[Paper Review] 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 Biomechanics0 citations

TL;DR

The paper develops a 3D fractional viscoelastic finite element model of brain white matter (axon + ECM) using a spring-pot approach, validates a VUMAT implementation, and extracts homogenized parameters as functions of axon volume fraction.

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.

Motivation & Objective

• Motivate a physically interpretable viscoelastic characterization of brain white matter beyond Prony-series models.
• Introduce a power-law (fractional) spring-pot representation for axons and ECM.
• Develop a 3D finite element framework with a VUMAT subroutine to simulate biphasic RVE behavior.
• Calibrate homogenized fractional viscoelastic parameters as functions of axon volume fraction across loading directions.

Proposed method

• Fit power-law parameters for axon and ECM from frequency-domain data using a logistic regression approach.
• Formulate a fractional viscoelastic constitutive model using the Riemann–Liouville derivative (spring-pot) and implement it in a 3D FEM VUMAT.
• Construct hexagonally packed RVEs of axons in ECM with periodic boundary conditions and apply six loading directions.
• Use an optimization workflow (Nelder–Mead) to recover homogenized parameters by minimizing RMSD between FEM and target stresses.
• Incorporate short-memory truncation to reduce history-dependent computation in explicit time integration.

Experimental results

Research questions

• RQ1 Can a fractional (power-law) viscoelastic model capture BWM microstructure effects better than Prony-series models?
• RQ2 How do homogenized spring-pot parameters vary with axon volume fraction and loading direction?
• RQ3 Is it feasible to implement and validate a 3D fractional viscoelastic VUMAT for BVW microstructures in Abaqus?
• RQ4 What is the accuracy and computational efficiency impact of memory truncation (short-memory) in explicit simulations?

Key findings

• The spring-pot coefficient (E_beta) varies linearly with axon volume fraction along the fiber direction and nonlinearly in transverse and shear directions.
• The power-law exponent beta shows a saturating exponential trend along the fiber direction and bi-logistic variation in transverse and shear directions.
• Homogenized E_beta and beta increase with higher axon volume fraction, reflecting directional dependence and microstructural influence on BWM mechanics.
• Two stiffening regimes are observed: a lower-rate stiffening at low axon content and a higher-rate stiffening as axonal content increases.
• A short-memory approach reduces computation time with acceptable accuracy (up to ~2-3% deviation) in quasi-static cases.
• The study claims novelty in a 3D fractional viscoelastic FE model of the corpus callosum and provides a thread-safe, faster implementation compared to existing methods.