Numerical analysis of human vertebrae using the finite cell method

Setting

Osteoporosis compromises bone strength, increasing the risk of vertebral fractures with severe health consequences. The development of an accurate and reliable patient-specific vertebral model would be of major clinical relevance, for both the prognosis of fractures and the investigation of implant systems. The finite cell method (FCM) is a high-order embedded domain approach which can handle complex geometry without the need for mesh generation [1, 2]. In this work, we apply the FCM to simulate the biomechanical behaviour of human vertebrae. The flexible nature of FCM regarding complex geometry and different types of geometric representation allows us to develop a robust simulation pipeline for patient-specific analysis.

Validation

In the first steps for the validation of the numerical results obtained using FCM, a study was carried out in cooperation with “Klinikum Rechts der Isar”, performing compression tests on fresh-frozen vertebral specimens. FCM models based on micro-CT scans were generated to simulate the mechanical behaviour of the vertebral bodies. Figure 1 depicts a micro-CT-based FCM mesh of a vertebral body. Preliminary results show very good agreement between the numerical results of the FCM models and digital volume correlation (DVC) measurements (see Figures 2 and 3). Next steps will focus on the experimental validation of FCM models based on clinical CT scans, and on multiple-segment models, which incorporate realistic boundary conditions. 

Figure 1: Micro-CT-based FCM mesh	Figure 2: Numerical approximation of displacements vs. measurements

Figure 3: Linear regression between numerically approximated (FCM) and experimental (DVC) local displacements within a vertebral body.

Simulation of vertebra-implant system

For the accurate simulation of vertebra-implant systems, the finite cell method is combined with a domain-coupling technique [3]. Using a separate FCM mesh for each subdomain, and weakly enforcing the interface conditions, the discretization can reproduce the weak discontinuity across the material interface. Moreover, the recently developed multi-level hp-refinement scheme [4] is employed to refine the FCM meshes in order to resolve stress concentration and local solution features. This combination of numerical techniques is able to efficiently handle material interface problems involving complex geometries, without the need for boundary-conforming mesh generation.

Figure 4: a) Finite cell mesh for the vertebral body	b) surface mesh for coupling interface	c) Finite cell mesh for the screws

Figure 5 depicts the results of an FCM simulation for a vertebral body with two posterior screws. Using separate FCM meshes for the screws and the vertebral bodies, and applying the multi-level hp-refinement scheme to resolve stress concentrations at the bone-implant interface, the numerical discretization yields mechanically sound results - similar to results reported for simulations using micro-FE models [5]. Further work will consider implants in multiple vertebral bodies (a mobile segment) for more realistic modelling of the boundary conditions. 

Figure 5: Equivalent Von Mises stresses within the screws and the vertebral body

References

1. Parvizian, J., Düster, A., Rank, E., 2007. Finite cell method. Computational Mechanics
2. Ruess, M., Tal, D., Trabelsi, N., Yosibash, Z., Rank, E., 2012. The finite cell method for bone simulations: verification and validation. Biomechanics and modeling in mechanobiology
3. Ruess, M., Schillinger, D., Özcan, A.I., Rank, E., 2014. Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Computer Methods in Applied Mechanics and Engineering
4. Zander, N., Bog, T., Elhaddad, M., Frischmann, F., Kollmannsberger, S., Rank, E., 2016. The multi-level hp-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes. Computer Methods in Applied Mechanics and Engineering
5. Ruffoni, D., Wirth, A.J., Steiner, J.A., Parkinson, I.H., Müller, R., van Lenthe, G.H., 2012. The different contributions of cortical and trabecular bone to implant anchorage in a human vertebra. Bone

Publications



• Elhaddad, M.; Zander, N.; Bog, T.; Kudela, L.; Kollmannsberger, S.; Kirschke, J.S.; Baum, T.; Ruess, M.; Rank, E. Multi-level hp-finite cell method for embedded interface problems with application in biomechanics
  International Journal for Numerical Methods in Biomedical Engineering 34 (4), pp. e2951, Wiley, 2017
  DOI: 10.1002/cnm.2951 - Status: Verlagsversion / published
  
  
  
  
  
  
  

• Elhaddad, M.; Kollmannsberger, S.; Valentinitsch, A.; Kirschke, J.S.; Ruess, M.; Rank, E. Micro-CT based finite cell analysis of vertebral bodies
  In: Engineering Mechanics Institute Conference 2017, San Diego, CA, USA, 2017
  
  
  
  
  
  
  

• Elhaddad, M.; Zander, N.; Jomo, J.; Kollmannsberger, S.; Kirschke, J. S.; Ruess, M.; Rank, E. Show abstract Adaptive discretizations for bone-implant systems using the finite cell method
  In: Engineering Mechanics Institute Conference 2016, Nashville, TN, USA, 2016
  
  
  
  In recent years, the finite element method has been increasingly used to study the biomechanics of bone-implant systems. Patient-specific models need to consider the individual geometry for every case. Typically, geometric models of the bone are reconstructed from medical images --- most commonly CT scans . Following segmentation, a finite element mesh has to be generated. This can be a simple voxel-based mesh, where a predefined number of voxels constitute one finite element in a grid that matches the voxel model. Alternatively, geometry-conforming meshing procedures can be used to generate an unstructured mesh. To this end, the surface points are extracted from the voxel model and converted to a solid model, which is then meshed into an unstructured grid. Similarly, the geometry of the implant is usually reconstructed from medical images, or provided by Computer-Aided-Design (CAD) models, and then incorporated in the finite element mesh. The adequate treatment of the bone-implant interface demands the spatial refinement of the finite element mesh at the interface for an accurate solution. For voxel-based finite elements, a very fine resolution is already necessary to resolve the implant's geometry. This leads to an extremely high number of degrees of freedom, as the whole grid has to be refined. On the other hand, geometry-conforming finite elements need more pre-processing (converting the bone's voxel model to a solid model), and complex meshing procedures to resolve geometries of the bone and the implant with graded refinement of the mesh towards the interface. In this contribution, we will use the finite cell method (FCM) in conjunction with a hierarchical refinement scheme to model a vertebral fixation system. The FCM discretization completely circumvents the mesh generation procedure, as the geometry is resolved on the integration level. Here, the flexibility of FCM in dealing with different types of geometric representations is demonstrated: the bone's geometry is represented by the voxel model from a CT-scan, whereas the implant's geometry is directly described by a CAD solid model. Both representations are directly used in the FCM framework, thereby substantially reducing the pre-processing effort. Furthermore, the hierarchical refinement scheme yields a higher accuracy at the bone-implant interface by adaptively refining the FCM grid.
  
  
  

• Elhaddad, M.; Zander, N.; Kollmannsberger, S.; Bauer, J.; Ruess, M.; Rank, E. Show abstract Loading simulation of spinal vertebrae using the finite cell method
  In: VI International Conference on Computational Bioengineering, Barcelona, Spain, 2015
  
  
  
  Osteoporosis compromises bone strength, increasing the risk of vertebral fractures with severe health consequences. The development of an accurate and reliable patient-specific vertebral model would be of major clinical relevance, for both the prognosis of fractures and the investigation of implant systems. In the case of spinal fusion, the peri-implant bone strains are of major interest. High forces are transmitted at the bone-implant interface and screw loosening is the major cause of surgical failure. In recent years, the finite element method has been increasingly applied to predict the biomechanical response of vertebrae. In general, quantitative CT-data is used to retrieve the geometric model and the inhomogeneous material properties. A finite element mesh is then generated for the numerical simulation. For the finite element analysis of a bone-implant system, the adequate treatment of the bone-implant interface demands the adaptive refinement of the finite element mesh for an accurate solution. In this contribution, the finite cell method (FCM), a high order embedded domain approach, is applied to simulate the biomechanical behavior of human vertebrae. The recently developed multi- level hierarchical refinement is used to enrich the FCM, allowing to resolve the highly inhomogeneous material properties of vertebrae. For the simulation of bone-implant systems, the hierarchic refinement is used to resolve the bone-implant interface appropriately. The feasibility of using FCM for predicting the biomechanical response of human vertebrae is demonstrated with numerical examples on both intact and fused vertebral segments. Furthermore, the inherent hierarchical nature of the FCM strongly supports verification and validation.
  
  
  

• Rank, E.; Elhaddad, M.; Zander, N.; Kollmannsberger, S.; Bauer, J.; Ruess, M. Show abstract Coupling scales and adaptive refinement for simulation of bone-fixation interfaces using the Finite Cell Method
  In: VI International Conference on Coupled Problems in Science and Engineering, Venice, Italy, 2015
  
  
  
  In recent years many studies on the numerical simulation of bone-implant systems have been presented. In general, a geometric model of a bone is derived from qCT-data, including its highly inhomogeneous material properties. Bone and implant are then meshed and simulated by finite elements. In the case of a fixation the zone of the bone close to the screw is of major interest. High forces are transmitted here and screw loosening is the major cause of surgical failure. As the scale of interest for this interface region may be several orders of magnitude smaller than that of the bone, an adaptive refinement of the finite element mesh is mandatory for an accurate solution. Whereas this is already quite demanding in cases of a single simulation, it becomes prohibitively expensive if a sequence of many simulations in order to optimize type, position and pre-stress of an instrumentation shall be performed.In this paper we will extend the recently developed Finite Cell Method, a high order embedded domain approach by a hierarchical refinement strategy in the regions of interest. Whereas the concept of the FCM completely relieves from the need to generate a finite element mesh, the hierarchical refinement yields an accuracy and a scale resolution which could not obtained by classical finite elements methods. We will discuss the basic properties of this adapted multi-scale method and demonstrate its feasibility for simulating bone-mechanics on the example of the fixation of a spinal segment.
  
  
  

Contact person

Mohamed Elhaddad