regulation of a physiological steady state, we built parsimonious AP models using experimentally supported kinetic parameters. The models further allowed us to test quantitative roles played by negative and positive regulators of the pathway in order to test hypotheses regarding their mechanisms of action, thus providing more insight into the complex regulation of AP.

Mathematical modeling of the glucagon challenge test.

A model for the homeostasis of glucose through the regulating hormones glucagon and insulin is described. It contains a subsystem that models the internalization of the glucagon receptor. Internalization is a mechanism in cell signaling, through which G-protein coupled receptors are taken from the surface of the cell to the endosome. The model is used to interpret data from a glucagon challenge test in which subjects have been under treatment with a novel glucagon receptor anti-sense drug which is aimed at reducing the number of receptors in the liver. It is shown how the receptor internalization results in tolerance of the blood glucose concentration to glucagon-induced hyperglycemia. We quantify the reduction of the number of receptors using the model and the data before and after treatment.

Impact of mathematical pharmacology on practice and theory: four case studies.

Drug-discovery has become a complex discipline in which the amount of knowledge about human biology, physiology, and biochemistry have increased. In order to harness this complex body of knowledge mathematics can play a critical role, and has actually already been doing so. We demonstrate through four case studies, taken from previously published data and analyses, what we can gain from mathematical/analytical techniques when nonlinear concentration-time courses have to be transformed into their equilibrium concentration-response (target or complex) relationships and new structures of drug potency have to be deciphered; when pattern recognition needs to be carried out for an unconventional response-time dataset; when what-if? predictions beyond the observational concentration-time range need to be made; or when the behaviour of a semi-mechanistic model needs to be elucidated or challenged. These four examples are typical situations when standard approaches known to the general community of pharmacokineticists prove to be inadequate.

Correction to: Modelling the delay between pharmacokinetics and EEG effects of morphine in rats: binding kinetic versus effect compartment models.

The original version of this article was published open access. Unfortunately, due to a technical issue, the copyright holder name in the online version (HTML and XML) is incorrectly published as "Springer Science+Business Media, LLC, part of Springer Nature 2018". Instead, it should be "The Author(s) 2018".

Modelling the delay between pharmacokinetics and EEG effects of morphine in rats: binding kinetic versus effect compartment models.

Drug-target binding kinetics (as determined by association and dissociation rate constants, kon and koff) can be an important determinant of the kinetics of drug action. However, the effect compartment model is used most frequently instead of a target binding model to describe hysteresis. Here we investigate when the drug-target binding model should be used in lieu of the effect compartment model. The utility of the effect compartment (EC), the target binding kinetics (TB) and the combined effect compartment-target binding kinetics (EC-TB) model were tested on either plasma (ECPL, TBPL and EC-TBPL) or brain extracellular fluid (ECF) (ECECF, TBECF and EC-TBECF) morphine concentrations and EEG amplitude in rats. It was also analyzed when a significant shift in the time to maximal target occupancy (TmaxTO) with increasing dose, the discriminating feature between the TB and EC model, occurs in the TB model. All TB models assumed a linear relationship between target occupancy and drug effect on the EEG amplitude. All three model types performed similarly in describing the morphine pharmacodynamics data, although the EC model provided the best statistical result. The analysis of the shift in TmaxTO (∆TmaxTO) as a result of increasing dose revealed that ∆TmaxTO is decreasing towards zero if the koff is much smaller than the elimination rate constant or if the target concentration is larger than the initial morphine concentration. The results for the morphine PKPD modelling and the analysis of ∆TmaxTO indicate that the EC and TB models do not necessarily lead to different drug effect versus time curves for different doses if a delay between drug concentrations and drug effect (hysteresis) is described. Drawing mechanistic conclusions from successfully fitting one of these two models should therefore be avoided. Since the TB model can be informed by in vitro measurements of kon and koff, a target binding model should be considered more often for mechanistic modelling purposes.

Impact of saturable distribution in compartmental PK models: dynamics and practical use.

We explore the impact of saturable distribution over the central and the peripheral compartment in pharmacokinetic models, whilst assuming that back flow into the central compartiment is linear. Using simulations and analytical methods we demonstrate characteristic tell-tale differences in plasma concentration profiles of saturable versus linear distribution models, which can serve as a guide to their practical applicability. For two extreme cases, relating to (i) the size of the peripheral compartment with respect to the central compartment and (ii) the magnitude of the back flow as related to direct elimination from the central compartment, we derive explicit approximations which make it possible to give quantitative estimates of parameters. In three appendices we give detailed explanations of how these estimates are derived. They demonstrate how singular perturbation methods can be successfully employed to gain insight in the dynamics of multi-compartment pharmacokinetic models. These appendices are also intended to serve as an introductory tutorial to these ideas.

Modelling the dynamics of polar auxin transport in inflorescence stems of Arabidopsis thaliana.

The polar transport of the plant hormone auxin has been the subject of many studies, several involving mathematical modelling. Unfortunately, most of these models have not been experimentally verified. Here we present experimental measurements of long-distance polar auxin transport (PAT) in segments of inflorescence stems of Arabidopsis thaliana together with a descriptive mathematical model that was developed from these data. It is based on a general advection-diffusion equation for auxin density, as suggested by the chemiosmotic theory, but is extended to incorporate both immobilization of auxin and exchange with the surrounding tissue of cells involved in PAT, in order to account for crucial observations. We found that development of the present model assisted effectively in the analysis of experimental observations. As an example, we discuss the analysis of a quadruple mutant for all four AUX1/LAX1-LAX3 influx carriers genes. We found a drastic change in the parameters governing the exchange of PAT channels with the surrounding tissue, whereas the velocity was still of the order of magnitude of the wild type. In addition, the steady-state flux of auxin through the PAT system of the mutant did not exhibit a saturable component, as we found for the wild type, suggesting that the import carriers are responsible for the saturable component in the wild type. In the accompanying Supplementary data available at JXB online, we describe in more detail the data-driven development of the model, review and derive predictions from a mathematical model of the chemiosmotic theory, and explore relationships between parameters in our model and processes and parameters at the cellular level.

Cross-membrane signal transduction of receptor tyrosine kinases (RTKs): from systems biology to systems pharmacology.

Receptor tyrosine kinases are high-affinity cell surface receptors for many polypeptide growth factors, cytokines, and hormones. They straddle the cell wall and play an important role in cross-membrane signalling. We present a two-component systems pharmacology model based on the local physiology and identify characteristic features of its dynamics. We thus present a transparent tool for studying the effects of drug intervention and ways of administration on cross-membrane signalling through these receptors.

Feedback modeling of non-esterified fatty acids in obese Zucker rats after nicotinic acid infusions.

This study investigates the impact of disease on nicotinic acid (NiAc)-induced changes in plasma concentrations of non-esterified fatty acids (NEFA). NiAc was given by constant intravenous infusion to normal Sprague-Dawley and obese Zucker rats, and arterial blood samples were taken for analysis of NiAc, NEFA, insulin and glucose plasma concentrations. The intravenous route was intentionally selected to avoid confounding processes, such as absorption, following extravascular administration. Data were analyzed using nonlinear mixed effects modeling (NONMEM, version VI). The disposition of NiAc in the normal rats was described by a two-compartment model with endogenous synthesis of NiAc and two parallel capacity-limited elimination processes. In the obese rats disposition was described by a one-compartment model with endogenous synthesis of NiAc and one capacity-limited elimination process. The plasma concentration of NiAc drove NEFA (R) turnover via an inhibitory drug-mechanism function acting on the formation of NEFA. NEFA turnover was described by a feedback model with a moderator distributed over a series of transit compartments, where the first compartment (M 1 ) inhibited the formation of R and the last compartment (M N ) stimulated the loss of R. All processes regulating plasma NEFA concentrations were assumed to be captured by the moderator function. Differences in the pharmacodynamic response of the two strains included, in the obese animals, an increased NEFA baseline, diminished rebound and post-rebound oscillation, and a more pronounced slowly developing tolerance during the period of constant drug exposure. The feedback model captured the NiAc-induced changes in NEFA response in both the normal and obese rats. Differences in the parameter estimates between the obese and normal rats included, in the former group, increases in R 0 , k in and p by 44, 41 and 78 %, respectively, and decreases in k out and γ by 64 and 84 %, respectively. The estimates of k tol and IC 50 were similar in both groups. The NiAc-NEFA concentration-response relationship at equilibrium was substantially different in the two groups, being shifted upwards and to the right, and being shallower in the obese rats. The extent of such shifts is important, as they demonstrate the impact of disease at equilibrium and, if ignored, will lead to erroneous dose predictions and, in consequence, poorly designed studies. The proposed models are primarily aimed at screening and selecting candidates with the highest potential of becoming a viable drug in man.

Challenges of a mechanistic feedback model describing nicotinic acid-induced changes in non-esterified fatty acids in rats.

Previously, we developed a feedback model to describe the tolerance and oscillatory rebound of non-esterified fatty acid (NEFA) plasma concentrations in male Sprague Dawley rats after intravenous infusions of nicotinic acid (NiAc). This study challenges that model, using the following regimens of intravenous and oral NiAc dosing in male Sprague Dawley rats (n = 95) to create different patterns of exposure: (A) 30 min infusion at 0, 1, 5 or 20 µmol kg(-1) body weight; (B) 300 min infusion at 0, 5, 10 or 51 µmol kg(-1); (C) 30 min infusion at 5 µmol kg(-1), followed by a stepwise decrease in rate every 10 min for 180 min; (D) 30 min infusion at 5 µmol kg(-1), followed by a stepwise decrease in rate every 10 min for 180 min and another 30 min infusion at 5 µmol kg(-1) from 210 to 240 min; (E) an oral dose of 0, 24.4, 81.2 or 812 µmol kg(-1). Serial arterial blood samples were taken for measurement of plasma NiAc and NEFA concentrations. The gradual decrease in infusion rate in (C) and (D) were also designed to test the hypothesis that a gradual reduction in NiAc plasma concentration may be expected to reduce or prevent rebound. The absorption of NiAc was described by parallel linear and non-linear processes and the disposition of NiAc by a two-compartment model with endogenous turnover rate and two parallel capacity-limited elimination processes. NEFA (R) turnover, which was driven by the plasma concentration of NiAc via an inhibitory drug-mechanism function acting on NEFA formation, was described by a feedback model with a moderator distributed over a series of transit compartments, where the first compartment (M 1) inhibited the formation of R and the last compartment (M N ) stimulated the loss of R. All processes regulating the plasma NEFA concentration were assumed to be captured by the moderator function. Data were analyzed using non-linear mixed effects modeling (NONMEM). The potency IC 50 of NiAc was 68 nmol L(-1), the fractional turnover rate k out 0.27 L mmol(-1) min(-1), and the turnover rate of moderator k tol 0.023 min(-1). The lower physiological limit of NEFA, which was modeled as a NiAc-independent release (k cap ) of NEFA into plasma, was estimated to 0.023 mmol L(-1) min(-1). The parameter estimates derived in this study were consistent with our previous estimates, suggesting that the model may be used for prediction of the NEFA response time-course following different modes and routes administration of NiAc or NiAc analogues. In order to avoid NiAc-induced NEFA rebound, a slow decline in the NiAc exposure pattern is needed at or below IC (50).

Application of a mechanism-based disease systems model for osteoporosis to clinical data.

A recently proposed mechanism-based disease systems model for osteoporosis (Schmidt et al., J Pharmacokinet Pharmacodyn 38:873-900, 2011) was applied to clinical data from post-menopausal women (n = 767) receiving various doses of the selective estrogen receptor modulator tibolone. Plasma bone-specific alkaline phosphatase activity and urinary N-telopeptide were used as biomarkers reflecting the activity of osteoblasts (bone forming cells) and osteoclasts (bone removing cells), respectively. These data were analyzed in conjunction with data on osteocalcin and on bone mineral density (BMD) (both lumbar spine and total hip), which reflect the activity of both cell types. While the dynamics of bone turnover markers changes rapidly, closely following changes in the activity of bone cells, changes in BMD are slower and have their own dynamics. Application of the mechanism-based disease systems model to the clinical data allowed for an adequate description of the data and yielded parameter estimates that are consistent with physiological values reported in the literature (Lemaire et al., J Theor Biol 229:293-309, 2004). The fitted model enabled characterization of (i) the critical time scales involved in disease progression, (ii) the dynamics of the system during onset and offset of the therapeutic intervention, and (iii) the distinction between responders and low-responders to tibolone treatment.

The quest for balance between capturing data and model complexity: A quantitative clinical pharmacology approach applied to monoclonal antibodies.

The main objective of this tutorial is to provide the readers with a roadmap of how to establish increasingly complex target-mediated drug disposition (TMDD) models for monoclonal antibodies. To this end, we built mathematical models, each with a detailed visualization, starting from the basic TMDD model by Mager and Jusko to the well-established, physiologically based model by Li et al. in a step-wise fashion to highlight the relative importance of key physiological processes that impact mAb kinetics and system dynamics. As the models become more complex, the question of structural and parameter identifiability arises. To address this question, we work through a trastuzumab case example to guide the modeler's choice for model and parameter optimization in light of the context of use. We leave the readers of this tutorial with a brief summary of the advantages and limitations of each model expansion, as well as the model source codes for further self-guided exploration and hands-on analysis.

Dynamics of target-mediated drug disposition: characteristic profiles and parameter identification.

In this paper we present a mathematical analysis of the basic model for target mediated drug disposition (TMDD). Assuming high affinity of ligand to target, we give a qualitative characterisation of ligand versus time graphs for different dosing regimes and derive accurate analytic approximations of different phases in the temporal behaviour of the system. These approximations are used to estimate model parameters, give analytical approximations of such quantities as area under the ligand curve and clearance. We formulate conditions under which a suitably chosen Michaelis-Menten model provides a good approximation of the full TMDD-model over a specified time interval.

An Extended Model Including Target Turnover, Ligand-Target Complex Kinetics, and Binding Properties to Describe Drug-Receptor Interactions.

Since the beginning of this century, target-mediated drug disposition has become a central concept in modeling drug action in drug development. It combines a range of processes, such as turnover, protein binding, internalization, and non-specific elimination, and often serves as a nucleus of more complex pharmacokinetic models. It is simple enough to comprehend but complex enough to be able to describe a wide range of phenomena and data sets. However, the complexity comes at a price: many parameters. In this chapter, we present an overview of the temporal development of the compounds involved after different types of drug doses and offer convenient handles for dissecting data sets in a sophisticated manner in order to estimate the values of these parameters, such as rate constants and pertinent concentrations.

Impact of enzyme turnover on the dynamics of the Michaelis-Menten model.

Enzymatic (metabolic rate) processes are traditionally modelled by means of Michaelis-Menten type reactions. The experimental setup is usually performed in vitro also denoted as a 'closed system'. In this paper we explore the impact of enzyme turnover on the classical Michaelis-Menten model by modifying it to include enzyme turnover, specifically through zeroth-order synthesis and first-order degeneration of the enzyme. It is shown how enzyme turnover significantly alters the dynamics of substrate, free- and bound enzyme, and impacts the rate with which substrate is converted to a metabolite P. Qualitative and quantitative estimates are derived for the effect of the parameters ksyn, kdeg and kcat on the dynamics of substrate, and free- and bound enzyme. The model integrates four distinct processes, each characterised with its own parameter(s): (i) substrate-enzyme binding, characterised by kon and koff; (ii) the catalytic process, characterised by kcat; (iii) simultaneous re-generation of free enzyme; and (iv) turnover of free enzyme, characterised by kdeg. The properties of the open Michaelis-Menten model have a direct bearing on the drug discovery process, the translation of data to the human situation and on explaining deviating clinical metabolic observations.

Feedback modeling of non-esterified fatty acids in rats after nicotinic acid infusions.

A feedback model was developed to describe the tolerance and oscillatory rebound seen in non-esterified fatty acid (NEFA) plasma concentrations following intravenous infusions of nicotinic acid (NiAc) to male Sprague-Dawley rats. NiAc was administered as an intravenous infusion over 30 min (0, 1, 5 or 20 µmol kg(-1) of body weight) or over 300 min (0, 5, 10 or 51 µmol kg(-1) of body weight), to healthy rats (n = 63), and serial arterial blood samples were taken for measurement of NiAc and NEFA plasma concentrations. Data were analyzed using nonlinear mixed effects modeling (NONMEM). The disposition of NiAc was described by a two-compartment model with endogenous turnover rate and two parallel capacity-limited elimination processes. The plasma concentration of NiAc was driving NEFA (R) turnover via an inhibitory drug-mechanism function acting on the formation of NEFA. The NEFA turnover was described by a feedback model with a moderator distributed over a series of transit compartments, where the first compartment (M (1)) inhibited the formation of R and the last compartment (M ( N )) stimulated the loss of R. All processes regulating plasma NEFA concentrations were assumed to be captured by the moderator function. The potency, IC (50), of NiAc was 45 nmol L(-1), the fractional turnover rate k ( out ) was 0.41 L mmol(-1) min(-1) and the turnover rate of moderator k ( tol ) was 0.027 min(-1). A lower physiological limit of NEFA was modeled as a NiAc-independent release (k ( cap )) of NEFA into plasma and was estimated to 0.032 mmol L(-1) min(-1). This model can be used to provide information about factors that determine the time-course of NEFA response following different modes, rates and routes of administration of NiAc. The proposed model may also serve as a preclinical tool for analyzing and simulating drug-induced changes in plasma NEFA concentrations after treatment with NiAc or NiAc analogues.

Coping with time scales in disease systems analysis: application to bone remodeling.

In this study we demonstrate the added value of mathematical model reduction for characterizing complex dynamic systems using bone remodeling as an example. We show that for the given parameter values, the mechanistic RANK-RANKL-OPG pathway model proposed by Lemaire et al. (J Theor Biol 229:293-309, 2004) can be reduced to a simpler model, which can describe the dynamics of the full Lemaire model to very good approximation. The response of both models to changes in the underlying physiology and therapeutic interventions was evaluated in four physiologically meaningful scenarios: (i) estrogen deficiency/estrogen replacement therapy, (ii) Vitamin D deficiency, (iii) ageing, and (iv) chronic glucocorticoid treatment and its cessation. It was found that on the time scale of disease progression and therapeutic intervention, the models showed negligible differences in their dynamic properties and were both suitable for characterizing the impact of estrogen deficiency and estrogen replacement therapy, Vitamin D deficiency, ageing, and chronic glucocorticoid treatment and its cessation on bone forming (osteoblasts) and bone resorbing (osteoclasts) cells. It was also demonstrated how the simpler model could help in elucidating qualitative properties of the observed dynamics, such as the absence of overshoot and rebound, and the different dynamics of onset and washout.

Comparisons of basic target-mediated drug disposition (TMDD) and ligand facilitated target removal (LFTR).

In the well-known model for basic Target-Mediated Drug Disposition (TMDD), drug binds to the target and the resulting drug-target complex is removed by a first order process, leading to loss of both drug and target. In the present note we study what happens when, instead, drug is returned to the free drug pool so that it can a new target molecule. What results is a mechanism in which the drug, here referred to as the ligand, facilitates the removal of the target,and then returns to the free ligand pool. Accordingly the process will be referred to as Ligand-Facilitated Target Removal (LFTR). It is shown through simulations and mathematical analysis how the two models differ and how their signature profiles typically appear. We also derive a useful parameter of both models, the in vivo potency EC50 (L50) which contains both ligand-target binding properties (kon,koff), target turnover (kdeg) and ligand-target complex kinetics (ke(RL)). Thus, this parameter contains a conglomerate of properties and is therefore potentially more informative about relevant (clinical) exposure than the binding affinity (Kd) alone. The derived potency parameter EC50 may therefore be used as a more robust ranking parameter among small and large drug molecules in drug discovery. Subsequently the LFTR model is applied to experimentally obtained literature data and the relevant parameters are estimated.

Impact of protein binding on receptor occupancy: a two-compartment model.

In this paper we analyse the impact of protein-, lipid- and receptor-binding on receptor occupancy in a two-compartment system, with proteins in both compartments and lipids and receptors in the peripheral compartment only. We do this for two manners of drug administration: a bolus administration and a constant rate infusion, both into the central compartment. We derive explicit approximations for the time-curves of the different compounds valid for a wide range of realistic values of rate constants and initial concentrations of proteins, lipids, receptors and the drug. These approximations are used to obtain both qualitative and quantitative insight into such critical properties as the distribution of the drug over the two compartments, the maximum receptor occupancy and the area under the drug-receptor complex curve. In particular we focus on assessing the impact of the dissociation constants, K(P), K(L) and K(R) of the drug with, respectively, the proteins, the lipids and the receptors, the permeability and the surface area of the membrane between compartments, and the rate the drug is eliminated from the system.