**4.3 Finite element models for offline FEA**

172 Haptics Rendering and Applications

interaction is simplified and optimized so that the number of elements and nodes is kept as small as possible. In the haptic interface, spines are represented by beam elements. A similar strategy of simulating spines with beam elements can be found in (Lafage et al., 2004). Figure 8 shows the construction of a rigid beam element FE model of a vertebra. Points A and B are the center points of the upper and lower endplates of the vertebral body. Points C and D are the left and rightmost points of the transverse processes. Point E is the rearmost point of the spinal process. Vertebrae are represented using these rigid beams. Point A and B are then connected to intervertebral discs. Point C, D and E are to be connected to specific ligaments. Figure 9(a) and (b) show side and front views of a section of the assemblage of

the spine finite element model and explanation is expressed in Figure 9(c).

Fig. 8. Structure of beam element model of a vertebra

Fig. 9. Composition of finite elements of the spine

Our vertebrae are considered rigid and only the intervertebral discs (IVDs) and ligaments are deformable. The beam element of IVDs is of cylinder type. The flexural moments of inertia (Iy, Iz) of beam elements for IVDs are calculated from the intervertebral discs' geometries. An analogy with the beam theory allows Young's modulus to be expressed as a The purpose of the offline FEA is to gain information on the stress/strain relationships of vertebrae and IVDs. The FE model should be sufficiently delicate and precise in order to achieve this. Normal vertebral body models have a cancellous core covered by a cortical shell of approximately 1.5mm thickness. The thickness of most IVDs is around 10mm. The nucleus pulposus of the IVD is modeled as an incompressible material comprising 40% of the total disc volume. The models of the nucleus, the annulus and the vertebral bodies are assembled with tie constraints between the interacting surfaces. Besides vertebral bodies and IVDs, ligaments play an important role in spinal biomechanics and stability. The seven intervertebral ligaments were incorporated and modeled with truss elements. The ribcage is currently not included in this FE model.

Since two FE models are employed in this system, their deformation under the same boundary conditions should be coherent. Therefore, the mechanical properties of spine FE models must be carefully tuned in order to ensure deformations from both models match. A similar approach of material property adjustment can be found in (Lafage et al., 2004). Since the offline FEA is considered as more accurate than the online FE simulator, its results can serve as a standard, meaning the mechanical properties of the beam element model should be adjusted according to the behaviour of a solid FE spine model. The iterative process of mechanical personalization is engaged in the commercial FEA package. Firstly, the offline element spine model is applied with a force of 40N on the T1 vertebra. The force direction is from the rear to the front. The displacement of each vertebra is recorded and serves as the template. Then, the same load and boundary conditions are applied to the online beam element spine model. Differences between the beam element model and the template solid element model are quantified. Mechanical characteristics of soft tissues of the beam element spine are then tuned manually and locally until the difference between the two simulations tend to less than 5 degrees for the vertebral orientation and 10mm for the vertebral bodyline. The mechanical properties to be tuned here include: the stiffness of ligaments; the Poisson's ratio, the shear stiffness and the torsional stiffness of intervertebral discs.
