Nonlinear viscoelastic characterization of the porcine spinal cord

Abstract

Although quasi-static and quasi-linear viscoelastic properties of the spinal cord have been reported previously, there are no published studies that have investigated the fully (strain-dependent) nonlinear viscoelastic properties of the spinal cord. In this study, stress relaxation experiments and dynamic cycling were performed on six fresh porcine lumbar cord specimens to examine their viscoelastic mechanical properties. The stress relaxation data were fitted to a modified superposition formulation and a novel finite ramp time correction technique was applied. The parameters obtained from this fitting methodology were used to predict the average dynamic cyclic viscoelastic behavior of the porcine cord. The data indicate that the porcine spinal cord exhibited fully nonlinear viscoelastic behavior. The average weighted RMSE for a Heaviside ramp fit was 2.8kPa, which was significantly greater (p < 0.001) than that of the nonlinear (comprehensive viscoelastic characterization (CVC) method) fit (0.365kPa). Further, the nonlinear mechanical parameters obtained were able to accurately predict the dynamic behavior, thus exemplifying the reliability of the obtained nonlinear parameters. These parameters will be important for future studies investigating various damage mechanisms of the spinal cord and studies developing high resolution finite elements models of the spine.

Skeletal muscle tensile strain dependence: Hyperviscoelastic nonlinearity

“…Calculated stress relaxation data for each strain increment (five steps each and nine specimens for a total of 45 stress relaxation steps) were fitted to a constitutive formulation for viscoelastic materials (Shetye et al, 2014; Van Loocke et al, 2008) (Equation 2, where S ( e G , t ) is the stress at time t and strain level e G , E ( t ) is the time dependent elastic modulus, and τ is an integration variable). A three-term linear Prony series (Ghoreishy, 2012; Maikos et al, 2008; Shetye et al, 2014; Troyer et al, 2012; Van Loocke et al, 2008) was utilized to describe the time dependent elastic modulus (Equation 3, where E 0 is the instantaneous modulus, E n are relaxation coefficients, and τ n are time constants).…”

“…A three-term linear Prony series (Ghoreishy, 2012; Maikos et al, 2008; Shetye et al, 2014; Troyer et al, 2012; Van Loocke et al, 2008) was utilized to describe the time dependent elastic modulus (Equation 3, where E 0 is the instantaneous modulus, E n are relaxation coefficients, and τ n are time constants). A nonlinear optimization ( lsqnonlin ) in Matlab applied the Levenberg-Marquardt unbounded local optimizing algorithm to fit experimental and model data.…”

“…A nonlinear optimization ( lsqnonlin ) in Matlab applied the Levenberg-Marquardt unbounded local optimizing algorithm to fit experimental and model data. This approach varied all seven of the material properties from Equation 3 to minimize the residuals between the model output and experimental data at each time point (Calvo et al, 2010; Kauer et al, 2002; Lei and Szeri, 2007; Shetye et al, 2014; Wheatley et al, 2015). Additional weight was given to the peak response to ensure a good fit throughout loading and initial relaxation by mutiplying the peak error by the total number of data points in each step (Wheatley et al, 2015).…”

“…Previous studies have applied a polynomial expression to identify the dependence of Prony series parameters on stretch level for compressed muscle (Van Loocke et al, 2009) and spinal cord (Shetye et al, 2014). However, we chose to employ a simple exponential function for the purpose of minimizing the number of optimized parameters while still maintaining a robust, non-linear function.…”

An Experimental and Numerical Investigation of the Mechanical Properties of Spinal Cords

“…Due to the limited availability of human specimens, testing techniques are generally limited to animal autopsy samples [23,24,25,26,27,28,29,30]. A pioneer in conducting research on the mechanical properties of the spinal cord was Tunturi in 1978 [23].…”