Achievement of Targeted Posterior Slope in the Medial Opening Wedge High Tibial Osteotomy: A Mathematical Approach

Abstract

High tibial osteotomy (HTO) for medial compartment knee osteoarthritis is preferred in the activity patient since it allows patients to return to sports and recreational activities similar to the preoperative level. The purpose of this study was to mathematically formulate medial and anteromedial opening gaps in the medial opening wedge HTO to achieve a targeted tibial posterior slope. The change of posterior slope angle was mathematically derived in terms of the medial and anteromedial opening gaps, and the medial opening angles. The derived equations were validated by comparing them with those directly measured by performing simulated HTOs. In the triangular geometries of osteotomy planes, measured from three-dimensional osteotomy models of 30 knee patients, the mean anteromedial, medial, and lateral included angles were 92.4°, 53.9°, and 33.7°, respectively, and the mean lateral–medial edge length was 53.3 mm. The ratio of the anteromedial opening gap to the posterior opening gap should be “sin(the medial included angle) × cos(the lateral included angle)/sin(the anteromedial included angle)” to maintain an intact posterior tibial slope angle. With the derived equations, surgeons can estimate the opening gaps and opening angles to get a targeted posterior tibial slope with a medial opening angle.

Appendices

Appendix A: Opening Gaps

Opening Gaps (G _M1 and G _A1) Due to the Medial Opening

The medial opening gap (G _M1) due to medial HTO opening is obtained by calculating the height $\overline{{(M - M^{\prime})}}$ of the triangle ∆LMM′:

$$ \begin{aligned} G_{M1} & = \overline{M - M'} \\ & = 2 \cdot l_{LM} \cdot \sin (\alpha /2) \\ \end{aligned} $$

(A.1)

The anteromedial opening gap (G _A1) due to medial HTO opening is also obtained by calculating the distance $\overline{A - A'} .$ However, $\overline{A - A'}$ cannot be directly calculated like G _M , since the opening angle of the anterior-lateral edge (γ) is unknown. Instead, A and A′ are projected posterior onto the lateral–medial edges of the proximal and distal cut-section triangles, which makes intersections T and T′. Here, T and T′ are posterior projections of A and A′, respectively. $\overline{T - T'}$ equals $\overline{A - A'} ,$ and l _LT  = l _LT′ = l _AL ·cosθ _L when ψ is zero. The anteromedial opening gap (G _A1) due to medial opening is then expressed.

$$ \begin{aligned} G_{A1} & = \overline{A - A'} \\ & = \overline{T - T'} \\ & = 2 \cdot l_{LT} \cdot \sin (\alpha /2) \\ & = 2 \cdot l_{AL} \cdot \cos \theta_{L} \cdot \sin (\alpha /2) \\ \end{aligned} $$

(A.2)

Opening Gaps (G _M2 and G _A2) Due to the Change of Posterior Slope Angle

The change in medial opening gap (G _M2) due to a change of posterior slope angle is zero since the rotation axis for posterior slop change is coincident with the medial–lateral edge of the osteotomy triangle (Fig. 6).

$$ G_{M2} = 0 $$

(A.3)

The change of anteromedial opening gap (G _A2) due to the change of posterior slope angle equals to two times l _AT′sin(ψ/2) (Fig. 7).

$$ \begin{aligned} G_{A2} & = 2 \cdot l_{A'T'} \cdot \sin (\psi /2) \\ & = 2 \cdot l_{AL} \cdot \sin \theta_{L} \cdot \sin (\psi /2) \\ \end{aligned} $$

(A.4)

Opening Gaps (G _M and G _A ) Due to the Medial Opening Angle and the Change of Posterior Slope Angle

Consequently, medial and anteromedial opening gaps are expressed in terms of the opening angle of the lateral–medial edge (α) and the change of the posterior slope angle (ψ). And l _AL  = l _LM  · sin θ _M /sin θ _A by the sine rule.

$$ \begin{aligned} G_{M} & = G_{M1} + G_{M2} \\ & = G_{M1} \\ & = \overline{M - M'} \\ & = 2 \cdot l_{LM} \cdot \sin (\alpha /2) \\ \end{aligned} $$

(A.5)

$$ \begin{aligned} G_{A} & = G_{A1} + G_{A2} \\ & = 2 \cdot l_{AL} \cdot \cos \theta_{L} \cdot \sin (\alpha /2) + 2 \cdot l_{AL} \cdot \sin \theta_{L} \cdot \sin (\psi /2) \\ & = 2 \cdot l_{AL} \cdot [\cos \theta_{L} \cdot \sin (\alpha /2) + \sin \theta_{L} \cdot \sin (\psi /2)] \\ & = 2 \cdot l_{LM} \cdot \sin \theta_{M} /{\sin \theta_{A}} \cdot [\cos \theta_{L} \cdot \sin (\alpha /2) + \sin \theta_{L} \cdot \sin (\psi /2)] \\ \end{aligned} $$

(A.6)

Since 2l _LM  = G _M /sin (α/2) from Eq. (A.5), the relationship between G _A and G _M is expressed.

$$ G_{A} = G_{M} \cdot \sin \theta_{M} /{\sin \theta_{A}} \cdot \left[ {\cos \theta_{L} + {\frac{{\sin \theta_{L} \cdot \sin (\psi /2)}}{\sin (\alpha /2)}}} \right] $$

(A.7)

Appendix B: Opening Angles

As the osteotomy wedge is opened, each edge of the proximal and distal cut-section triangles comes to have its own opening angle (Fig. 5), i.e., the opening angles of the lateral–medial edge (α), the medial-anterior edge (β), and the anterior-lateral edge (γ). α is calculated using Pythagoras’s theorem. $\overline{m - M'}$ equals to l _LM ·sin(α/2), and since G _M is twice $\overline{m - M'} ,$ α can be expressed.

$$ \alpha = 2 \cdot \sin^{ - 1} \left( {{\frac{{G_{M} }}{{2 \cdot l_{LM} }}}} \right) $$

(B.1)

In the same way, β and γ can be expressed:

$$ \begin{aligned} \beta & = 2 \cdot \sin^{ - 1} \left( {{\frac{{G_{M} - G_{A} }}{{2 \cdot l_{MA} }}}} \right) \\ & = 2 \cdot \sin^{ - 1} \left( {{\frac{{G_{M} - G_{A} }}{{2 \cdot l_{LM} \cdot \sin \theta_{L} /{\sin \theta_{A}} }}}} \right) \\ \end{aligned} $$

(B.2)

$$ \begin{aligned} \gamma & = 2 \cdot \sin^{ - 1} \left( {{\frac{{G_{A} }}{{2 \cdot l_{AL} }}}} \right) \\ & = 2 \cdot \sin^{ - 1} \left( {{\frac{{G_{A} }}{{2 \cdot l_{LM} \cdot \sin \theta_{M} /{\sin \theta_{A}} }}}} \right) \\ \end{aligned} $$

(B.3)