Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries.

Viscoelastic models can be used to better understand arterial wall mechanics in physiological and pathological conditions. The arterial wall reveals very slow time-dependent decays in uniaxial stress-relaxation experiments, coherent with weak power-law functions. Quasi-linear viscoelastic (QLV) theory was successfully applied to modeling such responses, but an accurate estimation of the reduced relaxation function parameters can be very difficult. In this work, an alternative relaxation function based on fractional calculus theory is proposed to describe stress relaxation experiments in strips cut from healthy human aortas. Stress relaxation (1 h) was registered at three incremental stress levels. The novel relaxation function with three parameters was integrated into the QLV theory to fit experimental data. It was based in a modified Voigt model, including a fractional element of order α, called spring–pot. The stressrelaxation predictionwas accurate and fast. Sensitivity plots for each parameter presented a minimum near their optimal values. Least-squares errors remained below 2%. Values of order α = 0.1–0.3 confirmed a predominant elastic behavior. The other two parameters of the model can be associated to elastic and viscous constants that explain the time course of the observed relaxation function. The fractional-order model integrated into the QLV theory proved to capture the essential features of the arterial wall mechanical response

Estimation of the viscoelastic properties of vessel walls using a computational model and Doppler ultrasound

Human arteries affected by atherosclerosis are characterized by altered wall viscoelastic properties. The possibility of noninvasively assessing arterial viscoelasticity in vivo would significantly contribute to the early diagnosis and prevention of this disease. This paper presents a noniterative technique to estimate the viscoelastic parameters of a vascular wall Zener model. The approach requires the simultaneous measurement of flow variations and wall displacements, which can be provided by suitable ultrasound Doppler instruments. Viscoelastic parameters are estimated by fitting the theoretical constitutive equations to the experimental measurements using an ARMA parameter approach. The accuracy and sensitivity of the proposed method are tested using reference data generated by numerical simulations of arterial pulsation in which the physiological conditions and the viscoelastic parameters of the model can be suitably varied. The estimated values quantitatively agree with the reference values, showing that the only parameter affected by changing the physiological conditions is viscosity, whose relative error was about 27% even when a poor signal-to-noise ratio is simulated. Finally, the feasibility of the method is illustrated through three measurements made at different flow regimes on a cylindrical vessel phantom, yielding a parameter mean estimation error of 25%.This work has been partially funded by the Industrial and Technological Development Centre (CDTI) within the CENIT Programme (CDTEAM Project), the EC @neurIST (IST-FP6-2004-027703) projects, by BQR INSA-Lyon and by the Italian/nMinistry of University & Research (COFIN-PRIN 2005)