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

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

TL;DR

This study considers a permeability-limited, four-compartment PBPK brain model that mimics human brain functionality in drug delivery and introduces PBPK-iPINN, a method for estimating drug-specific or patient-specific parameters and drug concentration profiles using inverse PINNs.

Abstract

Physics-Informed Neural Networks (PINNs) integrate machine learning with differential equations to solve forward and inverse problems while ensuring that predictions adhere to physical laws. Physiologically based pharmacokinetic (PBPK) modeling advances beyond classical compartmental approaches by employing a mechanistic, physiology-focused framework. Such models involve many unknown parameters that are difficult to measure directly in humans due to ethical and practical constraints. PBPK models are constructed as systems of ordinary differential equations (ODEs) and these parametric ODEs are often stiff, and traditional numerical and statistical methods frequently fail to converge. In this study, we consider a permeability-limited, four-compartment PBPK brain model that mimics human brain functionality in drug delivery. We introduce PBPK-iPINN, a method for estimating drug-specific or patient-specific parameters and drug concentration profiles using inverse PINNs. We also conducted parameter identifiability analysis to determines whether the parameters can be uniquely and reliably estimated from the available data. We demonstrate that, for the inverse problem to converge to the correct solution, the components of the loss function (data loss, initial condition loss, and residual loss) must be appropriately weighted, and the hyperparameters including the number of layers and neurons, activation functions, learning rate, optimizer, and collocation points must be carefully tuned. The performance of the PBPK-iPINN approach is then compared with established numerical and statistical methods. Accurate parameter estimation yields precise drug concentration-time profiles, which in turn enable the calculation of pharmacokinetic metrics. These metrics support drug developers and clinicians in designing and optimizing therapies for brain cancer.

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

TL;DR

This study considers a permeability-limited, four-compartment PBPK brain model that mimics human brain functionality in drug delivery and introduces PBPK-iPINN, a method for estimating drug-specific or patient-specific parameters and drug concentration profiles using inverse PINNs.

Abstract

Physics-Informed Neural Networks (PINNs) integrate machine learning with differential equations to solve forward and inverse problems while ensuring that predictions adhere to physical laws. Physiologically based pharmacokinetic (PBPK) modeling advances beyond classical compartmental approaches by employing a mechanistic, physiology-focused framework. Such models involve many unknown parameters that are difficult to measure directly in humans due to ethical and practical constraints. PBPK models are constructed as systems of ordinary differential equations (ODEs) and these parametric ODEs are often stiff, and traditional numerical and statistical methods frequently fail to converge. In this study, we consider a permeability-limited, four-compartment PBPK brain model that mimics human brain functionality in drug delivery. We introduce PBPK-iPINN, a method for estimating drug-specific or patient-specific parameters and drug concentration profiles using inverse PINNs. We also conducted parameter identifiability analysis to determines whether the parameters can be uniquely and reliably estimated from the available data. We demonstrate that, for the inverse problem to converge to the correct solution, the components of the loss function (data loss, initial condition loss, and residual loss) must be appropriately weighted, and the hyperparameters including the number of layers and neurons, activation functions, learning rate, optimizer, and collocation points must be carefully tuned. The performance of the PBPK-iPINN approach is then compared with established numerical and statistical methods. Accurate parameter estimation yields precise drug concentration-time profiles, which in turn enable the calculation of pharmacokinetic metrics. These metrics support drug developers and clinicians in designing and optimizing therapies for brain cancer.

Paper Structure

This paper contains 12 sections, 1 theorem, 32 equations, 12 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Physiologically-Based Four Compartment Brain Model
  3. System of Differential Equations
  4. Existence and Uniqueness
  5. Parameter Identifiability Analysis
  6. Methodology of Inverse PINN
  7. PINN Architecture of 4-Compartmental Brain Model
  8. Augmented Loss Function and Its Minimization
  9. Experimental Procedure
  10. A Step-by-Step Guide to Implementing a Four-Compartment Brain PINN in DeepXDE
  11. Numerical Results
  12. Conclusion

Key Result

Theorem 2.1

Let, A be an $n\times n$ constant matrix and let G be an n-dimensional vector function continuous on an interval I $\subseteq \mathbb{R}$. Pick $t_0 \in I$. Then the initial value problem has a unique solution on $I$, which is

Figures (12)

  • Figure 1: Schematically illustrates three possible categories of physical problems and associated available data.
  • Figure 2: Advancing pharmacokinetic modeling with neural network.
  • Figure 3: Schematic illustration of the 4 compartment brain model.
  • Figure 4: Julia code to perform local structural identifiability analysis
  • Figure 5: The inverse physics-informed neural network model starts with time as inputs and then the outputs goes to an optimization block where it minimizes the total loss by optimizing the parameters $\psi$ which includes system parameters $p$ and neural network parameters $\theta$.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 2.1: Existence and Uniqueness of the 4 Compartment Brain Model
  • proof