A simple method for quantitative evaluation of the missing area of the anterior glenoid in anterior instability of the glenohumeral joint

Abstract

Objective

The objective of this study was to describe and validate a simple method to quantitatively calculate the missing area of the anterior part of the glenoid in anterior glenohumeral instability.

Materials and methods

The calculations were developed from three-dimensional (3D)-reconstructed computerized tomography en face images of the glenoid with “subtraction” of the humeral head in 13 consecutive cases with known anterior glenohumeral joint instability diagnosed by history and clinical examination. The inferior portion of the glenoid was approximated to a true circle whose center was determined by means of a femoral head gauge. The eroded anterior area was calculated as the ratio between the depth (a perpendicular line from the center of the circle to the eroded edge of the anterior glenoid) and the radius of the inferior glenoid circle. This data was then compared to the results obtained by two additional different methods: direct computerized measurements of the missing area and direct computerized measurement of the ratio between the radius and depth, on two dimensional computed tomography (CT) en face view reconstructions of the glenoid.

Results

We provide a function that correlates the ratio between depth and radius of the inferior glenoid circle and the area of the missing anterior glenoid. The results obtained by three different methods were comparable. Simple trigonometric calculations showed that a 5% area defect corresponds to 0.8 (12.5%) of the radius of the inferior glenoid, while a 20% area defect corresponds to 0.5 (50%) of the same radius (Table 1).

Table 1 Results according to each different method

Conclusion

Using this simple method and the function provided, the eroded area of the anterior part of the glenoid in anterior glenohumeral instability can be calculated preoperatively using a 3D CT reconstruction of the glenoid with “subtraction” of the humeral head, obviating the need for sophisticated software to obtain this critical information for preoperative decision making.

Mathematical formulation

As the inferior part of the glenoid is circular and the leading edge of the remodeled glenoid area is linear, it is possible to calculate the relationship between the missing anterior glenoid area (s1) and the ratio between the radius of the circle (R) and the distance from the center of the circle to the leading edge of the missing area (d) (depth of the defect) as follows:

Using simple geometrical formulas [13] the area s1 (missing area) is

$$s1 = \frac{1}{2}R^2 \left( {\alpha - \sin \left( \alpha \right)} \right),$$

where α is the angle shown in the figure, 0 ≤ α ≤ π.

The depth d as a function of the circle radius and the angle is:

$$d = R\cos \left( {{\alpha \mathord{\left/{\vphantom {\alpha 2}} \right.\kern-\nulldelimiterspace} 2}} \right)$$

The ratio q between the affected area s1 and the total area of the circle is

$$q = \frac{{s1}}{{\pi R^2 }} = \frac{1}{\pi }\left( {\cos ^{ - 1} \left( {\frac{d}{R}} \right) - \frac{d}{R}\sqrt {1 - \left( {\frac{d}{R}} \right)^2 } } \right)$$

One can see that when d is 0.8 of the radius_, q = 5%. In other words, when the depth of the defect is 0.8 of the radius of the inferior glenoid circle the missing area is 5%.

Analogously, when d is 0.5 of the radius, q = 20%.