Patient-tailored plate for bone fixation and accurate 3D positioning in corrective osteotomy

Abstract

A bone fracture may lead to malunion of bone segments, which gives discomfort to the patient and may lead to chronic pain, reduced function and finally to early osteoarthritis. Corrective osteotomy is a treatment option to realign the bone segments. In this procedure, the surgeon tries to improve alignment by cutting the bone at, or near, the fracture location and fixates the bone segments in an improved position, using a plate and screws. Three-dimensional positioning is very complex and difficult to plan, perform and evaluate using standard 2D fluoroscopy imaging. This study introduces a new technique that uses preoperative 3D imaging to plan positioning and design a patient-tailored fixation plate that only fits in one way and realigns the bone segments as planned. The method is evaluated using artificial bones and renders realignment highly accurate and very reproducible (d _err < 1.2 ± 0.8 mm and φ _err < 1.8° ± 2.1°). Application of a patient-tailored plate is expected to be of great value for future corrective osteotomy surgeries.

Appendix

A patient-tailored plate is designed by virtually cutting the bone and temporary plate at a user-defined location and repositioning the distal plate segment using the correction matrix M _c. (Fig. 2e). The cross section of the plate (Fig. 6a, Plane 0) is positioned repetitively within the gap such that it smoothly runs from the proximal plate segment to the distal plate segment. This is achieved by extracting the angles of rotation (φ _x , φ _y , φ _z ) from the rotation matrix that orients the cross-sectional points in “Plane 0” to “Plane N” (Fig. 6a), and by linear interpolation of the rotation angles for intermediate planes. Positioning of these N planes within the gap is done using cubic Bezier interpolation between the starting point (P ₀ ), at the centroid of the cross-sectional points, and the end point (P ₃ = M _c P ₀ ). The control points of this Bezier curve (P ₁) and (P ₂) are positioned at P ₁ = P ₀ + cT ₁ and P ₂ = P ₃ + cT ₂, with T ₁ and T ₂ the average tangent vector of the (transformed) cross-sectional points in the direction as shown in Fig. 6c. These control points define the curvature of the Bezier path. With this definition of the Bezier parameters (P ₀, P ₁, P ₂, P ₃,), the centroids of the cross-sectional planes P(i) (i = [0, N]) follow a cubic Bezier curve:

$$ {\vec{\text{P}(}}i )= \left( {1 - \frac{i}{N}} \right)^{3} {\vec{\text{P}}}_{0} + 3\left( {1 - \frac{i}{N}} \right)^{2} \frac{i}{N}{\vec{\text{P}}}_{1} + 2\left( {1 - \frac{i}{N}} \right)\left( \frac{i}{N} \right)^{2} {\vec{\text{P}}}_{2} + \left( \frac{i}{N} \right)^{3} {\vec{\text{P}}}_{3,} \quad i \in [0,N] $$

(A.1)

A polygon mesh of the insert is created by tessellation between neighboring points (Fig. 6c).

Fig. 6

a Creation of a plate insert by copying the cross section (Plane 0) to intermediate planes [0, N] showing smoothly varying orientations. b The centroid of these planes follows a cubic Bezier curve defined by a starting point (P ₀), an end point (P ₃) and two control points (P ₁, P ₂). c Tessellation between consecutive points yields a smooth polygon mesh of the insert