Axonal beading or formation of multiple beads along an axon is characteristic of many brain pathological states like Alzheimer's, Parkinson's and traumatic injuries. Despite the many existing experimental studies, the underlying mechanisms of this shape instability remain still poorly understood. In this paper, we establish a combined theoretical and numerical framework to study the governing key factors of this morphological transformation. We develop a three-dimensional (3D) non-equilibrium large deformation thermodynamic model with two main parts: the central axoplasm which is considered as a polyelectrolyte hydrogel and the encapsulating cortical membrane which is modeled as an incompressible hyperelastic layer with surface energy and growing surface. The model constitutive and evolution equations are then extracted employing thermodynamic balance principles for both bulk and surface material points. It is shown that the second law of thermodynamics indicates that the axolemma growth rate is proportional to the membrane tension which is in perfect agreement with the available experimental findings. While the developed model is general and can be extended to cover other types of axonal beadings, for the sake of simplicity, here, we focus on osmotically driven axisymmetric beadings which are compressible viscoelastic periodic modulations. We solve the corresponding governing equations using the linear perturbation method. This perturbation analysis proves that: 1) the beading instability is a rate dependent phenomenon that is controlled by the axolemma growth, 2) the initially dominant beading waves (the fastest waves) might be replaced only by longer waves which are more stable and 3) the wavelength of the fastest beads should vary roughly linearly with the axonal radius. These main findings are all in good agreement with the existing experimental results. Finally, the finite element implementation of the model is also presented to verify the results of the linear stability analysis for slow waves. The obtained axisymmetric finite element results are in good agreement with the corresponding theoretical findings.

轴突珠状或沿轴突形成多个珠状是许多大脑病理状态的特征，例如阿尔茨海默病、帕金森病和外伤。尽管已有许多实验研究，但这种形状不稳定性的潜在机制仍然知之甚少。在本文中，我们建立了一个结合的理论和数值框架来研究这种形态转变的控制关键因素。我们开发了一个三维（3D）非平衡大变形热力学模型，有两个主要部分：中央轴浆（被认为是聚电解质水凝胶）和封装皮质膜（被建模为具有表面能和生长表面的不可压缩超弹性层） 。然后利用体材料点和表面材料点的热力学平衡原理提取模型本构方程和演化方程。结果表明，热力学第二定律表明轴膜生长速率与膜张力成正比，这与现有的实验结果完全一致。虽然开发的模型是通用的，并且可以扩展到涵盖其他类型的轴突珠，但为了简单起见，在这里，我们重点关注渗透驱动的轴对称珠，它们是可压缩的粘弹性周期性调制。我们使用线性摄动法求解相应的控制方程。该扰动分析证明：1) 串珠不稳定性是一种由轴突生长控制的速率依赖现象，2) 最初占主导地位的串珠波（最快的波）可能只会被更稳定的较长波取代，3)最快珠子的波长应与轴突半径大致线性变化。这些主要发现与现有的实验结果非常吻合。最后，还给出了模型的有限元实现，以验证慢波线性稳定性分析的结果。获得的轴对称有限元结果与相应的理论结果非常吻合。