What is the Geometric Rule that Guides Accurate Vertical Orientation of the Single-cut Inclined Osteotomy in a Combined Angulation-rotation Deformity of Long Bone?

Balachandar Gopalan

doi:10.5005/jp-journals-10080-1638

. 2025 Aug 18;20(1):11–16. doi: 10.5005/jp-journals-10080-1638

What is the Geometric Rule that Guides Accurate Vertical Orientation of the Single-cut Inclined Osteotomy in a Combined Angulation-rotation Deformity of Long Bone?

Balachandar Gopalan ^1,^✉

PMCID: PMC12445133 PMID: 40979928

Abstract

Background

Single-cut inclined osteotomy for angulation-rotation (A-R) deformity in long bone has a known transverse orientation, which is opposite to the direction of rotational deformity. The geometric rule(s) to guide the vertical orientation is hitherto unknown.

Materials and methods

Using cylinder-shaped non-hardening modelling clay, eight angular (coronal, sagittal and their combinations) and two rotational (internal and external) deformities yielding 16 A-R deformity pairs were simulated for a right-sided model. The magnitudes of A and R deformities were 45° each. Resultant magnitudes of vertical and transverse orientations of the single-cut were constant at 45° and 22.5°, respectively. Transverse rotational orientation of the cut was external for internal rotational deformity and, internal for external rotational deformity. Vertical orientation of the cut was ascending and descending for each of the 32 A-R deformity models. Outcome measure was visual contact between oblique cut surfaces.

Results

After ascending cut and derotational correction, the A-R deformities that maintained contact were varus-internal rotation, procurvatum-internal rotation, varus-procurvatum-internal rotation, varus-recurvatum-internal rotation, valgus-external rotation, recurvatum-external rotation, valgus-recurvatum-external rotation and valgus-procurvatum-external rotation. After descending cut and derotational correction, the A-R deformities that maintained contact were valgus-internal rotation, recurvatum-internal rotation, valgus-recurvatum-internal rotation, valgus-procurvatum-internal rotation, varus-external rotation, procurvatum-external rotation, varus-procurvatum-external rotation and varus-recurvatum-external rotation.

Conclusion

The geometric rules guiding the vertical orientation of single-cut inclined osteotomy in A-R deformity are:

Complementary A-R deformity requires an ascending osteotomy.
- – Varus and/or procurvatum with internal rotation.
- – Valgus and/or recurvatum with external rotation.
Compensatory A-R deformity requires a descending osteotomy.
- – Varus and/or procurvatum with external rotation.
- – Valgus and/or recurvatum with internal rotation.
In an A-R deformity with dissociative angular components, coronal plane deformity supersedes sagittal plane deformity in dictating the vertical orientation of the osteotomy. This is irrespective of the magnitude of coronal deformity.
- – Varus-recurvatum with internal or external rotation.
- – Valgus-procurvatum with internal or external rotation.

Clinical significance

The combination pattern of angular and rotational components (A-R) determines accurate vertical orientation of the cut.
Application of the geometric rules bypasses (1) complex calculations, (2) multiple trial-and-error methods and (3) expensive bone models.
These simple rules will enable surgeons to consider the appropriate inclined osteotomy for any A-R deformity in clinical practice.
The utility of 3D-printed models would be appropriate to improve the precision of the cut before surgery.

How to cite this article

Gopalan B. What is the Geometric Rule that Guides Accurate Vertical Orientation of the Single-cut Inclined Osteotomy in a Combined Angulation-rotation Deformity of Long Bone? Strategies Trauma Limb Reconstr 2025;20(1):11–16.

Keywords: Angulation rotation deformity, Clay model, Compensatory deformity, Complementary deformity, Geometric rule, Inclined osteotomy, Single-cut osteotomy, Transverse orientation, Vertical orientation

Introduction

The geometrical equivalent of a human long bone diaphysis is a cylinder that has 360 possible angular deformities. Angular deformities (A) in anatomically recognised frontal (C) and sagittal (S) planes are characterised as apex anterior (procurvatum/flexion), medial (valgus), posterior (recurvatum/extension), and lateral (varus). The remaining three-hundred-and-fifty-six “oblique plane” angular deformities fall under four categories, namely, apex antero-lateral (varus-procurvatum), postero-lateral (varus-recurvatum), postero-medial (valgus-recurvatum) and antero-medial (valgus-procurvatum). The torsional/rotational deformity (R) in the axial/transverse plane is either internal or external. The combination of eight angular and two rotational deformities yields 16 angulation-rotation (A-R) deformities with an inclined axis of rotation. All 720 A-R deformities in a long bone of any one side fall within the ambit of these 16 deformity patterns.

A-R deformities are conventionally categorised as “positive”, which comprise (varus, procurvatum and internal rotation) and “negative” (valgus, recurvatum and external rotation).^1–3 A complementary A-R deformity has either all positive or all negative individual components of angular and rotational deformities. In contrast, a compensatory A-R deformity has either positive angular with negative rotational or negative angular with positive rotational deformities. Complementary deformity indicates twisting along the longitudinal axis with progressive curvature resembling “C”. Compensatory deformity indicates twisting along the longitudinal axis in the opposite direction, resembling “S”. Both represent mental models that aid the surgeon in comprehension of the deformity in three dimensions. Furthermore, the angular deformities (A) can be either associative (varus-procurvatum, valgus-recurvatum) or dissociative (valgus-procurvatum, varus-recurvatum).

Single-cut inclined osteotomy and acute correction of A-R deformity is an accepted technique. Each inclined osteotomy is perpendicular to the inclined axis of rotation of a particular A-R deformity. Consequently, there are 16 specific inclined osteotomies. The goals of this osteotomy are (1) to obtain ellipsoid bone surfaces with a large contact area that slide against each other, (2) to correct deformities in all three dimensions, while (3) enabling compression fixation. The unique features that make this osteotomy specific to a particular A-R deformity are (1) the orientation in the transverse plane, (2) the magnitude of transverse orientation, which is known as “phi” angle and determined by the equation R/2 or half of the rotational deformity.^1,4,5 (3) the orientation in the vertical plane and (4) the magnitude of vertical orientation, which is known as “theta” angle and determined by the formula arctan [sin (A/2)/tan (R/2)].^1,4,5 Orientation of the osteotomy represents accuracy, while the magnitude of the osteotomy represents precision. Accuracy of osteotomy is the prerequisite for its precise execution.

In the last four decades, several publications have detailed the intricacies of single-cut inclined osteotomy for A-R deformity based on mathematical, graphical, geometrical and visual concepts.^1,4–9 Established osteotomies for varus-internal rotation deformity of the tibia in Blount's disease and the first metatarsal in hallux valgus require simple calculations without preoperative trials in bone models, but are not universally applicable to all A-R deformities.^10–12 Therefore majority of inclined osteotomies rely on complex methods and multiple trials and errors in models.^5,6,8,9,13 This has been attributed to difficulty in determining the accurate vertical orientation of the inclined single-cut. While the geometric rule for transverse orientation of inclined single-cut is external for an internal rotational deformity and vice versa, a similar rule that guides the vertical orientation of inclined osteotomy is hitherto unknown.^1,2,5–8 The author aimed to identify the geometric rule(s) that guided accurate vertical orientation of the single-cut inclined osteotomy in a specific A-R deformity.

Materials and Methods

The author simulated 32 (16 pairs) A-R deformities in cylinder-shaped non-hardening modelling clay measuring 55 × 10 mm and weighing 75 grams. The validity of its use as a preoperative template in hand surgery has been documented by Zolotov et al.¹⁴ The deformity is defined based on the human anatomical position of the upper and lower limbs. Anterior, posterior, medial and lateral surfaces were marked as A, P, M and L, respectively. Cephalad/superior and caudad/inferior ends were defined. Using a goniometer, rotations and angulations were simulated.

For rotational simulation, wooden sticks were placed perpendicular to the transverse plane of the cylindrical model in proximal and distal fragments. The distal aspect of the model was twisted along its long axis, internally and externally, to obtain a magnitude of 45° between the wooden sticks. Anatomical plane angular deformities of 45° magnitude were simulated with the appropriate centre of rotation of angulation (CORA) at the proximal-middle junction. For oblique plane angular deformities, 32° of coronal and sagittal angulations were created. This was intentional to yield a value of 45° for both combined angular magnitude (A = √S² + C²) and A-R deformity plane (arctan A/R). A right-sided varus-procurvatum-internal rotation deformity has been depicted (Fig. 1).

Figs 1A to D — Right-sided cylindrical clay model (orange colour) depicting 32° of varus (A) and procurvatum (B) with resultant apex anterolateral deformity of 45°. (C) There is internal rotation of 45° based on the angle between perpendiculars to proximal and distal fragments in axial view (D)

Determination of Plane of the Cut

The definition of angular deformity and the location of the apex were the first and crucial steps. This is best seen on the “maximum deformity” view. The oblique plane is identified by the formula arctan S/C. The CORA and transverse bisector line (tBL) along the angular correction axis (ACA) were marked (Fig. 2A). The plane of cut must be perpendicular to the plane of deformity. For a varus and valgus deformity, the cut begins anteriorly and exits posteriorly. For a procurvatum and recurvatum deformity, the cut begins medially and exits laterally. For an associative angular deformity, i.e., varus-procurvatum and valgus-recurvatum deformity, the cut begins anteromedially and exits posterolaterally. For a dissociative angular deformity, i.e., valgus-procurvatum and varus-recurvatum deformity, the cut begins anterolaterally and exits posteromedially.

Figs 2A to D — Plane of the cut is anteromedial, representing the “maximum deformity” view (A). Anatomical axis lines are drawn and their intersection is at the CORA with the tBL along the ACA. Axial view (B) shows a stick perpendicular to CORA and another stick, which is 22.5° (phi angle or half of rotational deformity) in external rotation. This represents the transverse orientation of the cut, which begins some distance distal to tBL, seen from frontal view (C). Anterolateral view representing “no deformity” view (D) depicts vertical orientation of the cut at 45° between the stick and transverse plane of the model (theta angle or arctan angulation/rotation)

Magnitude of Transverse Orientation of the Cut (Phi Angle)

This angle is determined by the equation R/2, or half of the rotational deformity.⁴ In the models, rotational deformity was 45°. Therefore, phi angle was 22.5° (Fig. 2B).

Direction of Transverse Orientation of the Cut

This is best seen on “axial” view. The wooden stick was rotated externally in the transverse plane (Fig. 2C).

Magnitude of Vertical Orientation of the Cut (Theta Angle)

This was calculated using the formula arctan A/R for deformities less than 45°, and arctan [sin (A/2)/tan (R/2)] for deformities greater than 45°.⁵ It represents the vertical orientation of the inclined cut concerning the transverse plane of the cylinder. In all the clay models, it was 45°.

Direction of Vertical Orientation of the Cut

This is best seen on the “no deformity” view (Fig. 2D). Two inclined cuts (ascending and descending) were performed for each deformity with a sharp-edged plastic knife. Ascending cuts always began in the distal fragment and exited through the proximal fragment, while a descending cut began in the proximal fragment and exited through the distal fragment. The fragments were derotated along the sliding surfaces to correct A-R deformity. The exact amount of derotation required can be calculated using the formula √A² + R². Our endpoint of correction was visual confirmation of good contact between cut surfaces as a tangible measure, which corroborates with published literature.¹ Each pair of 16 A-R deformity models was cut with an ascending and descending orientation yielding 32 different outcomes.

Results

After an accurate vertically oriented cut, the A-R deformity was corrected in all planes and there was complete contact between oblique surfaces (Fig. 3). After an inaccurate vertically oriented cut, the A-R deformity was corrected in all planes, but there was no contact between oblique surfaces (Figs 4A and B). If the surfaces were compressed, there was only angular deformity correction, but the rotational deformity became exaggerated (Figs 4C and D). The process was repeated for every model.

Figs 3A to C — After the ascending cut (distal anteromedial to proximal posterolateral), (A) frontal view shows correction of varus and internal rotation, (B) axial view confirms correction of internal rotation and (C) sagittal view shows procurvatum correction and good contact between the oblique cut surfaces

Figs 4A to D — Right-sided cylindrical clay model (yellow colour) with varus-procurvatum and internal rotation deformities of 45° each was cut with a similar method, except that the cut was descending (proximal anteromedial to distal posterolateral). (A) Sagittal view shows correction of procurvatum, but with no contact between oblique cut surfaces. The frontal view shows correction of varus and internal rotation deformity (B). If the oblique cut surfaces were made to contact each other, there is exaggeration of internal rotation deformity (C, D) while procurvatum in sagittal (C) and varus in frontal (D) views show complete correction

Sixteen out of 32 A-R deformities that maintained contact after an accurate cut are reported in Table 1. The results indicate that among the 16 A-R deformity patterns, six complementary A-R deformities (varus and/or procurvatum with internal rotation, valgus and/or recurvatum with external rotation) require an ascending cut, and six compensatory A-R deformities (varus and/or procurvatum with external rotation, valgus and/or recurvatum with internal rotation) require a descending cut. A caveat to these rules is regarding dissociative angular deformities (varus-recurvatum and valgus-procurvatum) in combination with internal and external rotational deformities. In these four A-R deformity patterns, coronal plane deformity supersedes sagittal plane deformity in dictating the vertical orientation of the cut in combination with rotational deformity. This is irrespective of the magnitude of coronal plane deformity.

Table 1.

Thirty-two right-sided A-R deformity models were cut with a 45° vertical orientation, ascending and descending. The transverse rotational orientation measuring 22.5° was external for internal rotational deformity and internal for external rotational deformity

Right side internal rotation deformity External rotation transverse orientation of cut	Vertical orientation
Varus	Ascending
Valgus	Descending
Procurvatum	Ascending
Recurvatum	Descending
Varus-procurvatum	Ascending
Valgus-recurvatum	Descending
Varus-recurvatum	Ascending
Valgus-procurvatum	Descending
Right side external rotation deformity Internal rotation transverse orientation of cut
Varus	Descending
Valgus	Ascending
Procurvatum	Descending
Recurvatum	Ascending
Varus-procurvatum	Descending
Valgus-recurvatum	Ascending
Varus-recurvatum	Descending
Valgus-procurvatum	Ascending

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Outcome measure was visual contact between the oblique cut surfaces in 16 models

Discussion

Literature relevant to the correction of A-R deformity, with particular reference to the vertical orientation of single-cut inclined osteotomy, is saturated with confounding concepts,^2,7 esoteric mathematical calculations, intricate planning methods, complex tools, unclear rationale for choosing the appropriate cut and lack of universal applicability.^{1,4,6,8–11,15,16} Most authors have utilised pre-osteotomy multiple trial-and-error cuts on 3D-printed models, plastic bone models glued with epoxy resin, bone models with a hinge which are expensive, and bananas which are imprecise models.^{2,3,5,8,9,13,17,18}

Paccola published his lookup table that recommended ascending (clockwise) cuts for right internal rotation and left external rotation deformities and descending (anticlockwise) cuts for left internal rotation and right external rotation deformities.⁷ Wallace et al.² modified the look-up table from Paccola by recommending that all internal rotation deformities need correction with ascending osteotomies, and all external rotation deformities need descending osteotomies, irrespective of the laterality of the limb.⁷ Notwithstanding a lack of clarity regarding the vertical inclination of the cut in both publications, they also did not elucidate the role of angular deformity in determining the verticality of the cut.^2,7

In 1989, Sangeorzan et al.^1,4 elaborated on the mathematical concepts of A-R deformity correction with single-cut inclined osteotomy. In 2002, Paley attempted to decode these mathematical concepts into a visual heuristic, by suggesting that the convex tilt of the saw blade in the transverse plane indicates an ascending cut and vice versa.⁵ The vertical orientation of the saw blade seems to be dependent on its transverse orientation, and this rule-of-thumb has been adopted without further inquiry.^5,17–19 It is logical to assume that when there is a rule for the transverse orientation of the saw blade, there must be an analogous rule that guides the vertical orientation of the blade. In 2008, Christian and Mercadante experimented with plastic bone models utilizing both these mathematical and visual concepts.¹³ Using these pre-existing mathematical, visual and experimental methods.^1,4,5,13 I have attempted to decode the geometric equivalent that underpins the vertical orientation of single-cut inclined osteotomy for A-R deformity. In that measure, the current study represents an evolution in our understanding of principles of A-R deformity correction.

All 16 A-R deformities with their respective inclined osteotomy parameters are listed in Table 2. A complementary A-R deformity will require an ascending cut which starts in the distal fragment and is reoriented in the transverse plane to the convexity of the angular deformity. While a compensatory A-R deformity will require a descending cut, which begins in the proximal fragment and is reoriented in the transverse plane to the concavity of the angular deformity. By negating the dependence of vertical orientation on transverse orientation of the saw blade, these rules stand in contrast to the accepted heuristic of Paley.⁵ The current study aimed to provide a comprehensive solution to the problem of A-R deformity by avoiding (1) complex calculations, (2) multiple trial-and-error methods and (3) expensive bone models. These simple rules will enable surgeons to consider the appropriate inclined osteotomy for any A-R deformity in clinical practice. The utility of 3D-printed models would be appropriate to improve the precision of the cut before surgery.

Table 2.

Chart providing geometric rules of inclined osteotomy for the 16 A-R deformity patterns in a right-sided long bone

Deformity	Pattern of deformity	Plane of osteotomy	Transverse orientation of osteotomy	Vertical orientation of osteotomy
IR + varus	C	A to P	External rotation	Ascending
IR + procurvatum	C	M to L	External rotation	Ascending
IR + varus + procurvatum	C	AM to PL	External rotation	Ascending
IR + varus + recurvatum	C	AL to PM	External rotation	Ascending
IR + valgus	S	A to P	External rotation	Descending
IR + recurvatum	S	M to L	External rotation	Descending
IR + valgus + recurvatum	S	AM to PL	External rotation	Descending
IR + valgus + procurvatum	S	AL to PM	External rotation	Descending
ER + varus	S	A to P	Internal rotation	Descending
ER + procurvatum	S	M to L	Internal rotation	Descending
ER + varus + procurvatum	S	AM to PL	Internal rotation	Descending
ER + varus + recurvatum	S	AL to PM	Internal rotation	Descending
ER + valgus	C	A to P	Internal rotation	Ascending
ER + recurvatum	C	M to L	Internal rotation	Ascending
ER + valgus + recurvatum	C	AM to PL	Internal rotation	Ascending
ER + valgus + procurvatum	C	AL to PM	Internal rotation	Ascending

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A, anterior; AL, anterolateral; AM, anteromedial; C, complementary; ER, external rotation; IR, internal rotation; L, lateral; M, medial; P, posterior; PL, posterolateral; PM, posteromedial; S, compensatory

In an A-R deformity, the angular component (A) determines the plane of the cut and the rotational component (R) determines the transverse orientation of the cut. While the combination pattern of angular and rotational components (A-R) determines vertical orientation of the cut, which has been unaddressed in literature until now. The author believes that these geometric rules are a valuable addition to the principles of deformity correction.

Conclusion

The geometric rules guiding the vertical orientation of single-cut inclined osteotomy in A-R deformity are:

Complementary A-R deformity requires an ascending osteotomy.
- – Varus and/or procurvatum with internal rotation.
- – Valgus and/or recurvatum with external rotation.
Compensatory A-R deformity requires a descending osteotomy.
- – Varus and/or procurvatum with external rotation.
- – Valgus and/or recurvatum with internal rotation.
In an A-R deformity with dissociative angular components, coronal plane deformity supersedes sagittal plane deformity in dictating vertical orientation of the osteotomy. This is irrespective of the magnitude of coronal deformity.
- – Varus-recurvatum with internal or external rotation.
- – Valgus-procurvatum with internal or external rotation.

Orcid

Balachandar Gopalan https://orcid.org/0000-0001-7202-5379

Footnotes

Source of support: Nil

Conflict of interest: None

References

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[B1] 1.Sangeorzan BJ, Sangeorzan BP, Hansen ST, Jr., et al. Mathematically directed single-cut osteotomy for correction of tibial malunion. J Orthop Trauma. 1989;4:267–275. doi: 10.1097/00005131-198912000-00001. [DOI] [PubMed] [Google Scholar]

[B2] 2.Wallace SJ, Patterson JT, Nork SE. Mathematically directed single-cut osteotomy. Medicina. 2022;58:971. doi: 10.3390/medicina58070971. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B3] 3.Schweinsberg LP, Ehrle A, Schweinsberg U, et al. Single-cut osteotomy for correction of a complex multiplanar deformity of the radius in a Shetland pony foal. Vet Surg. 2022;51:148–156. doi: 10.1111/vsu.13725. [DOI] [PubMed] [Google Scholar]

[B4] 4.Sangeorzan BP, Judd RP, Sangeorzan BJ. Mathematical analysis of single-cut osteotomy for complex long bone deformity. J Biomech. 1989;22:1271–1278. doi: 10.1016/0021-9290(89)90230-3. [DOI] [PubMed] [Google Scholar]

[B5] 5.Paley D. Principles of Deformity Correction. Berlin: Springer-Verlag; 2002. Combined angulation and rotation deformities. pp. 252–266. [Google Scholar]

[B6] 6.Youngman J, Raptis D, Al-Dadah K, et al. An accurate method of determining a single-plane osteotomy to correct a combined rotational and angular deformity. Strateg Trauma Limb Reconstr. 2015;10:35–39. doi: 10.1007/s11751-015-0222-6. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B7] 7.Paccola CAJ. A simplified way of determining the direction of a single-cut osteotomy to correct combined rotational and angular deformities of long bones. Rev Bras Ortop (English Ed) 2011;46:329–334. doi: 10.1016/s2255-4971(15)30206-8. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B8] 8.Meyer DC, Siebenrock KA, Schiele B, et al. A new methodology for the planning of single-cut corrective osteotomies of mal-aligned long bones. Clin Biomech. 2005;20:223–227. doi: 10.1016/j.clinbiomech.2004.09.017. [DOI] [PubMed] [Google Scholar]

[B9] 9.Dobbe JGG, Du Pré KJ, Kloen P, et al. Computer-assisted and patient-specific 3-D planning and evaluation of a single-cut rotational osteotomy for complex long-bone deformities. Med Biol Eng Comput. 2011;49:1363–1370. doi: 10.1007/s11517-011-0830-3. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B10] 10.Rab GT. Oblique tibial osteotomy for Blount's disease (Tibia vara) J Pediatr Orthop. 1988;8:715–720. doi: 10.1097/01241398-198811000-00018. [DOI] [PubMed] [Google Scholar]

[B11] 11.Rab GT. Oblique tibial osteotomy revisited. J Child Orthop. 2010;4:169–172. doi: 10.1007/s11832-009-0228-z. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B12] 12.Wagner P, Wagner E. Proximal rotational metatarsal osteotomy for Hallux Valgus (PROMO): Short-term prospective case series with a novel technique and topic review. Foot Ankle Orthop. 2018:3. doi: 10.1177/2473011418790071. [DOI] [PMC free article] [PubMed] [Google Scholar]

[B13] 13.Christian RW, Mercadante MT. A osteotomia em um plano oblíquo por corte único para correção de deformidades complexas diafisárias de ossos longos: um método para sua realização. Rev Bras Ortop. 2008;43:376–387. doi: 10.1590/s0102-36162008000900003. [DOI] [Google Scholar]

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PERMALINK

What is the Geometric Rule that Guides Accurate Vertical Orientation of the Single-cut Inclined Osteotomy in a Combined Angulation-rotation Deformity of Long Bone?

Balachandar Gopalan