Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity

Background: Quasilinear viscoelasticity (QLV) theory has been widely and successfully used to describe the time-dependent response of connective tissues. Difficulties remain, however, particularly in material parameter estimation and sensitivities. In this study, we introduce a new alternative: the fractional order viscoelasticity (FOV) theory, which uses a fractional order integral to describe the relaxation response. FOV implies a fractal-like tissue structure, reflecting the hierarchical arrangement of collagenous tissues. Method of Approach: A one-dimensional (1-D) FOV reduced relaxation function was developed, replacing the QLV 'box-spectrum' function with a fractional relaxation function. A direct-fit, global optimization method was used to estimate material parameters from stress relaxation tests on aortic valve tissue. Results: We found that for the aortic heart valve, FOV had similar accuracy and better parameter sensitivity than QLV, particularly for the long time constant ([tau]2). The mean (n=5).fractional order was 0.29, indicating that the viscoelastic response of the tissue was strongly fractal-like. Results summary: mean QLV parameters were C=0.079, [tau]1=0.004, [tau]2=76, and mean FOV parameters were [beta]=0.29, [tau]=0.076, and [rho]= 1.84. Conclusions: FOV can provide valuable new insights into tissue viscoelastic behavior. Determining the fractional order can provide a new and sensitive quantitative measure for tissue comparison. [DOI: 10.1115/1.1933900] Keywords: Fractional Calculus, Viscoelasticity, Nonlinear Elasticity, Aortic Valve, Fractals, Soft Tissue