Table 2.

Penalty calculation.

Constraint	Metric parameter	Penalty equation	Description
Minimum bone overlap	δd, δp (*)	${10}^{{100(A}_{\mathit{rel}}-A_{\min })}$	This penalty term increases exponentially once the overlap drops below a user-defined limit (Amin).With Arel the relative amount of bone overlap (range [0,1]) at the distal and proximal interfaces respectively
Maximum plane angulation	ρp, ρd (*)	$2^{\alpha -{\alpha }_{\mathit{max}}}+2^{-\alpha -{\alpha }_{\mathit{max}}}$	The steepness α is calculated as the angle between the plane normals nd, np and k representing the direction of the gravitation axis. Avoids steep osteotomies by increasing the metric exponentially when the steepness of either osteotomy plane exceeds αmax as set by the user;
Maximum tissue torsion	μp, μd (*)	$2^{\beta -{\beta }_{\mathit{max}}}+2^{-\beta -{\beta }_{\mathit{max}}}$	Increases exponentially when the proposed bone rotation β = [βd, βp] exceeds a predefined torsion limit βmax as set by the user
Plane position	τ	$100(n_a+n_b+n_c)$	Increases when either osteotomy plane passes beyond the plane constraints as set by the user. For this evaluation the cross section of the bone is first determined using the proximal and distal plane constraints. This provides two point sets p1 and p2 na is assigned the number of points out of p1 that are found distally from the distal plane constraint nb is assigned the number of points out of p2 proximally from the proximal plane constraint nc contains the number of points in the volumetric overlap that remains after osteotomizing a distal and proximal bone segment using the two osteotomy planes. If the osteotomies do not intersect, nc equals 0

In cases where the optimizer exceeds a given constraint the penalty largely increases the metric, effectively invalidating the current optimizer attempt.

(*) Metric parameter separately determined for distal and proximal plane, as indicated by the index (d, p).