The first nine load cycles preconditioned the tissue such that a stable response to loading could be achieved. Samples were considered properly preconditioned if the peak strain of the final preconditioning cycle was within 2% of the previous cycle. According to the method described by Fung,
14 a continuous exponential curve of the form σ(ε) =
A/
B(
e B ε − 1) was fit to the discrete stress (σ) versus strain (ε) data
(Fig. 2) .
14 For each sample, distinct loading, and unloading coefficients were optimized using a Levenberg-Marquardt algorithm such that a minimum of 90% of the variation in the data could be explained by the curve fit. In addition, a simplistic parameterization of modulus (
E) was reported as the derivative of the fitted loading curve at 1% strain, the upper end of what was identified by Downs et al.
15 to be a physiologic range of strains. Finally, the strain energy absorbed on a load cycle was reported. The area contained within the hysteresis loop delimited by the loading and unloading curve fits gives a measure of strain energy absorbed (
W) on a loading cycle in SI units of Joules per cubic meter. Practically, this parameter was determined by calculating the difference between integrals of the loading curve fit, σ
l (ε), and the unloading curve fit, σ
u (ε), up to the peak strain, ε
max \[W{=}\ {{\int}_{0}^{{\epsilon}_{\mathrm{max}}}}{\sigma}_{l}({\epsilon})d{\epsilon}{-}\ {{\int}_{0}^{{\epsilon}_{\mathrm{max}}}}{\sigma}_{u}({\epsilon})d{\epsilon}.\]