Asymptotic Analysis of a Target-Mediated Drug Disposition Model: Algorithmic and Traditional Approaches

An asymptotic description of a basic FcRn-regulated clearance mechanism and its implications for PBPK modelling of large antibodies

A basic FcRn-regulated clearance mechanism is investigated using the method of matched asymptotic expansions. The broader aim of the work is to obtain further insight on the mechanism, thereby providing theoretical support for future pharmacologically-based pharmacokinetic modelling efforts. The corresponding governing equations are first non-dimensionalised and the order of magnitudes of the model parameters are assessed based on their values reported in the literature. Under the assumption of high FcRn-binding affinity, analytical approximations are derived that are valid over the characteristic phases of the problem. Additionally, relatively simple equations relating clearance and AUC to physiological model parameters are derived, which are valid over the longest characteristic time scale of the problem. For lower to moderate doses clearance is effectively linear, whereas for higher doses it is nonlinear. It is shown that for all doses sufficiently high the leading-order approximation for the IgG concentration in plasma, over the longest characteristic time scale, is independent of the initial dose. This is because IgG that is in 'excess' of FcRn is eliminated over a time scale much shorter than that of the terminal phase. In conclusion, analytical approximations of the basic FcRn mechanism have been derived using matched asymptotic expansions, leading to a simple equation relating clearance to FcRn binding affinity, the ratio of degradation and FcRn concentration, and the volumes of the system.

Algorithmic multiscale analysis for the FcRn mediated regulation of antibody PK in human

A demonstration is provided on how algorithmic asymptotic analysis of multi-scale pharmacokinetics (PK) systems can provide (1) system level understanding and (2) predictions on the response of the model when parameters vary. Being algorithmic, this type of analysis is not hindered by the size or complexity of the model and requires no input from the investigator. The algorithm identifies the constraints that are generated by the fast part of the model and the components of the slow part of the model that drive the system within these constraints. The demonstration is based on a typical monoclonal antibody PK model. It is shown that the findings produced by the traditional methodologies, which require significant input by the investigator, can be produced algorithmically and more accurately. Moreover, additional insights are provided by the algorithm, which cannot be obtained by the traditional methodologies; notably, the dual influence of certain reactions depending on whether their fast or slow component dominates. The analysis reveals that the importance of physiological processes in determining the systemic exposure of monoclonal antibodies (mAb) varies with time. The analysis also confirms that the rate of mAb uptake by the cells, the binding affinity of mAb to neonatal Fc receptor (FcRn), and the intracellular degradation rate of mAb are the most sensitive parameters in determining systemic exposure of mAbs. The algorithmic framework for analysis introduced and the resulting novel insights can be used to engineer antibodies with desired PK properties.

Algorithmic asymptotic analysis: Extending the arsenal of cancer immunology modeling

The recent advances in cancer immunotherapy boosted the development of tumor-immune system models, with the aim to indicate more efficient treatments. Physical understanding is however difficult to be acquired, due to the complexity and the multi-scale dynamics of these models. In this work, the dynamics of a fundamental model formulating the interactions of tumor cells with natural killer cells, CD8⁺ T cells and circulating lymphocytes is examined. It is first shown that the long-term evolution of the system towards high-tumor or tumor-free equilibria is determined by the dynamics of an initial explosive stage of tumor progression. Focusing on this stage, the algorithmic Computational Singular Perturbation methodology is employed to identify the underlying mechanisms confining the system's evolution and the governing slow dynamics along them. These insights are preserved along different tumor-immune system and patient-dependent realizations. On top of these identifications, a novel reduced model is algorithmically constructed, which accurately predicts the dynamics of the system during the explosive stage and includes half of the parameters of the detailed model. The present analysis demonstrates the potential of algorithmic asymptotic analysis for acquiring physical understanding and for simplifying the complexity of cancer immunology models. Along with the current techniques on the field, this analysis can provide guidelines for more effective treatment development.

Computational singular perturbation analysis of brain lactate metabolism

Lactate in the brain is considered an important fuel and signalling molecule for neuronal activity, especially during neuronal activation. Whether lactate is shuttled from astrocytes to neurons or from neurons to astrocytes leads to the contradictory Astrocyte to Neuron Lactate Shuttle (ANLS) or Neuron to Astrocyte Lactate Shuttle (NALS) hypotheses, both of which are supported by extensive, but indirect, experimental evidence. This work explores the conditions favouring development of ANLS or NALS phenomenon on the basis of a model that can simulate both by employing the two parameter sets proposed by Simpson et al. (J Cereb. Blood Flow Metab., 27:1766, 2007) and Mangia et al. (J of Neurochemistry, 109:55, 2009). As most mathematical models governing brain metabolism processes, this model is multi-scale in character due to the wide range of time scales characterizing its dynamics. Therefore, we utilize the Computational Singular Perturbation (CSP) algorithm, which has been used extensively in multi-scale systems of reactive flows and biological systems, to identify components of the system that (i) generate the characteristic time scale and the fast/slow dynamics, (ii) participate to the expressions that approximate the surfaces of equilibria that develop in phase space and (iii) control the evolution of the process within the established surfaces of equilibria. It is shown that a decisive factor on whether the ANLS or NALS configuration will develop during neuronal activation is whether the lactate transport between astrocytes and interstitium contributes to the fast dynamics or not. When it does, lactate is mainly generated in astrocytes and the ANLS hypothesis is realised, while when it doesn't, lactate is mainly generated in neurons and the NALS hypothesis is realised. This scenario was tested in exercise conditions.

Algorithmic Analysis of Chemical Dynamics of the Autoignition of NH3–H2O2/Air Mixtures

The dynamics of a homogeneous adiabatic autoignition of an ammonia/air mixture at constant volume was studied, using the algorithmic tools of Computational Singular Perturbation. Since ammonia combustion is characterized by both unrealistically long ignition delays and elevated NO x emissions, the time frame of action of the modes that are responsible for ignition was analyzed by calculating the developing time scales throughout the process and by studying their possible relation to NO x emissions. The reactions that support or oppose the explosive time scale were identified, along with the variables that are related the most to the dynamics that drive the system to an explosion. It is shown that reaction H 2 O 2 (+M) → OH + OH (+M) is the one contributing the most to the time scale that characterizes ignition and that its reactant H 2 O 2 is the species related the most to this time scale. These findings suggested that addition of H 2 O 2 in the initial mixture will influence strongly the evolution of the process. It was shown that ignition of pure ammonia advanced as a slow thermal explosion with very limited chemical runaway. The ignition delay could be reduced by more than two orders of magnitude through H 2 O 2 addition, which causes only a minor increase in NO x emissions.

A new Michaelis-Menten equation valid everywhere multi-scale dynamics prevails

The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.

Geometric singular perturbation analysis of a dynamical target mediated drug disposition model

In this paper we present a mathematical analysis of a pharmacological ODE model for target mediated drug disposition (TMDD). It is known that solutions of this model undergo four qualitatively different phases. In this paper we provide a mathematical identification of these separate phases by viewing the TMDD model as a singular perturbed system. Our analysis is based on geometric singular perturbation theory and we believe that this approach systemizes-and sheds further light on-the scalings arguments used by previous authors. In particular, we present a novel description of the third phase through a distinguished solution of a nonlinear differential equation. We also describe the solution curve for large values of initial drug doses and recover, en route, a result by Aston et al. (J Math Biol 68(6):1453-1478, 2014) on rebounding using our alternative perturbation approach. Finally, from our main result we derive a new method for estimating the parameters of the system in the event that detailed data is available. Ideally our approach to the TMDD model should stimulate further research into applications of these methods to more complicated models in pharmacology.

Mathematical analysis and drug exposure evaluation of pharmacokinetic models with endogenous production and simultaneous first-order and Michaelis-Menten elimination: the case of single dose

Drugs with an additional endogenous source often exhibit simultaneous first-order and Michaelis-Menten elimination and are becoming quite common in pharmacokinetic modeling. In this paper, we investigate the case of single dose intravenous bolus administration for the one-compartment model. Relying on a formerly introduced transcendent function, we were able to analytically express the concentration time course of this model and provide the pharmacokinetic interpretation of its components. Using the concept of the corrected concentration, the mathematical expressions for the partial and total areas under the concentration time curve (AUC) were also given. The impact on the corrected concentration and AUC is discussed as well as the relative contribution of the exogenous part in presence of endogenous production. The present findings theoretically elucidate several pharmacokinetic issues for the considered drug compounds and provide guidance for the rational estimation of their pharmacokinetic parameters.

Target-mediated drug disposition with drug-drug interaction, Part I: single drug case in alternative formulations

Target-mediated drug disposition (TMDD) describes drug binding with high affinity to a target such as a receptor. In application TMDD models are often over-parameterized and quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are essential to reduce the number of parameters. However, implementation of such approximations becomes difficult for TMDD models with drug-drug interaction (DDI) mechanisms. Hence, alternative but equivalent formulations are necessary for QE or QSS approximations. To introduce and develop such formulations, the single drug case is reanalyzed. This work opens the route for straightforward implementation of QE or QSS approximations of DDI TMDD models. The manuscript is the first part to introduce DDI TMDD models with QE or QSS approximations.