Collagen fibers and their structural arrangement influence tissue tensile stiffness and strength. While a variety of modeling approaches incorporate collagen fibers explicitly as one of the components, due to the complexity of the fiber organization, aligned fibers are usually considered. In the pioneering work of Lanir [1], a constitutive relation for continuous fiber distributions was proposed, where the strain energy and stresses are obtained by angular integration (AI) of infinitesimal fractions of fibers aligned in a given direction. Lanir’s formulation has been successfully used to describe the mechanical behavior of a variety of tissues. In particular, Ateshian et al. [2] showed that large values of the tensile Poisson’s ratio for articular cartilage in tension and the low values observed in compression can be explained using a continuous angular distribution for the fibers. A disadvantage of the AI formulation is the large number of calculations required to evaluate the strains and stresses. On the other hand, Generalized Structure Tensors (GST) have been proposed to model tissues with continuously distributed collagen fibers [3,4]. These tensors are assumed to represent the three-dimensional distribution of the fibers. Once the tensor has been defined, the strain in the fibers can be readily obtained by multiplication with a strain tensor. The advantage of this approach is the small number of calculations required to obtain the strain energy and stresses of the fibers. As a result, this formulation can be efficiently implemented in numerical algorithms like finite elements. However, this approach is limited, as it is valid only when all of the fibers are in tension and when the fiber distribution is small [5]. A numerical comparison is required to quantify when an angular distribution can be considered acceptably small to justify using this more computationally efficient approach. The objective of this study is to numerically compare the AI and GST formulations to determine the range of values of angular distribution for which the GST approach can be accurately used.

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