A Mathematical Model of Regenerative Axon Growing along Glial Scar after Spinal Cord Injury

A major factor in the failure of central nervous system (CNS) axon regeneration is the formation of glial scar after the injury of CNS. Glial scar generates a dense barrier which the regenerative axons cannot easily pass through or by. In this paper, a mathematical model was established to explore how the regenerative axons grow along the surface of glial scar or bypass the glial scar. This mathematical model was constructed based on the spinal cord injury (SCI) repair experiments by transplanting Schwann cells as bridge over the glial scar. The Lattice Boltzmann Method (LBM) was used in this model for three-dimensional numerical simulation. The advantage of this model is that it provides a parallel and easily implemented algorithm and has the capability of handling complicated boundaries. Using the simulated data, two significant conclusions were made in this study: (1) the levels of inhibitory factors on the surface of the glial scar are the main factors affecting axon elongation and (2) when the inhibitory factor levels on the surface of the glial scar remain constant, the longitudinal size of the glial scar has greater influence on the average rate of axon growth than the transverse size. These results will provide theoretical guidance and reference for researchers to design efficient experiments.