Mandibular third molar (MTM) impaction is a significant global health concern, representing one of the most common developmental anomalies encountered in dental practice [1,2]. The condition, defined as the failure of a tooth to erupt into its normal functional position, is associated with a wide range of clinical pathologies, including pericoronitis, odontogenic cysts and tumors, distal caries in adjacent second molars, and root resorption [1,2]. The global prevalence of MTM impaction is substantial, though estimates vary. A 2016 systematic review and meta-analysis reported a worldwide prevalence of 24.4% [3], while a more recent 2024 meta-analysis involving a larger dataset found a higher prevalence of 36.9% per subject [4]. This discrepancy underscores the dynamic nature of MTM epidemiology and reinforces the need for population-specific data to guide clinical practice. Historically, the assessment of impacted MTMs has relied on descriptive classification systems. Foundational work by winter introduced a classification based on the angulation of the third molar relative to the second molar (e.g., mesioangular, distoangular, vertical, horizontal) [5]. Subsequently, Pell and Gregory developed a system to classify impaction based on the spatial relationship of the tooth to the ascending ramus of the mandible and its depth relative to the occlusal plane of the second molar [6]. While these systems are invaluable for describing an existing impaction and planning for surgical removal, they are inherently retrospective. The contemporary clinical challenge has shifted from merely describing impaction to proactively predicting the eruption potential of a developing MTM. Early prediction is critical for transitioning from reactive treatment to preventive management. This has led to the exploration of quantitative radiographic indices that can objectively measure the anatomical factors governing eruption. A key determinant of MTM eruption is the adequacy of the retromolar space—the area available between the distal surface of the second molar and the anterior border of the mandibular ramus [7]. Seminal research demonstrated the prognostic value of a simple space-to-width ratio, finding that a ratio of available space to tooth width of ≥1 was associated with a high probability of successful eruption [8]. This concept has been refined into various indices, such as the R1 and R2 ratios, which offer a simple, reproducible method for quantifying eruption potential using standard panoramic radiographs [1,8]. However, the predictive accuracy and optimal cut-off values for these indices may vary across different ethnic populations. This study, therefore, aimed to evaluate the diagnostic performance of the space-width ratios (R1 and R2) and to define precise, evidence-based cut-off values that can be readily integrated into clinical decision-making.

Study design and setting This cross-sectional analytical study was conducted at the Department of Oral Medicine and Radiology at NIMS Dental College & Hospital and Rajasthan Dental College, Jaipur, India. Data were collected from patients attending the outpatient departments between January 2022 and January 2024. Ethical considerations The study was conducted in full accordance with the ethical principles outlined in the Declaration of Helsinki. Ethical approval was obtained from the Institutional Ethics Committee of NIMS University. All participants were informed about the study's purpose and procedures, and written informed consent was obtained from each individual prior to their inclusion in the study. Participant selection A total of 350 subjects were enrolled in the study based on a purposive sampling method. Inclusion criteria: • Individuals aged between 18 and 40 years, of both genders. • Presence of one or more mandibular third molars, either erupted or impacted. • Willingness to participate and provide voluntary informed consent. Exclusion criteria: • History of previous orthodontic treatment. • History of surgical extraction of mandibular first or second molars. • Presence of any maxillofacial pathological conditions, such as cysts, tumors, or fractures. • Known developmental anomalies or syndromes affecting craniofacial growth. • Pregnancy. Sample From the 350 enrolled subjects, a total of 520 mandibular third molars were evaluated and divided into two groups of equal size: an impaction group (n = 260) and an eruption group (n = 260). Radiographic protocol and measurements All participants underwent standardized digital panoramic radiography (SIRONA Orthophos XG or Carestream 8100; 68 kVp, 11 mA, 18 seconds). The following parameters were measured on each panoramic radiograph [1,8]: • Mesiodistal Width (MDW): The maximum mesiodistal diameter of the MTM crown. • Lower Eruption Space–Ramus (LES-R): The linear distance from the distal surface of the mandibular second molar to the anterior border of the ascending ramus, measured along the occlusal plane. • Lower Eruption Space–Xi point (LES-Xi): The linear distance from the distal surface of the mandibular second molar to the Xi point (geometric center of the ramus). • Space-Width Ratio 1 (R1): R1=LES−R/MDW. • Space-Width Ratio 2 (R2): R2=LES−Xi/MDW. Outcome definition An MTM was "erupted" if in full functional occlusion and "impacted" if its eruption path was obstructed. The pattern of impaction was categorized according to winter’s classification [5]. Statistical analysis Data were analyzed using SPSS Statistics for Windows, Version 22.0 (IBM Corp., Armonk, NY). Independent samples t-tests, chi-square tests, and receiver operating characteristic (ROC) curve analysis were used. A p-value of less than 0.05 was considered statistically significant.

Demographics and basic characteristics The study included 350 subjects (220 males, 130 females). The highest frequency of impaction was in the 20–25-year age group (44.62%) (Table 1). Males constituted the majority in both the impaction (63.5%) and eruption (75.8%) groups. Bilateral presentation was common (Table 2). Table 1: Age-wise distribution of subjects by impaction and eruption status. Age Group (years) Impaction (n=260) Eruption (n=260) n (%) n (%) 18-20 85 (32.69%) 102 (39.23%) 20-25 116 (44.62%) 98 (37.69%) 25-30 48 (18.46%) 45 (17.31%) >30 11 (4.23%) 15 (5.77%) Table 2: Gender and side distribution of MTMs by impaction and eruption status. Characteristic Impaction (n=260) Eruption (n=260) n (%) n (%) Gender Male 165 (63.5%) 197 (75.8%) Female 95 (36.5%) 63 (24.2%) Side Right 128 (49.2%) 133 (51.2%) Left 132 (50.8%) 127 (48.8%) Pattern of impaction Mesioangular impaction was the most prevalent type (37.3%), followed by distoangular (33.8%) (Table 3). No significant association was found between impaction pattern and age, gender, or side (p>0.05). Table 3: Frequency and pattern of MTM impaction according to Winter's classification. Impaction Pattern Frequency (n) Percentage (%) Mesioangular 97 37.3% Distoangular 88 33.8% Vertical 60 23.1% Horizontal 15 5.8% Total 260 100.0% Key panoramic measurements Statistically significant differences were observed between the impacted and erupted groups for all measured parameters (Table 4). The mean values for LES-R, LES-Xi, R1, and R2 were significantly higher in the erupted group. The mean MDW was significantly larger in the impacted group. Table 4: Comparative analysis of panoramic measurements between impacted and erupted MTMs. Parameter Impaction Group (n=260) Eruption Group (n=260) p-value Mean ± SD Mean ± SD MDW (mm) 10.93 ± 1.06 10.56 ± 1.20 <0.001 LES-R (mm) 16.30 ± 2.13 19.66 ± 2.41 <0.001 LES-Xi (mm) 25.94 ± 2.49 28.63 ± 3.35 <0.001 R1 1.52 ± 0.23 1.87 ± 0.25 <0.001 R2 2.38 ± 0.27 2.75 ± 0.37 <0.001 ROC analysis and diagnostic performance ROC analysis results are presented in Table 5 and Figure 1. The ratio R1 demonstrated the best balance of sensitivity and specificity, yielding the highest overall diagnostic accuracy (78.3%). The linear measurement LES-R achieved the highest specificity (91.1%). Table 5: Diagnostic performance of predictive measurements from ROC analysis. Measurement AUC Cut-off Value Sensitivity (%) Specificity (%) Accuracy (%) R1 0.847 1.67 84.3 72.2 78.3 LES-R 0.858 19.45 mm 61.2 91.1 76.1 LES-Xi 0.746 27.25 mm 72.9 69.5 71.2 R2 0.758 2.57 67.1 72.2 69.7

This study provides robust evidence supporting the use of quantitative panoramic indices for predicting MTM eruption outcomes. The space-width ratio R1 emerged as the most accurate overall predictor, while the linear measurement LES-R offered exceptional specificity, making them clinically complementary tools. Interpretation and comparison with existing literature The demographic and impaction patterns in our cohort align with global trends. The peak impaction incidence in the 20–25-year age group is consistent with the MTM developmental timeline, as eruption has typically completed or failed by this stage [9]. Our finding that mesioangular impaction is the most common pattern (37.3%) is in agreement with a large-scale systematic review which reported a global prevalence of 41.17% for mesioangular impactions [3]. Similarly, the lack of a significant association between impaction patterns and gender or side corroborates the findings of that same meta-analysis [3]. The core contribution of this study is the validation and establishment of predictive cut-off values. The finding that a larger MTM crown width (MDW) is significantly associated with impaction reinforces the logical premise that a larger tooth requires more space. The derived indices—R1 and LES-R—provide a standardized method for evaluating this critical relationship. The cut-off values determined in this study can be contextualized by comparing them with findings from other regional populations (Table 6). Table 6: Comparative Analysis of Predictive Cut-Off Values for MTM Impaction from Regional Studies. Study Population Index Cut-off Value for Impaction Risk Present Study (2024) Indian (Jaipur) R1 < 1.67 LES-R < 19.45 mm Asha & Shrestha (2020) [1] Nepalese R1 < 1.31 LES-R < 18.70 mm Qamruddin et al. (2012) [10] Pakistani Space/Width Ratio < 1.0 AER-7 (similar to LES-R) < 13 mm This comparison reveals both consistencies and variations. All studies affirm that a smaller space-to-width ratio is a primary risk factor. However, specific cut-off values differ, highlighting the importance of population-specific reference values. The general principle that a ratio below 1 indicates a high likelihood of impaction is supported, as our mean R1 for the impacted group was 1.52, well below the mean of 1.87 for the erupted group [8]. Clinical and practical implications The findings have direct clinical implications. The strength of the R1 and LES-R indices lies in their practicality. Panoramic radiography is a ubiquitous, low-cost, and low-radiation imaging modality [11]. Clinicians can easily perform the measurements and calculate the R1 ratio, providing an objective, evidence-based adjunct to subjective judgment. Based on our results, a two-step diagnostic process can be proposed. The R1 ratio serves as an excellent primary screening tool. An R1 value of <1.67 should raise a significant red flag for impaction risk. In borderline cases, the LES-R measurement becomes highly valuable. Due to its exceptional specificity (91.1%), an LES-R value of <19.45 mm acts as a powerful confirmatory test. This approach enables more precise patient counseling and timely intervention planning. Strengths and limitations The primary strength of this study is its large sample size (520 MTMs). However, its cross-sectional design does not allow for longitudinal tracking. Second, the study relies on 2D panoramic radiography, which is subject to magnification, distortion, and superimposition of structures [11]. It cannot fully capture complex buccolingual or root anatomy. In high-risk cases, three-dimensional imaging with cone-beam computed tomography (CBCT) remains the gold standard for detailed surgical planning [11,12]. Recommendations for future research Future research should include multi-center, longitudinal studies to validate these indices across wider populations. Incorporating 3D imaging like CBCT would allow for more sophisticated predictive models. Furthermore, investigations should aim to integrate these anatomical predictors with other potential factors, such as genetics and functional occlusal forces, to create a more comprehensive risk model.

The space-width ratio R1, derived from standard panoramic radiographs, serves as a robust and accurate predictor for MTM eruption potential. With a validated cut-off of 1.67, this index provides a superior balance of sensitivity and specificity. The linear measurement LES-R, with a cut-off of 19.45 mm, acts as a highly specific adjunct. The routine integration of these simple, objective indices into clinical workflows can empower clinicians to make earlier, evidence-based decisions, moving the paradigm from reactive treatment to proactive risk mitigation and reducing the significant morbidity associated with impacted MTMs. Disclosures Conflicts of interest In compliance with the ICMJE uniform disclosure form, all authors declare no financial relationships with any organizations that might have an interest in the submitted work in the preceding three years and no other relationships or activities that could appear to have influenced the submitted work.