Abstract
Purpose
Identifying the elbow flexion–extension (F–E) movement axis is important for placing a hinged elbow external fixator. An X-ray fluoroscopy-based method is widely used in clinical practice, exposing the patient and surgeons to high doses of radiation. Additionally, the accuracy and repeatability of the fluoroscopy-based method are very low and affected by subjective factors.
Methods
To solve this problem, an X-ray-free method based on kinematics analysis was proposed to identify the elbow F–E movement axis, and a navigation system was built to guide the placement of the elbow external fixator.
Results
Our X-ray-free navigation method is more repeatable than the current X-ray fluoroscopy method used clinically. Both our algorithm and the NIST (National Institute of Standards and Technology) algorithm showed high accuracy and repeatability to identify the axis.
Conclusions
The method proposed in this study is very promising to avoid a large dose of X-ray radiation and increases the repeatability and performance of identifying the elbow F–E movement axis for the placement of the hinged elbow external fixator.
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Abbreviations
- F–E:
-
Flexion–extension
- \(\hbox {X}_W \hbox {Y}_W \hbox {Z}_W \) :
-
Coordinate system of the positioning system
- \(\hbox {X}_H \hbox {Y}_H \hbox {Z}_H \) :
-
Coordinate system of the humerus tracker
- \(T_{H\rightarrow W} \) :
-
Space transformation from \(\hbox {X}_H \hbox {Y}_H \hbox {Z}_H \) to \(\hbox {X}_W \hbox {Y}_W \hbox {Z}_W \)
- \(P_W \) :
-
Point set of the position of the ulna tracker in reference to \(\hbox {X}_W \hbox {Y}_W \hbox {Z}_W \)
- \(P_H \) :
-
Point set of the position of the ulna tracker in reference to \(\hbox {X}_H \hbox {Y}_H \hbox {Z}_H \)
- \(P_{Hi} \) :
-
The \(i\hbox {th}\) point of dataset \(P_H \)
- \({P_{H}}^{\prime }\) :
-
Point set of the projection of \(P_H \) to plane \(\beta \)
- \({P_{Hi}}^{\prime }=[{P_{Hi}^x }~{P_{Hi}^y }~{P_{Hi}^z }{]}\) :
-
The \(i\hbox {th}\) point of dataset \(P_H ^{{\prime }}\)
- \({{{\varvec{n}}}}_r \) :
-
Orientation vector of the elbow flexion–extension axis
- \(\beta \) :
-
The plane on which \(P_H ^{{\prime }}\) distributes
- PC1:
-
The first principal component of \(P_H \)
- PC2:
-
The second principal component of \(P_H \)
- PC3:
-
The third principal component of \(P_H \)
- \(E[\cdot {]}\) :
-
Mathematical expectation
- \(\mathrm{Cov} \left( \cdot \right) \) :
-
Covariance matrix
- \({\varvec{\alpha }} _\mathrm{min} \) :
-
The eigenvector of the \(\mathrm{Cov}\left( {P_H } \right) \) smallest eigenvalue
- \(P_m =[{P_m^x }~ {P_m^y }~ {P_m^z }{]}\) :
-
The centroid of \(P_H \)
- \(P_a =[{P_a^x }~ {P_a^y }~{P_a^z }{]}\) :
-
The position (centre) of the elbow F–E axis
- N :
-
The number of points of dataset \(P_H \)
- S :
-
The degree of the simulated arc circle
- \({\upsigma }\) :
-
The standard deviation of the noise added to the simulated arc circle
- \(P_\mathrm{E} \) :
-
The enter-point of the axis pin in the model test
- \(P_\mathrm{O} \) :
-
The outpoint of the axis pin in the model test
- \(l^{r}\) :
-
The real axis of the simulated F–E movement
- \(l_j\,\left( {j=1,2,\ldots ,5} \right) \) :
-
The calculated axis of the \(j\hbox {th}\) F–E movement of the elbow model or specimen
- \(l^{m}\) :
-
The mean optimal axis of the five F–E movements of the elbow model or specimen
- \(C_j\,\left( {j=1,2,\ldots ,5} \right) \) :
-
The calculated position (centre) of the \(j\hbox {th}\) elbow model or specimen movement
- \(E_\mathrm{sa}\) :
-
The axis angle errors of the simulated kinematics data
- \(E_\mathrm{st}\) :
-
The axis position (centre) errors of the simulated kinematics data
- \(\mathrm{Ang} (A,B)\) :
-
The angle between line A and line B
- Dist (A, P):
-
The distance from point P to line A or Circle A
- \(E_\mathrm{t} \) :
-
The mean axis position (rotation centre) error in the model or specimen experiment
- \(E_\mathrm{a} \) :
-
The mean axis orientation error in the model or specimen experiment
- \(E_\mathrm{r} \) :
-
The distance from the kinematics data \(P_H \) to the circle fitted in the model or specimen experiment
- n :
-
The sample size of the statistical comparison
- T :
-
T is the duration of placing the axis pin under navigation
- T1:
-
The time of acquiring kinematics data
- T2:
-
The duration of placing the axis pin under navigation
- A1:
-
Our algorithm
- A2:
-
NIST algorithm
- ROM1:
-
Range of specimen F–E movement after fixator placement
- ROM2:
-
The maximal range of motion of the elbow after external fixator placement
- NIST:
-
National Institute of Standards and Technology
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Acknowledgements
This work was supported by the China Major R and D Plan (No. 2016YFC0105800) and National Natural Science Foundation of China (61361160417, 81271735).
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Song, J., Ding, H., Han, W. et al. An X-ray-free method to accurately identify the elbow flexion–extension axis for the placement of a hinged external fixator. Int J CARS 13, 375–387 (2018). https://doi.org/10.1007/s11548-017-1680-8
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DOI: https://doi.org/10.1007/s11548-017-1680-8