Accuracy of an image matching technique for assessing knee alignment during the stance phase of gait using single-plane anteroposterior radiography

Abstract

BACKGROUND

Accurate measurement of bone alignment of the knee during walking provides ideal clinical information for diagnosis and treatment of knee joint disorders. To bring this ideal closer to clinical reality, we developed an image matching technique to measure the three-dimensional (3D) position of bones using anteroposterior radiography during the stance phase of treadmill walking.

OBJECTIVE

To develop and validate an image matching method for evaluation of 3D knee alignment using anteroposterior radiography of artificial femoral and tibial bones.

METHODS

The 3D position of each bone was recovered by minimizing the difference between the projected outline and the contour of the bone in the anteroposterior radiograph. The true value of the position was measured using a 3D coordinate measuring machine.

RESULTS

The mean values ± standard deviation and root mean squares (RMS) of translation errors were within −1.6 ± 2.1 mm and 2.6 mm, respectively, for femur, and 2.1 ± 1.9 mm and 2.8 mm for tibia. The mean values ± standard deviation and RMS of errors in rotation were within 0.3 ± 0.7° and 0.7°, respectively, for femur, and −0.3 ± 0.9° and 0.9°, respectively, for tibia.

CONCLUSION

Our method is suitable for evaluating 3D knee alignment on anteroposterior radiography.

Keywords

Knee alignment 2D to 3D image matching anteroposterior radiography pose estimation

1. Introduction

Assessment of the lower extremities using radiography is critical for diagnosis and treatment of knee joint disorders [1–4]. As the knee is a weight-bearing joint, static standing radiographs are used in clinical practice to evaluate coronal alignment parameters such as the mechanical axis (MA) and the relative angle between the femur and tibia [5]. However, these parameters do not fully represent the conditions of dynamic loading experienced at the knee during functional activities involving the lower limbs [6]. In particular, because walking is the most basic load-bearing movement in daily living, one of the ideal clinical assessments is to measure knee alignment during walking. Although motion capture systems using body surface markers have been used to evaluate lower limb alignment during the complete gait cycle, bone alignment is difficult to evaluate because of skin movement artifact [7]. This drawback also applies to markerless motion capture methods [8,9]. In response to this difficulty, methods using mobile radiography platforms that allow for tracking of dynamic activity such as level walking have been developed over the past two decades [10–14]. However, these methods require special equipment and a large radiation-shielded work space, making them unsuitable for assessing a large number of cases in general clinical facilities.

Anteroposterior (AP) radiographs of the knee acquired in the single-legged standing posture have been used as an alternative to assess gait alignment [15,16]. However, they only provide static evaluations and the posture does not always reproduce the alignment present in a dynamic gait. In addition, elderly patients and patients with knee disease often have difficulty maintaining a stable single-legged standing posture, and therefore the reproducibility of the true gait is expected to be poorer in such patients [17]. Furthermore, because the methods are applied only to a two-dimensional (2D) image, they cannot evaluate the 3D alignment of the knee. Therefore, a simple 3D method for evaluating knee alignment during the stance phase of gait would be of great importance in clinical practice. Such a method could potentially be accomplished by a 2D to 3D image matching algorithm applied to AP radiographs of the knee obtained during treadmill walking. The purpose of this study was therefore to develop a method for 3D evaluation of knee alignment using AP radiography, and to investigate the accuracy of the method using artificial femoral and tibial bones.

2. Materials and methods

2.1. Ground truth 3D alignment

Experiments were performed on Sawbone^® (Pacific Research Laboratories Inc., USA) artificial femur and tibia models. Three spherical acrylic markers (20 mm diameter) were attached to the femur and tibia. Computed tomography (CT) of the artificial femur and tibia with these markers attached was acquired on a Revolution CT scanner (GE Medical Systems, Chicago, USA) with 1.0-mm slice thickness and 0.78 × 0.78 mm pixel size. Femoral and tibial 3D surface models were reconstructed using ZedView modeling software (ZedView, LEXI, Tokyo, Japan). A coordinate system creator (ModelViewer, LEXI, Tokyo, Japan) was used to define the local coordinate systems of the femoral and tibial surface models according to the central coordinates of the markers attached to the surface models. Each artificial bone was securely fixed to an acrylic holder attached to a 3D goniometer stage (TS-C642, Chuo-Seiki, Tokyo, Japan). The range of rotation of the goniometer stage was ±5° in each orthogonal axis, with an accuracy of 0.07°. The 3D goniometer stage was mounted on a camera calibration frame that had 40 spherical steel markers. The central coordinates of the steel markers were measured using a 3D coordinate measuring machine (UPMC-550, Zeiss, Germany) to define the world coordinate system (WCS), as shown in Fig. 1. This machine measures 3D coordinates of the subject using a probe with an accuracy of 1.0 μm. The goniometer stage rotated about each orthogonal axis over a range of around 4°, resulting in translation of up to approximately 8 mm and rotation of up to approximately 4° in terms of the local coordinate system of the bones. A total of nine positions were set for each bone model within these ranges of translation and rotation. At each position, the central coordinates of the three acrylic makers attached to the artificial bone were measured using the 3D coordinate measuring machine to create the local coordinate system and the relative position of bone with respect to the WCS in terms of the parameters for six-degrees of freedom (DOF); that is, three translations along the x-, y-, and z-axes, and three rotations about the x-, y-, and z-axes of the WCS. These parameter values were used as the ground truth. In this configuration, the translation along the y-axis corresponded to an out-of-plane translation, and the rotation about the y-axis was an in-plane rotation.

Fig. 1.

Experimental apparatus for measuring the ground truth alignment of the artificial bones. The tibial artificial bone was securely fixed to an acrylic holder attached to a 3D goniometer stage. A 3D coordinate measuring machine was used to measure 3D coordinates of the markers using a probe. The three axes of the world coordinate system (WCS) are indicated as X_w, Y_w, and Z_w.

2.2. Two-dimensional to 3D image matching

A radiography system (DR-300, Shimadzu, Kyoto, Japan) was used to capture frontal X-ray images of the artificial bones fixed to the goniometer stage and camera calibration frame, as they had been set for the 3D coordinate measuring machine. The X-ray source to detector distance was 1.5 m (Fig. 2). This radiography system was capable of capturing images at a frame rate of 6 s⁻¹. The nine positions reproduced on the 3D goniometer stage were radiographed and images were digitized at a resolution of 1512 × 1512 pixels (covering an area of 373 × 373 mm²) with a 16-bit gray-scale depth using a flat panel detector. An X-ray tube calibration was performed using the calibration frame to determine the projection matrix [18]. The projection matrix provided 3D positioning of the focus of the X-ray source and the image plane with respect to the WCS embedded in the calibration frame, enabling projections of 3D objects to be replicated on the image plane following X-ray exposure. Approximately 30 points surrounding the outer edge of one-third of the distal part of the femur and one-third of the proximal part of the tibia were manually selected based on the radiographic imaging area of the knee joint obtained in clinical practice. The projected outline points of the 3D surface model were the finite edge points of the 2D shadow created from the projections of all visible triangular surfaces forming the 3D surface model [19].

Fig. 2.

Set-up for anteroposterior X-ray imaging of the artificial bones.

For the i-th point of the object edge points p_i, the closest point belonging to the model projection line was defined as q_i. The distance between the two points was then summed over all object edge points and subsequently normalized by the total number of points, N.

F = \sum_{i = 1}^{N} | p_{i} - q_{i} | / N .

(1)

As F is arameters of the 3D surface model, its position can be determined by minimizing it. For this image matching process, the local coordinate system of the artificial bones was redefined from the acrylic marker-based system to an anatomical one [20]. The femoral x-axis was defined as the line connecting the centers of spheres representing the medial and lateral posterior femoral condyles (positive medially), and the midpoint of this line was defined as the origin. The femoral y-axis was the normal vector of the plane containing the three points: the center of the femoral head and the centers of the two spheres representing the posterior femoral condyles. The anterior direction was defined as positive. The femoral z-axis was defined as the cross product of the x-axis and y-axis. For the tibia, the z-axis was defined as a line connecting the midpoint of the medial and lateral top of the talar dome and the midpoint of the tibial eminence (positive proximally). The tibial y-axis was defined as a line drawn perpendicular to the z-axis, running from the z-axis to the mediolateral center of the insertion of the posterior cruciate ligament (considered positive in the anterior direction). The intersection of the y-axis and z-axis was considered the origin. The tibial x-axis was the cross product of the y-axis and z-axis (Fig. 3). The ground truth bone position was also transformed from the acrylic marker-based coordinate system to the anatomical one.

Fig. 3.

Local coordinate systems of femur and tibia defined using the center points of three markers (a) are transformed to those defined using anatomical landmarks (b).

Using the downhill simplex algorithm [21], the minimization procedure started from an initial set of 6-DOF parameters that were randomly chosen from within ±5.0° and ±5.0 mm of the ground truth, and the minimization was terminated if either the number of iterations exceeded 500 or the relative change in F was below 0.00005 (Fig. 4). The computation was performed using custom-made software running on a Windows 11 PC (Core i9-9900X CPU, 3.5 GHz, 64 GB RAM).

Fig. 4.

Image matching using projected outlines of the 3D surface models and bone edge points selected at the distal femur (a) and proximal tibia (b).

3. Results

The mean values, standard deviation, and root mean squares (RMS) of errors in the six-DOF parameters of the femur and tibia determined by the image matching method are listed in Table 1. The running time for minimization was about 60 s for each alignment estimation. The mean values, standard deviation, and RMS of errors in translation along the x- and z-axes were within 0.1 ± 0.5 mm, and 0.5 mm, respectively, for the femur, and within 0.1 ± 0.3 mm and 0.5 mm, respectively, for the tibia. However, estimation of the y-axis translations, which were almost parallel to the out-of-plane direction of the imaging system, was less accurate, with a mean value and standard deviation of −1.6 ± 2.1 mm and RMS of 2.6 mm for the femur, and a mean value and standard deviation of 2.1 ± 1.9 mm and RMS of 2.8 mm for the tibia. The mean values, standard deviations, and RMS of errors in rotation about the y-axis were within 0.0 ± 0.1° and 0.1° for the femur and tibia. The rotations about the x- and z-axes, which were almost parallel to the x- and z-axes of the WCS, were less accurate and precise for both bones. For the femur, the mean value, standard deviation, and RMS of errors were 0.1 ± 0.6° and 0.5° for x-axis rotation, and 0.3 ± 0.7° and 0.7° for z-axis rotation. For the tibia, the mean value, standard deviation, and RMS of errors were 0.3 ± 0.9° and 0.9° for x-axis rotation, and −0.3 ± 0.9° and 0.9° for z-axis rotation.

Table 1.
Mean values, standard deviations, and root mean squares (RMS) of errors in the six degrees-of-freedom parameters of femur and tibia

Translation (mm) Rotation (°)

x y z x y z

Mean −0.1 −1.6 −0.1 0.1 0.0 0.3

Femur SD 0.3 2.1 0.5 0.6 0.1 0.7

RMSE 0.3 2.6 0.5 0.5 0.1 0.7

Mean 0.1 2.1 −0.1 0.3 −0.0 −0.3

Tibia SD 0.1 1.9 0.3 0.9 0.1 0.9

RMSE 0.1 2.8 0.3 0.9 0.1 0.9

		Translation (mm)	Rotation (°)
	Mean	−0.1	−1.6	−0.1	0.1	0.0	0.3
Femur	SD	0.3	2.1	0.5	0.6	0.1	0.7
	RMSE	0.3	2.6	0.5	0.5	0.1	0.7
	Mean	0.1	2.1	−0.1	0.3	−0.0	−0.3
Tibia	SD	0.1	1.9	0.3	0.9	0.1	0.9
	RMSE	0.1	2.8	0.3	0.9	0.1	0.9

4. Discussion

This study developed and validated an image matching technique for the measurement of 3D knee alignment using single-plane frontal radiography. A 3D goniometer stage and 3D coordinate measuring machine were used to determine the ground truth positions of artificial femur and tibia bones. The accuracy obtained was comparable to that of studies using single-plane radiographs applied to natural knees. Fregly et al. [22] performed a theoretical investigation of the accuracy of 3D position estimation under ideal conditions, using synthetic images to eliminate error sources such as blurred bone edges and image distortion. Their precision was 0.20 mm for in-plane translation, 3.1 mm for out-of-plane translation, and 0.45° for the overall rotations. Scarvell et al. [23] reported the accuracy of an image registration technique using a digitally reconstructed radiograph (DRR) obtained from CT data and fluoroscopy. Their technique showed a standard deviation of the error of 0.38 mm for in-plane translation, 0.65 mm for out-of-plane translation, and 0.42° for rotation. Ishimaru et al. [24] also investigated the accuracy of a DRR-based method and found RMS errors of 0.2 mm and 0.2° for patellar bone. Other researchers used actual internal bone contours as well as bone edges, and found precision for the position of the femur relative to the tibia of 0.45 mm [25] and 1.2 mm [26] for in-plane translation, 4.0 mm [26] for out-of-plane translation, and 0.66° [25] and 0.8° [26] for overall rotations.

These previous studies validated image matching techniques using single-plane radiography, showing it to be adequate for high accuracy measurement of the 3D position of the knee in vivo. However, the limited size of view of the stationary imaging system restricted its use for capturing the whole gait cycle. For this reason, moving imaging platforms were developed to track knee motion during gait cycles over level ground, stair climbing, and ramp walking. Although these systems allow the measurement of complete cycles during several kinds of daily activities, the requirement for specialized equipment and a large radiation-shielded work space can cause difficulty with accommodating them in ordinary clinical facilities.

We therefore proposed our method to capture coronal knee alignment during treadmill walking, aiming between static single-legged standing radiography and moving fluoroscopy for tracking the whole gait cycle. Our proposed method offers advantages over single-legged standing radiography because it enables the complete treadmill gait cycle to be captured, and offers advantages over moving fluoroscopy systems because no special equipment and no large radiation-shielded space are needed. In addition, in our method, the test knee is free from contralateral leg occlusion, which can occur during sagittal imaging using moving fluoroscopy. These advantages make our method feasible for use in general clinical facilities with conventional fluoroscopy systems, with the only additional equipment required being an off-the-shelf treadmill device and software for bone model reconstruction and image registration.

The present study had several limitations. First, the validation experiments were carried out on bone models. On clinical X-ray images, there may often be an ambiguous bone edge due to surrounding soft tissues, which was not present with the bone models. However, because the present image matching method requires only 30 contour points at most, and these points can be selected from clear areas, we consider that the effect of such an ambiguous bone edge on accuracy will be small. Second, only stationary positions of bone were evaluated. Because our method is intended for the capture of knee alignment during treadmill walking, motion-related blurring, which was not considered in this study, may be a potential source of error in alignment estimation. To avoid this, the walking speed may be restricted to be lower than that of a healthy population [27], and even to that of patients who have undergone total knee arthroplasty [28]. Nevertheless, a lower walking speed is considered suitable for elderly subjects and patients with knee disease, to ensure safety.

5. Conclusion

In conclusion, we developed an image matching technique to measure 3D knee alignment using single-plane frontal radiography. We validated the accuracy of our method using femur and tibia models. The mean values, standard deviations, and RMS of the error when estimating translations were within −1.6 ± 2.1 mm and 2.6 mm for femur, and 2.1 ± 1.9 mm and 2.8 mm for tibia. The mean values, standard deviations, and RMS of errors when estimating rotation were within 0.3 ± 0.7° and 0.7° for femur and within −0.3 ± 0.9° and 0.9° for tibia.

Footnotes

Acknowledgements

We thank Edanz () for editing a draft of this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.

Funding

The authors report no funding.

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