Advanced Computational Framework for the Automatic Analysis of the Acetabular Morphology from the Pelvic Bone Surface for Hip Arthroplasty Applications

Abstract

2D- and 3D-based innovative methods for surgical planning and simulation systems in orthopedic surgery have emerged enabling the interactive or semi-automatic identification of the clinical landmarks (CL) on the patient individual virtual bone anatomy. They enable the determination of the optimal implant sizes and positioning according to the computed CL, the visualization of the virtual bone resections and the simulation of the overall intervention prior to surgery. The virtual palpation of CL, highly dependent upon the examiner’s expertise, was proved to be time consuming and to suffer from considerable inter-observer variability. In this article, we propose a fully automatic algorithmic framework that processes the pelvic bone surface, integrating surface curvature analysis, quadric fitting, recursive clustering and clinical knowledge, aiming at computing the main parameters of the acetabulum. The performance of the method was evaluated using pelvic bone surfaces reconstructed from CT scans of cadavers and subjects with pathological conditions at the hip joint. The repeatability error of the automated computation of acetabular center, size and axis parameters was less than 1 mm, 0.5 mm, and 1.5°, respectively. The computed parameters were in agreement (<1.5 mm; <0.5 mm; <3.0°) with the corresponding reference parameters manually identified in the original datasets by medical experts. According to our results, the proposed method is put forward to improve the degree of automation of image/model-based planning systems for hip surgery.

Appendix: Mean-Shifted Curvature

Let $N_{\text{F}} \left( v \right)$ and ${\textbf{n}}\left( f \right)$ be a set of all triangular faces adjacent to vertex $v \in V$, with V the set of the h mesh vertices, and the normal direction to the face f, respectively. The vertex normal ${\textbf{n}}\left( v \right)$ is defined as the weighted average of the normal directions $\left\{ {{\textbf{n}}\left( f \right)} \right\}$ of its adjacent faces. Considering ${\textbf{c}}\left( f \right)$ the centroid of the face f, the vertex normal can be computed as:

$$ {\textbf{n}}\left( v \right) = \frac{{\sum\limits_{{f \in N_{\text{F}} }} {w\left( f \right){\textbf{n}}\left( f \right)} }}{{\left\| {\sum\limits_{{f \in N_{\text{F}} }} {w\left( f \right){\textbf{n}}\left( f \right)} } \right\|}}\quad {\text{with }}w\left( f \right) = \frac{1}{{\left\| {{\textbf{c}}\left( f \right) - v} \right\|}} $$

(A1)

For $v_{j} \in N_{V} \left( v \right)$, the set of n vertices adjacent to v (1-ring), a unit tangent t _j associated to v _j is defined as the normalization of the projection of $v_{j} - v$ on the tangent plane of v. The normal curvature of v along t _j can be then approximated by:

$$ k_{n} \left( {{\textbf{t}}_{j} } \right) = - \frac{{\langle v_{j} - v,{\textbf{n}}\left( {v_{j} } \right) - {\textbf{n}}\left( v \right)\rangle }}{{\left\| {v_{j} - v} \right\|}}\quad {\text{for}}\,{\textbf{t}}_{j} \left( {j = 1,2, \ldots ,m} \right) $$

(A2)

Without loss of generality, let $k_{n} \left( {{\mathbf{t}}_{1} } \right)$ be the maximum among all these values. Denoting $\theta_{j}$ the angle between t _j and t ₁, $k_{n} \left( {{\mathbf{t}}_{j} } \right)$ can be then expressed as

$$ k_{n} \left( {{\mathbf{t}}_{j} } \right) = a\cos^{2} \theta_{j} + b\cos \theta_{j} \sin \theta_{j} + c\sin^{2} \theta_{j} $$

(A3)

where $a = k_{n} \left( {{\mathbf{t}}_{1} } \right)$ and b and c can be estimated by least square fitting. The Gaussian $k_{G}$, the mean $k_{m}$, and the two principle curvatures k ₁ and k ₂ at the vertex v can be computed as:

$$ k_{G} = ac - \frac{1}{4}b^{2} ,\;k_{m} = \frac{1}{2}\left( {a + c} \right),\;k_{1,2} = k_{m} \pm \sqrt {k_{m}^{2} - k_{G} } $$

(A4)

In order to amplify the difference between convex and concave regions the following curvature combination is utilized:

$$ k_{c} = \frac{2}{\pi }{ \arctan }\left( {\frac{{k_{1} + k_{2} }}{{k_{1} - k_{2} }}} \right) $$

(A5)

The mean-shifted curvature refinement takes its root from the general mean-shift algorithm.4,5 Given a sample point of a feature space, a mean-shift algorithm firstly estimates the density of the feature space, then evaluates the gradient of the density function, and finally moves the sample points along their gradient direction. A standard mean-shift algorithm is generally guaranteed to be convergent. The mean-shifted curvature refinement40 in this case involves two components in the feature space $\Upomega$, namely the 2D manifold, embedded in 3D space, and the 1D curvature, as $\Upomega = \left\{ {\left( {v,k_{c} \left( v \right)} \right):v \in V,k_{c} \in K} \right\}$.

A generalized mean-shift density function is defined as:

$$ d\left( {v,k_{c} \left( v \right)} \right) = \frac{1}{{ha^{2} b}}\sum\limits_{{v_{j} \in V}} {F_{a} \left( {\frac{{v - v_{j} }}{a}} \right)} F_{b} \left( {\frac{{k_{c} \left( v \right) - k_{c} \left( {v_{j} } \right)}}{b}} \right) $$

(A6)

where a and b are constants representing the bandwidth, F _a and F _b are two kernel functions and h in the number of vertices of the mesh. In order to simplify the density function and without losing generality, the kernel F _a can be setup to a flat function, i.e., $F_{a} \left( x \right) = 1\quad {\text{for}}\left\| x \right\| \le t\,{\text{and}}\,0$ otherwise. Aiming at updating the curvature of a vertex with that of its 1-ring vertices leads to set t be 1. By setting a = 2, (Eq. A6) simplifies further in:

$$ d\left( {k_{c} \left( v \right)} \right) = \frac{1}{4hb}\sum\limits_{{v_{j} \in N_{V} \left( v \right)}} {F_{b} \left( {\frac{{k_{c} \left( v \right) - k_{c} \left( {v_{j} } \right)}}{b}} \right)} $$

(A7)

Setting $F_{b} \left( x \right) = pf_{b} \left( {\left\| x \right\|^{2} } \right)$, with p a normalization factor assuring that $F_{b} \left( x \right)$ integrates to 1, the gradient of the curvature density of Eq. (A7) assumes the following shape:

$$ \nabla d\left( {k_{c} \left( v \right)} \right) = p\frac{{m\left( {k_{c} \left( v \right)} \right)}}{{2hb^{3} }}\sum\limits_{{v_{j} \in N_{V} \left( v \right)}} {g_{b} } \left( {\frac{{\left\| {k_{c} \left( v \right) - k_{c} \left( {v_{j} } \right)} \right\|^{2} }}{b}} \right) $$

(A8)

where $g_{b} \left( x \right) = - F_{b}^{\prime } \left( x \right)$ and

$$ m\left( {k_{c} \left( v \right)} \right) = - k_{c} \left( v \right) + \frac{{\sum\limits_{{v_{j} \in N_{V} \left( v \right)}} {k_{c} \left( {v_{j} } \right)g_{b} \left( {\left\| {\frac{{k_{c} \left( v \right) - k_{c} \left( {v_{j} } \right)^{2} }}{b}} \right\|} \right)} }}{{\sum\limits_{{v_{j} \in N_{V} \left( v \right)}} {g_{b} \left( {\frac{{\left\| {k_{c} \left( v \right) - k_{c} \left( {v_{j} } \right)} \right\|^{2} }}{b}} \right)} }} $$

(A9)

is the curvature mean-shift at the vertex v, which does not depend on the number of vertices h in the mesh. The curvature $k_{c} \left( v \right)$ is iteratively refined as $k_{c}^{t + 1} \left( v \right) = k_{c}^{t} \left( v \right) + m\left( {k_{c}^{t} \left( v \right)} \right)$ until $\left| {k_{c}^{t + 1} \left( v \right) - kctv} \right.$ is less than a user specified threshold $\in$. By convenience, the Epanechnikov kernel $F_{b} \left( x \right) = \left( {0.75\left( {1 - x^{2} } \right)} \right)$, proposed successfully in kernel smoothing,4 was utilized in this development. The value b and $\in$ were heuristically setup equal to 10⁻⁵ and 10⁻⁴, respectively. From 5 to 10 iterations were sufficient to refine the curvature at all the mesh vertices.