Low-dimensional neural ODEs and their application in pharmacokinetics.

Machine Learning (ML) is a fast-evolving field, integrated in many of today's scientific disciplines. With the recent development of neural ordinary differential equations (NODEs), ML provides a new tool to model dynamical systems in the field of pharmacology and pharmacometrics, such as pharmacokinetics (PK) or pharmacodynamics. The novel and conceptionally different approach of NODEs compared to classical PK modeling creates challenges but also provides opportunities for its application. In this manuscript, we introduce the functionality of NODEs and develop specific low-dimensional NODE structures based on PK principles. We discuss two challenges of NODEs, overfitting and extrapolation to unseen data, and provide practical solutions to these problems. We illustrate concept and application of our proposed low-dimensional NODE approach with several PK modeling examples, including multi-compartmental, target-mediated drug disposition, and delayed absorption behavior. In all investigated scenarios, the NODEs were able to describe the data well and simulate data for new subjects within the observed dosing range. Finally, we briefly demonstrate how NODEs can be combined with mechanistic models. This research work enhances understanding of how NODEs can be applied in PK analyses and illustrates the potential for NODEs in the field of pharmacology and pharmacometrics.

Clinically practical pharmacometrics computer model to evaluate and personalize pharmacotherapy in pediatric rare diseases: application to Graves' disease.

Objectives: Graves' disease (GD) with onset in childhood or adolescence is a rare disease (ORPHA:525731). Current pharmacotherapeutic approaches use antithyroid drugs, such as carbimazole, as monotherapy or in combination with thyroxine hormone substitutes, such as levothyroxine, as block-and-replace therapy to normalize thyroid function and improve patients' quality of life. However, in the context of fluctuating disease activity, especially during puberty, a considerable proportion of pediatric patients with GD is suffering from thyroid hormone concentrations outside the therapeutic reference ranges. Our main goal was to develop a clinically practical pharmacometrics computer model that characterizes and predicts individual disease activity in children with various severity of GD under pharmacotherapy. Methods: Retrospectively collected clinical data from children and adolescents with GD under up to two years of treatment at four different pediatric hospitals in Switzerland were analyzed. Development of the pharmacometrics computer model is based on the non-linear mixed effects approach accounting for inter-individual variability and incorporating individual patient characteristics. Disease severity groups were defined based on free thyroxine (FT4) measurements at diagnosis. Results: Data from 44 children with GD (75% female, median age 11 years, 62% receiving monotherapy) were analyzed. FT4 measurements were collected in 13, 15, and 16 pediatric patients with mild, moderate, or severe GD, with a median FT4 at diagnosis of 59.9 pmol/l (IQR 48.4, 76.8), and a total of 494 FT4 measurements during a median follow-up of 1.89 years (IQR 1.69, 1.97). We observed no notable difference between severity groups in terms of patient characteristics, daily carbimazole starting doses, and patient years. The final pharmacometrics computer model was developed based on FT4 measurements and on carbimazole or on carbimazole and levothyroxine doses involving two clinically relevant covariate effects: age at diagnosis and disease severity. Discussion: We present a tailored pharmacometrics computer model that is able to describe individual FT4 dynamics under both, carbimazole monotherapy and carbimazole/levothyroxine block-and-replace therapy accounting for inter-individual disease progression and treatment response in children and adolescents with GD. Such clinically practical and predictive computer model has the potential to facilitate and enhance personalized pharmacotherapy in pediatric GD, reducing over- and underdosing and avoiding negative short- and long-term consequences. Prospective randomized validation trials are warranted to further validate and fine-tune computer-supported personalized dosing in pediatric GD and other rare pediatric diseases.

Computing optimal drug dosing with OptiDose: implementation in NONMEM.

Determining a drug dosing recommendation with a PKPD model can be a laborious and complex task. Recently, an optimal dosing algorithm (OptiDose) was developed to compute the optimal doses for any pharmacometrics/PKPD model for a given dosing scenario. In the present work, we reformulate the underlying optimal control problem and elaborate how to solve it with standard commands in the software NONMEM. To demonstrate the potential of the OptiDose implementation in NONMEM, four relevant but substantially different optimal dosing tasks are solved. In addition, the impact of different dosing scenarios as well as the choice of the therapeutic goal on the computed optimal doses are discussed.

OptiDose: Computing the Individualized Optimal Drug Dosing Regimen Using Optimal Control.

Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic-pharmacodynamic models in substantially different scenarios with various routes of administration by solving an optimal control problem. The aim is to compute a control that brings the underlying system as closely as possible to a desired reference function by minimizing a cost functional. In pharmacokinetic-pharmacodynamic modeling, the controls are the administered doses and the reference function can be the disease progression. Drug administration at certain time points provides a finite number of discrete controls, the drug doses, determining the drug concentration and its effect on the disease progression. Consequently, rewriting the cost functional gives a finite-dimensional optimal control problem depending only on the doses. Adjoint techniques allow to compute the gradient of the cost functional efficiently. This admits to solve the optimal control problem with robust algorithms such as quasi-Newton methods from finite-dimensional optimization. OptiDose is applied to three relevant but substantially different pharmacokinetic-pharmacodynamic examples.

Delay differential equations based models in NONMEM.

Delay differential equations (DDEs) are commonly used in pharmacometric models to describe delays present in pharmacokinetic and pharmacodynamic data analysis. Several DDE solvers have been implemented in NONMEM 7.5 for the first time. Two of them are based on algorithms already applied elsewhere, while others are extensions of existing ordinary differential equations (ODEs) solvers. The purpose of this tutorial is to introduce basic concepts underlying DDE based models and to show how they can be developed using NONMEM. The examples include previously published DDE models such as logistic growth, tumor growth inhibition, indirect response with precursor pool, rheumatoid arthritis, and erythropoiesis-stimulating agents. We evaluated the accuracy of NONMEM DDE solvers, their ability to handle stiff problems, and their performance in parameter estimation using both first-order conditional estimation (FOCE) and the expectation-maximization (EM) method. NONMEM control streams and excerpts from datasets are provided for all discussed examples. All DDE solvers provide accurate and precise solutions with the number of significant digits controlled by the error tolerance parameters. For estimation of population parameters, the EM method is more stable than FOCE regardless of the DDE solver.

Modeling of levothyroxine in newborns and infants with congenital hypothyroidism: challenges and opportunities of a rare disease multi-center study.

Modeling of retrospectively collected multi-center data of a rare disease in pediatrics is challenging because laboratory data can stem from several decades measured with different assays. Here we present a retrospective pharmacometrics (PMX) based data analysis of the rare disease congenital hypothyroidism (CH) in newborns and infants. Our overall aim is to develop a model that can be applied to optimize dosing in this pediatric patient population since suboptimal treatment of CH during the first 2 years of life is associated with a reduced intelligence quotient between 10 and 14 years. The first goal is to describe a retrospectively collected dataset consisting of 61 newborns and infants with CH up to 2 years of age. Overall, 505 measurements of free thyroxine (FT4) and 510 measurements of thyrotropin or thyroid-stimulating hormone were available from patients receiving substitution treatment with levothyroxine (LT4). The second goal is to introduce a scale/location-scale normalization method to merge available FT4 measurements since 34 different postnatal age- and assay-specific laboratory reference ranges were applied. This method takes into account the change of the distribution of FT4 values over time, i.e. a transformation from right-skewed towards normality during LT4 treatment. The third goal is to develop a practical and useful PMX model for LT4 treatment to characterize FT4 measurements, which is applicable within a clinical setting. In summary, a time-dependent normalization method and a practical PMX model are presented. Since there is no on-going or planned development of new pharmacological approaches for CH, PMX based modeling and simulation can be leveraged to personalize dosing with the goal to enhance longer-term neurological outcome in children with the rare disease CH.

Target-Mediated Drug Disposition Model for Bispecific Antibodies: Properties, Approximation, and Optimal Dosing Strategy.

Bispecific antibodies (BsAbs) bind to two different targets, and create two binary and one ternary complex (TC). These molecules have shown promise as immuno-oncology drugs, and the TC is considered the pharmacologically active species that drives their pharmacodynamic effect. Here, we have presented a general target-mediated drug disposition (TMDD) model for these BsAbs, which bind to two different targets on different cell membranes. The model includes four different binding events for BsAbs, turnover of the targets, and internalization of the complexes. In addition, a quasi-equilibrium (QE) approximation with decreased number of binding parameters and, if necessary, reduced internalization parameters is presented. The model is further used to investigate the kinetics of BsAb and TC concentrations. Our analysis shows that larger doses of BsAbs may delay the build-up of the TC. Consequently, a method to compute the optimal dosing strategy of BsAbs, which will immediately create and maintain maximal possible TC concentration, is presented.

Delayed logistic indirect response models: realization of oscillating behavior.

Indirect response (IDR) models are probably the most frequently applied tools relating the effect of a signal to a baseline response. A response modeled by such a classical IDR model will always return monotonously to its baseline after drug administration. We extend IDR models with a delay process, i.e. a retarded response state, that leads to oscillating response behavior. First, IDR models with a first-order production and second-order loss term based on the famous logistic equation are constructed. Second, a delay process similar to the delayed logistic equation is included. Relations of the classical IDR model with our extended IDR model concerning response and model parameters are revealed. Simulations of typical response profiles are presented and data fitting of a model for leptin and cholesterol dynamics after administration of methylprednisolone is performed. The influence of the delay parameter on the other model parameters is discussed.

Facilitate Treatment Adjustment After Overdosing: Another Step Toward 21st-Century Medicine.

Overdosing occurs frequently because of prescription errors in neonates, infants, children, adolescents, and adults. Currently there is no quantitative approach that can be used by clinicians to adjust dosing so that toxic drug concentrations can be brought back to levels observed with safe and efficacious therapeutic doses. We present a mathematical solution that offers the time between last overdosing and next therapeutic dose to achieve therapeutic drug concentrations as soon as possible. To facilitate applications of this solution in clinical practice, a minimal amount of information has to be provided, and no simulations are necessary to compute the optimal waiting time. For educational purposes, we provide access to an online decision support tool for overdosing situations (Time to next Dose Calculator) that (1) computes the waiting time after accidental overdosing in patients with normal elimination and (2) computes the waiting time and adjusted reference dosing for patients with abnormal elimination. This user-friendly online tool will help clinicians to quickly adjust a dosing schedule in overdosing situations to mitigate risk for negative clinical consequences.

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.

Target mediated drug disposition with drug-drug interaction, Part II: competitive and uncompetitive cases.

We present competitive and uncompetitive drug-drug interaction (DDI) with target mediated drug disposition (TMDD) equations and investigate their pharmacokinetic DDI properties. For application of TMDD models, quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are necessary to reduce the number of parameters. To realize those approximations of DDI TMDD models, we derive an ordinary differential equation (ODE) representation formulated in free concentration and free receptor variables. This ODE formulation can be straightforward implemented in typical PKPD software without solving any non-linear equation system arising from the QE or QSS approximation of the rapid binding assumptions. This manuscript is the second in a series to introduce and investigate DDI TMDD models and to apply the QE or QSS approximation.

Assessment of non-linear combination effect terms for drug-drug interactions.

Drugs interact with their targets in different ways. A diversity of modeling approaches exists to describe the combination effects of two drugs. We investigate several combination effect terms (CET) regarding their underlying mechanism based on drug-receptor binding kinetics, empirical and statistical summation principles and indirect response models. A list with properties is provided and the interrelationship of the CETs is analyzed. A method is presented to calculate the optimal drug concentration pair to produce the half-maximal combination effect. This work provides a comprehensive overview of typically applied CETs and should shed light into the question as to which CET is appropriate for application in pharmacokinetic/pharmacodynamic models to describe a specific drug-drug interaction mechanism.

Distributed transit compartments for arbitrary lifespan distributions in aging populations.

Transit compartment models (TCM) are often used to describe aging populations where every individual has its own lifespan. However, in the TCM approach these lifespans are gamma-distributed which is a serious limitation because often the Weibull or more complex distributions are realistic. Therefore, we extend the TCM concept to approximately describe any lifespan distribution and call this generalized concept distributed transit compartment models (DTCMs). The validity of DTCMs is obtained by convergence investigations. From the mechanistic perspective the transit rates are directly controlled by the lifespan distribution. Further, DTCMs could be used to approximate the convolution of a signal with a probability density function. As example a stimulatory effect of a drug in an aging population with a Weibull-distributed lifespan is presented where distribution and model parameters are estimated based on simulated data.

Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations.

In pharmacokinetics/pharmacodynamics (PKPD) the measured response is often delayed relative to drug administration, individuals in a population have a certain lifespan until they maturate or the change of biomarkers does not immediately affects the primary endpoint. The classical approach in PKPD is to apply transit compartment models (TCM) based on ordinary differential equations to handle such delays. However, an alternative approach to deal with delays are delay differential equations (DDE). DDEs feature additional flexibility and properties, realize more complex dynamics and can complementary be used together with TCMs. We introduce several delay based PKPD models and investigate mathematical properties of general DDE based models, which serve as subunits in order to build larger PKPD models. Finally, we review current PKPD software with respect to the implementation of DDEs for PKPD analysis.

Solution and implementation of distributed lifespan models.

We consider a population where every individual has a unique lifespan. After expiring of its lifespan the individual has to leave the population. A realistic approach to describe these lifespans is by a continuous distribution. Such distributed lifespan models (DLSMs) were introduced earlier in the indirect response context and consist of the mathematical convolution operator to describe the rate of change. Therefore, DLSMs could not directly be implemented in standard PKPD software. In this work we present the solution representation of DLSMs with and without a precursor population and an implementation strategy for DLSMs in ADAPT , NONMEM and MATLAB . We fit hemoglobin measurements from literature and investigate computational properties.

General relationship between transit compartments and lifespan models.

Transit compartment models (TCM) are important tools in pharmacokinetic/pharmacodynamic (PKPD) modeling. In this work we investigate the relationship between TCMs with arbitrary initial values and lifespan models (LSM) with non-constant past and constant lifespan. We show that the total population in all transit compartments converges to a LSM, if the number of compartments n tends to infinity. The key to obtain this result is to establish a relationship between the initial values of the TCM and the non-constant past of the LSM. We apply the result to two already published PKPD models.

Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice treated with an anti-GM-CSF antibody.

Collagen-induced arthritis (CIA) in mice is an experimental model for rheumatoid arthritis, a human chronic inflammatory destructive disease. The therapeutic effect of neutralizing the cytokine granulocyte-macrophage colony-stimulating factor (GM-CSF) by an antibody was examined in the mouse disease in a view of deriving a pharmacokinetic/pharmacodynamic (PKPD) model. In CIA mice the development of disease is measured by a total arthritic score (TAS) and an ankylosis score (AKS). We present a multi-response PKPD model which describes the time course of the unperturbed and perturbed TAS and AKS. The antibody acts directly on GM-CSF by binding to it. Therefore, a compartment for the cytokine GM-CSF is an essential component of the mathematical model. This compartment drives the disease development in the PKPD model. Different known properties of arthritis development in the CIA model are included in the PKPD model. Firstly, the inflammation, driven by GM-CSF, dominates at the beginning of the disease and decreases after some time. Secondly, a destructive (ankylosis) part evolves in the TAS that is delayed in time. In order to model these two properties a delay differential equation was used. The PKPD model was applied to different experiments with doses ranging from 0.1 to 100 mg/kg. The influence of the drug was modeled by a non-linear approach. The final mathematical model consists of three differential equations representing the compartments for GM-CSF, inflammation and destruction. Our mathematical model described well all available dosing schedules by a simultaneous fit. We also present an equivalent and easy reformulation as ordinary differential equation which grants the use of standard PKPD software.

Modeling of tumor growth and anticancer effects of combination therapy.

Combination therapies are widely used in the treatment of patients with cancer. Selecting synergistic combination strategies is a great challenge during early drug development. Here, we present a pharmacokinetic/pharmacodynamic (PK/PD) model with a smooth nonlinear growth function to characterize and quantify anticancer effect of combination therapies using time-dependent data. To describe the pharmacological effect of combination therapy, an interaction term was introduced into a semi-mechanistic anticancer PK/PD model. This approach enables testing of a pharmacological hypothesis with respect to an anticipated pharmacological synergy of drug combinations, such as an assumed pharmacological synergy of complementary inhibition of a particular signaling pathway. The model was applied to three real data sets derived from preclinical screening experiments using xenograft mice. The suggested model fitted well the observed data from mono- to combination-therapy and allowed physiologically meaningful interpretation of the experiments. The tested drug combinations were assessed for their ability to act as synergistic modulators of tumor growth inhibition by the interaction parameter psi. The presented approach has practical implications because it can be applied to standard xenograft experiments and may assist in the selection of both optimal drug combinations and administration schedules. The unique feature of the presented approach is the ability to characterize the nature of combined drug interaction as well as to quantify the intensity of such interactions by assessing the time course of combined drug effect.