Download Appendix : Finite element model design

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Appendix : Finite element model design
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Parametric models are created in two steps: first the vertebral body is designed with pedicles,
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and then the posterior elements were adjusted and joined to the pedicles. The appendix
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presents the creation of the subject-specific vertebral body with pedicles.
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Fourteen independent geometric parameters were fully automatically computed from the 3D
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reconstructions using algorithms designed with Matlab softwares (Matlab 7.5.0 (R2007b),
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The MathWorks, Inc, Massachussets, USA). First, anterior height (Ha), posterior height (Hp),
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left and right lateral heights (Hg and Hd respectively) and mean height (H) which corresponds
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to the distance between the centres of the superior and the inferior endplates, were assessed
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(Figure A1 and Figure A2). Second, three transversal planes were defined: middle plane (z=0
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in the local vertebral frame), superior and inferior least-square planes of the vertebral
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endplates. Widths and depths were assessed using these three planes: Lm, pm_max et pm_min in
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the middle plane, Ls, ps_max, ps_min et Li, pi_max et pi_min in the superior and inferior planes
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(Figure A2 and Figure A3).
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The finite element model of the vertebral body was based on the following parameters:
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- 6 parameters in the transversal planes: Lm, pm_min, Li, pi_min, Ls, ps_min
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- 3 posterior concavities: cm = pm_max – pm_min, ci = pi_max - pi_min, cs = ps_max -ps_min
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- 3 heights: mean height H, sagittal plane and frontal plane height ratio (R AP = Ha/Hp, and RDG
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= Hd/Hg).
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- 2 angular parameters (ANG_A et ANG_P) defined the anterior part and the posterior part of
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the vertebral body. They were fixed to π/6 regarding generic 3D reconstructions of thoracic
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and lumbar vertebrae. (Figure A3).
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Pedicles were designed with 4 parameters defined as follow:
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- Hcentrum_ped = 3/5*H: Z-axis location of the centres of the pedicles (3/5 of the middle height
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H)
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- ANG_PED = π/4, defining the angular position of the centres of the pedicles relatively to the
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X-axis. (Figure A3).
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- Hped = 0.60*H: pedicle height (60% of the vertebral body height in the middle plane)
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- Lped = 0.20*Lm: pedicle width (20% of the vertebral body width in the middle plane)
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The values were chosen regarding the generic 3D reconstructions of thoracic and lumbar
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vertebrae.
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Finally, 5 cortical bone thicknesses were defined:
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- ec ant: anterior thickness (anterior region defined using ANG_A)
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- ec post: posterior thickness (posterior region defined using ANG_P)
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- ec G et ec D: left and right lateral thicknesses.
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- ec ped : thickness for the pedicles.
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In the study, all the thicknesses were fixed to 3 mm, which corresponds to the thickness of
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samples used to derive the apparent mechanical properties of cortico-cancellous bone 24.
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In parallel to the geometric parameters, 5 meshing parameters were defined:
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- N: number of nodes on perimeter of the vertebral body. In this study, N=40.
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- NH: number of nodes on the vertebral body height. In this study, NH=13.
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- N_PED_H: number of nodes on the pedicle height. In this study, N_PED_H = 5.
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- N_PED_L : number of nodes on the pedicle width. In this study, N_PED_L = 5.
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- N_PED_p : number of nodes on the pedicle depth. In this study, N_PED_p = 7.
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Values were chosen to find a balance between a short computation time and an accurate
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reproduction of the stress and strain fields in the vertebral body and in the pedicles.
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