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Mesh Morphing and Response Surface Analysis: Quantifying Sensitivity of Vertebral Mechanical Behavior

Abstract

Vertebrae provide essential biomechanical stability to the skeleton. In this work novel morphing techniques were used to parameterize three aspects of the geometry of a specimen-specific finite element (FE) model of a rat caudal vertebra (process size, neck size, and end-plate offset). Material properties and loading were also parameterized using standard techniques. These parameterizations were then integrated within an RSM framework and used to produce a family of FE models. The mechanical behavior of each model was characterized by predictions of stress and strain. A metamodel was fit to each of the responses to yield the relative influences of the factors and their interactions. The direction of loading, offset, and neck size had the largest influences on the levels of vertebral stress and strain. Material type was influential on the strains, but not the stress. Process size was substantially less influential. A strong interaction was identified between dorsal–ventral offset and dorsal–ventral off-axis loading. The demonstrated approach has several advantages for spinal biomechanical analysis by enabling the examination of the sensitivity of a specimen to multiple variations in shape, and of the interactions between shape, material properties, and loading.

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Acknowledgments

Funding for this work was provided by the Canadian Institutes of Health Research (CIHR).

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Correspondence to Ian A. Sigal.

Appendix

Appendix

The coefficients of the metamodels (Table A1) can be assembled into a function, such as

$$ { \log }_{ 10} \left( {\text{Median von Mises stress}} \right) = 1. 70 8 + 0.0 4 7\times {\text{A}} - 0.00 3\times {\text{B}} \cdots - 0. 1 9 1\times {\text{B}} \times {\text{D}} - 0.0 4 5\times {\text{B}}^{ 2} \cdots + 0. 1 9 8\times {\text{E}}^{ 2}. $$

where A = Process size, B = Offset, etc. all coded from −1 to +1. Predictions are in MPa for the stresses and in percentages for the strains. Note that the coding extends to the material model, although only the extreme values −1 and +1 are meaningful. Some of the elements included in the equations contribute little to the metamodel, and were included for completeness. If those elements are not included the equations are simpler and predict essentially the same values. See Table 2 for measures of the quality of fit. To simplify comparison we report strain absolute values.

Table A1 Coefficients of the metamodel (the polynomial functions fit to each) of the responses

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Sigal, I.A., Whyne, C.M. Mesh Morphing and Response Surface Analysis: Quantifying Sensitivity of Vertebral Mechanical Behavior. Ann Biomed Eng 38, 41–56 (2010). https://doi.org/10.1007/s10439-009-9821-z

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  • DOI: https://doi.org/10.1007/s10439-009-9821-z

Keywords

  • Morphing
  • Response surface methods
  • Parameterization
  • Sensitivity analysis
  • Vertebra
  • Biomechanics
  • Finite element modeling