📄 pharmacology and toxicology

Segmented regression to streamline semi-mechanistic pharmacodynamic modelling of antibiotic effects

This paper proposes a three-step workflow utilizing segmented regression on time-kill assay data to efficiently derive concentration-effect relationships and streamline the development of robust semi-mechanistic pharmacodynamic models for antibiotics, as demonstrated with colistin against *Pseudomonas aeruginosa*.

Original authors: Pham, A. D., van Os, W., Brink, J., van Hasselt, J. G. C., Aulin, L. B. S.

Published 2026-11-04

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CC BY 4.0

Original authors: Pham, A. D., van Os, W., Brink, J., van Hasselt, J. G. C., Aulin, L. B. S.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Antibiotics are the tools we use to stop bacteria from multiplying and causing infection, but bacteria are not passive targets. They are living organisms that can adapt, change, and sometimes survive the very drugs meant to kill them. To understand how to use these medicines effectively, scientists study the battle between a drug and a bacterium in a controlled setting, often using a method called a time-kill assay. In this experiment, researchers mix bacteria with different amounts of an antibiotic and watch what happens over time, counting how many bacteria are alive at each step. The goal is to build a mathematical picture of this interaction, a model that can predict how a specific dose will perform inside a human body. However, these bacterial populations are rarely uniform. Often, a drug kills the weak ones quickly, but a tougher, more resistant group survives and begins to grow back. Because scientists can only see the total number of bacteria, they cannot easily tell which ones are dying and which are surviving, making it difficult to build an accurate model without guessing too much.

A team of researchers at the Leiden Academic Centre for Drug Research in the Netherlands has developed a new, more straightforward way to build these models. Instead of trying to fit a complex equation to the entire messy curve of bacterial growth and death all at once, they broke the problem into three clear steps. First, they looked at the data from time-kill experiments involving the antibiotic colistin and a strain of Pseudomonas aeruginosa, a common and stubborn bacterium. They separated the timeline into two distinct phases: the initial period where the drug was killing the bacteria, and the later period where the surviving bacteria started to multiply again. By treating these two phases separately, they could calculate the exact speed of the kill and the speed of the regrowth for each drug concentration without getting lost in the complexity of the whole system.

Once they had these separate speeds, the researchers moved to the second step, which was to figure out how the drug concentration changed those speeds. They tested different mathematical shapes to see which one best described the relationship between the amount of drug and the rate of killing or regrowth. They found that the killing effect followed a specific pattern where the speed increased with the drug dose until it hit a limit, while the regrowth phase showed that the surviving bacteria were much harder to kill than the original group. This confirmed that the bacteria were not just one uniform group, but that a resistant subpopulation had emerged. This step allowed them to identify the most likely rules governing the bacteria's behavior before they ever tried to build the final, complex model.

In the final step, the team took the rules they had discovered and the specific numbers they had calculated and plugged them into a full, semi-mechanistic model. This is a type of computer simulation that tries to mimic the real biological processes, including how the bacteria grow, how the drug kills them, and how the resistant group takes over. Because they had already done the hard work of figuring out the basic relationships in the first two steps, the final model was much easier to build and more reliable. It successfully described the entire 24-hour timeline of the experiment, capturing both the rapid initial drop in bacteria and the slower, stubborn regrowth that followed. The numbers the model produced matched closely with the simpler calculations from the earlier steps, proving that this step-by-step approach works.

The researchers suggest that this three-step workflow offers a more efficient and robust way to develop these models compared to traditional methods, which often require endless trial and error. By separating the phases and estimating the slopes first, they reduced the need for guesswork and made the models less sensitive to initial assumptions. While this specific study focused on one type of bacteria and one antibiotic, the method is designed to be applied to other drugs and bacteria as well. The ultimate goal is to help doctors choose better dosing regimens that maximize the chance of curing an infection while minimizing the risk of the bacteria becoming resistant, ensuring that these vital medicines remain effective for as long as possible.

Technical Summary: Segmented Regression to Streamline Semi-Mechanistic Pharmacodynamic Modelling of Antibiotic Effects

Problem Statement
The development of semi-mechanistic pharmacodynamic (PD) models for antibiotics, typically informed by static time-kill (STK) assays, faces significant challenges regarding practical identifiability and computational efficiency. These models often require inferring the dynamics of unobserved bacterial subpopulations (e.g., susceptible and resistant strains) from total bacterial counts. This inference leads to sensitivity to initial parameter estimates, parameter bias, and a burdensome iterative process of model testing to select appropriate structures and concentration-effect relationships. Furthermore, estimating unobserved maximum effects within limited sampling ranges often results in practical identifiability issues.

Methodology
The authors propose a three-step workflow designed to decouple the estimation of concentration-effect relationships from the complex fitting of full semi-mechanistic models. The workflow is demonstrated using 24-hour STK data of colistin against a Pseudomonas aeruginosa isolate (PA11976).

Step 1: Segmented Regression: Log-transformed STK data are segmented into two distinct phases: an initial bacterial kill phase and a subsequent regrowth phase. The segmentation point is empirically determined by the time of minimum colony-forming units (CFU). Linear regression is applied to each phase to estimate concentration-specific slopes representing the net rate of kill or growth.
Step 2: Concentration-Effect Relationship Identification: The regression-derived slopes from Step 1 are pooled to fit empirical concentration-effect relationships. Three models are evaluated for each phase: linear, $E_{max}$ , and sigmoidal $E_{max}$ . Model selection is guided by visual inspection, parameter precision, and the Akaike Information Criterion (AIC). This step identifies the most suitable mathematical description of drug effect for both the kill and regrowth phases.
Step 3: Semi-Mechanistic PD Model Development: The relationships and parameter estimates identified in Step 2 are incorporated as initial estimates and structural constraints into a full semi-mechanistic PD model. This final model utilizes a two-subpopulation framework (susceptible $S$ and resistant $R$ ) governed by ordinary differential equations (ODEs). The kill phase informs the drug effect on the $S$ subpopulation, while the regrowth phase informs the drug effect on the $R$ subpopulation, linked by a transition rate ( $k_{SR}$ ).

Key Results

Phase Segmentation: The segmented regression successfully captured kill and regrowth slopes across all tested colistin concentrations.
Concentration-Effect Dynamics: The analysis revealed that the regrowth phase exhibited consistently weaker drug effects compared to the initial kill phase. Specifically, the regrowth phase showed lower slope values, lower $E_{max}$ , and higher $EC_{50}$ estimates, indicating a less susceptible subpopulation.
Model Selection: For the kill phase, a sigmoidal $E_{max}$ model provided the best fit (lowest AIC), while the regrowth phase was best described by a standard $E_{max}$ model.
Final Model Performance: The resulting semi-mechanistic model adequately described the bacterial dynamics across all concentrations. The parameters estimated in the full ODE model were comparable to those derived in Step 2, validating the workflow's ability to provide robust initial estimates. The model successfully captured the initial rapid killing by the susceptible population and the delayed regrowth driven by the emergence of the less-susceptible population.

Significance and Claims
The paper claims that this three-step workflow enables a more efficient and robust identification of concentration-effect relationships, thereby reducing the model-testing burden and computational complexity associated with developing semi-mechanistic antibiotic PD models. By extracting features directly from observed data before fitting complex ODEs, the approach mitigates issues related to sensitivity to initial estimates and practical identifiability. The authors position this method as a tool to streamline the development of models that are critical for optimizing dosing regimens and understanding resistance development, without asserting that the workflow identifies the single "optimal" biological model structure, but rather supports the efficient incorporation of concentration-effect relationships into such structures.

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