Prediction variability in physiologically based pharmacokinetic modeling of tissue disposition under deep uncertainty

This study quantifies how deep uncertainty in parameter predictions and model assumptions significantly impacts the variability of tissue-specific pharmacokinetic outcomes in physiologically based pharmacokinetic (PBPK) models, revealing substantial prediction discrepancies particularly for lipophilic, protonated molecules.

The Gist

Imagine you are a chef trying to invent a new recipe for a super-delicious cake. You have a list of ingredients (the chemical structure of a new drug) and a cookbook (the PBPK model) that tells you how those ingredients will behave in the oven (the human body).

The goal of this paper is to figure out: How much can we trust the cookbook's predictions when we don't have the exact ingredients in front of us, and we have to guess their properties?

Here is the story of the research, broken down into simple concepts:

1. The Setup: The "Virtual Kitchen"

In the world of drug discovery, scientists use computers to test thousands of potential medicines before ever making them in a lab. They use PBPK models (Physiologically Based Pharmacokinetic models). Think of these models as a high-tech GPS for drugs. Once you tell the GPS where the drug starts (your mouth or a vein), it predicts exactly where the drug will go in the body, how long it will stay there, and how much of it will reach the target (like a tumor or an infected cell).

Usually, this GPS works great if you give it exact measurements of the drug (like its weight, how oily it is, how acidic it is). But in the "virtual kitchen," scientists often don't have the real drug yet. They have to use AI guesses (Machine Learning) to estimate those measurements.

2. The Problem: The "Fuzzy Compass"

The problem is that AI guesses aren't perfect. They are like a fuzzy compass.

• If the compass says "North," it might actually be North-North-East.
• If the compass says "10 miles," it might be 12 miles.

The researchers asked: If we feed these "fuzzy" guesses into our drug GPS, does the final destination prediction become a total mess? Or does the GPS stay reliable enough to tell us which drugs are worth testing?

3. The Experiment: Testing Four Different Maps

The researchers took four different versions of the drug GPS (four different mathematical models) and tested them against real-world data first to make sure they worked. Then, they introduced "fuzziness" (uncertainty) to the input data to simulate what happens when we rely on AI guesses.

They created 10,000 fake drugs (pseudomolecules) and ran them through these four maps, adding random errors to the inputs to see how the predictions wobbled.

4. The Big Discovery: The "Lipophilic Proton" Trap

The results revealed a fascinating and tricky pattern:

• Most of the time, the maps agreed. For most drugs, even with fuzzy inputs, all four models pointed to roughly the same destination.
• But for a specific type of drug, the maps went crazy. They found a "danger zone" in the chemical world: Drugs that are both very oily (lipophilic) and very charged (protonated).

Think of these drugs like greasy magnets.

• Because they are oily, they love to stick to fatty tissues (like adipose tissue).
• Because they are charged, they love to stick to specific proteins in the blood.

When the models tried to predict where these "greasy magnets" would go, the four different maps started arguing with each other. One map said, "It will stay in the fat!" Another said, "It will rush to the liver!" A third said, "It will get stuck in the muscle!"

5. Why Did They Disagree? (The "Rulebook" Differences)

The researchers dug into why the maps disagreed. It turned out the models had different rulebooks for how to handle these greasy magnets:

• Model A assumed that if a molecule is charged, it can't stick to fat.
• Model B assumed that if a molecule is charged, it can still stick to fat, but only if it's really oily.
• Model C had a special "calibration" step that tweaked the numbers based on past data, which made it very stable but maybe too rigid.

When the input data was fuzzy (the AI guesses were slightly off), these small differences in the rulebooks got amplified. It's like four people trying to navigate a foggy forest. If they all agree on the path, they are fine. But if one person thinks "North is left" and another thinks "North is right," and the fog is thick, they will end up in completely different places.

6. The Takeaway: Don't Trust a Single Map

The main lesson from this paper is a warning for drug developers:

1. Uncertainty is real: When we use AI to guess drug properties, the final prediction isn't a single number; it's a range of possibilities.
2. Watch out for "Greasy Magnets": If a new drug candidate is both oily and charged, different models might give you wildly different answers. You can't just pick one model and trust it blindly.
3. Better inputs matter: To get a better prediction, we don't just need better maps (models); we need better compasses (more accurate AI predictions for things like how oily or charged the drug is).

The Bottom Line

This study is like a safety check for the future of drug discovery. It tells us that while our computer models are powerful, they have blind spots. If we are designing a drug that is a "greasy magnet," we need to be extra careful, run multiple models, and understand that our predictions might have a bigger margin of error than we thought. It's a call to be humble about our predictions and to keep improving the tools we use to guess the properties of new medicines.

Technical Summary

1. Problem Statement

Physiologically based pharmacokinetic (PBPK) models are increasingly used in virtual screening workflows to predict drug behavior from chemical structures. These workflows often rely on Machine Learning (ML) or Quantitative Structure-Activity Relationship (QSAR) models to generate input parameters (e.g., partition coefficients, clearance, protein binding) for PBPK simulations.

However, this introduces deep epistemic uncertainty from two sources:

1. Structural Uncertainty: PBPK models are abstractions of physiology; different mathematical formulations (e.g., how they handle ionization or lipid binding) may misrepresent biological processes for certain chemical classes.
2. Parametric Uncertainty: QSAR predictions for virtual compounds contain significant errors (Mean Absolute Error) compared to experimental data.

The core problem is that it is unclear how these uncertainties propagate through different PBPK frameworks. Specifically, it is unknown whether different models yield consistent predictions for virtual compounds, which regions of chemical space are most prone to model disagreement, and which input parameters drive the most variance.

2. Methodology

The authors conducted a comprehensive uncertainty quantification study comparing four distinct PBPK models:

1. This Model: A modified Rodgers & Rowland (R&R)-Lukacova framework developed by the authors.
2. Mathew Model: An R&R-Lukacova based model (Mathew et al.).
3. Pearce (Uncalibrated): A Schmitt-based model (Pearce et al.).
4. Pearce (Calibrated): The same Schmitt-based model with empirical corrections applied to tissue partitioning coefficients ($K_p$).

Key Steps:

• Fidelity Validation: First, the models were validated against 1,854 experimental data points (rat tissue partitioning coefficients $K_p$ and human steady-state volume of distribution $V_{ss}$) using measured input parameters to establish baseline predictive fidelity.
• Synthetic Dataset Generation: A dataset of $10^4$ "pseudomolecules" was generated by correlated random sampling of six key physicochemical properties (Molecular Mass, $f_u$, $CL_{int}$, $\log D$, donor/acceptor $pK_a$) based on distributions from the Obach dataset.
• Monte Carlo Propagation: For each pseudomolecule, 1,000 Monte Carlo realizations were run. Input parameters were perturbed according to the typical Mean Absolute Error (MAE) of published QSAR models to simulate deep uncertainty.
• Outcome Metrics: The study tracked four PK outcomes: $V_{ss}$, maximum unbound concentration ($C_{max,u}$), area under the curve ($AUC_{u0-\infty}$), and time above a threshold ($T_{0.1\mu M}$) in muscle intracellular water.
• Clustering & Sensitivity Analysis:
  • Model Agreement: Wasserstein distances were calculated between the output distributions of the four models. K-means clustering grouped molecules based on agreement patterns.
  • Global Sensitivity Analysis: A Sobol' variance-based analysis (using the Kucherenko estimator for correlated inputs) was performed to quantify how much each input parameter contributed to output variance.

3. Key Contributions

• Quantification of Deep Uncertainty: The study moves beyond simple error metrics to quantify how parametric uncertainty (typical of QSAR predictions) propagates into PK outcome variability across different model structures.
• Identification of "Danger Zones" in Chemical Space: The authors identified a specific subset of chemical space (highly lipophilic, protonated molecules) where different PBPK models exhibit severe prediction discordance.
• Mechanistic Insight into Model Disagreement: The paper dissects why models disagree, linking specific mathematical assumptions (e.g., how ionization affects phospholipid binding) to prediction variance.
• Impact of Calibration: The study demonstrates that while empirical calibration improves nominal fidelity, it can constrain model flexibility and alter sensitivity to input parameters, potentially masking structural deficiencies.

4. Key Results

A. Model Fidelity vs. Uncertainty

• All four models showed broadly similar fidelity to experimental data when using measured inputs (accuracy within 2-fold error was comparable for most ion classes).
• However, under parameter uncertainty, prediction variance diverged significantly. The coefficient of variation for PK statistics ranged from $10^{-6}$ to 31 depending on the molecule and model.
• The Mathew model exhibited the highest variance and the most outliers, particularly for zwitterions and bases.

B. Chemical Space Clustering

• Clustering analysis revealed two distinct groups:
  • Cluster 2 (92.5% of molecules): Models agreed well (high concordance).
  • Cluster 1 (7.5% of molecules): Models showed significant discordance.
• Characteristics of Cluster 1: These molecules were exclusively bases (78%) and zwitterions (22%), characterized by high lipophilicity ($\log D$) and high protonation (high acceptor $pK_a$).
• In this region, the Mathew model was the primary outlier, disagreeing with the other three models on $V_{ss}$, $C_{max,u}$, and $T_{0.1\mu M}$.

C. Sources of Disagreement

• The discordance in Cluster 1 stems from structural assumptions regarding how ionization affects lipid binding.
  • Mathew Model: Uses $\log P$ (partition coefficient) and assumes only neutral species partition into neutral lipids. For highly protonated molecules, this effectively excludes lipid binding, leading to massive variance when $pK_a$ and $\log D$ interact.
  • Other Models: Use $\log D$ (distribution coefficient) and Schmitt's correlations, allowing ionized species to associate with acidic phospholipids, resulting in more stable predictions for this chemical class.

D. Sensitivity Analysis Findings

• Mathew Model: Variance in $V_{ss}$ was driven almost entirely by parameter interactions (correlated sensitivity), specifically between $\log D$ and $pK_a$. This implies that improving the prediction of a single parameter (e.g., just $\log D$) will not reduce uncertainty; all interacting parameters must be improved simultaneously.
• Pearce Model: Calibration reduced the influence of $\log D$ on variance but increased the sensitivity to fraction unbound ($f_u$).
• Dynamic Metrics: For $AUC_u$ and $T_{0.1\mu M}$, intrinsic clearance ($CL_{int}$) became a dominant driver of variance, particularly after calibration in the Pearce model, suggesting calibration can "gate-keep" the relevance of hepatic metabolism.

5. Significance and Implications

• Risk in Virtual Screening: Relying on a single PBPK model for virtual screening of novel chemical entities is risky, especially for lipophilic, ionizable compounds. The choice of model abstraction can lead to qualitatively different conclusions about drug exposure.
• QSAR Priorities: The study highlights that for models like Mathew's, reducing uncertainty requires improving the joint prediction of multiple correlated parameters ($\log D$ and $pK_a$), whereas for other models, improving specific inputs like $f_u$ or $\log D$ yields direct variance reduction.
• Calibration Trade-offs: While calibration improves fit to known drugs, it may reduce a model's ability to extrapolate to new chemical spaces or alter the sensitivity of the model to physiological parameters (like metabolism), potentially leading to false confidence in predictions.
• Guidance for Modelers: The authors recommend avoiding "binarized" assumptions (e.g., excluding lipid binding for ionized species) and suggest that modelers should explicitly test for prediction variability in regions of chemical space characterized by high lipophilicity and ionization.

In conclusion, the paper provides a critical framework for assessing the reliability of ML-informed PBPK workflows, demonstrating that structural model assumptions can amplify parametric uncertainty, leading to divergent predictions that could misguide early-stage drug discovery decisions.