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. 2021 Jul 19;11:14731. doi: 10.1038/s41598-021-94141-4

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Definition of the parameters involved in optimizing translational alignment of an affected bone with a proximally aligned target bone (green). (a) The direction of the gravitation axis of the target bone is represented by unit vector k. During the optimization procedure the distal and proximal plane positions are defined by (d₁, d₂), as measured along the gravitation axis of the bone. The large planes define the search range for the osteotomy planes and are set by the user. The orientation of a distal search plane (n_d) is quantified by azimuth and elevation angles (φ_d, ψ_d) and the rotation of the distal segment is equal to β_d (see Fig. 1). The proximal plane orientation and bone rotation follow by calculation (R_p = R_cR_d⁻¹ and Eqs. 3 and 4). (b) In-plane translations of the distal and proximal bone segments are defined in the sideward and upward (x, y) direction of the bone cross section. The shaded surface areas represent the amount of bone overlap. (c) Vector e represents the residual translation error of the distal bone segment and is used to calculate the error in the length and transverse direction (see text). The mean of the nearest-neighbor distances (red lines) between mesh points of the middle-proximal bone assembly and the target bone (green) is used to control alignment of these bone segments (e_m in Eq. 5).