22.1 Input Parameters

Consider, for example, a femur subjected to loads on the femoral head, as shown in Fig. 22.1. We may be interested in the deformation or stresses/strains in the bone in the proximal region. We briefly discuss the key input parameters essential for modeling a problem such as this in the following paragraphs and these are considered in greater detail in the following sections.

1. 
   

   The geometry: We need to know the shape and the internal structure (e.g., volumetric regions with cancellous and cortical bone). For FE analysis, the geometry is subdivided into simpler shapes (e.g., tetrahedrons or hexahedrons) called elements.

   
   
2. 
   

   Material properties: We need to know, for instance, the elastic properties of bone in different regions of the femur. Further, if the femur is expected to be loaded beyond its elastic capacity, the simulation program will require postelastic properties as well. Furthermore, it is known that bone does not deform immediately on application of loads; its response is time dependent. For some problems, it may be essential to prescribe properties that can describe such time-dependent behavior. It is apparent that the computer code being employed for analysis needs to be capable of accepting and interpreting this complex range of properties.

   
   
3. 
   

   Boundary conditions: We need to specify how the object being analyzed is supported or restrained from undergoing rigid body motions (i.e., move without deforming). It is important to note that in the absence of any restraints, the body will simply shoot off in space when subjected to forces. Restraints should be such that the object cannot undergo rigid body translations or rotations. For the femur subjected to forces on the femoral head shown in Fig. 22.1, a typical approach involves restraining it completely in all directions at a distal transverse plane as shown. Clearly, this is not the in vivo support condition, which is far more complex. However, this approach is capable of providing good answers to many clinical questions. Similar support conditions are often employed when conducting physical laboratory experiments.

   
   
4. 
   

   Loads: The direction, magnitude, and location of loads need to be specified for a computational analysis. For the example of the femur being considered, the value and direction of the loads as well as the surface region of the femoral head on which these loads act will change with the physiological activity being undertaken. Computational biomechanics offers the opportunity to conduct an evaluation for a wide range of load cases, which is not possible in a laboratory setting.

22.2 Output Results

Once the input has been specified, the FE analysis can be initiated. What can one expect to get out of it? The commonly sought output parameters are deformations, stresses, and strains. Computational biomechanics can provide this information at practically any location. Moreover, it permits evaluation of all components of displacements stresses and strains. These details are impossible to obtain from biomechanical experiments.

It is important to give the output a sanity check before examining the more complex output parameters. Superimposed undeformed and deformed shapes of the geometry (before and after loading) provide the simplest check as normally it is quite easy to visualize how deformation is likely to happen. A second simple check is to examine the reactions at the restrained points in all directions. The algebraic sum of the reactions needs to be equal to the load applied.

Stresses and strains constitute the key features of the output. It is important to remember that each of these has six components. With respect to stress, we have normal stresses σ_x, σ_y and σ_z, and shear stresses τ_xy, τ_yz, and τ_zx. Similarly for strain, we have normal strains ε_x, ε_y, and ε_z, and shear strains γ_xy, γ_yz, and γ_zx. In these, x, y, and z refer to the coordinate directions chosen when providing geometric data. While these values are useful (e.g., we may be interested in finding out how load in a particular direction affects normal stresses in another direction), it is important to remember that these values vary with the choice of the axes; another modeler doing exactly the same problem with a different set of axes will get different absolute values for these coordinate-dependent quantities. It is more useful to examine invariant quantities (i.e., those that do not depend on the choice of the coordinate system). It is always possible to transform the coordinate system at each point such that the stress (or strain) only has normal components and no shear components. These three normal components are called principal stresses (or strains) and are provided by all commercial codes. The principal values provide the largest, an intermediate, and the smallest normal stresses/strains, and are very useful to examine. As per the commonly used sign convention, tension is positive and compression is negative, so maximum principal stress contours can be used to indicate regions experiencing tensile stresses and minimum principal stress contours for compressive stresses. It is, however, important to remember that in general the directions of principal values vary from point to point.

Another stress invariant often used in literature is the von Mises stress (σ_v). This is simply a scalar quantity constructed using stress components as

While von Mises stress is a good indicator of “how much” stress the object is experiencing, it is not able to tell whether these are tensile or compressive. The von Mises stress is a very good predictor of yielding of many metals and is used as their yield criterion. It is not a good indicator for bone yielding or failure, though it has been used in this way (further discussed later in this chapter). As a result, many researchers criticize production of von Mises stress contours for bone. However, von Mises stresses provide a good measure of stress state in bone but should be used in conjunction with principal stress values.

While stress and strain are related, it is important to note that for triaxial problems, the two can provide quite distinct results. Consider, for example, a cube subjected to uniformly distributed compressive load in one direction. As a result, it will experience compressive strains in the direction of loading and tensile strains in orthogonal directions. However, while it will have compressive stresses in the direction of loading, stresses in the orthogonal directions will be zero. Therefore, it is a good idea to examine not only stresses but also strains.

The output can provide a large variety of information, depending on the type of analysis being conducted. It is useful to appraise the variety of output parameters; some may provide much better answers than others on the behavior being examined.

22.3 Defining the Geometry

Biomechanical systems have a complex geometry. However, advances in imaging technologies followed by automated software procedures to convert these images to FE meshes have made defining the geometry of biomechanical systems much easier. For biomechanical analysis, the three-dimensional (3D) geometry is typically developed from computed tomography (CT) or magnetic resonance imaging scans of either cadaveric bone samples or that of real patients. These are then segmented to define regions and provide boundaries for different segments (e.g., soft tissue and bone) of the object. Although research into more reliable automated segmentation techniques is ongoing, it is now possible to create a fairly accurate representation of 3D bone geometry (depending on the resolution of the scan), which can be converted to a FE mesh using specialized packages such as ScanIP (Simpleware Ltd., Exeter, UK) or Mimics (Materialise, Leuven, Belgium). Many FE packages will also generate the mesh from 3D geometry provided to them as a computer-aided design (CAD) file. Such files are often available for implants from their manufacturers.

Similarly, by using high-resolution, µCT, or µ-magnetic resonance imaging scans, even the geometry of the bone microstructure can be constructed. The typical process is illustrated in Fig. 22.2. It is, however, important to note that a computational analysis of the whole bone with microlevel resolution (as shown in Fig. 22.2) is rarely conducted, though this is not impossible with the high speed and parallel computing resources available today. For the whole bone analysis, the bone material is mostly assumed to be a solid continuum. The effect of porosity and microstructure is incorporated by varying the material properties of this solid. This aspect is further discussed in the next section.

In recent years, “patient-specific modeling” has been receiving increasing attention. 1 In clinical practice, CT or magnetic resonance imaging scans are not regularly obtained for operations involving joint replacement or trauma. In the absence of volumetric images, patient-specific geometry cannot be readily constructed, though there has been some research on the development of 3D models from planar radiographs. 2

Imaging technologies have made it possible to generate high-quality 3D geometries. As a consequence, expectations from computational modeling have increased. Unfortunately, it is not always realized that geometry alone does not constitute a model; it requires other input parameters (discussed in the following sections) whose specification can be challenging.