Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Application of Fractional Order Derivatives for Characterizing Brain White Matter Viscoelasticity

P. Pasupathy, J. G. Georgiadis, A. A Pelegri

TL;DR

This study addresses the physically interpretable modeling of brain white matter viscoelasticity by adopting a power-law, fractional framework. It estimates power-law parameters $(\kappa,\beta)$ from frequency-domain data, formulates a fractional constitutive law, and implements a 3D spring-pot within a thread-safe Abaqus VUMAT to simulate hexagonally packed axons in ECM under periodic boundary conditions. An optimization workflow yields homogenized $E_\beta$ and $\beta$ as functions of axon volume fraction across six loading directions, revealing linear and nonlinear trends and a bi-logistic fit for off-fiber directions that indicate two stiffening regimes. The work demonstrates substantial computational gains through module-based Fortran code and short-memory truncation, linking microstructural organization to macroscopic mechanics and offering a framework for region-specific brain tissue modeling in aging and disease contexts, all while noting key limitations and the need for richer data to constrain parameters.

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.

On the Application of Fractional Order Derivatives for Characterizing Brain White Matter Viscoelasticity

TL;DR

This study addresses the physically interpretable modeling of brain white matter viscoelasticity by adopting a power-law, fractional framework. It estimates power-law parameters from frequency-domain data, formulates a fractional constitutive law, and implements a 3D spring-pot within a thread-safe Abaqus VUMAT to simulate hexagonally packed axons in ECM under periodic boundary conditions. An optimization workflow yields homogenized and as functions of axon volume fraction across six loading directions, revealing linear and nonlinear trends and a bi-logistic fit for off-fiber directions that indicate two stiffening regimes. The work demonstrates substantial computational gains through module-based Fortran code and short-memory truncation, linking microstructural organization to macroscopic mechanics and offering a framework for region-specific brain tissue modeling in aging and disease contexts, all while noting key limitations and the need for richer data to constrain parameters.

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.
Paper Structure (12 sections, 28 equations, 18 figures, 2 tables)

This paper contains 12 sections, 28 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Logistic Regression Model for Power-Law
  4. Governing Equations
  5. Fractional Viscoelastic Constitutive Material Model
  6. Finite Element Model
  7. Model Validation
  8. Results
  9. Discussion
  10. Efficient Computing
  11. Limitations
  12. Conclusions

Figures (18)

  • Figure 1: A schematic representation of the brain examined through different cut planes eilbes2025neuroanatomical. (a) Sagittal cut of the brain depicting the CC of the brain containing aligned myelinated axons with volume fraction (vf): 0.4 - 0.6 suzuki_estimation_2016 and, anterior commissure, vf: 0.33 - 0.37 edwards_microstructural_2024. (b) Coronal cut depicting white matter and corona radiata containing highly tortuous myelinated axons propagating outward with a vf: 0.25 huang_high-gradient_2020.
  • Figure 2: An algorthmic worflow for developing a fractional viscoelastic model of 3D biphasic unit cells representing axon and glia.
  • Figure 3: Evolution of the cost function over iterations for determining the power-law model parameters for (a) axons and (b) ECM.A total of 1500 iterations were performed to ensure convergence and stability of the minimization process.
  • Figure 4: Storage and loss moduli predicted by the power-law model for (a) axons and (b) ECM as functions of frequency.
  • Figure 5: Left, Top:A schematic of a periodic geometry of axons in an ECM. Right, Top: A FE model of the hexagonally packed RVE of two axons embedded in an ECM, Bottom: RVEs with different volume fractions [Clockwise: 0.2, 0.3, 0.5, 0.7].
  • ...and 13 more figures