Estimation of radiographic joint space of the trapeziometacarpal joint with computed tomographic validation

Abstract.

Purpose

Joint space width (JSW) is a common metric used to evaluate joint structure on plain radiographs. For the hand, quantitative techniques are available for evaluation of the JSW of finger joints; however, such techniques have been difficult to establish for the trapeziometacarpal (TMC) joint. This study aimed to develop a validated method for measuring the radiographic joint space of the healthy TMC joint.

Approach

Computed tomographic scans were taken of 15 cadaveric hands. The location of a JSW analysis region on the articular surface of the first metacarpal was established in 3D space and standardized in a 2D projection. The standardized region was applied to simulated radiographic images. A correction factor was defined as the ratio of the CT-based and radiograph-based joint space measurements. Leave-one-out validation was used to correct the radiograph-based measurements. A t-test was used to evaluate the difference between CT-based and corrected radiograph-based measurements ($\alpha =0.05$).

Results

The CT-based and radiograph-based measurements of JSW were $3.61\pm 0.72\mathrm{mm}$ and $2.18\pm 0.40\mathrm{mm}$, respectively. The correction factor for radiograph-based joint space was $1.69\pm 0.41$. Before correction, the difference between the CT-based and radiograph-based joint space was 1.43 mm [95% CI: $0.99-1.86\mathrm{mm}$; $p<0.001$]. After correction, the difference was $-0.11\mathrm{mm}$ [95% CI: $-0.63-0.41\mathrm{mm}$; $p=0.669$].

Conclusions

Corrected measurements of radiographic TMC JSW agreed well with CT-measured JSW. With in-vivo validation, the developed methodology has potential for automated and accurate radiographic measurement of TMC JSW.

1. Introduction

The trapeziometacarpal (TMC) joint at the base of the thumb is defined by the articulation between the trapezium and first metacarpal. Plain radiography is commonly used to evaluate the joint with respect to the presence of common musculoskeletal disorders,¹ such as osteoarthritis, for which the TMC joint is the most commonly affected joint of the wrist.^2,3 Radiographic evaluation can be performed using established scoring systems, such as the Kellgren–Lawrence classification.⁴ Structural degradation, such as joint space narrowing, can be included within the metrics of these scoring systems.⁵ The use of specialized radiographic views generated from standardized hand orientations, in conjunction with prescribed beam angles, can highlight different articulations of the trapezium and aid in the joint evaluation;¹ however, the determination of quantifiable parameters can be challenging due to the spatial complexities of the two-dimensional radiographic projection of the three-dimensional joint when interpreted visually by an observer.

The most important radiographic criterion used in surgical decision making for osteoarthritis is joint space narrowing, or the loss of joint space width (JSW), which reflects the degradation of cartilage.⁶ While easy to quantify in other joints, the complexities in the visualization of JSW on radiographic projections of the TMC joint may arise due to the saddle-shaped joint structure,⁷ rendering difficult the identification of reference features from which to quantify JSW. Conversely, methods to identify such features for JSW measurement of the finger joints have been performed for the healthy metacarpophalangeal, proximal interphalangeal and distal interphalangeal joints, respectively.^8–10 Similarly established methods have also been utilized for the analysis of arthritic finger joints,^11–14 providing valuable insight in the radiographic evaluation of JSW in these joints. Given that the TMC joint is the most commonly affected wrist joint by osteoarthritis, and more commonly affected than most hand joints,^15,16 an analogous method for the quantification of radiographic TMC JSW can similarly aid in the radiographic evaluation of the joint health.

Computed tomography (CT) is an imaging modality that can readily evaluate the complex TMC joint structure in three dimensions and allow quantification of the TMC JSW. The definition of the TMC JSW can vary depending on the nature of the study. These definitions can include proximity maps, showing the distribution of the joint spacing within a given distance tolerance⁷ or singular values typically given as the minimum distance between the articulating joint surfaces.¹⁷ TMC loss of JSW is believed to progress from one side of the joint to another over time.^6,18 Independent of progression, the two-dimensional radiographic visualization derived from the three-dimensional joint structure is difficult. There may exist a relationship between the two visualizations that, through the identification of corresponding features, can be used to approximate the TMC JSW using radiographic measurement alone.

Comparative studies evaluating the respective outcomes for CT and radiographic analysis of various anatomical aspects of the hand and wrist have been performed.^19,20 These studies often focus on highlighting the advantages or disadvantages of one imaging modality with respect to the other. Although radiography is more commonly used and more readily available, CT can have greater analytical capability. Advancements in the evaluation of TMC joint health can utilize the advantages of both imaging modalities by combining the visual utility of radiographic imaging with the analytical prowess of CT for the development of a measurement method for radiographic TMC JSW. One optimal solution would be a standardized method to assess TMC JSW in a manner that can correct for the added information absent without a CT scan. The purpose of this study was to formulate and validate a methodology for measuring the radiographic JSW of the TMC joint. The hypothesis was that radiograph-based TMC JSW would underestimate CT-based TMC JSW for standardized regions of analysis.

2. Methods

Fifteen arm cadaveric specimens (seven females, $70.0\pm 7.0$ years; 8 males, $51.1\pm 8.9$ years) transected at the mid-humerus were placed onto a custom fixation device, with the forearm secured using a series of constraints. The four fingers were secured around a cylinder in a grasping orientation. The specimens and device were placed inside a cone beam CT machine (CurveBeam AI, Hatfield, Pennsylvania, United States) with the wrist aligned with the center axis of the bore. For each specimen, the thumb was positioned in one of two orientations. One orientation was neutral, with no external manipulation. The other orientation was in approximately 45 deg of extension. The extension orientation was maintained with the application of a 1 N traction load applied using a finger trap and pulley-system [Fig. 1(a)]. Six specimens were aligned in the neutral orientation and nine specimens were aligned in the extension orientation. A CT image was taken for each specimen. The reconstructed anatomy [Fig. 1(b)] and a simulated radiographic image [Fig. 1(c)], generated from the CT imaging data using the postprocessing software CubeVue (CurveBeam AI, Hatfield, Pennsylvania, United States), were exported for each specimen. The radiographic image was produced in the posterior-anterior view.

Fig. 1.

CT imaging of the cadaveric hand specimens. (a) Representative example of a specimen secured within the CT machine with the thumb angled at approximately 45 deg of extension. (b) The reconstructed anatomy from the CT scan. (c) The simulated posterior-anterior radiographic image.

The articular surfaces of the exported 3D models of the metacarpal and trapezium were manually identified [Fig. 2(a)] using MeshMixer (Autodesk, San Rafael, California, United States). MATLAB (MathWorks, Natick, Massachusetts, United States) was used to fit a fifth order polynomial to the articular surfaces of the metacarpal and trapezium; and to approximate the location of the saddle point of the first metacarpal²¹ [Fig. 2(a)]. A sphere of radius 2 mm was defined centered at the saddle point of the metacarpal articular surface. The intersection of this sphere with the metacarpal polynomial articular surface defined a boundary [Fig. 2(a)].

Fig. 2.

CT-based TMC joint space definition and calculation. (a) First metacarpal with manually identified articular surface, saddle point, and boundary formed by the intersection of the articular surface and sphere of radius 2 mm. (b) Example of the minimum distance solution calculated for each grid point within the boundary.

A $100\times 100$ grid was defined within the domain of the metacarpal polynomial articular surface and constraints were imposed so that the grid points to be analyzed did not exceed the boundary. The collection of points within the boundary defined the JSW analysis region. For each of the points within the JSW analysis region, the minimum distance [Fig. 2(b)] to the trapezium articular surface $D(x,y)$, was determined by solving the unconstrained minimization of Eq. (1). Equation (1) describes the distance from a point within the JSW analysis region on the metacarpal polynomial articular surface $(x_M,y_M,z_M)$ to an arbitrary point on the trapezium polynomial articular surface $(x_T,y_T,z_T)$. The average of these individual JSW values was defined as the CT-based JSW and was calculated for the 15 specimens

$$D(x,y)=\min \left(\sqrt{(x_M-x_T{)}^2+(y_M-y_T{)}^2+(z_M-z_T(x_T,y_T){)}^2}\right).$$ (1)

A second sphere [Fig. 3(a)] was defined centered at the centroid of the metacarpal articular surface with a radius $R=120\mathrm{mm}$. At a discrete number of points (1681) along the spherical surface, a plane of projection was defined with unit normal vector $\hat{\mathbf{R}}$. Every point of the metacarpal surface, and the points within the JSW analysis region, were projected onto each of the planes. The projections were visually inspected and the analysis projection was determined [Fig. 3(b)]. The analysis projection was defined as the projection that resembled the radiographic projection of the first metacarpal in the simulated x-ray image.

Fig. 3.

Standardization of the joint space analysis region in 2D. (a) First metacarpal joint with surrounding spherical region for surface projections, representative 3D planes of projection and 2D projections of the first metacarpal. (b) Selected analysis projection with projected joint space analysis region (green) and third order polynomial $f(x)$, outlining the articular surface of the first metacarpal. (c) Standardization of the projected joint space analysis region based on articular surface arc length.

A series of points [$p_{\alpha }$, $p_{\beta }$] were manually selected along the articular surface of the analysis projection. Depending on the orientation of the analysis projection, $p_{\alpha }$ and $p_{\beta }$ represented the most radial and ulnar points of the articular surface projection. A third order polynomial $f(x)$, was fit to these points [Fig. 3(b)]. Three points were determined from the projection of the joint space analysis region. The first point was the center point of the region. The second and third points were the points with the minimum and maximum coordinates, with respect to $x$, in the projection $(x,y)$ coordinate system. The minimum distance of these three points to $f(x)$ was determined. The corresponding points along $f(x)$ were designated $p_0$, $p_a$, and $p_b$, respectively [Fig. 3(c)]. The locations of $p_0$, $p_a$, and $p_b$ were standardized with respect to the arc length of $f(x)$. The total arc length $S$ of $f(x)$ was calculated using Eq. (2), where $x_{\alpha }$ and $x_{\beta }$ were the $x$-coordinates of $p_{\alpha }$ and $p_{\beta }$, respectively. $S_{\alpha 0}$ was defined as the arc length between points $p_{\alpha }$ and $p_0$. $S_{0a}$ and $S_{0b}$ were defined as the arc length between points $p_0$ and $p_a$ and between points $p_0$ and $p_b$, respectively. The location of $p_0$ along $f(x)$ was standardized with respect to $S_{\alpha 0}$ and $S$, i.e., $S_{\alpha 0}/S$. The location of $p_a$ along $f(x)$ was standardized with respect to $S_{0a}$ and $S$, i.e., $S_{0a}/S$. The location of $p_b$ along $f(x)$ was standardized with respect to ${\mathrm{S}}_{0\mathrm{b}}$ and $\mathrm{S}$, i.e., $S_{0b}/S$

$$S={\int }_{x_{\alpha }}^{x_{\beta }}{[1-{\left(\frac{\mathrm{df}(\mathrm{x})}{\mathrm{dx}}\right)}^2]}^{\frac{1}{2}}\mathrm{dx}.$$ (2)

The radiographic TMC JSW was calculated for each specimen. The simulated radiographic images for each specimen were analyzed [Fig. 4(a)]. The simulated radiographs were enhanced near the region of the TMC joint to provide clear visualization of the TMC joint radiographic projection [Fig. 4(b)]. A series of points was manually selected along the radiographic articular surfaces of the metacarpal and trapezium; and a third order polynomial was fit to each [Fig. 4(c)]. The arc length of the polynomial defining the metacarpal articular surface was calculated and the radiographic locations of $p_0$, $p_a$, and $p_b$ were determined based on the previously established standardized definitions. The domain between $p_a$ and $p_b$ was divided into 101 equally spaced points. The minimum distance between these points and the polynomial defining the radiographic trapezium articular surface was calculated [Fig. 4(c)]. The radiograph-based TMC JSW for each specimen was defined as the average of these distance values. A specimen-specific correction factor was defined for each specimen as the ratio of the CT-based JSW and the radiograph-based JSW.

Fig. 4.

Radiograph-based JSW measurement procedures. (a) The simulated posterior-anterior radiographic image of a representative specimen was generated. (b) The projection region was focused at the TMC joint to provide an unobstructed visualization of the joint. (c) A third order polynomial was fit to the metacarpal and trapezium articular surfaces, the locations of $p_0$, $p_a$ and $p_b$ were estimated and the minimum distance between the projected articular surfaces was calculated.

Validation was performed using the leave-one-out method using fifteen iterations. For a given validation iteration, one specimen was designated as the test specimen. The remaining specimens were designated as the training specimens. The radiograph-based JSW of the test specimen was calculated as previously described, where the radiographic locations of $p_0$, $p_a$, and $p_b$ were determined based on the average of the standardized definitions for the training specimens. The radiograph-based JSW for the test specimen was corrected by multiplying the JSW value by the average of the specimen-specific correction factors for the training specimens. A two-tailed t-test was used to compare the CT-based and radiograph-based JSW; and the CT-based and corrected radiograph-based JSW for all specimens, the female specimens and the male specimens. The statistical significance was set at 0.05.

3. Results

The mean radiograph-based TMC JSW was less than the mean CT-based TMC JSW for all female and male specimens (Fig. 5). The difference between the means of the CT-based JSW and the radiograph-based JSW for all specimens was 1.43 mm [95% CI: $0.99-1.86\mathrm{mm}$; $p<0.001$]. The difference between the means of the CT-based and radiograph-based JSW was 1.77 mm [95% CI: $1.09-2.45\mathrm{mm}$; $p<0.001$] for the female specimens and 1.12 mm [95% CI: $0.65-1.59\mathrm{mm}$; $p<0.001$] for the male specimens.

Fig. 5.

TMC JSW results for the CT-based, radiograph-based and corrected radiograph-based measurements (mean and standard deviations) for all female and male specimens.

The leave-one-out validation procedure increased the magnitude of the radiograph-based TMC JSW. The mean correction factor was $1.69\pm 0.41$ for all specimens. The mean correction factor was $1.79\pm 0.23$ and $1.60\pm 0.53$ for the female and male specimens, respectively. The difference between the means of the CT-based and the corrected radiograph-based JSW for all specimens was $-0.11\mathrm{mm}$ [95% CI: $-0.63-0.41\mathrm{mm}$; $p=0.669$]. The difference between the means of the CT-based and the corrected radiograph-based JSW was 0.17 mm [95% CI: $-0.65-0.99\mathrm{mm}$; $p=0.657$] for the female specimens and $-0.36\mathrm{mm}$ [95% CI: $-0.99-0.28\mathrm{mm}$; $p=0.248$] for the male specimens.

4. Discussion and Conclusion

This study formulated a method to measure the JSW of the TMC joint using simulated radiography. The method was formulated using TMC JSW measurements derived from three-dimensional TMC joint models developed from the CT scans of fifteen cadaveric specimens. The location of a JSW analysis region was defined and standardized based on the two-dimensional articular surface length of the first metacarpal for general applicability to an arbitrary radiographic image. Applying the results of the standardization in the leave-one-out validation routine showed that the established methodology was able to generate estimates of the radiograph-based TMC JSW that agreed well with the CT-based JSW measurements.

The definition of three-dimensional TMC JSW used in this study was unique in comparison to previous studies, which tend to use the proximity of the entire articulating joint surface.^7,22 In a radiographic projection of the TMC joint, the complexities of the articular surface are not visible; therefore, a more localized region of three-dimensional JSW analysis was required to identify a corresponding region of analysis in the two-dimensional space. Confining the JSW analysis to a constrained region around the approximate location of the saddle point of the first metacarpal allowed for the standardization of the JSW definition that would not be substantially influenced by extraneous properties of the articular surfaces. The saddle point of the first metacarpal has been used in previous studies²¹ and was chosen as the center of the JSW analysis region because this point represents a critical point on the surface, which is not local extrema while remaining agnostic to the differing ways that arthritis may progress across the joint over time. This methodology also provided a relatively homogenous region for JSW analysis.

Comparisons between three-dimensional and two-dimensional JSW measurements have been a continuing topic for both CT and radiographic imaging. Quantitative studies that investigate various anatomical measurements of the hand, such as the canal diameter of the distal phalanx²³ and alignment of the carpal bones¹⁹ show good agreement between CT and radiographic measurement. Qualitative comparisons have also been performed with respect to detection of osteoarthritis.²⁰ These studies can use analogous two-dimensional features, which can be identified both on the radiographic images and the two-dimensional CT slices. The current study uniquely identifies a single three-dimensional feature, the JSW analysis region from the three-dimensional anatomical geometry of the first metacarpal, and then approximates the standardized location in the corresponding radiographic image. The radiographic location of the JSW analysis region was standardized based on the arc length of the two-dimensionally projected articular surface of the first metacarpal. Standardizing using length allows for directional independence of the region location and minimizes operator subjectivity, requiring only an outlined identification of the articular surface. It is also a step toward a more automated future analysis.

The radiograph-based JSW was calculated from the minimum distance between the radiographic articular surfaces of the metacarpal and trapezium, discretized within the limits of the radiographic JSW analysis region. The method of measuring the radiograph-based JSW is similar to methods used for more commonly measured hand joints, such as those of the four fingers, where measurement techniques can use distance values between the proximal and distal margins of the opposing articular surfaces.^8,24 For the TMC joint, the distance between the proximal and distal margins of the radiographic projection underestimates the three-dimensional JSW, as the depth of the articular surfaces cannot be identified on the radiographic projection. The underestimation of the radiograph-based JSW necessitated the definition of a correction factor, which describes the multiplicative increase required to equate the radiograph-based and CT-based JSW measurements. Identification of such a correction factor is, in itself, a goal of this work.

The leave-one-out validation procedure performed well. There were no significant differences between the CT-based JSW and the corrected radiograph-based JSW measurements for all specimens, the female specimens or the male specimens. The difference between the CT-based and radiograph-based JSW measurements represents the systematic error. After correction, the magnitude of the error was reduced to 0.11 mm. Magnitudes of three-dimensional TMC JSW can be on the order of less than 1 mm to greater than 3 mm,^17,25 depending on the articular surface location between which the measurements are made; meaning that the error observed in this study is well below the measurement scale for the TMC JSW. Additionally, the error observed for the corrected estimates of radiograph-based TMC JSW were similar to the errors observed for radiographic measurements of other hand joints; such as the errors resulting from measurements of cadaver derived phalangeal JSW during progressive increasing of the joint space.²⁴ Progressive joint space increasing of an MCP model showed that computerized radiographic JSW measurements can be accurate to within of 0.018 mm.²⁶ Continued improvement of the accuracy of the radiograph-based TMC JSW measurement method can be accomplished through further research, potentially through optimizing hand orientation. Hand orientation can have an effect on JSW measurement. In a cadaveric study, flexion angle was shown to vary the measurement of metacarpophalangeal JSW.²⁷ Similar observations may exist for the TMC joint and suggests that an optimal thumb orientation may exist for where the difference between radiograph-based and CT-based JSW can be minimized.

Thumb TMC osteoarthritis can be classified in various stages, each accompanied by increasing severity of joint space narrowing.^28,29 Studies have investigated joint space narrowing in diverse cohorts.^15,30 The proposed measurement method of radiographic TMC JSW can be used to complement these qualitative measures by first providing a standardized means to measure the radiographic JSW of the healthy TMC joint. Future work will validate the method for measurement of the healthy TMC JSW in-vivo using common radiographic hand orientations.¹ Validation of the method for measuring the in-vivo JSW of arthritic TMC joints would be performed in subsequent studies, providing quantitative correlates for qualitative joint space narrowing scores; such as those that have been developed for the finger joints using hand radiographs of patients with arthritis.^12,31,32

This study has several limitations. First, the methodology was developed computationally using methods to simulate the projection of the first metacarpal. This approach cannot exactly mimic what would result from an actual x-ray projection, meaning that the radiographic location of the joint space analysis region could not be standardized using a true x-ray projection. This limitation was minimized by performing the standardization using the computational projection, which most closely resembled the simulated x-ray projection. Second, the method was validated using simulated radiographic images generated from CT scans of cadaveric specimens, which limited the projection manipulation; meaning that the quality of the radiographic view of the TMC joint could not be guaranteed for every specimen. This limitation was overcome by using a focused region around the TMC joint that allowed for the radiographic projection to be clearly visualized for measurement. In-vivo, coordinated CT and radiographic imaging, combined with standardized hand orientations³³ and additional CT-based analyses, which can consider variables such as joint orientation in the utilization of such views,³⁴ can be used to increase the radiographic image quality of the TMC joint. The use of specialized views in the surgical evaluation of TMC osteoarthritis argues that this type of standardization of radiographs is possible, and this has been demonstrated in other joints as well.³⁵ Finally, the method was developed and validated using cadaveric specimens. Further validation utilizing in-vivo imaging from a larger sample size can strengthen the standardization and improve the accuracy and applicability of the developed radiographic TMC JSW measurement method.

Biographies

David Jordan, MS, PhD, is a postdoctoral research associate at the University of Arizona, College of Medicine, Department of Orthopaedic Surgery. He received his BS degree in mechanical engineering from Louisiana State University in 2016 and his MS and PhD degrees in mechanical engineering from the University of Pittsburgh in 2018 and 2021, respectively. His current research interests are in the imaging and biomechanical analysis of the hand and wrist.

John Elfar, MD, is board-certified in orthopaedic surgery, sports medicine and hand surgery. He is a professor and chair of the Department of Orthopaedic Surgery at the University of Arizona. He specializes in orthopedic surgery, and orthopedic sports medicine. This includes sports-related injuries and non-sports-related injuries. Most of Dr. Elfar’s practice centers on the upper extremity, shoulder, elbow, wrist, hand, and fingers. He also specializes in complex orthopedic reconstructions.

Chian K. Kwoh, MD, is board-certified in internal medicine and rheumatology. He is a professor of Medicine and Medical Imaging in the University of Arizona College of Medicine, director of the UA Arthritis Center, chief of the Division of Rheumatology, and holds the Charles A.L. and Suzanne M. Stephens Endowed Chair in Rheumatology. His research interests focus on the evaluation of imaging biomarkers of the development and/or progression of knee and hand osteoarthritis.

Zong-Ming Li, PhD, is the William and Sylvia Rubin Chair of Orthopedic Research and a professor of Orthopedic Surgery and Biomedical Engineering at the University of Arizona. He is a Fellow of the American Institute for Medical and Biological Engineering, the American Society of Biomechanics, the Asia-Pacific Artificial Intelligence Association, the Institute of Electrical and Electronics Engineers, and the American Society of Mechanical Engineering.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Code and Data Availability

Data are available upon request.