Impact Force, Polar Gap and Modal Parameters Predict Acetabular Cup Fixation: A Study on a Composite Bone

Abstract

The balanced initial fixation of an implant makes up a crucial condition for its long-term survival. However, the quantification of initial fixation is no easy task and, to date, only qualitative assessments can be made. Although the concept of measuring fixation by means of vibration analysis is already widely used in dental implantology, the rigorous application of this method for the assessment of the fixation of femoral and acetabular components remains a challenge. Moreover, most studies on this subject have tended to focus solely on the femoral stem even though acetabular cup fixation is also important and even more difficult with respect to qualitative measurement. This study describes a comprehensive experiment aimed at assessing acetabular cup fixation. Fixation is expressed in terms of the impact force and polar gap variables, which are correlated with the modal properties of the acetabular implant during the various insertion stages. The predictive capabilities of modal frequencies and frequency functions were investigated by means of surrogate models based on the Gaussian process and functional principal component analysis. The prediction accuracy of the proposed models was in the range 82–94%. The results indicate that natural frequencies, reduced frequency, impact force and polar gap features provide great potential in terms of the prediction of implant fixation.

Appendix A

A.1 Coherence Function

The coherence function $\gamma ^{2}$ between output Y and input X is defined as:

$$\begin{aligned} \gamma ^{2}(f) = \frac{G_{xy}(f)G_{xy}^{*}(f)}{G_{xx}(f)G_{yy}(f)} \end{aligned}$$

(2)

where $G_{xy}(f)$ is the cross-spectral function and $G_{xx}, G_{yy}$ are auto-spectral power functions. The function is strictly real and positive with values of 0–1.

A.2 Modal Assurance Criterion

The MAC criterion is defined as a real-value function of two modal shape vectors $\phi _{1}, \phi _{2}$:

$$\begin{aligned} MAC(\phi _{1}, \phi _{2}) = \frac{\vert \phi _{1}\phi _{2}^{*}\vert ^{2}}{\phi _{1}\phi _{1}^{*}\phi _{2}\phi _{2}^{*}} \end{aligned}$$

(3)

The MAC function has a range of 0–1.

A.3 Spectral Evolution During the Insertion of the Implant

Typical bone-implant assembly frequency responses at different insertion steps. Only 20% of the samples are shown. The number of samples per one run varied from 50 to 60 (Fig. 8).

Figure 8

An illustrative frequency response over the insertion steps. The vertical lines correspond to the identified mode and its frequency. 5–7 modes were successfully detected in the range 0–2000 Hz.

A.4 Reproducibility Test

The shape similarity of any two curves $f_{1}(x),\, f_{2}(x)$ was measured by dot product metric defined as:

$$\begin{aligned} \theta = \frac{\big (f_{1},\, f_{2}\big )}{\vert \vert f_{1}\vert \vert \vert \vert f_{2}\vert \vert } \end{aligned}$$

(4)

where $\big (\cdot , \cdot \big )$ is an inner product defined at $L^{2}$ function space:

$$\big (f_{1},\, f_{2}\big ) = \int f_{1}f_{2}{\text{d}}x$$

(5)

The denominator represents a norm:

$$\vert \vert f\vert \vert = \sqrt{\big (f, f\big )}$$

(6)

This metric has a range of −1–1. The discrete data was smoothed and represented by b-spline approximation functions (fda package, R-project) (Fig. 9 and Table 5).55

Figure 9

The repeated measurement of the bone–implant response for the bone samples; bone #1: 3 runs, bone #2: 10 runs.

Table 5 The similarity metric for model variables (implants #1 and #2).

A.5 Inelastic Bone Deformation

Tests on inelastic bone deformation due to repeated use and the post-fracture stiffness of bone sample #1 (Fig. 10 and Table 6).

Figure 10

Reproducibility test on the reinsertion of implant #1 into bone sample #2. The post-fracture stiffness of bone sample #1.

Table 6 Test for the force difference due to inelastic bone deformation: Mann–Whitney U test ($p^{*}\,<\,0.05$).

A.6 Material Properties of the Composite Bone

The material from which the model was constructed comprised a composite mixture of short glass-fibres and poly-acrylate resin (outer cortical shell) and solid foam (inner core) (Table 7).

Table 7 Material properties of sawbone.

A.7 Iterative Removing of Irrelevant Features

If the length-scale l is large with respect to the others then the corresponding feature is deemed irrelevant. Such irrelevant features were iteratively removed so as to maintain the optimal level of model performance. The threshold was estimated according to a decrease in model accuracy ${ {f=\{R^{2}, {\text{MSE}}, \sigma _{\text{e}}\}}}$ of more than 5% following the removal of the feature (Fig. 11).

Figure 11

Iterative removing of features.