Corrigendum for the paper ‘Nonlinear elasticity of biological tissues with statistical fibre orientation’

Abstract

We made an error in the numerical example simulating the unconfined compression of a cylindrical cartilage sample, given in §5.2. In articular cartilage, the collagen fibres are close to lying on the transverse plane in the superficial zone and close to aligned with the axial direction in the deep zone. These two opposite situations correspond to model (A) and model (B), respectively, as described in §3.4 and figure 1. In both cases, the limit for the concentration parameter b approaching zero yields isotropy, which is attained in the middle zone. In our example, we imposed b to vary linearly with the coordinate running from the bone – cartilage interface to the articular surface: from 5 to 0 (model (B)) in the deep zone, and from 0 to 5 (model (A)) in the superficial zone. Also, in order to reproduce the experiment of Park et al. [1], we ‘cut’ a 0.5 mm layer of the deep zone, so that the concentration parameter b was varying from 2.85 to 0 in the deep zone. Because of an error in a sign in our code, the fibre distribution in the cylindrical sample of §5.2 is upside-down, i.e. model (A) was used in place of model (B), and vice versa. As a result of this error, things went such that a 0.5 mm layer of superficial (rather than deep) zone was ‘cut’. This implies that the contour plot shown in figure 4 is only roughly but not exactly upside-down, and the force response in figure 3 is not accurate because the zonal subdivision (deep-middle-superficial) is not as desired. Furthermore, the comparison of the radial displacement of the lateral boundary in figure 4 with the experiments of Fortin et al. [2] was not appropriate, as the radial displacements in their tests were relative to a cartilage sample attached to a layer of subchondral bone, which constrains radial displacement at the bone – cartilage interface. While these errors do not affect the validity of the model nor the robustness of the numerical implementation, we felt it was important to correct them.