CMINNs: Compartment Model Informed Neural Networks -- Unlocking Drug Dynamics

TL;DR

An innovative approach that enhances PK and integrated PK-PD modeling by incorporating fractional calculus or time-varying parameter(s), combined with constant or piecewise constant parameters is proposed, which has the potential to streamline drug development by improving the prediction of drug behavior in complex biological systems and shedding light on cancer cell death mechanisms.

Abstract

In the field of pharmacokinetics and pharmacodynamics (PKPD) modeling, which plays a pivotal role in the drug development process, traditional models frequently encounter difficulties in fully encapsulating the complexities of drug absorption, distribution, and their impact on targets. Although multi-compartment models are frequently utilized to elucidate intricate drug dynamics, they can also be overly complex. To generalize modeling while maintaining simplicity, we propose an innovative approach that enhances PK and integrated PK-PD modeling by incorporating fractional calculus or time-varying parameter(s), combined with constant or piecewise constant parameters. These approaches effectively model anomalous diffusion, thereby capturing drug trapping and escape rates in heterogeneous tissues, which is a prevalent phenomenon in drug dynamics. Furthermore, this method provides insight into the dynamics of drug in cancer in multi-dose administrations. Our methodology employs a Physics-Informed Neural Network (PINN) and fractional Physics-Informed Neural Networks (fPINNs), integrating ordinary differential equations (ODEs) with integer/fractional derivative order from compartmental modeling with neural networks. This integration optimizes parameter estimation for variables that are time-variant, constant, piecewise constant, or related to the fractional derivative order. The results demonstrate that this methodology offers a robust framework that not only markedly enhances the model's depiction of drug absorption rates and distributed delayed responses but also unlocks different drug-effect dynamics, providing new insights into absorption rates, anomalous diffusion, drug resistance, peristance and pharmacokinetic tolerance, all within a system of just two (fractional) ODEs with explainable results.

Figures (13)

• Figure 1: CMINNs workflow. We consider two types of compartment models in pharmacology: pharmacokinetic models and pharmacokinetic-pharmacodynamic models. When the model is represented by a system of two ODEs, we apply the PINNs/fPINNs framework to introduce additional flexibility in capturing complex dynamics. This enhances fitting accuracy, particularly for models with reduced parameters or non-exponential decay dynamics, where traditional methods fail to achieve satisfactory accuracy and insights. For multi-compartment models, the workflow proceeds to a compartment reduction step, reformulating the model to optimize accuracy and provide deeper insights into drug dynamics. This allows for comparative analysis of different drug behavior and ensures the best possible fit with a generalized model.
• Figure 2: PINNs block in CMINNs method. The physics-informed neural network model starts from an input scaling layer that normalizes all values to the order of 1. If the model is complicated we use the feature layer to add exponential terms within the neural networks. Then, as the output of neural network model we obtain the solution of the system of ODEs along with the time-varying parameter(s). The output goes to an optimization block where it minimizes the physical loss as well as the data loss and updates the neural network parameters. We employ adaptive weights for the physics loss, where the coefficient $\lambda_{ode}$ is updated simultaneously with the neural network parameters $\theta$ and the compartment model's constant parameters $p$.
• Figure 3: fPINNs Block in CMINNs method. The abbreviation “AD” refer to automatic differentiation. In this work, we use the Finite Difference Method (FDM) and FBDF for discretization of fractional order derivative to compute numerical solutions. In this model, the neural network output is passed to the AD block for integer-order derivative computation and to the FDM block for fractional-order differentiation. The computed derivatives are then incorporated into the equation to calculate the physics loss and simultaneously update the fractional compartment model's constant parameters $p$ and $\alpha$, along with the neural network parameters $\theta$.
• Figure 4: Impact of $k_1$ and $k_2$ on tumor growth dynamics \ref{['eq:pkpd']}. In the top panel, $k_2$ is set to $6 \times 10^{-4}$$\text{ml} \cdot {ng}^{-1} \cdot \text{day}^{-1}$, while in the bottom panel, $k_1$ is set to 1 $day^{-1}$. The figure highlights the pharmacodynamic model's behavior following a single drug dose administered on day 13. Parameter $k_1$ represents the rate of drug-induced cell death. In contrast, $k_2$ quantifies drug potency, indicating how effectively the drug inhibits tumor growth. The figure reveals that increasing $k_2$ enhances treatment efficacy, resulting in a greater reduction in tumor size. In contrast, variations in $k_1$ yield only minor changes in the curve's overall shape and final slope.
• Figure 5: Model 1. Comparison of the results between PINNs and fPINNs. From top to bottom: the inferred time-varying parameter $k(t)$ in the PINNs method Eq. \ref{['eq:pinn_general']}, the PINNs and fPINNs solutions for the first compartment, and the PINNs and fPINNs solutions for the second compartment. Total concentration of released gentamicin ($C_0$) and concentration of released gentamicin at time $t$ ($C_t$) are shown. $C_{\text{gel},t}$ represents the concentration of gentamicin remaining in the hydrogel at time $t$, while $C_{\text{gel},0}$ denotes the initial concentration of gentamicin inside the hydrogel. Data source: miskovic2023two.