Determination of dynamically adapting anisotropic material properties of bone under cyclic loading
Introduction
Bone adapts to the loading conditions imposed by the musculoskeletal system. The macro-structure of bone consists of an outer surface, cortical or compact bone, an inner spongy structure, and cancellous or trabecular bone. The tissue, whether compact or cancellous, is minimized in mass for its function with changes in the loading cycle affecting orientation of structure (Currey, 2006).
The observations on the structure of bone date back to Julius Wolff, Hermann Meyer and Karl Culmann (Wolff, 1892). Wolff hypothesized that (I) trabeculae tend to be oriented along the principal stress directions and (II) bone tissue is not present where there are high shear loads. However, the relative orientations of the trabecular network were also observed to occur in a wide range of directions often differing from Wolff’s statements (Cowin, 2001). This work attempts to demonstrate the compatibility of Wolff’s hypothesis and Cowin’s observations.
Several groups describe bone remodelling on the basis of Wolff and Roux’s findings (Roux, 1885). Steeb et al. (2005) showed that the orientation of randomly oriented elements could be adjusted relatively to the direction of the principal stresses by rotation. The research groups of Huiskes, Nackenhorst and Carter have used development theories to describe addition and removal of bone material along specific directions (Weinans et al., 1992, Nackenhorst, 2005, Beaupré et al., 1990). Additionally, Huiskes et al. simulated cancellous bone remodelling in terms of osteoblast build up of bone and osteclast resorption of bone as a function of stresses (Mullender et al., 1994, Mullender and Huiskes, 1995, Huiskes et al., 2000, Tanck et al., 2001, Ruimermann et al., 2005, Ruimermann and Huiskes, 2005). Weinkamer et al. (2004) formulated the development of bone remodelling as a consequence of mechanical conditions.
Coelho et al. (2009) presented a method to simulate the development of bone density by calculations on the micro- and macro-scale. The micro-level takes into account the trabecular structure as well as bone surface density. At the macro-scale the bone is treated as a continuum material. Jang and Kim (2010) and Kowalczyk (2010) presented studies which simulate the structural changes in compact and trabecular bone, also by calculating on the macro- and the micro-scale.
The goal of the research presented here is to develop the global effective elastic constants of cancellous bone for inclusion in finite element (FE) simulations, and not to describe the development of the inner structure of cancellous bone. There are twenty-one global effective elastic constants that are the coefficients of a fourth-order-tensor. A methodology similar to the one presented here is often applied in plasticity theories, where appropriately twenty-one parameters of Hill-anisotropy are described to change over time. The assumption here is that the optimal growth direction of a trabecula is parallel to a time dependent principal stress. In the present theory, before the material parameters reach the optimal value, the loading situation changes and new values will be considered optimal. Therefore, the actual orientation of the trabecula may not be parallel to the current principal stress, but the momentary stiffness matrix is, nevertheless, assumed to be the predicted one.
Our long term goal is to calculate the material parameters for bone under realistic loading conditions to validate this theory. The novelty of this theory is that only a small number of material parameters are necessary to simulate a bone with anisotropic material behaviour. Two simple models are presented to show how the theory works.
Section snippets
Materials and methods
To calculate the changes of material parameters of bone it was assumed that (a) bone adapts to loading conditions; (b) the trabeculae tended to orient parallel to the directions of the momentary principal stresses, which are orthogonal to one another; and (c) loading changes with time.
For the FE simulation the bone specimen was considered to be filled with reference cubes (Fig. 1a), which are disconnected, each representing the vicinity of a Gaussian point. The length of each edge of the
Results
The results of the first simple example of one reference cube show the dynamic changes of the twenty-one material parameters under cyclic loading calculated with the present theory. The four orientations (Fig. 2a) were always those the cube tended to reach. Before the cube could reach these final orientations and the trabeculae reach these final thicknesses, the loading is changed and the cube started to get into the next final position. The development of the twenty-one material parameters
Discussion
The theory presented in this paper is based on the fundamentals of Wolff’s hypotheses, viewed as a momentary tendency only. Thus, the micro-structure of the cancellous bone does not necessarily form the orthogonal axes of an orthotropic material, and may not always be co-axial to any final state of stresses.
If the cyclic loading processes are neglected, especially in cases where in one direction there is no average compressive or tensile stress, strong differences between predictions and in
Conflict of interest statement
None declared.
Acknowledgements
The author thanks the German Research Foundation for funding this work in the Collaborative Research Centre 599.
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