Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Analysis of Intracranial Aneurysm Haemodynamics Altered by Wall Movement

Computational fluid dynamics is intensively used to deepen our understanding of aneurysm growth and rupture in an attempt to support physicians during therapy planning. Numerous studies assumed fully rigid vessel walls in their simulations, whose sole haemodynamics may fail to provide a satisfactory criterion for rupture risk assessment. Moreover, direct in vivo observations of intracranial aneurysm pulsation were recently reported, encouraging the development of fluid-structure interaction for their modelling and for new assessments. In this work, we describe a new fluid-structure interaction functional setting for the careful evaluation of different aneurysm shapes. The configurations consist of three real aneurysm domes positioned on a toroidal channel. All geometric features, employed meshes, flow quantities, comparisons with the rigid wall model and corresponding plots are provided for the sake of reproducibility. The results emphasise the alteration of flow patterns and haemodynamic descriptors when wall deformations were taken into account compared with a standard rigid wall approach, thereby underlining the impact of fluid-structure interaction modelling.

Thermodynamic Restrictions in Linear Viscoelasticity

The thermodynamic consistency of linear viscoelastic models is investigated. First, the classical Boltzmann law of stress-strain is considered. The kernel (Boltzmann function) is shown to be consistent only if the half-range sine transform is negative definite. The existence of free-energy functionals is shown to place further restrictions. Next, the Boltzmann function is examined in the unbounded power law form. The consistency is found to hold if the stress functional involves the strain history, not the strain-rate history. The stress is next taken to be given by a fractional order derivative of the strain. In addition to the constitutive equations involving strain-rate histories, finding a free-energy functional, consistent with the second law, seems to be an open problem.

A Strain-Rate Dependent Constitutive Model for Göttingen Minipig Cerebral Arteries

Computational simulations of traumatic brain injury (TBI) are commonly used to advance understanding of the injury-pathology relationship, tissue damage thresholds, and design of protective equipment such as helmets. Both human and animal TBI models have developed substantially over recent decades, partially due to the inclusion of more detailed brain geometry and representation of tissues like cerebral blood vessels. Explicit incorporation of vessels dramatically affects local strain and enables researchers to investigate TBI-induced damage to the vasculature. While some studies have indicated that cerebral arteries are rate-dependent, no published experimentally based, rate-sensitive constitutive models of cerebral arteries exist. In the present work, we characterize the mechanical properties of axially failed porcine arteries, both quasi-statically (0.01 s-1) and at high rate (<100 s-1), and propose a rate-sensitive model to fit the data. We find that the quasi-static and high-rate stress-stretch curves become significantly different (p>0.05) above a stretch of 1.23. We additionally find a significant change in both failure stretch and stress as a result of strain rate. The stress-stretch curve is then modeled as a Holzapfel-Gasser-Ogden material, with a Prony series added to capture the effects of viscoelasticity. Ultimately, this paper demonstrates that rate dependence should be considered in the material properties of cerebral arteries undergoing high strain-rate deformations and provides a ready-to-use model for finite element implementation.

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

We introduce a data-driven fractional modeling framework for complex materials, and particularly bio-tissues. From multi-step relaxation experiments of distinct anatomical locations of porcine urinary bladder, we identify an anomalous relaxation character, with two power-law-like behaviors for short/long long times, and nonlinearity for strains greater than 25%. The first component of our framework is an existence study, to determine admissible fractional viscoelastic models that qualitatively describe linear relaxation. After the linear viscoelastic model is selected, the second stage adds large-strain effects to the framework through a fractional quasi-linear viscoelastic approach for the nonlinear elastic response of the bio-tissue of interest. From single-step relaxation data of the urinary bladder, a fractional Maxwell model captures both short/long-term behaviors with two fractional orders, being the most suitable model for small strains at the first stage. For the second stage, multi-step relaxation data under large strains were employed to calibrate a four-parameter fractional quasi-linear viscoelastic model, that combines a Scott-Blair relaxation function and an exponential instantaneous stress response, to describe the elastin/collagen phases of bladder rheology. Our obtained results demonstrate that the employed fractional quasi-linear model, with a single fractional order in the range α = 0.25–0.30, is suitable for the porcine urinary bladder, producing errors below 2% without need for recalibration over subsequent applied strains. We conclude that fractional models are attractive tools to capture the bladder tissue behavior under small-to-large strains and multiple time scales, therefore being potential alternatives to describe multiple stages of bladder functionality.

Mathematical model of atherosclerotic aneurysm

Atherosclerosis is a major cause of abdominal aortic aneurysm (AAA) and up to 80% of AAA patients have atherosclerosis. Therefore it is critical to understand the relationship and interactions between atherosclerosis and AAA to treat atherosclerotic aneurysm patients more effectively. In this paper, we develop a mathematical model to mimic the progression of atherosclerotic aneurysms by including both the multi-layer structured arterial wall and the pathophysiology of atherosclerotic aneurysms. The model is given by a system of partial differential equations with free boundaries. Our results reveal a 2D biomarker, the cholesterol ratio and DDR1 level, assessing the risk of atherosclerotic aneurysms. The efficacy of different treatment plans is also explored via our model and suggests that the dosage of anti-cholesterol drugs is significant to slow down the progression of atherosclerotic aneurysms while the additional anti-DDR1 injection can further reduce the risk.

Fractional viscoelastic models for power-law materials

Soft materials often exhibit a distinctive power-law viscoelastic response arising from broad distribution of time-scales present in their complex internal structure. A promising tool to accurately describe the rheological behaviour of soft materials is fractional calculus. However, its use in the scientific community remains limited due to the unusual notation and non-trivial properties of fractional operators. This review aims to provide a clear and accessible description of fractional viscoelastic models for a broad audience and to demonstrate the ability of these models to deliver a unified approach for the characterisation of power-law materials. The use of a consistent framework for the analysis of rheological data would help classify the empirical behaviours of soft and biological materials, and better understand their response.

Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model

In this communication, the single element version of the fractional Maxwell model (single-FMM or Scott–Blair model) is adopted to quantify the observed behavior of the linear interfacial dilational viscoelasticity. This mathematical tool is applied to the results obtained by capillary pressure experiments under low-gravity conditions aboard the International Space Station, for adsorption layers at the hydrocarbon/water interface. Two specific experimental sets of steady-state harmonic oscillations of interfacial area are reported, respectively: a drop of pure water into a Span-80 surfactant/paraffin-oil matrix and a pure n-hexane drop into a C13DMPO/TTAB mixed surfactants/aqueous-solution matrix. The fractional constitutive single-FMM is demonstrated to embrace the standard Maxwell model (MM) and the Lucassen–van-den-Tempel model (L–vdT), as particular cases. The single-FMM adequately fits the Span-80/paraffin-oil observed results, correctly predicting the frequency dependence of the complex viscoelastic modulus and the inherent phase-shift angle. In contrast, the single-FMM appears as a scarcely adequate tool to fit the observed behavior of the mixed-adsorption surfactants for the C13DMPO/TTAB/aqueous solution matrix (despite the single-FMM satisfactorily comparing to the phenomenology of the sole complex viscoelastic modulus). Further speculations are envisaged in order to devise combined FMM as rational guidance to interpret the properties and the interfacial structure of complex mixed surfactant adsorption systems.

Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials

Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an equation is established as the governing equation for the dynamic displacement of the viscoelastic pipeline. In order to solve this class of equations, a numerical method based on shifted Legendre polynomials is presented for the first time. The method is effective and accurate after the numerical example verifying. Numerical results show that how dynamic properties are influenced by internal fluid velocity, force excitation, and variable fractional order through the proposed method. More importantly, the numerical method has shown great potentials for dynamic problems with the high precision model.

Detection of arterial wall abnormalities via Bayesian model selection

Patient-specific modelling of haemodynamics in arterial networks has so far relied on parameter estimation for inexpensive or small-scale models. We describe here a Bayesian uncertainty quantification framework which makes two major advances: an efficient parallel implementation, allowing parameter estimation for more complex forward models, and a system for practical model selection, allowing evidence-based comparison between distinct physical models. We demonstrate the proposed methodology by generating simulated noisy flow velocity data from a branching arterial tree model in which a structural defect is introduced at an unknown location; our approach is shown to accurately locate the abnormality and estimate its physical properties even in the presence of significant observational and systemic error. As the method readily admits real data, it shows great potential in patient-specific parameter fitting for haemodynamical flow models.

Respiratory resistance and reactance in adults with sickle cell anemia: Part 2-Fractional-order modeling and a clinical decision support system for the diagnosis of respiratory disorders

BACKGROUND

A better understanding of sickle cell anemia (SCA) and improvements in drug therapy and health policy have contributed to the emergence of a large population of adults living with this disease. The mechanisms by which SCA produces adverse effects on the respiratory system of these patients are largely unknown. Fractional-order (FrOr) models have a high potential to improve pulmonary clinical science and could be useful for diagnostic purposes, offering accurate models with an improved ability to mimic nature. Part 2 of this two-part study examines the changes in respiratory mechanics in patients with SCA using the new perspective of the FrOr models. These results are compared with those obtained in traditional forced oscillation (FOT) parameters, investigated in Part 1 of the present study, complementing this first analysis.

METHODOLOGY/PRINCIPAL FINDINGS

The data consisted of three categories of subjects: controls (n = 23), patients with a normal spirometric exam (n = 21) and those presenting restriction (n = 24). The diagnostic accuracy was evaluated by investigating the area under the receiver operating characteristic curve (AUC). Initially, it was observed that biomechanical changes in SCA included increased values of fractional inertance, as well as damping and hysteresivity (p<0.001). The correlation analysis showed that FrOr parameters are associated with functional exercise capacity (R = -0.57), pulmonary diffusion (R = -0.71), respiratory muscle performance (R = 0.50), pulmonary flows (R = -0.62) and airway obstruction (R = 0.60). Fractional-order modeling showed high diagnostic accuracy in the detection of early respiratory abnormalities (AUC = 0.93), outperforming spirometry (p<0.03) and standard FOT analysis (p<0.01) used in Part 1 of this study. A combination of machine learning methods with fractional-order modeling further improved diagnostic accuracy (AUC = 0.97).

CONCLUSIONS

FrOr modeling improved our knowledge about the biomechanical abnormalities in adults with SCA. Changes in FrOr parameters are associated with functional exercise capacity decline, abnormal pulmonary mechanics and diffusion. FrOr modeling outperformed spirometric and traditional forced oscillation analyses, showing a high diagnostic accuracy in the diagnosis of early respiratory abnormalities that was further improved by an automatic clinical decision support system. This finding suggested the potential utility of this combination to help identify early respiratory changes in patients with SCA.