used that coincides with the largest textural variations in the spongy bone image. Since bone density, the fractal dimension, and lacunarity are orthogonal metrics, they can be used for multidimensional clustering to better discriminate between degrees of osteoporosis (Table 9.1) [74].

9.3.5 Computer Modeling of Biomechanical Properties

Up to this point, Sect. 9.3 was primarily concerned with the empirical relationship between image properties and bone strength. Most notably X-ray attenuation, which is directly related to bone density, can be linked to bone strength. The texture analysis methods introduced in the previous sections aim at extracting information from biomedical images of trabecular bone that are independent from average density and therefore provide additional information. A less empirical approach is the modeling of bone biomechanical properties with finite-element models (FEM). In finite-element analysis, an inhomogeneous object is subdivided into a large number of small geometrical primitives, such as tetrahedrons or cuboids. Each element is considered homogeneous with defined mechanical properties (stress– strain relationship). External forces and displacements are applied by neighboring elements. External boundary conditions can be defined. Those include spatial fixation and external forces. The entire system of interconnected finite elements is solved numerically, and the forces, the shear tensor and the displacement for each element are known. Finite-element models also allow time-resolved analysis, providing information on motion and the response to time-varying forces.

The use of FEM for skeletal bone became popular in the late 1970s and has been extensively used to relate skeletal variation to function (for a general overview, see [76]). Since then, literally dozens of studies have been published each year where FEM were used for the functional understanding of bone, predominantly in the spine. Because of the large volume of available literature, we will focus on spinal vertebrae as one pertinent example. One of the most fundamental questions that can be approached with finite-element analysis is the contribution of compact bone to the overall weight-bearing capacity of bone. Studies by Rockoff et al. and Vesterby et al. indicate a major load-bearing contribution of the cortical shell [15, 77].

Two main approaches exist. A general vertebral model with representative geometry can be designed and used to study general spine biomechanics. Conversely, the geometry of individual vertebrae can be extracted from volumetric images (CT or MRI), approximated by finite elements, and subjected to load and deformation analysis. Although the second approach holds the promise to improve the assessment of the individual fracture risk, it has not found its way into medical practice, mainly because of the computational effort and because of uncertainties about the influence of the finite-element subdivision, material property assignment, and the exact introduction of external forces [78]. The two representative approaches are shown in Fig. 9.10. The early model by Lavaste et al. [79] was generated from global X-ray based measurements, including the width and height of the vertebral

Fig. 9.10 Finite-element approximations of the spine. (a): Early parametric model of a vertebral body (adapted from [79]). The vertebral body is generated from X-ray measurements, such as height, width, and curvature. (b): Comprehensive finite-element model of the lumbar spine and the computed stress magnitude (adapted from [80] with permission to reprint through the Creative Commons License)

body, the diameter of its waist, and the length of the vertebral processes. As such, it is a semi-individual model that reflects gross measurements combined with a generalized shape model. Individual vertebral bodies can be combined to form a semi-individualized model of the spine or segments of the spine [79]. The recent model by Kuo et al. was segmented from volumetric high-resolution CT images (0.35 mm pixel size), and different material properties were assigned to spongy bone, the cortical shell, the endplates, and the intervertebral discs. Such a model may easily contain 20,000 elements per vertebra, and it accurately reflects the geometry of an individual spine.

The assignment of material properties is an ongoing question. In many models, including the two examples presented above, the spongiosa is modeled as a homogeneous material – in the example of Kuo et al., spongy bone is assigned a Young’s modulus of 100 MPa compared to cortical bone with 12,000 MPa. However, spongy bone may have a larger local variation of its mechanical strength than the model allows. General models for spongy bone include hexagonal or cubic stick models [81, 82]. Specific models have been developed for spongy bone, often based on micro-CT or micro-MRI images that can resolve individual trabeculae [83,84]. Once again, the model strongly depends on the accurate representation of the geometry and the local material properties. To determine the load-bearing capacity of an entire vertebra, the spongiosa model needs to be appropriately connected to the cortical shell and the endplates. Furthermore, detailed models of the spongiosa cannot presently be used in a clinical setting, because whole-body scanners do not provide the necessary microscopic resolution to build a detailed model of the trabeculae.