Understanding the Behavior of Systems Pharmacology Models Using Mathematical Analysis of Differential Equations: Prolactin Modeling as a Case Study

Dynamical systems analysis as an additional tool to inform treatment outcomes: The case study of a quantitative systems pharmacology model of immuno-oncology

Quantitative systems pharmacology (QSP) proved to be a powerful tool to elucidate the underlying pathophysiological complexity that is intensified by the biological variability and overlapped by the level of sophistication of drug dosing regimens. Therapies combining immunotherapy with more traditional therapeutic approaches, including chemotherapy and radiation, are increasingly being used. These combinations are purposed to amplify the immune response against the tumor cells and modulate the suppressive tumor microenvironment (TME). In order to get the best performance from these combinatorial approaches and derive rational regimen strategies, a better understanding of the interaction of the tumor with the host immune system is needed. The objective of the current work is to provide new insights into the dynamics of immune-mediated TME and immune-oncology treatment. As a case study, we will use a recent QSP model by Kosinsky et al. [J. Immunother. Cancer 6, 17 (2018)] that aimed to reproduce the dynamics of interaction between tumor and immune system upon administration of radiation therapy and immunotherapy. Adopting a dynamical systems approach, we here investigate the qualitative behavior of the representative components of this QSP model around its key parameters. The ability of T cells to infiltrate tumor tissue, originally identified as responsible for individual therapeutic inter-variability [Y. Kosinsky et al., J. Immunother. Cancer 6, 17 (2018)], is shown here to be a saddle-node bifurcation point for which the dynamical system oscillates between two states: tumor-free or maximum tumor volume. By performing a bifurcation analysis of the physiological system, we identified equilibrium points and assessed their nature. We then used the traditional concept of basin of attraction to assess the performance of therapy. We showed that considering the therapy as input to the dynamical system translates into the changes of the trajectory shapes of the solutions when approaching equilibrium points and thus providing information on the issue of therapy.

Importance of Stability Analysis When Using Nonlinear Semimechanistic Models to Describe Drug-Induced Hematotoxicity

Stability analysis, often overlooked in pharmacometrics, is essential to explore dynamical systems. The model developed by Friberg et al.¹ to describe drug‐induced hematotoxicity is widely used to support decisions across drug development, and parameter values are often identified from observed blood counts. We use stability analysis to study the parametric dependence of stable and unstable solutions of several Friberg‐type models and highlight the risks associated with system instability in the context of nonlinear mixed effects modeling. We emphasize the consequences of unstable solutions on prediction performance by demonstrating nonbiological system behaviors in a real case study of drug‐induced thrombocytopenia. Ultimately, we provide simple criteria for identifying parameters associated with stable solutions of Friberg‐type models. For instance, in the original Friberg model, we find that stability depends only on the parameter that governs the feedback from peripheral cells to progenitors and provide the exact range of values that results in stable solutions.

Estimating parameters of nonlinear dynamic systems in pharmacology using chaos synchronization and grid search

Bridging fundamental approaches to model optimization for pharmacometricians, systems pharmacologists and statisticians is a critical issue. These fields rely primarily on Maximum Likelihood and Extended Least Squares metrics with iterative estimation of parameters. Our research combines adaptive chaos synchronization and grid search to estimate physiological and pharmacological systems with emergent properties by exploring deterministic methods that are more appropriate and have potentially superior performance than classical numerical approaches, which minimize the sum of squares or maximize the likelihood. We illustrate these issues with an established model of cortisol in human with nonlinear dynamics. The model describes cortisol kinetics over time, including its chaotic oscillations, by a delay differential equation. We demonstrate that chaos synchronization helps to avoid the tendency of the gradient-based optimization algorithms to end up in a local minimum. The subsequent analysis illustrates that the hybrid adaptive chaos synchronization for estimation of linear parameters with coarse-to-fine grid search for optimal values of non-linear parameters can be applied iteratively to accurately estimate parameters and effectively track trajectories for a wide class of noisy chaotic systems.

Modeling of prolactin response following dopamine D2 receptor antagonists in rats: can it be translated to clinical dosing?

Prolactin release is a side effect of antipsychotic therapy with dopamine antagonists, observed in rats as well as humans. We examined whether two semimechanistic models could describe prolactin response in rats and subsequently be translated to predict pituitary dopamine D₂ receptor occupancy and plasma prolactin concentrations in humans following administration of paliperidone or remoxipride. Data on male Wistar rats receiving single or multiple doses of risperidone, paliperidone, or remoxipride was described by two semimechanistic models, the precursor pool model and the agonist–antagonist interaction model. Using interspecies scaling approaches, human D₂ receptor occupancy and plasma prolactin concentrations were predicted for a range of clinical paliperidone and remoxipride doses. The predictions were compared with corresponding observations described in literature as well as with predictions from published models developed on human data. The pool model could predict D₂ receptor occupancy and prolactin response in humans following single doses of paliperidone and remoxipride. Tolerance of prolactin release was predicted following multiple doses. The interaction model underpredicted both D₂ receptor occupancy and prolactin response. Prolactin elevation may be deployed as a suitable biomarker for interspecies translation and can inform the clinical safe and effective dose range of antipsychotic drugs. While the pool model was more predictive than the interaction model, it overpredicted tolerance on multiple dosing. Shortcomings of the translations reflect the need for better mechanistic models.

Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models

Drug dosing regimen can significantly impact drug effect and, thus, the success of treatments. Nevertheless, trial and error is still the most commonly used method by conventional pharmacometric approaches to optimize dosing regimen. In this tutorial, we utilize four distinct classes of quantitative systems pharmacology models to introduce frequency-domain response analysis, a method widely used in electrical and control engineering that allows the analytical optimization of drug treatment regimen from the dynamics of the model.

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.

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.

Revealing the Neuroendocrine Response After Remoxipride Treatment Using Multi-Biomarker Discovery and Quantifying It by PK/PD Modeling

To reveal unknown and potentially important mechanisms of drug action, multi-biomarker discovery approaches are increasingly used. Time-course relationships between drug action and multi-biomarker profiles, however, are typically missing, while such relationships will provide increased insight in the underlying body processes. The aim of this study was to investigate the effect of the dopamine D2 antagonist remoxipride on the neuroendocrine system. Different doses of remoxipride (0, 0.7, 5.2, or 14 mg/kg) were administered to rats by intravenous infusion. Serial brain extracellular fluid (brainECF) and plasma samples were collected and analyzed for remoxipride pharmacokinetics (PK). Plasma samples were analyzed for concentrations of the eight pituitary-related hormones as a function of time. A Mann-Whitney test was used to identify the responding hormones, which were further analyzed by pharmacokinetic/pharmacodynamic (PK/PD) modeling. A three-compartment PK model adequately described remoxipride PK in plasma and brainECF. Not only plasma PRL, but also adrenocorticotrophic hormone (ACTH) concentrations were increased, the latter especially at higher concentrations of remoxipride. Brain-derived neurotropic factor (BDNF), follicle stimulating hormone (FSH), growth hormone (GH), luteinizing hormone (LH), and thyroid stimulating hormones (TSH) did not respond to remoxipride at the tested doses, while oxytocin (OXT) measurements were below limit of quantification. Precursor pool models were linked to brainECF remoxipride PK by E_max drug effect models, which could accurately describe the PRL and ACTH responses. To conclude, this study shows how a multi-biomarker identification approach combined with PK/PD modeling can reveal and quantify a neuroendocrine multi-biomarker response for single drug action.