LAPP Leiden – Mathematicians Meet Molecules: Characterization of the dynamics of drug-binding models #SWI2026

Introduction

This problem is in the field of pharmacometrics and concerns a question about mathematical modelling of target-mediated drug disposition. Pharmacometrics is the quantitative analysis of drug-related data using mathematical and statistical models to understand how drugs affect the body and disease. Models describe in a simplified fashion what happens to the drug after dosing (called pharmacokinetics) and what effect the drug has on the body (called pharmacodynamics). Usually, models describe the subject’s body as a set of compartments that the drug can traverse. Compartments can, for example, represent blood or plasma, organs or tissues, or be a more abstract representation of a combination of several of these. Other compartments may be used to describe the drug after metabolization or binding. The drug typically goes through 4 processes: absorption, distribution and metabolization, and elimination (ADME).

Models take the form of ordinary differential equations (ODE), with parameters whose values are typically unknown and estimated based on the data gathered in studies. The simplest approach to modelling ADME is by linear ODE’s where all drug flows are assumed to be proportional to concentration. One of the main challenges in pharmacometrics is to fit the model parameters on data collected from the subjects.

Target mediated drug disposition

Each drug has a specific target in the body, which evokes the desired effect of the drug. For many drugs the target is a receptor, which is a protein located on the cell surface. After entering the body, by administration of the drug, the drug can (reversibly) bind its target. After binding, the drug-receptor complex is internalized and can be destroyed inside the cell. This creates a non-linear elimination process because the amount of receptor is limited and so the binding can be saturated at high levels of drug. Next to the nonlinear elimination pathways, the drug is also eliminated via a linear elimination pathway, e.g., by filtration by the kidneys. This pathway is usually slower than the nonlinear pathway.

This is modelled by a so-called target mediated drug disposition model (TMDD), illustrated in Figure 1.

 

Drug is dosed via a depot compartment, representing e.g. the digestive system or an administration via injection into the skin. It is then linearly absorbed into the central compartment, representing blood. From there, it can distribute to a peripheral compartment (for example, an organ or fat tissue), be linearly eliminated, or bind to a receptor to form a complex. The binding is reversible, and the complex internalizes and is then destroyed. The receptor (or target) is synthesized and degraded in a turnover process, where the receptor is formed with a constant formation rate, and degraded linearly, i.e., in proportion to the amount of receptor).

 

Figure 2 shows time profiles for total (i.e., free + bound) drug plasma concentration. The profiles are non-linear because of the saturation of the binding. The concentration first increases because of absorption inflow, then reaches a peak, followed by a decline when elimination becomes the dominant process.

 

Problem statement

In some parameter regions and for some dose levels, a TMDD model may exhibit a second peak, as illustrated by Figure 3. We would like to understand why and when this happens. More specifically:

• What is the mechanism causing the second peak? Which processes (absorption, distribution, etc.) are essential for this to happen?
• What is the parameter region where this happens?
• How high is the second peak (relative to the first peak or the dose), and can it be higher than the first one?
• There are alternative mechanisms that may cause a second peak, such as recycling or reuptake from the gastrointestinal tract (entero-hepatic circulation) or a baseline variation during the day (circadian rhythm). Is it possible to identify the mechanism from the time profile? Ideally, it would be possible to do this with discrete, noisy data.