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Computational Modeling of the Spine

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Shape Analysis in Medical Image Analysis

Part of the book series: Lecture Notes in Computational Vision and Biomechanics ((LNCVB,volume 14))

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Abstract

Modeling of the human spine requires the extension from single object modeling to object ensembles. The spine consists of a constellation of vertebrae where the individual vertebrae show a complex shape. While most neighbouring vertebrae look very similar, their shape changes significantly along the spine. Due to these challenges, more sophisticated model formulations are needed that go beyond shape modeling of vertebrae. In this article, we combine several high-level models of the spine into one common framework. The individual vertebrae are represented as a set of models covering shape, gradient and appearance information as well as relative location and orientation. By encoding further anatomical information into the shape representation of the individual vertebrae, e.g., important anatomical regions or significant landmarks, clinically relevant parameters can be easily derived from the shape models. The spine is expressed as a sequence of rigid transformations between vertebrae and different statistical methods can be used to cover the variability of spinal curvatures. For selected applications that are vertebra labelling in limited field of view scans and segmentation in both CT and MRI, we show how this comprehensive framework can be used for an automatic image interpretation of medical images of the spine. Furthermore, the problem of change assessment for osteoporotic fracture detection is tackled with this framework as an example for CAD.

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Author information

Authors and Affiliations

  1. Philips Research Europe-Hamburg, Röntgenstrasse 24-26, 22335 , Hamburg, Germany

    Tobias Klinder & Cristian Lorenz

  2. Ecole Polytechnique de Montreal, 6079, Succursale Centre-Ville, Montreal, QC, H3C 3A7, Canada

    Samuel Kadoury

Authors
  1. Tobias Klinder
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  2. Samuel Kadoury
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  3. Cristian Lorenz
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Corresponding author

Correspondence to Tobias Klinder .

Editor information

Editors and Affiliations

  1. GE Healthcare and University of Western Ontario, London, Ontario, Canada

    Shuo Li

  2. Departamento de Engenharia Mecânica, Universidade do Porto, Porto, Portugal

    João Manuel R. S. Tavares

Appendix

Appendix

The conversion between the two representations of rotation is described by Rodrigues’ formula:

$$\begin{aligned} R = I + \sin (\theta ) C({\mathbf {n}}) + (1-\cos (\theta ))C({\mathbf {n}})^2 \end{aligned}$$
(32)

with \(C({\mathbf {n}}) = \left[ \begin{array}{ccc} 0 &{} -n_z &{} n_y \\ n_z &{} 0 &{} -n_x \\ -n_y &{} n_x &{} 0 \\ \end{array} \right] \).

The inverse map (from a rotation matrix to a rotation vector) is given by the following equations [8]:

$$\begin{aligned} \theta = \arccos \left( \frac{tr(R)-1}{2}\right) \ \ \ \mathrm{{and}} \ \ \ {C(\mathbf {n})} = \frac{R-R^T}{2\sin (\theta )}. \end{aligned}$$
(33)

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Klinder, T., Kadoury, S., Lorenz, C. (2014). Computational Modeling of the Spine. In: Li, S., Tavares, J. (eds) Shape Analysis in Medical Image Analysis. Lecture Notes in Computational Vision and Biomechanics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-03813-1_11

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  • DOI: https://doi.org/10.1007/978-3-319-03813-1_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03812-4

  • Online ISBN: 978-3-319-03813-1

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