## 4 Discussion  
In this study, the angle sensor values were used as the true values, and the range of motion for the radial abduction, palmar abduction, and pronation of the thumb was estimated based on a markerless video of thumb movements using MediaPipe and machine learning. The accuracy of the estimations was evaluated, and it was found that high estimation accuracy was achieved. In this study, thumb movement was measured using a conical model operating under the assumption that the trajectories of the CMC, MP, and IP joints lie on the same straight line and a concentric sphere. Numerous studies have measured thumb range of motion in two dimensions [7, 8, 17, 18]. In fact, the current guidelines from the International Society for Surgery of the Hand and the American Academy of Orthopaedic Surgeons rely on two-dimensional data for quantifying thumb range of motion [2, 36, 37]. However, the CMC joint of the thumb is generally considered a single joint that permits biaxial motion and  
Figure 23: Feature importance and SHAP values for LightGBM for radial abduction. Figure 24: Feature importance and SHAP values for LightGBM for palmar abduction. Figure 25: Feature importance and SHAP values for LightGBM for pronation. functions within three dimensions [6; 7; 8; 9; 10]. Given this three-dimensional functionality of the thumb, studies in which three-dimensional models were used to measure its range of motion have also been reported. [PERSON] et al. applied the spherical representation method [38], originally used for measuring shoulder joint range of motion, to the thumb. They used marker-based motion capture systems with markers attached to the fingers to measure the thumb's range of motion. The measurements were made while keeping the MP and IP joints fixed. The thumb was assumed to move conically, with the base of the thumb forming an elliptical trajectory. This model also assumed that the CMC joint, MP joint, IP joint, and the tip of the thumb align to form a straight line during movement. Furthermore, various studies have used the cone model to measure the thumb's range of motion. The authors of these studies utilized different measurement techniques, including markers, infrared cameras, and electromagnetic sensors, to approximate thumb motion as conical and quantify its range of motion [14; 15; 16; 17; 18]. In recent years, advancements in technology have led to reports on hand movement analysis using markerless methods. Since Google announced MediaPipe in 2020 [20; 21], studies have also attempted to measure hand range of motion using MediaPipe. However, when images of specific protocol positions captured with a smartphone were analyzed using MediaPipe alone, the average deviation from the goniometer was \(-2.21\pm 9.29^{\circ}\), indicating low accuracy [23]. In our previous study, thumb abduction movement was analyzed using a conical model, with the true value obtained using a protractor. High estimation accuracy was achieved by combining MediaPipe with machine learning techniques [19]. However, the use of a protractor introduced issues regarding accuracy and reproducibility due to inter-examiner variability [3; 4; 5]. Moreover, reliance on a protractor prevented the evaluation of thumb pronation, which involves axial rotation. The Kapanji score is a clinical evaluation method used to assess pronation [39]. It is a simple and widely used tool in clinical settings for evaluating thumb opposition [40; 41; 42]. However, this score does not provide an accurate quantitative evaluation; it merely classifies movements using numerical values [28]. Three-dimensional CT scans are accurate in measuring angles but come with high costs and involve invasive procedures. Optical motion capture systems enable dynamic and three-dimensional measurements without being invasive, although they necessitate large-scale, specialized setups and involve complex operations. Conversely, angle sensors are small, wearable, affordable, and easy to use and provide non-invasive, three-dimensional measurements [28; 43; 44]. [PERSON] et al.