Quantitative Analysis of Molecular Transport in the Extracellular Space Using Physics-Informed Neural Network

TL;DR

A novel approach to quantitatively analyze the molecular transport within the ECS by solving an inverse problem derived from the advection-diffusion equation (ADE) using a physics-informed neural network (PINN).

Abstract

The brain extracellular space (ECS), an irregular, extremely tortuous nanoscale space located between cells or between cells and blood vessels, is crucial for nerve cell survival. It plays a pivotal role in high-level brain functions such as memory, emotion, and sensation. However, the specific form of molecular transport within the ECS remain elusive. To address this challenge, this paper proposes a novel approach to quantitatively analyze the molecular transport within the ECS by solving an inverse problem derived from the advection-diffusion equation (ADE) using a physics-informed neural network (PINN). PINN provides a streamlined solution to the ADE without the need for intricate mathematical formulations or grid settings. Additionally, the optimization of PINN facilitates the automatic computation of the diffusion coefficient governing long-term molecule transport and the velocity of molecules driven by advection. Consequently, the proposed method allows for the quantitative analysis and identification of the specific pattern of molecular transport within the ECS through the calculation of the Peclet number. Experimental validation on two datasets of magnetic resonance images (MRIs) captured at different time points showcases the effectiveness of the proposed method. Notably, our simulations reveal identical molecular transport patterns between datasets representing rats with tracer injected into the same brain region. These findings highlight the potential of PINN as a promising tool for comprehensively exploring molecular transport within the ECS.

Figures (17)

• Figure 1: Illustration of the ECS. (a) An electron microscope image of the hippocampal region in the mouse brain. The red area denotes the ECS, which is an irregular, tortuous narrow space. (b) The schematic diagram of molecular transport within the ECS. The transport of molecules faces various obstructions and is likely driven by both advection and diffusion.
• Figure 2: The framework of the proposed method. The initial step in our methodology involves acquiring MRI data at various time points $t_0, t_1, t_2, \cdots, t_{T-1}, t_T$. Subsequently, we perform image preprocessing to eliminate unnecessary regions and align images through displacement registration. Using the preprocessed data, we develop a physics-informed neural network designed to solve the inverse problem arising from the advection-diffusion equation. The primary function of the network is to automatically determine the diffusion coefficient $D$ and velocity $v$. As a result, we can calculate the Péclet number, identifying the specific mode of molecular transport within the ECS. (a) MRI data obtained at different time points $t_0, t_1, t_2, \cdots, t_{T-1}, t_T$. (b) MRI data after image preprocessing. (c) A physics-informed neural network.
• Figure 3: The framework of PINN for solving the problem (\ref{['eq1-2']}) and (\ref{['eq2-2']}). A multi-layer perceptron (MLP) is constructed to approximate the solution C(x,t) in (\ref{['eq1-2']}), while optimizing the network by minimizing a loss function that incorporates the advection-diffusion equation as a constraint.
• Figure 4: Example MRI (coronal plane) obtained at different time points (i.e., $t=0, 20, 30, 50, 60, 70, 80, 90, 120, 180$ mins) from the dataset-1. Note that at $t=0$ min, the image was acquired before the tracer injection.
• Figure 5: Example MRI (coronal plane) obtained at different time points (i.e., $t=0, 30, 40, 50, 60, 90, 130, 150, 210, 240$ mins. Note that at $t=0$ min, the image was acquired before the tracer injection.) from the dataset-2.