Mathematical Modelling of Neuroblast Chemotaxis Migration towards the Olfactory Bulb

TL;DR

The paper develops a PDE model for neuroblast migration along the rostral migratory stream toward the olfactory bulb in a realistic 2D brain domain, where migration is driven by a chemoattractant gradient of a specially designed OB function $\mathcal{O}$ and modulated by anisotropic diffusion in the corpus callosum. The OB attractant is obtained from a steady elliptic problem with diffusion coefficient $\mu_O$, while the neuroblast density $u$ satisfies a convection–reaction equation with parameters $\Lambda=(\alpha, \beta, \gamma, \chi, \sigma)$ and a tunable time weight $\tau$, solved numerically using upwind discontinuous Galerkin methods to preserve the maximum principle. The authors calibrate the model to experimental rodent data by a two-stage optimization (initial-condition fit and evolution fit) using grid search, random forest regression, and L-BFGS-B, achieving a close quantitative match and revealing parameter ranges that reproduce the observed RMS trajectory. This framework provides a tractable, validated starting point for more advanced 3D models and for exploring neuroblast migration in damaged or injured brain tissue, with a robust numerical methodology suitable for complex brain geometries.

Abstract

This article is devoted to the mathematical modeling of migration of neuroblasts, precursor cells of neurons, along the pathway they usually follow before maturing. This pathway is determined mainly by chemotaxis and the heterogeneous mobility of neuroblasts in different regions of the brain. In numerical simulations, the application of novel discontinuous Galerkin methods allows to maintain the properties of the continuous model such as the maximum principle. We present some successful computer tests including parameter adjust to fit real data from rodent brains.

Figures (10)

• Figure 1: Composition of confocal images showing neuroblasts migrating from the SVZ, to the OB through the RMS. To obtain this image, mice were given three intraperitoneal injections of the thymidine analogue BrdU in one day and were sacrificed 3 hours after the last BrdU injection. Brain were isolated and 30 µM sagittal sections were obtained with the use of a cryostat. Sections were then processed for immunohistochemistry to detect BrdU (red) and the neuroblast marker DCX (green). The scale bar represents 100 µm. Abbreviations: subventricular zone (SVZ); corpus callosum (CC); rostral migratory stream (RMS); olfactory bulo (OB)
• Figure 2: Isolines of the olfactory bulb attraction function, obtained as solution of \ref{['esquema_DG_OlfBulb']} with the parameters described in Section \ref{['sec:experimental-param']}. The olfactory bulb stands out for the concentric isolines on the left. The corpus callosum is located in the center, in the region with no isolines.
• Figure 3: Right: mesh of a virtual rodent brain. Left: zoom around the RMS. Triangles defining the CC, SVZ and NZ are shown in black, red and green color, respectively
• Figure 4: Qualitative behavior of the numerical solution from the stationary model.
• Figure 5: Numerical solution for the initial condition from regression parameters.

Theorems & Definitions (1)

• Proposition 3.1