# A Refined Approach to Segmenting and Quantifying Inter-Fracture Spaces in Facial Bone CT Imaging
## 1 Introduction  
Facial bone fractures are most commonly caused by assaults (44-61%), traffic accidents (15.8%), and falls (15%) [1, 2, 3]. Patients who suffer facial trauma must receive appropriate treatment based on whether their facial bones are fractured. Failure to properly treat fractured facial bones or delaying treatment for more than two weeks can result in complications such as nasal rupture and enophthalmos. Therefore, early detection of facial bone fractures is crucial. The complexity and difficulty in diagnosing facial bone fractures arisefrom various factors, particularly due to the various symptoms and radiographic findings depending on a fracture's location, making accurate diagnosis challenging [4; 5; 6]. Many studies have been conducted to mitigate these challenges by employing engineering techniques to detect fractures. Most research focuses on using computer graphics and machine learning to detect fractures in specific areas of the bones within medical imaging [7; 8; 9; 10; 11]. However, these studies primarily aim at detecting fractured areas, with limited research proposing segmentation of the fractured skeleton [12; 13]. The clinical benefit of identifying spaces between fractured bones lies in aiding radiologists in understanding the complex interplay between fracture patterns and the grades of injury, which is essential for developing individualized treatment plans that maximize aesthetic and functional outcomes while minimizing complications [14]. Moreover, no studies have explored segmenting the spaces between bones caused by fractures. Accurate segmentation and quantification of spaces between fractured facial bones are crucial for several reasons. First, precise segmentation aids in understanding the spatial relationships and severity of fractures, which directly influences treatment planning and surgical interventions. Quantifying the inter-fracture spaces allows for objective assessment of fracture severity, enabling more consistent and standardized clinical decision making. Furthermore, traditional diagnostic approaches rely heavily on subjective interpretation by clinicians, which can introduce variability. Automated segmentation minimizes such inconsistencies, providing reproducible and reliable measurements, which are critical for evaluating patient outcomes. In addition, quantification of fracture spaces can assist in predicting post-surgical complications and monitoring healing progress, thereby improving long-term patient management [15]. This limitation is due to the segmentation in medical imaging traditionally applied to organs or lesions, where the shape and contour of the target are intuitively distinguishable. One such method is the graph cut algorithm, a popular rule-based segmentation approach which formulates segmentation as a graph optimization problem, where pixels or voxels are represented as nodes in a graph and edges define the likelihood of connectivity between nodes based on image intensity or other features [16]. While graph cut methods have been successful in segmenting well-defined structures in medical images [17], their effectiveness diminishes when applied to more complex tasks, like identifying spaces between fractured bones due to the irregular shapes and varying intensities of such regions. Similarly, segmentation of COVID-19-infected regions in chest CT images has also been proposed [18; 19; 20], but delineating objects with unclear boundaries remains a significant challenge. Moreover, despite the development of patented [21] algorithms tailored to medical imaging, such as methods specifically designed for enhancing segmentation of soft tissues or detecting tumors, these approaches still face significant challenges when applied to fracture-related spatial analysis. From a different perspective, the Dice similarity coefficient, a common metric for evaluating segmentation performance, is highly sensitive to the volume of the segmented region.