Keywords

15.1 Introduction

Total hip arthroplasty (THA) is one of the most frequent orthopedic surgical interventions. Proper positioning, in particular angulation of the acetabular cup, is essential for improving the success of total hip arthroplasty. Previous studies [1,2,3,4,5,6] demonstrate that higher rates of pelvis osteolysis and component migration have all been well-associated with the malpositioning of the acetabular component, and surgical experience indicates that improper orientation of the acetabular component in terms of anteversion and inclination is the major cause of dislocation. As the risk of dislocation is significantly higher in those who have already experienced dislocation or after revision surgery [6], obtaining proper cup orientation during primary surgery is crucial.

Optimal ranges for angular cup position in terms of anteversion and inclination of the acetabular component have been extensively debated in the literature. Several so-called safe zones have been suggested. Lewinnek et al. described a safe zone of 5∘–25∘ for anteversion and 30∘–50∘ for inclination [5]. They found that acetabular cups placed outside this safe zone were approximately four times as likely to dislocate. Consequently optimal cup positioning requires that the surgeon attains adequate and reproducible angulations of the acetabular component with respect to the patient’s individual pelvic morphology.

State-of-the-art mechanical guides, which are used in the vast majority of THAs, are easy to handle but cannot be registered to the individual pelvic morphology. Angular orientation is gained from reference objects in space-fixed coordinates and may be corrected/optimized through the surgeon’s expert knowledge. A study conducted by DiGioia et al. [7], in which they used navigation technology to evaluate the performance of mechanical guides, found that 78% of the inserted cups would have been implanted outside the safe zone as suggested by Lewinnek et al. [5].

The search for alternatives was unsuccessful until modern optoelectronic tracking robotic technology was introduced to the field of orthopedics. Previously, a variety of image-based and image-free so-called navigation systems have been introduced for THA [8,9,10,11,12,13], which for the first time allowed co-registration of the patient’s pelvic morphology. Despite encouraging results of smaller clinical trials and an early widespread enthusiasm, some drawbacks of navigation technology have been identified, which prevent their widespread use in clinical routine. Current criticism focuses on (a) significant investments required for acquisition and running costs for maintenance and use (training of users, additional operating room (OR) time, disposable markers, etc.); (b) all navigation systems proposed to date, no matter what image modality is used, have difficulty when the THA is operated in lateral approach, which is the approach of choice for more than 80% of THAs worldwide [13]; (c) the steep learning curve and the system complexities, which may result in 10–20% failure cases [12]; (d) optoelectronic tracking technology used in most navigation systems requires a straight line of sight, which is often difficult to maintain during surgery without paying additional efforts and time; (e) CT-based systems require CT scan, usually not performed for diagnosis, generating unnecessary radiation exposure to the patient and significant additional cost; (f) for image-free navigation systems, significant intra- and interobserver variability caused by variations during digitization of the anatomical landmarks, especially the one in the pubic region, was observed [14]; and (g) fluoroscopy-based navigation systems [12] have the advantage of eliminating a CT scan and achieving an equivalent accuracy. However, such a technology requires calibration of the image intensifier, and it was judged to be too cumbersome and time-consuming to intraoperatively manipulate the C-arm device for multiple image acquisition [12].

Recently, several groups [15,16,17] described methods to use the constant direction of the force of gravity as a reference in THA. Asayama et al. [15, 16] introduced a three-direction indicator to control intraoperative pelvic motion during THAs. The three-direction indicator incorporates a digital compass with two goniometers, as well as a pendulum and target apparatus. It allows for controlling only pelvic motion by measuring the three-dimensional (3D) angle formed by the gravitational direction and the Steinmann pin inserted into the iliac bone to fix the direction indicator. No control of acetabular cup placement was considered with this device. Echeverri et al. [17] described a gravity-assisted system to control both the pelvic motion and the acetabular component placement. Like any other mechanical guide, this system is simple to use, but it is also highly flawed. This is due to the fact that the alignment system developed by Echeverri et al. was placed in a fixed orientation relative to the shaft of a cup placement instrument, which in the best case scenario can be understood as being calibrated with respect to the morphology of a fixed pelvis without considering the morphological difference between the future pelvis to be navigated and the fixed pelvis used for calibration. It simply does not work due to the inherent morphological variations in human being. A recent simulation study of this device on 48 patient data revealed a maximum anteversion error of as high as 15∘ [18].

To address the limitations in the existing gravity-based systems, we developed a new gravity-assisted navigation system termed as “patient-specific, gravity-assisted navigation system” or “PS-GANS” in abbreviation. Conceptually it combines the accuracy of computer-generated patient-specific morphology information with an easy-to-use mechanical guide, which effectively uses natural gravity as the angular reference. Unlike the existing gravity-based systems, our system allows for calibration of the mechanical guide with respect to the patient-specific morphology, which is obtained by using a statistical shape model-based 2D-3D reconstruction technique that can generate a scaled, patient-specific 3D shape model of the pelvis from a single conventional anteroposterior (AP) X-ray radiograph. The reconstructed 3D shape model facilitates a reliable and accurate co-registration of the mechanical guide with the patient’s anatomy in the operating theater.

15.2 Materials and Methods

15.2.1 Notations

Throughout the paper, we always establish a local coordinate system of a rigid body on a local reference plane of the entity. Thus, without explicitly stating, we always name the local coordinate system after the local reference plane. Furthermore, a vector v that is defined in a local coordinate system X will be noted as v X. But if we would like to know the axis v of a local coordinate system X in another local coordinate system Y, we will note it as v X Y. A rigid body transformation from a local coordinate system X to another local coordinate system Y will be noted as T X Y. The inverse of this transformation will be recorded as T Y X. As in most of the time, we are only interested in knowing the orientation of a vector in a local coordinate system; knowing the rotational part R X Y of the rigid body transformation T X Y is enough for our purpose.

15.2.2 System Overview

Our system requires three bull’s-eye bubble levels, as shown in Fig. 15.1. The first one is called witness bubble level that is fixed on the iliac crest and is designed together with the one on the pelvic positioning device to place the pelvis to strict lateral decubitus. As soon as the pelvis is placed at strict lateral decubitus by using the pelvic positioning device, one can adjust a standard clamp to set the witness level’s bubble to the center and move away the pelvic positioning device (see below for details). The witness level then acts as a witness, identifying the strict lateral decubitus position of the pelvis throughout the operation. The third one is called the instrument bubble level that is placed on a mechanical guide that is rigidly attached to the cup placement instrument for controlling cup orientation to the desired anteversion and inclination. Please keep it in mind that all the passive markers appearing in Fig. 15.1 are only for our validation purpose and are not required to use our system.

Fig. 15.1
figure 1

An overview of the PS-GANS system

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15.2.3 Coordinate Systems

Before we describe the details about how the patient-specific system calibration is done, we would like to first present a summary of all four local reference planes as well as their associated local coordinate systems that will be used in the calibration. See Fig. 15.2a for an overview.

Fig. 15.2
figure 2

(a) Schematic view of the relationship between the APP, the IRP, and the IAP (IDP) when the cup is placed in the desired orientation. (b) The mechanical guide attached to the cup placement instrument and the definition of the IDP

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Jamaraz et al. [10] introduced the anterior pelvic plane (APP) concept for measuring anteversion and inclination of the acetabular cup in their computer-assisted acetabular cup placement system. The APP is a reference plane of the human pelvis and, thus, allows the exact definition of a corresponding 3D local coordinate system as shown in Fig. 15.2a. It is based on three landmarks: bilateral anterior superior iliac spines (ASIS) and the geometric center of two pubic tubercles. The APP x-axis points to the patient’s operating side, parallel with the line between the iliac spine points. The y-axis points inferior. The angular orientation of the acetabular component can be directly put into relation to the APP.

It is difficult, if it is not impossible, to locate the orientation of the APP without using a positional tracking device, largely due to the difficulty in mechanically aligning the geometric center of two pubic tubercles. In this work, we propose and use a new reference plane that is called intraoperative reference plane (IRP), which is defined by two lines that can be mechanically aligned with the design of our system: the line connecting the bilateral ASISs (we named it as the ASIS line) and the line CA from the cup center of the operating side to the ASIS of the operating side (we named it as the CA line). Similar to how we establish a 3D local coordinate system on the APP, we also establish a 3D local coordinate system on the IRP, as shown in Fig. 15.2a, where we translate the origins of both coordinate systems to the acetabular center of the operating side. As we are only interested in the orientation of the acetabular component, such a translation does not affect our analysis and computation below. The x-axis of the IRP local coordinate system has the same orientation as the x-axis of the APP local coordinate system, while the y-axis of the IRP local coordinate system is chosen to be a vector that is inside the IRP and perpendicular to the x-axis. The z-axis of the IRP local coordinate system can be computed from the cross product of the x-axis and the y-axis of the IRP local coordinate system.

The so-called instrument design plane (IDP) is a plane that is defined by the design of the mechanical guide as shown in Fig. 15.2b. Physically, it is defined by the instrument axis and a curved metal rod called the pitch pointer, as shown in Fig. 15.2b. The pitch pointer is calibrated to be always inside the IDP and to be freely rotated around a fixed axis at the distal end of the cup placement instrument. We could establish a local coordinate system on the IDP as follows. The origin of this local coordinate system is chosen to be the center of the attached cup; the x-axis is chosen to be the instrument axis, and the y-axis is defined as a vector that is inside the IDP and perpendicular to the x-axis. The z-axis of the IDP local coordinate system can be computed from the cross product of the x-axis and the y-axis of the IDP local coordinate system.

Given a desired orientation of the cup (e.g., a typically desired orientation of the cup is 20∘ anteversion and 45∘ inclination with respect to the APP), we can construct a virtual instrument axis with respect to the APP of the pelvis using the method introduced by Murray [19], and we call its direction as V I APP. This axis together with the CA line defines the instrument alignment plane (IAP), to which the IDP should be aligned in order to place the cup in the desired orientation using the method described below. Thus, similar to how we define a local coordinate system on the IDP, we also establish a local coordinate system on the IAP (see Fig. 15.2a for a schematic view of how the local coordinate system of the IAP is established). More specifically, we take the virtual instrument axis as the x-axis of the IAP local coordinate system. The y-axis is defined as a vector that is inside the IAP and perpendicular to the x-axis.

15.2.4 System Calibration

System calibration here means to define the orientation of the instrument bubble level as shown in Fig. 15.2b with respect to the local coordinate system of the IDP for a given pelvis whose morphology is known (the exact morphological information that our system requires will be described below), so that when all system requirements are satisfied (see below for the details about our system requirements) and when the bubble of the instrument level is oriented to the center, the axis of the cup placement instrument should be aligned with the virtual instrument axis that is constructed according to the desired orientation of the cup.

Without loss of generality, let’s assume that the x-axis of the IRP of the given pelvis is \([\begin{array}{*{20}c} 1&0&0\\ \end{array} ]^{T}\), the y-axis of the IRP is \([\begin{array}{*{20}c} 0&1&0\\ \end{array} ]^{T}\), and the z-axis is \([\begin{array}{*{20}c} 0&0&1\\ \end{array} ]^{T}\). As the morphology of this pelvis is given, we assume that we know the angle θ between its APP and its IRP, and we further assume that we know the orientation of the CA line in the local coordinate system of the IRP, which is defined as CA IRP. Using angle θ, we can find the rotation between the local coordinate system of the IRP and the local coordinate system of the APP, R APP IRP, and the inverse rotation R IRP APP as well. With rotation matrix R APP IRP, one can transform the vector V I APP from the local coordinate system of APP to the local coordinate system of IRP and denote it as V I IRP.

$$\displaystyle{ V I^{IRP} = R_{ APP}^{IRP}(\theta ) \cdot V I^{APP} }$$
(15.1)

When our system would be used for the navigation of the cup placement, additionally we require that (A) the pelvis should be placed in strict lateral decubitus, which means that the ASIS line should be parallel to the constant direction of the force of gravity (but with opposite direction). This is realized intraoperatively by using the pelvic positioning device and the witness level, as shown in Fig. 15.3a, following the procedure introduced by Echeverri et al. [17]; and (B) the pitch pointer should touch on the ASIS of the operating side of the pelvis, as shown in Fig. 15.3b. As only a thin layer of soft tissue exists on top of the ASIS of the operating side, this landmark can be easily palpatable by a surgeon [13].

Fig. 15.3
figure 3

(a) This image shows how to use the pelvic positioning device to place the pelvis in strict lateral decubitus and then to set the witness level at zero; (b) this image illustrates the touch of the pitch pointer on the ASIS of the operating side during cup placement navigation

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According to the requirement (A), the constant direction G of the force of gravity in the IRP coordinate system can now be represented as:

$$\displaystyle{ G^{IRP} = [\begin{array}{*{20}c} -1&0&0\\ \end{array} ]^{T} }$$
(15.2)

When the cup would be placed in the desired orientation by the cup placement instrument and at the same time when the requirement (B) is satisfied, the IDP would be aligned with the IAP (see Fig. 15.2a for details). Thus, at this moment, the orientations of the axes of the local coordinate system of the IDP (or the IAP, as the IDP is aligned with the IAP) with respect to the local coordinate system of the IRP of the pelvis are:

$$\displaystyle{ \left \{\begin{array}{*{20}c} x_{IDP}^{IRP} = V I^{IRP} \\ z_{IDP}^{IRP} = \frac{V I^{IRP}\times CA^{IRP}} {\left \vert V I^{IRP}\times CA^{IRP}\right \vert } \\ y_{IDP}^{IRP} = z_{IDP}^{IRP} \times x_{IDP}^{IRP}\\ \end{array} \right. }$$
(15.3)

where “×” means the cross product of two vectors.

And the rotation from the local coordinate system of the IRP to the local coordinate system of the IDP is:

$$\displaystyle{ R_{IRP}^{IDP} = [R_{ IDP}^{IRP}]^{T} = [\begin{array}{*{20}c} x_{ IDP}^{IRP}&y_{ IDP}^{IRP}&z_{ IDP}^{IRP} \\ \end{array} ]^{T} }$$
(15.4)

We thus can transform the constant direction of the force of gravity from the local coordinate system of the IRP to the local coordinate system of the IDP:

$$\displaystyle{ G^{IDP} = R_{ IRP}^{IDP} \cdot G^{IRP} }$$
(15.5)

We could then further compute the three angles between G IDP with all three axes of the local coordinate system of the IDP. Given an arbitrary fixation point on the cup placement instrument, these three angles will uniquely determine an alignment direction along which the instrument bubble level should be placed such that when the bubble is placed to the center by orienting the cup placement instrument and its attached level and when the above two requirements are satisfied, the cup will be placed in the desired orientation. This principle has been used to design a mechanical guide as shown in Figs. 15.2b and 15.3b. The mechanical guide has an intraoperatively exchangeable steel block with a set of pre-manufactured holes, where each hole defines an alignment orientation along with which the instrument bubble level should be placed. Intraoperatively, according to the desired cup orientation and the patient-specific morphological information, the surgeon can choose the right steel block with the correctly oriented hole to place the instrument bubble level.

15.2.5 2D-3D X-ray Radiograph Reconstruction-Based Morphological Information Derivation

As clearly indicated in the above calibration procedure, the system calibration is a patient-specific task. Given a desired cup orientation, the exact decomposition of the constant direction of the force of gravity with respect to the three axes of the local coordinate system of the IDP depends on two patient-specific morphological parameters: (a) the angle θ between the APP and the IRP of the pelvis and (b) the orientation of the CA line in the local coordinate system of the IRP of the pelvis. Both parameters can be easily obtained from a CT or a MRI scan. However, these have the disadvantages that they are expensive, time-consuming, and/or induce high-radiation doses to the patient. More importantly, they are not part of the standard treatment loop of every patient in clinical routine. In this paper, we propose to use a statistically deformable 2D-3D reconstruction technique [20], which can reconstruct a scaled, patient-specific 3D model from a single conventional AP pelvic X-ray radiograph based on a statistical shape model of the pelvis. The reconstructed model can then be used to extract all the required morphological parameters. Figure 15.4 shows one example of applying this technique to reconstruct a 3D surface model of the pelvis from a conventional AP pelvic X-ray radiograph. Another example of the single image-based 2D-3D reconstruction of a pelvis used in our experiment which has different morphology from the one shown in Fig. 15.4 is presented in Fig. 15.5. As we are only interested in the angular or orientational information, a scaled, patient-specific 3D model will be accurate enough for our purpose.

Fig. 15.4
figure 4

One convention AP pelvic X-ray radiograph of a pelvis used in our experiment (left) and the model reconstructed from the radiograph (right)

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Fig. 15.5
figure 5

Another example of the single image-based 2D-3D reconstruction of a pelvis used in our experiment which has different morphology from the one shown in Fig. 15.4. Left: a convention AP pelvic X-ray radiograph of the pelvis; right: the model reconstructed from the radiograph

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15.3 Experiments and Results

We designed and conducted two studies on placing seven cups to four pelvises with different morphologies (four left sides and three right sides) to validate the accuracy of the present system. As all the pelvises were dry bones, we implemented an image-free navigation system following the principles introduced by Dorr et al. [13] to get the ground truth measurement for each experiment. Every time when the bubble of the instrument level is placed at the center, we recorded the measurements of the image-free navigation system. For all the experiments, the desired cup orientation is set to be 45∘ inclination and 20∘ anteversion.

For the first study, we acquired one conventional AP X-ray radiograph for each pelvis. The morphological information extracted from a surface model that was reconstructed from the X-ray radiograph of the associated pelvis was used to calibrate our system. This study was designed to validate the accuracy of the present system in placing the acetabular cups to different pelvises with different morphologies. For this purpose, every time when the bubble of the instrument level was placed at the center and when both system requirements were satisfied, the recorded measurements from the image-free navigation system were compared to the desired cup orientation. Table 15.1 summarizes the placement errors where an average accuracy of 2. 1 ± 0. 7∘ was found for inclination and an average accuracy of 1. 2 ± 1. 4∘ was found for anteversion.

Table 15.1 Difference between the desired cup orientation and the cup orientation actually achieved by the present system
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The second study was designed to validate how sensitive the present system is to the orientation of the pelvis with respect to the X-ray table during image acquisition. For this purpose, one pelvis (B_04) was chosen, and the pelvis was placed in different orientations with respect to the X-ray plate. Starting from an initial position, which we defined as the 0∘ position, we tilted the pelvis around the acetabular center line in one direction with an incremental interval of 5∘ until 20∘, and at each orientation we acquired one X-ray radiograph. We thus obtained five X-ray radiographs of the same pelvis. We then reconstructed a surface model of the pelvis from each X-ray radiograph and used the reconstructed model to derive morphological parameters for the instrument calibration. Based on the calibration, we performed the similar experiments as we did in the first study. Table 15.2 shows the placement errors when different X-ray radiographs were used to derive the patient-specific morphological parameters.

Table 15.2 Difference between the desired cup orientation and the actually achieved one when different X-ray radiographs were used
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15.4 Discussions and Conclusions

In this paper, we presented a patient-specific, gravity-assisted navigation system for high-precision placement of acetabular cup for THA operated in lateral approach. It starts with a 2D-3D reconstruction of a scaled, patient-specific 3D surface model of the pelvis from one conventional AP pelvis X-ray radiograph. The reconstructed 3D model facilitates a reliable and accurate co-registration of a gravity-assisted mechanical guide with the patient’s anatomy in the operating theater. We validated the accuracy of our system by conducting experiments on placing seven cups to four pelvises with different morphologies. The experimental results demonstrated the efficacy of the present system.

The rationale of using the measurements of an image-free navigation system as the ground truth in our experiment should be discussed. Previously, several studies [21,22,23] have suggested that CT-based solutions seem to be the most reliable method for noninvasive postoperative assessment of the acetabular cup orientation with experienced and trained observers. Probably this is true for those studies where there are no direct bone access to the anatomical landmarks that are required to precisely calculate the postoperative cup orientation. In such a situation, all the required landmarks have to be digitized percutaneously, which lead to errors in determining the cup orientation. In contrast, in the present study, all pelvises used in our experiment are dry bones. We can thus do direct bone digitization with our image-free navigation system, which may result in more accurate ground truth than the CT-based method according to what have been reported by Lin et al. [24].

Our system offers several advantages in comparison to other existing systems. First, instead of using a positional tracker, whose price ranges from several thousand Euros to dozens of thousand Euros, our system uses a mechanical guide with bull’s eye-bubble level indicators, taking advantage of the constant direction of the natural gravity force as a globally available reference for acetabular cup placement. Second, unlike most previously introduced mechanical alignment units, our system allows for a calibration with respect to the patient’s individualized morphology. Furthermore, in our system the patient-specific morphological information is derived from a 3D surface model of the pelvis that is reconstructed from a conventional AP X-ray radiograph using a statistically deformable 2D-3D registration approach. No CT/MRI scan is required. Our system is completely integrated with the standard treatment protocol.