3D Geometric Shape Reconstruction for Revision TKA and UKA Knees Using Gaussian Process Regression

Abstract

Revision knee surgery is complicated by distortion of previous components and removal of additional bone, potentially causing misalignment and inappropriate selection of implants. In this study, we reconstructed the native femoral and tibial surface shapes in simulated total/unicompartmental knee arthroplasty (TKA/UKA) for 20 femurs and 20 tibias using a statistical inference method based on Gaussian Process regression. Compared to the true geometry, the average absolute errors (mean absolute distances) in the prediction of resected femur bones in TKA, medial UKA, and lateral UKA were 1.0 ± 0.3 mm, 1.0 ± 0.3 mm, and 0.8 ± 0.2 mm, respectively, while those in the prediction of tibia resections in the corresponding surgeries were 1.0 ± 0.4 mm, 0.8 ± 0.2 mm, and 0.7 ± 0.2 mm, respectively. Furthermore, it was found that the prediction accuracy depends on the size and gender of the resected bone. For example, the prediction accuracy for UKA cuts was significantly better than that for TKA cuts (p < 0.05). The female and male cuts were often overfit and underfit, respectively. The data indicated that this reconstruction approach can be a viable option for planning of revision surgeries, especially when contralateral anatomy is pathological or cannot be available.

Appendices

Appendix A: Formulations of Gaussian Process and Gaussian Process Regression

Gaussian Process

Mathematically, a Gaussian Process (GP) is used to model a vector-valued random variable, such as the 3D deformation field ($u$) of a shape group, which is a function of the spatial coordinates ($x\in {\mathbb{R}}^{3}$)¹:

$$u\left(x\right)={\left[{u}^{1}\left(x\right) {u}^{2}\left(x\right) {u}^{3}\left(x\right)\right]}^{T} \sim GP(\mu , k)$$

(A1)

where the subscripts (1, 2, 3) of $u$ represent its three components for the 3D case. $\mu (x)\in {\mathbb{R}}^{3}$ is the mean deformation, and $k(x,x{^{\prime}})\in {\mathbb{R}}^{3\times 3}$ is the covariance function, which is also termed as the kernel.

In terms of the discrete deformation field (as the reference surface mesh, ${\Gamma }_{R}$, in nature consisting of a set of points, is the feasible region, i.e., $x\in {\Gamma }_{R}$), the kernel of the GP can be converted to a covariance matrix ($\Sigma$), when reducing the GP to a multivariate normal distribution that models scalar-valued random variables. By singular value decomposition of $\Sigma$ (i.e., principal component analysis), eigenvalues (${\lambda }_{i}$) and eigenvectors (${\varphi }_{i}$) can be obtained, and the deformation field can be expressed as a linear combination of $\sqrt{{\lambda }_{i}}$ and ${\varphi }_{i}(x)$^15,44:

$$u(x)=\mu (x)+{\sum }_{i=1}^{n}{\alpha }_{i}\sqrt{{\lambda }_{i}}{\varphi }_{i}(x)$$

(A2)

where ${\alpha }_{i} \sim N(\mathrm{0,1})$ are random variables which follow the standard normal distribution, and $n$ is the number of principal components. It can be noted that Eq. A2 is the formulation of the well-known SSM.^15,44 Therefore, the SSM is a special application of the GP model (Eq. A1), when the discrete deformation field was considered. In this study, the SSMs were also created with an intent to visualize shape variations, where either femur or tibia SSM included all principal components ($n=51$, equal to the sample size minus 1).

Gaussian Process Regression

Gaussian process regression (GPR) is a Bayesian inference method designed to make predictions incorporating prior knowledge (kernels) and to provide uncertainty measures over predictions.²⁹ Technically, GPR computes a posterior GP model, ${u}_{p} \sim GP({\mu }_{p}, {k}_{p})$, which is the conditional distribution of the prior GP model (Eq. A1), given partial observations ($\tilde{u }$). Considering the uncertainty of observations, we have assumed that:

$$\tilde{u }=u\left({y}_{i}\right)+\epsilon$$

(A3)

where ${y}_{i}\in {\mathbb{R}}^{3}$, $i=1,\cdots ,m$ are the coordinates of observations. $\epsilon \sim N(0, {\sigma }^{2}{I}_{3m\times 3m})$ is the random noise, which follows a normal distribution; where ${\sigma }^{2}$ is the variance controlling the degree of uncertainty, and ${I}_{3m\times 3m}$ represents the identity matrix with the dimension of $3m\times 3m$.

As a merit, the posterior GP model has a closed-form (explicit) solution, and the posterior mean (${\mu }_{p}$) and kernel (${k}_{p}$) can be formulated as:

$${\mu }_{p}\left(x\right)=\mu \left(x\right)+\Sigma (x,Y){\left(\Sigma \left(Y,Y\right)+{\sigma }^{2}{I}_{3m\times 3m}\right)}^{-1}(\tilde{u }-\mu \left(Y\right))$$

(A4)

$${k}_{p}\left(x,{x}^{^{\prime}}\right)=k\left(x,{x}^{^{\prime}}\right)- \Sigma (x,Y){\left(\Sigma \left(Y,Y\right)+{\sigma }^{2}{I}_{3m\times 3m}\right)}^{-1}\Sigma (Y,x{^{\prime}})$$

(A5)

where $Y={[{y}_{1}^{1} {y}_{1}^{2} {y}_{1}^{3} \cdots {y}_{m}^{1} {y}_{m}^{2} {y}_{m}^{3}]}^{T}$. In particular, the mean of the posterior GP model, ${\mu }_{p}$, is the solution to GPR, as it has the maximal probability of the conditional distribution. Please refer to more mathematical details of GPR in the application of shape completion, which have been documented previously.¹

Appendix B: Rigid and Non-rigid Iterative Closest Point Algorithms

See Figs.

Figure B1

The pseudo code of the rigid iterative closest point (ICP) algorithm.

B1 and

Figure B2

The pseudo code of the non-rigid iterative closest point (ICP) algorithm.

B2.