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PBPK-iPINNs: Inverse Physics-Informed Neural Networks for Physiologically Based Pharmacokinetic Brain Models

This study introduces PBPK-iPINN, an inverse Physics-Informed Neural Network framework that effectively estimates unknown parameters and drug concentration profiles in stiff, permeability-limited physiologically based pharmacokinetic brain models, outperforming traditional numerical methods to support optimized brain cancer therapies.

Original authors: Charuka D. Wickramasinghe, Krishanthi C. Weerasinghe, Pradeep K. Ranaweera, Nelum S. S. M. Hapuhinna

Published 2026-02-24

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CC0 1.0

Original authors: Charuka D. Wickramasinghe, Krishanthi C. Weerasinghe, Pradeep K. Ranaweera, Nelum S. S. M. Hapuhinna

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out how a specific medicine travels through a very complex, crowded city (the human brain) to reach a specific destination (a tumor). You can't just watch every car (drug molecule) move in real-time because the city is too big, and it's illegal to put cameras everywhere (ethical constraints).

Instead, you have a map (a mathematical model) that describes how traffic flows between different neighborhoods (compartments like blood, brain tissue, and fluid). But here's the problem: you don't know the exact speed limits, the size of the roads, or how many cars get stuck at traffic lights for this specific medicine. You only have a few blurry photos of traffic at certain times.

This is the challenge the paper tackles. Here is the simple breakdown of their solution:

1. The Old Way: Guessing and Checking

Traditionally, scientists try to solve this by using standard math tools (like numerical methods). Think of this like trying to navigate the city by manually calculating every single turn and stoplight.

The Problem: The brain's traffic system is "stiff." This means some parts change super fast while others change slowly. Standard math tools often get confused, crash, or give up because the calculations are too messy. They also struggle when the data is noisy or incomplete.

2. The New Way: The "Physics-Aware" GPS (PBPK-iPINN)

The authors created a new tool called PBPK-iPINN. To understand this, imagine a super-smart GPS driver who doesn't just look at the map, but also knows the laws of physics.

The "Brain" (Neural Network): This is a computer program that learns by trial and error, like a student taking a test.
The "Physics" (The Rules): Usually, if a student guesses wrong, they just get a bad grade. But this student has a special rulebook (the laws of mass balance and fluid dynamics). If the student guesses a route that violates the laws of physics (e.g., cars appearing out of thin air), the rulebook immediately corrects them.
The "Inverse" Part: Usually, you use a map to predict where traffic will go (Forward). Here, they are doing the reverse (Inverse). They have the blurry photos of traffic (data) and the rulebook (physics), and they are asking the AI: "What must the speed limits and road sizes be for this traffic pattern to exist?"

3. How It Works (The Analogy)

Imagine you are trying to bake a cake but you don't know the exact amount of flour, sugar, or baking powder. You only have the final cake and a recipe book that says, "If you mix these ingredients, they must react in a specific way."

The AI Chef: The AI tries different amounts of ingredients.
The Taste Test (Data Loss): It compares the cake it made to the real cake. If it's too sweet, it adjusts the sugar.
The Chemistry Test (Physics Loss): Even if the cake tastes okay, the AI checks the recipe book. Did it use enough baking powder to make it rise? If the chemistry doesn't add up, the AI knows the guess is wrong, even if the taste is close.
The Result: By balancing the taste test and the chemistry rules, the AI eventually figures out the exact recipe (the drug parameters) that created that specific cake.

4. Why This Matters

The authors tested this on a model of how a cancer drug (Abemaciclib) moves through the brain.

The Win: Their new AI method was incredibly accurate. It found the "recipe" (parameters) with an error so small it's like measuring the width of a human hair on a football field.
Comparison: When they compared their AI to traditional math methods (like the "manual calculation" approach), the AI was much better at handling the messy, "stiff" parts of the problem where the old methods failed.

5. The Real-World Impact

Why should you care?

Better Cancer Treatment: Brain cancer is hard to treat because drugs struggle to cross the "blood-brain barrier" (the city's security checkpoint).
Personalized Medicine: This tool can help doctors figure out exactly how a specific drug will behave in a specific patient's brain.
Faster Drug Development: Instead of running thousands of expensive and time-consuming lab experiments to guess how a drug works, scientists can use this "Physics-Aware GPS" to simulate and optimize treatments much faster.

In a nutshell: The paper introduces a smart AI that learns the rules of physics while looking at data. It uses this superpower to reverse-engineer how drugs move through the brain, helping doctors design better, safer treatments for brain cancer without needing to guess in the dark.

1. Problem Statement

The paper addresses the challenge of estimating unknown parameters in Physiologically Based Pharmacokinetic (PBPK) models, specifically for drug delivery to the human brain.

Context: PBPK models use systems of Ordinary Differential Equations (ODEs) to simulate drug absorption, distribution, metabolism, and excretion (ADME) across physiological compartments.
The Challenge:
- Inverse Problems: Determining drug-specific (e.g., clearance rates) and patient-specific (e.g., tissue volumes) parameters from observed concentration data is an inverse problem.
- Stiffness: The ODE systems in PBPK models are often "stiff," causing traditional numerical solvers (e.g., Runge-Kutta) and statistical optimization methods (e.g., Maximum Likelihood) to fail to converge or become computationally expensive.
- Data Scarcity & Noise: Direct measurement of parameters in humans is limited by ethical and practical constraints, leading to sparse or noisy data.
- Identifiability: It is often unclear if parameters can be uniquely determined from the available data (structural and practical identifiability).

2. Methodology

The authors propose PBPK-iPINN, a framework utilizing Inverse Physics-Informed Neural Networks (iPINNs) to solve these inverse problems.

A. The Mathematical Model

System: A 4-compartment permeability-limited brain model (Brain Blood, Brain Mass, Cranial CSF, Spinal CSF).
Equations: The model is defined by a system of 4 coupled ODEs based on mass balance principles, incorporating parameters for blood flow, permeability-surface area products, and transporter clearances.
Existence/Uniqueness: The authors mathematically prove the existence and uniqueness of the solution for the forward problem using linear ODE theory.

B. Identifiability Analysis

Before applying the neural network, the authors performed:

Structural Identifiability: Using the Julia library StructuralIdentifiability, they determined which parameters could theoretically be uniquely estimated. They found that fixing certain system parameters (like flow rates) allowed the remaining drug-specific and volume parameters to become locally identifiable.
Practical Identifiability: Using the bootstrapping method (via MONOLIX software), they analyzed the correlation matrix of parameters. They found low correlations (mostly $|r| < 0.26$ ) between estimated parameters, confirming that the selected parameters could be reliably estimated from the data.

C. The iPINN Architecture

The core innovation is the use of a Deep Neural Network (DNN) as a surrogate model that respects physical laws.

Inputs: Time ( $t$ ).
Outputs: Drug concentrations in the 4 compartments ( $C_{bb}, C_{bm}, C_{ccsf}, C_{scsf}$ ).
Trainable Variables:
1. Network Parameters ( $\theta$ ): Weights and biases of the neural network.
2. System Parameters ( $p$ ): The unknown physical parameters (e.g., volumes $V$ , unbound fractions $f_u$ ) are treated as external trainable variables optimized simultaneously with the network weights.
Loss Function: The network is trained to minimize a composite loss function:
$L = \lambda_{Data} L_{data} + \lambda_{ODE} L_{ODE} + \lambda_{IC} L_{IC}$
- $L_{data}$ : Mean squared error between predicted and observed concentration data.
- $L_{ODE}$ : Residual of the ODE system (enforcing mass balance and physical laws).
- $L_{IC}$ : Error in satisfying initial conditions.
- Weighting: The authors manually tuned the weights ( $\lambda$ ) to balance the convergence of data fitting vs. physical consistency.

D. Implementation

Library: Implemented using DeepXDE (Python).
Optimization: Used the Adam optimizer initially, followed by L-BFGS for fine-tuning.
Activation Functions: Tested ReLU, Sigmoid, Tanh, and Sine. Tanh and Sigmoid performed best for this stiff system.
Data: Synthetic data generated from the Simcyp Simulator (gold standard PBPK software) for a 10mg oral dose of abemaciclib (a cancer drug) in a virtual population.

3. Key Contributions

Novel Framework: First application of iPINNs to a 4-compartment permeability-limited PBPK brain model for simultaneous parameter estimation and concentration profiling.
Handling Stiffness: Demonstrated that iPINNs can successfully converge on stiff ODE systems where traditional numerical/statistical methods struggle.
Parameter Identifiability Workflow: Integrated a rigorous workflow combining structural (Julia) and practical (bootstrapping) identifiability analysis to ensure the inverse problem is well-posed before training.
Hyperparameter Sensitivity: Provided a detailed analysis of how loss weighting, network depth, and activation functions impact the convergence of PBPK inverse problems.

4. Results

The study compared PBPK-iPINN against two established methods:

SAEM: Stochastic Approximation Expectation-Maximization (implemented in MONOLIX).
DE: Differential Evolution (implemented in R).

Key Findings:

Accuracy: PBPK-iPINN achieved parameter estimation errors on the order of $10^{-8}$ for volume parameters ( $V_{bb}, V_{bm}, V_{ccsf}, V_{scsf}$ $V_{bb}, V_{bm}, V_{ccs f}, V_{scs f}$ ), significantly outperforming SAEM and DE.
- Example: For $V_{bb}$ , PINN error was $9.34 \times 10^{-9}$ , while SAEM error was $5 \times 10^{-5}$ .
Concentration Profiles: The drug concentration-time profiles predicted by iPINN closely matched the reference solution and observed data across all four compartments.
Robustness: Even with parameters like $f_{ub}$ and $\lambda_{ccsf}$ showing moderate correlation, the iPINN provided physiologically acceptable estimates.
Computational Cost: While iPINN training required significant time (approx. 54,977 seconds for 5 million iterations) compared to traditional methods, it successfully solved a problem where traditional methods often fail to converge or require extensive manual tuning.

5. Significance and Impact

Drug Development: The ability to accurately estimate patient-specific and drug-specific parameters enables the calculation of critical pharmacokinetic metrics (Cmax, Tmax, AUC, Half-life) without invasive sampling.
Brain Cancer Therapy: This framework supports the optimization of dosing regimens for brain tumors, where crossing the blood-brain barrier is a major hurdle.
Generalizability: The methodology is adaptable to any multi-compartment PBPK model, offering a solution for complex systems where in vitro experiments are insufficient or traditional numerical solvers fail due to stiffness.
Future Directions: The authors plan to extend this approach to Metabolic Flux Analysis (MFA), which involves even larger systems (>60 ODEs, >80 parameters) where traditional convergence is a known bottleneck.

In conclusion, the paper establishes PBPK-iPINN as a powerful, physics-consistent alternative to traditional numerical methods for solving inverse problems in complex pharmacokinetic modeling, offering superior accuracy in parameter estimation for stiff biological systems.