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TECHNICAL PAPERS: Soft Tissue

# Fractional Order Viscoelasticity of the Aortic Valve Cusp: An Alternative to Quasilinear Viscoelasticity

[+] Author and Article Information
Todd C. Doehring1

Department of Biomedical Engineering, Lerner Research Institute,  The Cleveland Clinic Foundation, Cleveland, OH 90027

Alan D. Freed2

Department of Biomedical Engineering, Lerner Research Institute,  The Cleveland Clinic Foundation, Cleveland, OH 90027

Evelyn O. Carew

Department of Biomedical Engineering, Lerner Research Institute,  The Cleveland Clinic Foundation, Cleveland, OH 90027

Ivan Vesely3

Department of Biomedical Engineering, Lerner Research Institute,  The Cleveland Clinic Foundation, Cleveland, OH 90027

1

Correspondence and reprint requests to: Todd C. Doehring, Ph.D., Department of Biomedical Engineering/ND20, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195. Telephone: (216) 445-3411; fax: (216) 444-9198; electronic mail: tcdoe@bme.ri.ccf.org

2

Also at the NASA Glenn Research Center, Cleveland, OH 44195.

3

Also at the Saban Research Institute of Childrens Hospital Los Angeles, Los Angeles, CA.

J Biomech Eng 127(4), 700-708 (Jan 21, 2005) (9 pages) doi:10.1115/1.1933900 History: Received September 24, 2003; Revised January 21, 2005

## Abstract

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 $(τ2)$. 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 $C=0.079$, $τ1=0.004$, $τ2=76$, and mean FOV parameters were $β=0.29$, $τ=0.076$, and $ρ=1.84$. 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.

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## Figures

Figure 1

Spring and dashpot representations of QLV (serial) and FOV (fractional) models. QLV can be represented by a number of Kelvin–Zener solids connected in series (a), while FOV can be represented by a fractal-type tree model (b) with varying breadth and depth depending on the fractional order β. Note that in the special case β=1, the FOV model reduces to a single Kelvin–Zener solid.

Figure 2

Plots of FOV and QLV model fits and data from a typical specimen for (a) the entire relaxation time, (b) the QLV fit expanded to the first 5s, and (c) the FOV fit. Also shown are plots of the pointwise rms errors for each model fit. The apparent “noise” in the model fit results from using actual pointwise stretch data to compute the stress response, rather than an idealization.

Figure 3

Means and standard deviations of the rms errors for each model. The “∗” indicates a significant difference between QLV and FOV (t test, p<0.05, n=5).

Figure 4

Plots of the cyclic response data and model predictions for a typical specimen for (a) all 20cycles, and expanded to the first three cycles [(b) and (c)]. Also shown are the pointwise rms errors for both methods (d).

Figure 5

Sensitivity plots for each parameter. Parameters were varied around their optimal values, and the LS error was computed using Eq. 9.

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