脊髓损伤再生轴突生长的数学模型和三维格子波尔兹曼法数值模拟

A Mathematical Model and 3D Lattice Boltzmann Simulation for the Axonal Growth of Regeneration in An Injured Spinal Cord

中枢神经系统损伤是当今社会最具破坏力的疾病之一,虽然已经有办法使损伤后残存的神经元出芽,但如何保证处于萌芽状态的再生轴突继续生长直至与远端的靶细胞正确连接,是困扰至今的难题。为探讨中枢神经损伤所形成的胶质瘢痕和其所诱导的抑制因子对再生轴突生长进程的影响,根据轴突生长速度与其微环境中影响因子的浓度梯度成比例的原理,以脊髓损伤为背景构建数学模型,并采用格子波尔兹曼法进行三维数值模拟。数值试验中的主要观察指标为:1)当微环境中轴突生长抑制因子释放率和促进因子释放率一定时,胶质瘢痕的轴向厚度对轴突生长速率的影响,并跟踪记录生长锥所经过路线上的抑制因子浓度和促进因子浓度;2)当胶质瘢痕的轴向厚度一定时,抑制因子释放率和促进因子释放率对轴突生长速率的影响,并跟踪记录生长锥所经过路线上的抑制因子浓度和促进因子浓度。结果表明:1)胶质瘢痕的轴向厚度越大、抑制因子的释放率越强,轴突生长速率越小;2)轴突生长速率本质上取决于生长锥所在位置抑制因子浓度与促进因子浓度的比值,当该比值平均小于某个阈值时,再生轴突能够顺利生长并与靶细胞成功对接。为正确设计有关动物试验提供了理论参考。

Axonal growth is inevitably be subject to certain inhibition even if in development of the nervous system. A key to extension of the axons in development is that the inhibitors are in balance in concentration to the promotors. Speculatively, if this balance was reached in the environment of glial scar after spinal cord injury, the growth of regenerated axons would be successful, wouldn＇t it？ In order to answer this question, a mathematical model was constructed based on the principle of chemotaxis of cells and on the known related experimental data, and a three-dimensional lattice Boltzmann method was employed for the numerical simulation. The simulating tests were divided into two groups, i.e. 1） when the ratio of inhibitor release rate to the promotors＇ was a constant, the axonal growth rates varying with the glial scar thickness in the axial direction were simulated and the chemical concentrations located at the moving growth cones were probed; 2） when the glial scar thickness was a constant, the axonal growth rates varying with the ratios of inhibitor release rate to the promotors＇ were simulated and the chemical concentrations located at the moving growth cones were probed. The results show that 1） the larger the thickness of giial scar as well as the stronger the inhibitor release rate, the smaller the axonal growth rate will be; 2） axon growth rate essentially depends on the ratio of the inhibitor concentration to the promotors＇ located at the growth cones, and when the average ratio is less than a certain threshold, the regenerated axons will grow smoothly and will reach to their target cells successfully. The results support the speculation.