# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1
## 3 Numerical Results and Discussion
### Discussion on Model Limitations  
_Creep._ The deformations of bone under the load are time dependent not only due to osteoporosis but also due to creep [42; 43; 44]. Therefore, the long-term critical force or long-term strength of the bone is less than the short-term critical force or short-term strength. In case of the long-term deformation, long term deformation of the bone subject to long-term loading can be modelled properly using well-developed viscoelasticity or creep theories developed in the mechanics of materials provided that the characteristics of the bone is known, for example, modulus of elasticity, compliance function, or creep function. However, in the case of osteoporosis, the common viscoelasticity theories are not suitable due to fact that, in the classic viscoelasticity theories, the derivatives of the modulus of elasticity and the creep functions with respect to the time are nonnegative. In other words, the modulus of elasticity is a non-decreasing function with respect to time. However, in the case of osteoporosis, these conditions are violated and the long term analysis of the bone deformations when the osteoporosis simultaneously takes place is a much more complicated problem that also demands profound theoretical reconsiderations of the existing creep or viscoelasticity theories. The raised issue demands separate profound investigations which is not possible in the present study. _Anisotropy._ Trabeculae bone tissues are strongly anisotropic materials with individual mechanical and topological properties for each patient. Good known representative volume elements (RVE) and homogenization-based methods [45] require topology of trabeculae, which needs micro-CT or micro-MRI scans. In the case of lumbar vertebra, they are impossible or very expansive. Therefore, modelling of mechanical behaviour of patient-specific lumbar vertebra is the challenge and we needs some indirect methods for evaluating mechanical properties of patient-specific lumbar vertebra trabeculae. [PERSON]'s law [46] helps to understand the anisotropy of mechanical properties of bone, and from theoretical point of view, we can still investigate simplified trabecular structures arranged in the direction of external stresses as a suitable mechanical model if we want to understand mechanical behaviour of trabeculae inside a vertebral body. On the other hand, it is experimentally observed fact  
Figure 12: Load dependency graph. Displacement in horizontal direction at point A (Figure 11). that, in verteb L1, trabeculae arrangement is remodelled to withstand maximal compression loads. So, choosing of numerical compression tests is logical too. Yield criteriaIn general, lumbar vertebrae are strongly anisotropic plastic biological tissue. Therefore, failure criteria should be based on yield surface, which can be expressed as a Taylor expansion [47]  
\[\left(\sum_{ik}a_{ik}\sigma_{ik}\right)^{a}+\left(\sum_{pqmn}a_{pqmn}\sigma_{pq} \sigma_{mn}\right)^{\beta}+\left(\sum_{rstlmn}a_{rstlmn}\sigma_{rs}\sigma_{tl} \sigma_{mn}\right)^{\gamma}\leq 1 \tag{15}\]  
where \(a_{ik},a_{pqmn},a_{rstlmn}\) are strength tensors of different order. [PERSON] [48] proposed yield criterion for isotropic materials. [PERSON] [49] extended the [PERSON] yield criterion of isotropic materials to anisotropic materials. Later [PERSON] [50] extended this criterion for anisotropic materials to a unidirectional lamina. It is easy to show that von Mises or Tsai-Hill failure criteria can be derived form Goldenbar-Kopnov failure criteria. Two-dimensional failure propagation simulations by [PERSON] et al. [51] showed that the [PERSON] failure criterion did not capture the failure type, location, or propagation direction nearly as well as the Tsai-Hill criterion. Both the von Mises and Tsai-Hill failure criteria severely underpredicted the amount of displacement needed to produce initial failure in the porcine abdominal aortas. Therefore, using of von Mises or Tsai-Hill failure criteria would overestimate lumbar vertebra fracture risk.