[论文解读] On the Application of Fractional Order Derivatives for Characterizing Brain White Matter Viscoelasticity

P. Pasupathy, J. G. Georgiadis+1 more|arXiv (Cornell University)|Jan 26, 2026

Automotive and Human Injury Biomechanics被引用 0

一句话总结

本文开发了一个基于分数阶粘弹性的三维有限元脑白质模型（含轴突与ECM）采用弹簧-阻尼器模型，验证了VUMAT实现，并将均匀化参数随轴突体积分数函数化提取。

ABSTRACT

Conventional viscoelastic characterization of brain white matter (BWM), typically described using Prony series models, remains a largely empirical representation that is difficult to interpret physically. Growing evidence suggests that BWMviscoelasticity follows power-law behavior. Under the assumptions of linear viscoelasticity and causality, a power-law model in the frequency domain yields a fractional viscoelastic model in the time domain. A fractional viscoelastic constitutive model for the axon and extracellular matrix (ECM) is implemented via a Fortran VUMAT subroutine. A biphasic periodic finite element (FE) model of hexagonally packed representative volume elements (RVEs) of axons embedded in an ECM is constructed in Abaqus under quasi-static loading. The inverse problem of extracting homogenized material properties is solved using an optimization workflow. The model predicts that the springpot coefficient, which determines the solid-fluid behavior and, the power-law exponent, which encodes information about the underlying tissue architecture, follows a bi-logistic function along the transverse normal and shear directions. The nonlinear variation of the parameters reveals two distinct stiffening stages: a lower rate at low axon volume fractions, followed by a higher rate as increased axonal content reinforces the RVE. To our knowledge, this study is the first to propose and implement a 3D fractional viscoelastic FE model of the corpus callosum of BWM in the time domain. The thread-safe implementation of the VUMAT achieves significantly faster performance than existing approaches. The results reveal nonlinear variation in material parameters, directional dependence of BWM mechanics, and the complex interplay among microstructural elements.

研究动机与目标

• 推动对脑白质物理可解释的粘弹性表征，超越Prony系列模型。
• 引入轴突与ECM的幂律（分数阶）弹簧-阻尼器表示。
• 开发带VUMAT子程序的三维有限元框架以模拟双相RVEs的行为。
• 在不同加载方向下，将均质化的分数阶粘弹性参数标定为随轴突体积分数的函数。

提出的方法

• 利用频域数据通过逻辑回归方法拟合轴突和ECM的幂律参数。
• 使用Riemann–Liouville导数（弹簧-阻尼器）来建立分数阶粘弹性本构，并在三维FEM VUMAT中实现。
• 构建轴突在ECM中的六边格点堆积RVEs，施加周期性边界条件并应用六个加载方向。
• 采用优化流程（Nelder–Mead）通过最小化FEM应力与目标应力之间的RMSD来恢复均质化参数。
• 在显式时间积分中引入短时记忆截断以降低历史依赖计算。

实验结果

研究问题

• RQ1 分数阶（幂律）粘弹性模型是否比Prony系列模型更能捕捉脑白质微观结构的影响？
• RQ2 均质化的弹簧-阻尼系数如何随轴突体积分数和加载方向变化？
• RQ3 在Abaqus中实现并验证一个针对BVW微观结构的三维分数阶粘弹性VUMAT是否可行？
• RQ4 显式仿真中的记忆截断（短时记忆）对精度和计算效率的影响有多大？

主要发现

• 弹簧-阻尼系数(E_beta)在纤维方向随轴突体积分数呈线性变化，在横向和剪切方向呈非线性变化。
• 幂律指数beta在纤维方向呈饱和指数趋势，在横向和剪切方向呈双逻辑变化。
• 均质化的E_beta和beta随轴突体积分数增大而增大，体现了方向性依赖和微观结构对脑白质力学的影响。
• 观察到两种硬化机制：低轴突含量时的低速硬化和轴突含量增加时的高速硬化。
• 短时记忆方法在准静态情形下可减小计算时间且保持可接受的精度（偏差约2-3%）。
• 研究声称在胼胝体的三维分数阶粘弹性有限元模型方面具有新颖性，并给出相较现有方法更安全、速度更快的实现。