Investigating a continuous shear strain function for depth-dependent properties of native and tissue engineering cartilage using pixel-size data.

A previously developed novel imaging technique for determining the depth dependent properties of cartilage in simple shear is implemented. Shear displacement is determined from images of deformed lines photobleached on a sample, and shear strain is obtained from the derivative of the displacement. We investigated the feasibility of an alternative systematic approach to numerical differentiation for computing the shear strain that is based on fitting a continuous function to the shear displacement. Three models for a continuous shear displacement function are evaluated: polynomials, cubic splines, and non-parametric locally weighted scatter plot curves. Four independent approaches are then applied to identify the best-fit model and the accuracy of the first derivative. One approach is based on the Akaiki Information Criteria, and the Bayesian Information Criteria. The second is based on a method developed to smooth and differentiate digitized data from human motion. The third method is based on photobleaching a predefined circular area with a specific radius. Finally, we integrate the shear strain and compare it with the total shear deflection of the sample measured experimentally. Results show that 6th and 7th order polynomials are the best models for the shear displacement and its first derivative. In addition, failure of tissue-engineered cartilage, consistent with previous results, demonstrates the qualitative value of this imaging approach.