Mathematically Directed Single-Cut Osteotomy

Abstract

A mathematically directed osteotomy (MDO) is a surgical planning technique for correcting long bone deformities. Using a mathematically derived osteotomy plane, the single-cut correction simultaneously addresses angular deformity, axial malrotation, and minor shortening. This review describes an MDO's indications for use, defines its input and output variables, includes the required graphs for osteotomy planning, and provides intraoperative tips and tricks for successful execution. Finally, the authors present a digital MDO calculator to simplify the complex computations and allow for more precise planning.

Keywords: deformity; malunion; mathematically directed single-cut osteotomy.

Figure 1

(a) The mathematical model describes the deformed bone as two cylinders of equal radii whose axes intersect at the level of deformity to define an elliptical plane. These cylinders are “bent” and rotated relative to each other in space. (b) Tibial model with proximal metaphyseal deformity demonstrating projections of the true deformity in the coronal and sagittal views. Angle A is the angulation of the true deformity measured on the maximum deformity view. Angle α, i.e., the orientation angle, is the difference between the coronal plane view and the maximum deformity view [2].

Figure 2

(a) Representation of the osteotomy angle theta (θ), as illustrated by Sangeorzan et al. [1]. It describes how steep to make the osteotomy based on the sagittal view. It is referenced from a plane transverse to the long axis of the bone at the level of the deformity. θ is always measured distally. If θ is greater than 90°, the osteotomy plane goes from an ascending to a descending cut. (b) Corresponding graph that calculates θ based on the true angular deformity (A) and the rotational deformity T [1].

Figure 3

(a) Representation of the starting point angle phi (φ), as illustrated by Sangeorzan et al. [1]. It describes how far to rotate the osteotomy plane based on the axial plane. It is referenced from the midline of the long axis. By convention, it is reported as a number less than 180°. A positive number rotates the osteotomy plane laterally and a negative number rotates it medially. (b) Corresponding graph that calculates φ based on the orientation angle (α) and the rotational deformity T. For the purposes of the graphical calculations, the relative internal rotational deformity is positive (+) and the relative external rotational deformity is negative (−) [1].

Figure 4

Osteotomy technique demonstrated on a 3D printed model. (a) A sterile goniometer or metal triangles are used to precisely execute the osteotomy plane based on the angles θ and φ. (b) Multiple K-wires are placed along the planned osteotomy to confirm the correct osteotomy plane and serve as a cutting guide for the saw blade. (c) Osteotomy completion and removal of the guide K-wires before rotational correction and fixation.

Figure 5

A 3D-printed model before (a,b) and after (c,d) a deformity correction with a mathematically directed single-cut osteotomy. It allows for the simultaneous correction of angular and rotational deformity around one osteotomy plane. Note the change in the relative K-wire positions with the correction of the internal rotational deformity on the axial views. This 3D model represents the clinical example in case 2, where the calculation and execution of the mathematically directed osteotomy are described.

Figure 6

(a) The wrong orientation (ascending or descending) of the osteotomy plane will worsen a rotational deformity rather than correct it. To confirm the appropriate osteotomy position, Paccola developed a simple table to determine the appropriate direction of the osteotomy inclination. This is visualized on the no deformity view. The center of rotation of the osteotomy plane is located at the apex of the deformity in the center of the bone. (b) Modified from Paccola, this table can be used to verify the appropriate osteotomy plane inclination. It takes into account the limb laterality and the relative torsional deformity to be corrected [23].

Figure 6

(a) The wrong orientation (ascending or descending) of the osteotomy plane will worsen a rotational deformity rather than correct it. To confirm the appropriate osteotomy position, Paccola developed a simple table to determine the appropriate direction of the osteotomy inclination. This is visualized on the no deformity view. The center of rotation of the osteotomy plane is located at the apex of the deformity in the center of the bone. (b) Modified from Paccola, this table can be used to verify the appropriate osteotomy plane inclination. It takes into account the limb laterality and the relative torsional deformity to be corrected [23].

Figure 7

See Case 1 description for details. (a–e) demonstrate how to calculate the true angular deformity (A) and the orientation angle (α) from coronal and sagittal plane radiographs.

Figure 7

See Case 1 description for details. (a–e) demonstrate how to calculate the true angular deformity (A) and the orientation angle (α) from coronal and sagittal plane radiographs.

Figure 8

See Case 2 description for details of a tibial malunion. (a–h) calculate the variables θ (Osteotomy Angle) and φ (Starting Point Angle) to derive the mathematically directed osteotomy. Intraoperative and postoperative radiographs reveal successful correction of the deformity and osteotomy union.

Figure 8

See Case 2 description for details of a tibial malunion. (a–h) calculate the variables θ (Osteotomy Angle) and φ (Starting Point Angle) to derive the mathematically directed osteotomy. Intraoperative and postoperative radiographs reveal successful correction of the deformity and osteotomy union.

Figure 8

See Case 2 description for details of a tibial malunion. (a–h) calculate the variables θ (Osteotomy Angle) and φ (Starting Point Angle) to derive the mathematically directed osteotomy. Intraoperative and postoperative radiographs reveal successful correction of the deformity and osteotomy union.

Figure 9

An example of the mathematically derived osteotomy calculator. This tool was developed for ease of use and improved accuracy and precision for MDO planning. Additionally, the program digitally plots the results in a similar format to the original graphs (as seen in cases 1 and 2).