Measuring anatomical acetabular cup orientation with a new X-ray technique

Abstract

Proper alignment of the acetabular cup component is one of the most important requisites for a successful long-term outcome in total hip replacement. However, measurement and indication of cup orientation in an anatomical pelvic reference system is very difficult. We propose a new C-arm-based X-ray technique for determining the values for inclination and anteversion of the acetabular cup component. The proposed method is validated by computer simulation and sources of error are evaluated. The method predicts an accuracy of better then 5° for determination of anteversion of the cup.

• Total hip replacement
• hip navigation
• acetabulum
• inclination
• anteversion

Introduction

In orthopedic surgery, proper placement of the acetabular hip component is judged to be a prerequisite for a successful long-term outcome in total hip arthroplasty (THA), from a biomechanical viewpoint [1–4]. Therefore, exact positioning of the acetabular cup is one of the greatest challenges for the surgeon performing THA. Cup orientation is usually indicated by the two angles of inclination and anteversion, which represent the values for abduction and rotation of the cup with respect to the sagittal and coronal planes, respectively (Figure 1). Values of 45 ± 10° for inclination and 15 ± 10° for anteversion have been postulated to represent the so-called safe zone for cup positioning [5], [6]. Different types of mathematical definition have been specified for these angles [6] with respect to the various anatomical reference planes [7]. To quantify the outcome of surgery in clinical routine, the angles are usually determined from planar radiographic images, which represent 2D projections of the 3D geometry of the pelvis. As for angles in 3D space, proper alignment of the patient and adjustment of the X-ray beam is necessary to ensure that the radiographically projected planes are in the mathematically required orientation [1],[8–12]. In clinical practice, two different X-ray images are acquired to determine the two angles of interest, and this usually requires repositioning the patient.

Figure 1. Definition of the APP concept (a) and radiological inclination (RI) and anteversion (RA) (b). The pelvic reference plane (xy plane) is defined by the iliac anterior superior spines (SIAS) and the pubic spine (SP). [Color version available online].

Beyond the problem of patient alignment, there are other shortcomings, since these angles have to be indicated with respect to an anatomical coordinate system [6], [13]. In particular, since the pelvic orientation is not usually known, it cannot be accounted for in the patient's alignment. Thus, there are two main issues facing the orthopedic surgeon: How to adjust properly the alignment of the cup, and how to judge its alignment postoperatively. With regard to the first of these concerns, computational navigation systems have been developed in recent years to guide the surgeon while implanting the cup [5],[14–16]. Regardless of whether these systems work with computer tomography (CT) images or with CT-free techniques, anatomical landmarks on the pelvis must be registered intraoperatively to define the anatomical coordinate system. The orientation of the cup is then adjusted with an acetabulum reamer, guided by a tracking system with reference to the acquired anatomical coordinate system.

Despite the accuracy of navigation systems, proper palpation of the landmarks constrains exact alignment of the cup. It therefore remains desirable to measure cup alignment postoperatively for control purposes. However, none of the conventional radiographic methods of evaluation take into account the anatomical coordinate system of the pelvis. Using computer simulations, measuring errors of up to 10° have been reported for anteversion, depending on the amount of pelvic tilt [17], which ranged from approximately 20° of reclination to 10° of inclination [18].

Using an X-ray C-arm technique, it is possible to acquire the necessary radiographs without repositioning the patient, as the X-ray beam can be rotated in a defined manner. We therefore present a mathematical formalism for a new X-ray technique to determine cup inclination and anteversion by means of two radiographic images, which takes pelvic orientation into account. For this purpose, radiographic projections of acetabular cups were simulated with computer software, and the formulae were subsequently analyzed by simulation as well.

Methods

Mathematically, the orientation of the cup is represented by the normal vector of the cup plane with respect to an anatomical coordinate system of the pelvis [6], [7] (see appendix). In the Anterior Pelvic Plane (APP) concept, the two anterior superior iliac spines (SIAS) and the pubic symphysis define the pelvic reference plane (Figure 1a). The z-coordinate of this system lies in the direction normal to this plane (Figure 1b). Inclination and anteversion of the acetabular cup can thus be defined as the azimuth and polar angle of the normal vector in a polar coordinate system of the pelvic system (equation A1 in appendix). In accordance with the definitions of Murray [7], we used the radiological anteversion (RA) and inclination (RI), respectively, for the calculations.

The appearance of conventional radiographic X-ray images of acetabular cups was simulated in MatLab (The Mathworks, Inc., Natick, MA). Therefore, the cups as well as their circular rims were constructed as surface structures. Projection of the cups onto the frontal plane, which is a plane perpendicular to the X-ray beam, was simulated and visualized (Figure 2). By means of 3D rotation matrices (Robot Toolbox [19]), every cup orientation with respect to the frontal plane could be adjusted as desired. The origin of the coordinate system was chosen to be in the origin of the hemispherical cups. The x-axis joins the hip centers and is assumed to be the axis of rotation for the pelvic plane with respect to the frontal plane (pelvic tilt).

Figure 2. Frontal view of the hemispherical cup for 45° inclination (RI) and different values of anteversion (RA), pelvic inclination (α) and pelvic rotation (β). White lines mark the elliptical rim of the cup as well as its principal axis. The black line indicates the projected normal vector of the cup. (a) No anteversion and no pelvic inclination. (b) Anteversion RA of 15° and no pelvic tilt. (c) Anteversion RA of 15° and pelvic reclination (α = −10°). (d) Anteversion RA of 15° with pelvic reclination (α = −10°) and simulated pelvic rotation in the frontal plane (δ = 5°). The values for the angle between the horizontal line and the principal axis (RI-Line) are also depicted. [Color version available online].

In clinical practice, an elliptical outline can be identified in the radiographic images representing the rim of the acetabular cup. The inclination of the principal axis with respect to a horizontal line usually serves as the inclination value for the acetabular cup. Therefore, the principal axes of the projected rims of the half spheres were identified numerically as the maximum distance to the origin. This line we call the RI-Line (Figure 2). Its value differs from the inclination value when pelvic tilt has to be accounted for [17]. The orientation of the cup can be characterized by rotations of its normal vector. Its projection into the frontal plane is obtained from its x- and y-coordinates (see appendix). In this way, we were able to compare the geometric findings of a cup's projection in the X-ray images with the projection of the normal vector into the frontal plane.

We first simulated an X-ray photograph of the acetabular cups from a patient lying supine in the frontal plane (Figure 3a). Pelvic tilt (angle α) in the sagittal plane pelvis was accounted for, as well as pelvic rotation in the frontal plane (angle δ). Using a C-arm X-ray system, the plane of the X-ray projection can be changed without repositioning the patient. We therefore simulated the rotation of the X-ray beam around the longitudinal axis of the patient (angle β). Mathematically, the rotated frontal plane can be simulated by a subsequent coordinate transformation around the y-axis (see appendix).

Results

Since the normal vector of the cup characterizes the cup's orientation unambiguously, we first wanted to clarify the interrelations between the measurable RI-Lines from the ellipses and the projection of the normal vector in the frontal plane. We therefore varied the values for pelvic tilt, pelvic rotation and rotation of the X-ray beam for numerous combinations. In all simulations, the RI-Lines of the cups were perpendicular to the projection of the normal vector in the frontal plane (Figure 2). Hence, by measuring the RI-Line we could obtain the same information about cup alignment as from the projection of the cup's normal vector.

In general, the alignment of the patient's pelvis deviates slightly from the ideal alignment, which is imposed by the coordinate system of the X-ray system. That is, the pelvis is tilted with respect to the frontal plane around the x-axis and rotated in the frontal plane around the z-axis. In the appendix, we show that, for pelvic tilt alone, the value for the angle of the RI-Line not only depends on the true value for anatomic inclination, RI, but also on the value for the cup's anatomic anteversion, RA (equation A4b in appendix). Thus, by arranging two different projected views of the cups, it should be possible to determine the real values for inclination and anteversion with reference to the pelvic coordinate system by just measuring the corresponding RI-Lines from the projected images of the cups.

We therefore derived a formula for calculating the respective values for the RI-Lines for defined pelvic alignment: The first view simulates ordinary anterior-posterior (AP) X-ray images of the pelvis including pelvic tilt (α) and pelvic rotation (δ). The second view is achieved by simulating a rotation (β) of the X-ray beam around the longitudinal axis. The resulting formula is a simple system of two linear equations, whose solution reveals the normal vector (equation A10 in appendix).

To evaluate the obtained formula, we simulated numerous orientations of the cups with defined values for RI and RA, determined the RI-Lines numerically, and recalculated the values by means of the coordinate of the rotated normal vector. In all cases, we obtained exact equivalence of the values for RA and RI, thus verifying the validity of the formulae for both sides.

Based on these findings, we propose a new X-ray technique to quantify the angle of inclination and anteversion in clinical practice. From an initial standard AP X-ray image, the RI-Line (RI2) of the investigated hip should be determined, as well as the orientation of the pelvis (δ) in the frontal plane. A second X-ray image should be acquired with a defined rotation of the X-ray beam around the longitudinal axis of the patient (β) without repositioning the patient. Again, the RI-Line (RI3) of the corresponding hip cup is determined. In a supplementary measurement, the pelvic tilt (α) has to be determined by means of a pelvic inclinometer [18].

The presented mathematical approach requires measurement of five angular values in clinical practice. Since all measurements have associated errors, we determined in additional simulations the influence of variations in the angles mentioned. For this purpose, we simulated the cup projection for a cup lying in the safe zone (i.e., RI = 45° and RA = 15°). With the help of the RI-Lines thus obtained (RI2 = 49° and RI3 = 52.5°; see Figures 3a and 3b), we recalculated the values for RI and RA using equation A10 (see appendix) and varied each of the five angular parameters by ± 2° independently of the others (i.e., while keeping the others constant). The differences between these recalculated values and the initial values were determined. Of the five parameters, the values for the RI-Lines showed the greatest impact upon the calculated values. Table I summarizes the results for all varied parameters. It can be seen that the value for RI3 is the most critical of these parameters. Here, we found an error of 2.3° for RA for an angular variation of 1° (Figure 3c). In general, the influence on the values for inclination was less than that on the anteversion values (compare center and right columns of Table I). As for the influence of measurement errors within the safe zone, we found the worst case to represent an inclination of 55° and an anteversion of 25°. For this particular case, we found a deviation of 3.1° from the real value for anteversion when the RI-Line of the second image was falsely measured by 1° of deviation.

Figure 3. Projected images of simulated cups in AP view (a) and rotated view (b). Both views represent the two images necessary to determine the RI-Lines (RI2 and RI3). The simulated values for inclination (RI), anteversion (RA), pelvic tilt (α), pelvic rotation (δ) and X-ray rotation (β) are indicated in the title lines. (c) Plot of the calculated values for anteversion RA when the RI-Line (RI3 value) is varied by ± 2°. [Color version available online].

Table I.  Influence of measurement errors on the calculated values for anteversion RA and inclination RI. In the first column are the varied angular parameters. For example, a deviation of 1° in the value for the pelvic rotation (δ) results in an error of 0.95° in the calculation of cup inclination RI. (The orientation of the cup was simulated by RI = 45°, RA = 15°, pelvic reclination of −10°, pelvic rotation of 2°, and X-ray rotation of 40°.).

Parameter	Error RA (°/°)	Error RI (°/°)
RI2	1.8	1.22
RI3	2.3	0.3
α	0.71	0.19
β	0.244	0.04
δ	0.123	0.95

Discussion

Postoperative measurement of acetabular cup alignment is a necessity in orthopedics, even when using a navigation system to guide cup positioning. Clinically, the orientation of the cup is indicated by the angular values of inclination and anteversion, which are determined with respect to an anatomical coordinate system of the pelvis. Thus, the measurement of inclination and anteversion by conventional radiographic methods is always subject to erroneous measurement, because, in general, the orientation of the pelvis cannot be accounted for exactly. Values for pelvic tilt ranging from 20° of reclination to 10° of inclination, as measured with an ultrasound coordinate measuring system, have been reported [18].

We have proposed a new method for determining the anatomical values for inclination and anteversion by using two X-ray pictures, measuring pelvic tilt with an inclinometer, and measuring pelvic orientation on the basis of the X-ray image.

For another study conducted in our clinic, we developed a so-called pelvic inclinometer to measure the pelvic tilt of a patient lying supine on the X-ray table [18]. Measurement of pelvic rotation in the frontal plane is usually accomplished by drawing a line tangential to the two most inferior points of the pelvis. Hence, pelvic orientation can be easily determined in routine clinical practice. Using a C-arm X-ray technique, two X-ray images of the patient's pelvis can be obtained without repositioning the patient, which is a prerequisite to calculating 3D geometrical information concerning this interesting structure.

The proposed method can easily be performed in clinical orthopedic practice, where measuring angles on X-ray images is a routine task. The error attributable to mis-measurement is limited to approximately 2–3° for the typical values in the safe zone. Working carefully, an accuracy of approximately 2° should be achievable when measuring either pelvic rotation or orientation of the RI-Line. In this case, the proposed method attains a precision of approximately 5°.

Appendix

For all cup orientations we simulated the projection of the cup onto the frontal plane, which is defined as a plane perpendicular to the X-ray beam. The values of inclination and anteversion, RI and RA, respectively, are defined according to the radiological notation introduced by Murray [7] in an anatomical coordinate system that is oriented by the pelvic plane (APP concept). In this system, the normal vector of the cup, , appears as follows (Figure 1):The values for RI and RA can be calculated from the components of the normal vector:Thus, according to equation A2a, RI can be measured in the frontal plane from the projected line of the normal vector. The pelvis is generally tilted with respect to the frontal plane around the hip rotation centers so that the pelvic plane is no longer parallel to the frontal plane. Mathematically this corresponds to a rotation around the x-axis by an angle α. The undisturbed normal vector ₀ of the cup is therefore rotated to ₁:From the components of the rotated normal vector we can calculate a disturbed value for the angle of inclination, RI1:Rewriting this equation yieldsBecause the patient's alignment generally suffers in the longitudinal axis as well, the pelvis is not only tilted around the x-axis, but also around the z-axis. In a similar manner, the rotation of the pelvis in the frontal plane can be characterized by a consecutively rotation around the z-axis (rotation angle δ):With regard to equation A3, we obtainThe value for the newly disturbed value for inclination, RI2, gives the following (Figure 3a):Rewriting equation A6 yields a linear equation for the components of the undisturbed normal vector, ₀:To obtain a second radiographic projection of the pelvis, either the patient's alignment or the direction of the X-ray beam must be changed. The digital X-ray apparatus used in the C-arm technique is able to rotate the X-ray, along with the detecting sensor, in a defined manner around the longitudinal axis to obtain another projected view of the pelvis. Again, mathematically this corresponds to a supplementary rotation around the y-axis in the frontal plane (rotation angle β).As in the preceding cases, for the newly projected angle of inclination, RI3, we obtain the following equation (Figure 3b):Equations A7 and A9 represent a system of linear equations for the unknown values of the components n_x, n_y and n_z of the vector ₀, respectively, for the unknown values of inclination and anteversion.From equation A10 we can determine the direction of the undisturbed normal vector ₀ as the eigenvector of the matrix A corresponding to the eigenvalue 0. The correct direction of this vector has to be arranged separately for the left or right hip according to the anatomical arrangement. For instance, the y-component must point in the caudal direction.

Since the specification of inclination and anteversion in orthopedics does not differ for the left and right cups, equation A1 holds for the normal vector of the left hip and must be adapted when performing calculations for the right hip.