Allometric Scaling of Calcaneal Bone Stiffness in Young Athletes: An Exploratory Study of a Size-Free Index

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However, conventional indicators such as Bone Mineral Density (BMD) and the Osteo Sono-Assessment Index (OSI) are influenced by body size, limiting comparisons between sexes and sports disciplines. This study aimed to develop and validate a body size–independent indicator of calcaneal stiffness using allometric scaling in young athletes. Methods: This cross-sectional study included 295 athletes aged 10–30 years (153 males, 142 females). OSI was measured using quantitative ultrasound, and lean body mass (LBM) was assessed by bioelectrical impedance analysis. Using the scaling exponent (b = 0.199) derived from the log–log regression between OSI and LBM, the size-free OSI (sf_OSI) was calculated as sf_OSI = ln(OSI) − 0.199 × ln(LBM). Sex and sport differences were examined using Welch’s t-test and linear models, with muscle strength evaluated by including knee extension torque as a covariate. Results: Females showed significantly higher OSI values than males (3.31 ± 0.42 vs. 3.18 ± 0.42, p = 0.009), with an even difference for sf_OSI (0.45 ± 0.12 vs. 0.36 ± 0.11, p < 0.001, Hedges’ g = 0.79). Sex (η² = 0.036) and age group (η² = 0.152) exerted independent effects. Sport-specific analysis showed greater female predominance in Court-Jump (+10.3%) and Field-Sprint (+9.5%) sports, with no significant difference in combat sports. Adjustment for knee extension torque reduced the sex difference by up to 13% (ΔR² ≤ 0.04). Conclusions: sf_OSI appears to be a body size–adjusted indicator of calcaneal stiffness that clearly delineates sex- and sport-specific patterns. Although it demonstrated exploratory utility as a monitoring metric, direct causal or fracture risk associations were not established. sf_OSI may serve as a practical, noninvasive tool for monitoring sport-specific bone adaptation and stress fracture risk in field settings. In this study, females consistently exhibited higher sf_OSI values, particularly in Court-Jump and Field-Sprint sports. These sex differences were not fully explained by muscle strength, suggesting that sf_OSI is a useful parameter for evaluating sport-specific bone adaptation and stress fracture risk in young athletes. OSI bone strength allometry athletes sex difference Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction Bone health is essential for maintaining athletic performance and preventing stress fractures. During growth periods and among female athletes, maintaining an appropriate balance between bone mass development and energy availability is crucial. A recent International Olympic Committee (IOC) consensus statement identified Relative Energy Deficiency in Sport (REDs) as a major factor contributing to impaired bone development and reduced bone strength, emphasizing the need for standardized bone assessment using dual-energy X-ray absorptiometry (DXA) [ 1 , 2 ]. Although allometric scaling has gained increasing attention in physiology and biomechanics, its application to bone ultrasound metrics remains limited. However, DXA-based measurement of bone mineral density (BMD) is limited by high cost and radiation exposure, limiting its suitability for routine field screening. Quantitative ultrasound (QUS) is a simple, noninvasive technique widely used for bone assessment [ 3 – 6 ]. The Osteo Sono-Assessment Index (OSI) obtained from QUS reflects bone mechanical properties and has been applied to assess bone development during adolescence and early adulthood, and to predict fracture risk. A comprehensive understanding of bone health requires quantifying bone mass and evaluating the efficiency of bone adaptation to mechanical loading and strength development [ 7 ]. Based on this concept, allometric adjustment using lean body mass (LBM) has been proposed to derive body size–independent indices [ 8 , 9 ]. This approach minimizes the confounding effects of body size, enabling more accurate comparisons between sexes and sports. Moreover, previous studies have reported associations between habitual mechanical loading or impact frequency and calcaneal QUS parameters [ 10 ], suggesting that bone strength development depends on daily physical loading. Nevertheless, most previous studies have focused on specific sports or age groups, and few have compared sex- and sports-specific characteristics in a cross-sectional design. Furthermore, residual sex differences and sports-specific bone adaptations that persist after adjusting for body size and muscle strength remain sufficiently elucidated. Recent evidence indicates that athletes participating in high-impact, multidirectional sports have 5–30% greater bone mass or strength than those in low-impact sports [ 11 ]; however, the extent to which these effects relate to sex or sport characteristics remains unclear. This study aimed to investigate sex- and sports-related differences in relative calcaneal stiffness among young athletes using the size-free OSI (sf_OSI) derived from allometric adjustment. Additionally, the study explored the extent to which these sex differences could be explained by body composition–related factors, such as muscle strength, thereby providing insight into the structural and functional basis of bone adaptation. Methods Participants and data collection This study used data obtained from routine medical checkups conducted among representative athletes participating in the National Sports Festival (a nationwide multi-sport competition in Japan) in Wakayama Prefecture. A total of 980 athletes (414 males and 566 females, aged 12–66 years) who underwent medical evaluations between January 2012 and February 2023 were initially enrolled. For individuals with multiple entries, only the first record was retained. Participants aged 31 years were excluded, resulting in 666 eligible participants (303 males and 363 females). After excluding those with missing data for OSI, height, weight, LBM, or age, 446 participants (213 males and 233 females) remained. Subsequently, athletes who participated in predefined high-impact sports were selected (n = 315; 163 males and 152 females). After excluding unclassified sports, 295 athletes (153 males and 142 females) were included in the final analysis. The study protocol was approved by the Ethics Committee of Wakayama Medical University (Approval No. 4042). Written informed consent was obtained from all participants. Written consent was obtained from both the athletes and their parents or guardians for those younger than 18 years of age. All data were anonymized prior to analysis. Measurements Basic attributes: Sex, date of birth, and sporting discipline were recorded. Sport classification: Based on previous studies [ 12 , 13 ], high-impact sports were categorized into four groups. Court-Jump : basketball, handball, and volleyball Field-Sprint : track and field (short- and middle-distance), soccer, and rugby Racket-Strike : fencing, tennis, soft tennis, baseball, and badminton Combat kendo, judo, boxing, wrestling, karate, sumo, and taekwondo Body composition: LBM (kg) was measured using a multifrequency bioelectrical impedance analyzer (TANITA BC-118E, Tanita Corp., Japan). The Standard mode was used for teenage athletes, and the Athlete mode was used for athletes in their twenties. All measurements were performed in the morning under both fasting and resting conditions. Muscle strength: The isokinetic knee extension peak torque (60°/s) of the right leg was measured using the BIODEX System 4 (Biodex Medical Systems, USA). The maximum torque was divided by the body mass to obtain relative muscle strength (Nm/kg). Bone strength: Calcaneal bone stiffness was assessed using an ultrasound bone densitometer (AOS-100SA; Aloka, Japan). The right heel was used to eliminate the influence of leg dominance. Measurements were performed at approximately the same time of day whenever possible to minimize diurnal variation [ 4 – 6 ]. Statistical analysis Continuous variables are presented as the mean ± standard deviation (SD). Sex differences were examined using Welch’s t-test, and effect sizes were expressed as Hedges’ g [ 14 ]. For allometric adjustment, the scaling exponent b = 0.199 (95% CI: 0.132–0.266), derived from a log–log ordinary least squares (OLS) regression between the OSI and LBM, was used [ 8 , 9 ]. The sf_OSI was calculated as: sf_OSI = ln(OSI) − 0.199 × ln(LBM). Sensitivity analyses were conducted using Deming regression (error ratio = 1) [ 15 ] and the zero-correlation method, both yielding comparable results (Supplementary Table S1 ). The effect of the LBM measurement mode (standards/athletes) was not significant and had minimal influence on the main analysis. Pearson’s product–moment correlation coefficients were calculated for the relationships between raw OSI and LBM and between sf_OSI and ln(LBM). Two-way analysis of variance (ANOVA) was used to examine the effects of sex and age group (teen = ≤ 19 years, adult = ≥ 20 years) on sf_OSI, with partial η² representing effect size. Differences among sports categories were analyzed using linear models adjusted for age and measurement mode, and least-squares means were computed. Multiple comparisons were corrected using the Benjamini–Hochberg procedure to control the false discovery rate (FDR) [ 16 ]. Furthermore, knee extension torque (absolute, body mass–normalized, and LBM-normalized values) was sequentially added to the models to evaluate the attenuation of sex differences, changes in Akaike’s Information Criterion (ΔAIC), and adjusted determination coefficients (ΔR²). All statistical analyses were performed using EZR ( graphical interface for R version 4.2.2) [ 17 ], with a two-tailed significance level set at p < 0.05. Result 1. Descriptive characteristics The participants’ characteristics are summarized in Table 1 . Table 1 Participant characteristics by gender Variable Female (n = 142) Male (n = 153) p-value Hedges’ g Age (years) 17.2 ± 4.5 17.5 ± 5.0 0.639 — Height (cm) 160.9 ± 6.4 170.9 ± 7.7 < 0.001 — Body mass (kg) 52.3 ± 8.2 60.6 ± 13.0 < 0.001 — Body mass index (kg/m 2 ) 20.1 ± 2.3 21.1 ± 3.0 < 0.001 — Body fat (%) 22.3 ± 4.6 13.0 ± 4.0 < 0.001 — Lean body mass (kg) 40.4 ± 5.3 52.4 ± 10.1 < 0.001 — OSI 3.31 ± 0.42 3.18 ± 0.42 0.009 0.31 Size-free OSI (sf_OSI) 0.45 ± 0.12 0.36 ± 0.11 < 0.001 0.79 Knee ext torque (Nm) 127.0 ± 26.8 172.9 ± 50.0 < 0.001 — Knee torque/body mass (Nm/kg) 2.43 ± 0.33 2.84 ± 0.46 < 0.001 — Knee torque/lean body mass (Nm/kg) 3.13 ± 0.43 3.26 ± 0.54 0.022 — Abbreviations: OSI, osteo-sono-assessment index; LBM, lean body mass. Note : Values are presented as mean ± standard deviation. Sex differences were examined using Welch’s t-test, and Hedges’ g was calculated for the OSI and sf_OSI. sf_OSI was calculated as ln(OSI) − 0.199 × ln(LBM). Males had significantly greater height, body mass, lean body mass (LBM), and knee extension torque than females, whereas females had a higher body fat percentage. The OSI values were significantly higher in females than in males ( p = 0.009), and the sex difference persisted after adjusting for body size using the sf_OSI ( p < 0.001; Hedges’ g = 0.79). 2. Scaling validation The scaling exponent obtained from the log–log ordinary least squares (OLS) regression between OSI and LBM was b = 0.199 (95% confidence interval [CI]: 0.132–0.266), demonstrating a positive allometric relationship (Fig. 1 A). After allometric adjustment, no significant correlation was observed between sf_OSI and ln(LBM), indicating that the dependence on body size was effectively removed (Fig. 1 B). Sensitivity analyses using Deming regression (b = 0.292) and the zero-correlation method (b = 0.199) yielded comparable estimates, and the main conclusions were consistent across methods (Supplementary Table S1 ). The correlation profile used to determine the optimal scaling exponent based on the zero-correlation method is shown in Supplementary Figure S1 . 3. Sex/age differences Two-way ANOVA revealed significant effects of sex ( p < 0.01, Hedges’ g = 0.79) and age ( p < 0.001) on sf_OSI, whereas the sex × age interaction was not significant ( p = 0.22) (Table 3 , Fig. 2 ). Table 3 Regression models for size-free OSI (sf_OSI) Model Covariates βSexF (95% CI, p) βStrength (95% CI, p) βln(LBM) (95% CI, p) Adj. R² ΔR² AIC M0 Gender 0.090 (0.064–0.116), < 0.001 – – 0.134 – −445.846 M1 +Age (cubic polynomial) – – – 0.245 0.111 −483.404 M2 +Strength (Knee ext torque) 0.103 (0.073–0.133), < 0.001 0.00035 (− 0.00002–0.00072), 0.064 – 0.252 0.006 −484.905 M2 + LBM +ln(LBM) (sensitivity) 0.095 (0.059–0.132), < 0.001 0.00047 (− 0.00004–0.00098), 0.069 −0.054 (− 0.204–0.097), 0.482 0.250 −0.001 – Dependent variable: size-free OSI (sf_OSI). sf_OSI was defined as ln(OSI) − b ·ln(LBM), with b = 0.199 (95% CI: 0.132–0.266). M0–M2 represent the primary models that by design do not include ln(LBM). The M2 + LBM model was a sensitivity analysis; inclusion of ln(LBM) slightly reduced the adjusted R² (Δ = −0.001), and ln(LBM) was not significant ( p = 0.482). Females consistently showed higher sf_OSI values (β ≈ 0.09–0.10, p < 0.001). Knee extension strength exhibited a weak positive trend ( p ≈ 0.06–0.07). These results confirm that sf_OSI was adequately adjusted for body size, allowing robust inference across models. Post hoc comparisons showed that female teenagers had significantly higher sf_OSI than male teenagers, and female adults also exhibited higher sf_OSI than male adults (both FDR-adjusted p < 0.05). The effect size for age (partial η² = 0.152) was larger than that for sex (partial η² = 0.036). 4. Sport differences Across all sports categories, females exhibited higher sf_OSI values than males (Table 2 , Fig. 3 ). Table 2 LS-means of size-free OSI (sf_OSI, b = 0.199) by sport and gender before adjustment for muscle strength Sportstype Contrast Estimate SE 95% CI (Lower, Upper) % Difference p-value p_adj (BH) Combat Female − Male 0.067 0.042 −0.015, 0.149 + 6.9% 0.109 0.109 Court-Jump Female − Male 0.098 0.024 0.051, 0.146 + 10.3% < 0.001 < 0.001 Field-Sprint Female − Male 0.090 0.025 0.041, 0.140 + 9.5% < 0.001 < 0.001 Racket-Strike Female − Male 0.062 0.020 0.022, 0.102 + 6.4% 0.002 0.003 Note: Values represent least-squares mean differences (female–male) in size-free OSI (sf_OSI, b = 0.199) adjusted for age and Tanita measurement mode. Positive values indicate higher sf_OSI in females.Percentage differences were calculated as 100 × (exp(Estimate) − 1). p -values were corrected for multiple comparisons using the Benjamini–Hochberg false discovery rate (FDR) method, and the gender × sport interaction was not significant (Type II ANOVA, p = 0.642); therefore, sport-specific comparisons are presented as descriptive results and should be interpreted with caution, as shown in Fig. 3 (forest plot), in which effect sizes and 95% confidence intervals are illustrated. Sex differences were most pronounced in Court-Jump (+ 10.3%) and Field-Sprint (+ 9.5%) sports (both p (adj) < 0.001), whereas no significant difference was observed in Combat sports ( p (adj) = 0.109). Linear model analysis identified significant main effects of sex ( p < 0.001), sports category ( p < 0.001), and age ( p < 0.001), whereas the measurement mode was not significant ( p = 0.88). The sex × sport interaction was also non-significant ( p = 0.65), indicating no statistically meaningful differences in the magnitude of sex effects across sports. For reference, the sports- and sex-specific percentiles of sf_OSI are presented in Supplementary Table S4. 5. Muscle strength adjustment The inclusion of knee extension peak torque at 60°/s in the regression model attenuated the sex difference in sf_OSI by up to 13%, with only a minimal increase in explanatory power (ΔR² ≤ 0.04). These findings indicate that the contribution of muscle strength to the observed sex differences in sf_OSI is limited. The change in explanatory power (ΔR²) after adding muscle strength to the sf_OSI models across sport categories is shown in Fig. 4 . Discussion This study is among the first to examine sex- and sports-specific differences in young athletes using a relative calcaneal stiffness index adjusted for body size (sf_OSI). The key findings are summarized as follows: First, females consistently demonstrated higher sf_OSI values than males. Second, this sex difference was particularly pronounced in Court-Jump and Field-Sprint sports, whereas it was smaller in combat sports. Third, the sex difference persisted even after adjustment for knee extension muscle strength. Consistent with established knowledge on body composition differences, female athletes in this study exhibited lower LBM than males, consistent with previous research showing greater skeletal muscle mass and LBM in men in both the general and athletic populations [ 18 , 19 ]. Nevertheless, the size-adjusted sf_OSI revealed a higher relative calcaneal stiffness in females. This finding indicates that sex differences in bone strength cannot be fully explained by muscle mass alone, suggesting that structural adaptations driven by sports-specific mechanical loading—such as frequent and multidirectional impacts—may independently contribute to bone strength [ 11 , 20 ]. Indeed, peripheral quantitative computed tomography (pQCT) studies in female athletes have shown that muscle-derived joint moments and loading modalities, categorized as impact or odd-impact, are associated with differences in bone geometry [ 11 ]. These findings are consistent with our observations. From a methodological perspective, this study demonstrated the utility of allometric adjustment using a scaling exponent derived from log–log regression to minimize body size dependence inherent in traditional BMD or unadjusted QUS indices. The near-zero correlation between sf_OSI and ln(LBM) confirmed the effective removal of the body size bias. Furthermore, the sequential inclusion of knee extension torque reduced the sex difference by up to 13%, with minimal improvement in model fit. These findings indicate that neither muscle mass nor simple torque measurements sufficiently explain the higher relative bone stiffness observed in females. This pattern also aligns with mechanostat theory [ 21 ], which posits that impact and multidirectional loading can stimulate structural bone adaptation beyond the influence of muscle-derived forces. Additionally, the hormonal milieu, bone geometry, and movement characteristics specific to adolescence and young adulthood, such as landing and cutting maneuvers, may have contributed to these sex differences. Future studies should incorporate direct measures of mechanical loading, such as ground reaction forces, jump tests, and accelerometer-based impact frequency analyses, to elucidate these mechanisms more clearly. Regarding sport specificity, the relative sex differences were largest in Court-Jump and Field-Sprint sports and smallest in combat sports. Although the interaction was not statistically significant and should therefore be interpreted cautiously, the trend toward higher relative bone stiffness in sports involving jumping and multidirectional impact is consistent with findings from intervention studies and meta-analyses [ 22 , 23 ]. In contrast, combat sports often involve endurance-based training and rapid weight reduction, which increases the risk of stress fractures, particularly in female athletes [ 23 , 24 ]. Thus, future studies should consider risk stratification that integrates sport-specific loading characteristics with nutritional and weight-management factors. The sf_OSI is a useful indicator for visualizing sex- and sports-specific differences in bone properties. QUS is noninvasive, radiation-free, inexpensive, and portable, making it easily applicable in school-based health examinations and athletic training environments. From a clinical perspective, combining QUS with body composition analysis enables straightforward onsite assessment of both bone and muscle status, facilitating early screening for stress fracture risk and relative energy deficiency in sports (REDs) [ 5 , 6 , 25 ] among adolescent athletes. The sf_OSI’s independence from body size enables individualized risk assessment that considers personal physique- and sports-specific characteristics, thereby facilitating tailored preventive interventions such as nutritional guidance and training adjustments. Thus, QUS-based sf_OSI evaluation represents a safe and practical alternative to radiation-based techniques such as DXA or pQCT, offering a reliable means of repeated bone health monitoring in field settings. However, because this study was cross-sectional, the causal relationship between sf_OSI and fracture risk could not be determined. Future longitudinal studies are warranted to verify the predictive validity of the sf_OSI and establish its clinical and practical utility. Limitation This study has some limitations. First, due to its cross-sectional design, causal inferences cannot be made. Second, the nutritional status, hormonal environment, and biological maturity (e.g., Mirwald [ 26 ] and Moore [ 27 ] methods) were not directly assessed. Third, muscle strength was measured only unilaterally, which may not have fully captured sport-specific asymmetry. Fourth, the integration of multiple sports categories might have introduced within-group heterogeneity. Additionally, standard imaging techniques such as DXA or pQCT were not employed for validation. Future longitudinal and multifaceted studies are warranted to verify the validity of the sf_OSI and establish its clinical and practical utility in both research and field settings. Conclusion This study demonstrated that the size-free Osteo Sono-Assessment Index (sf_OSI), derived through allometric adjustment, can serve as an exploratory indicator of relative calcaneal stiffness, independent of body size. Females consistently exhibited higher sf_OSI values with sport-specific patterns observed; however, these sex differences could not be fully explained by muscle mass or knee extension strength. The sf_OSI offers a novel framework for assessing the qualitative aspects of bone adaptation and may be applicable to bone health monitoring and stress fracture risk assessment in young athletes. Nevertheless, because this study was cross-sectional, causal relationships remain uncertain, and maturity status and mechanical loading exposure were not sufficiently evaluated, leaving the possibility of residual confounding. The sf_OSI provides a new framework for evaluating relative bone stiffness in athletes. Future prospective studies should determine its predictive value for stress fractures and its usability in sports rehabilitation programs. List of abbreviations AIC, Akaike’s Information Criterion; ANOVA, analysis of variance; BMD, bone mineral density; CI, confidence interval; DXA, dual-energy X-ray absorptiometry; FDR, false discovery rate; LBM, lean body mass; OLS, ordinary least squares; OSI, Osteo Sono-Assessment Index; pQCT, peripheral quantitative computed tomography; QUS, quantitative ultrasound; REDs, Relative Energy Deficiency in Sport; SD, standard deviation; sf_OSI, size-free Osteo Sono-Assessment Index. Declarations Ethics approval and consent to participate The study protocol was approved by the Ethics Committee of the Wakayama Medical University (Approval No. 4042) and was conducted in accordance with the principles of the Declaration of Helsinki. Written informed consent was obtained from all participants, and for those under 18 years of age, consent was also obtained from their parents or legal guardians. Availability of data and materials The datasets generated and/or analyzed during the current study contain personal information and are therefore not publicly available. The data are securely stored at Wakayama Medical University and are available from the corresponding author on reasonable request. Competing interests The authors declare no conflict of interest. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Author’s contributions RN conceived and designed the study, analyzed and interpreted the data, and drafted the manuscript. KG contributed to data interpretation and critically revised the manuscript. MB and KK supervised the study and provided critical feedback on the manuscript. All authors read and approved the final version of the manuscript. Acknowledgement We thank the staff of the Genki Development Institute and the Satellite Clinic and the athletes for their cooperation in this study. References Mountjoy M, Ackerman KE, Bailey DM, Burke LM, Constantini N, Hackney AC, et al. 2023 International Olympic Committee’s (IOC) consensus statement on Relative Energy Deficiency in Sport (REDs). Br J Sports Med. 2023;57:1073–97. 10.1136/bjsports-2023-106994 . Dallman J, Herda A, Cleary CJ, Morey T, Diederich A, Vopat BG, et al. A brief review of the literature for published dual-energy x-ray absorptiometry protocols for athletes. Sports Health. 2024;16:735–43. 10.1177/19417381231208204 . De Souza MJ, Nattiv A, Joy E, Misra M, Williams NI, Mallinson RJ, et al. Female athlete triad coalition consensus statement expert panel. 2014 female athlete triad coalition consensus statement on treatment and return to play of the female athlete triad. Br J Sports Med. 2014;48:289. 10.1136/bjsports-2013-093218 . Alwis G, Rosengren B, Nilsson JA, Stenevi-Lundgren S, Sundberg M, Sernbo I, et al. Normative calcaneal quantitative ultrasound data as an estimation of skeletal development in Swedish children and adolescents. Calcif Tissue Int. 2010;87:493–506. Li C, Sun J, Yu L. Diagnostic value of calcaneal quantitative ultrasound in the evaluation of osteoporosis in middle-aged and elderly patients. Med (Baltim). 2022;101:e28325. 10.1097/MD.0000000000028325 . Métrailler A, Hans D, Lamy O, Gonzalez Rodriguez E, Shevroja E. Heel quantitative ultrasound (QUS) predicts incident fractures independently of trabecular bone score (TBS), bone mineral density (BMD), and FRAX: the OsteoLaus Study. Osteoporos Int. 2023;34:1401–9. 10.1007/s00198-023-06728-4 . Daly RM. The effect of exercise on bone mass and structural geometry during growth. Med Sport Sci. 2007;51:33–49. 10.1159/000103003 . Warton DI, Wright IJ, Falster DS, Westoby M. Bivariate line-fitting methods for allometry. Biol Rev Camb Philos Soc. 2006;81:259–91. 10.1017/S1464793106007007 . Packard GC. On the use of logarithmic transformations in allometric analyses. J Theor Biol. 2009;257:515–8. 10.1016/j.jtbi.2008.10.016 . discussion 519 – 21. Patel H, Woods L, Teesdale-Spittle P, Dennison E. A cross-sectional study of the relationship between recreational sporting activity and calcaneal bone density in adolescents and young adults. Calcif Tissue Int. 2021;109:218–26. 10.1080/00913847.2021.1903819 . Jonvik KL, Torstveit MK, Sundgot-Borgen J, et al. Do we need to change the guideline values for determining low bone mineral density in athletes? J Appl Physiol (1985). 2022;132:1320–2. 10.1152/japplphysiol.00851.2021 . Nikander R, Kannus P, Rantalainen T, Uusi-Rasi K, Heinonen A, Sievänen H. Cross-sectional geometry of weight-bearing tibia in female athletes subjected to different exercise loadings. Osteoporos Int. 2010;21:1687–94. 10.1007/s00198-009-1101-0 . Nikander R, Sievänen H, Uusi-Rasi K, Heinonen A, Kannus P. Loading modalities and bone structures at nonweight-bearing upper extremity and weight-bearing lower extremity: a pQCT study of adult female athletes. Bone. 2006;39:886–94. 10.1016/j.bone.2006.04.005 . Hedges LV, Olkin I. Statistical methods for meta-analysis. Orlando, FL: Academic; 1985. p. 369. Linnet K. Estimation of the linear relationship between the measurements of two methods with proportional errors. Stat Med. 1990;9:1463–73. 10.1002/sim.4780091210 . Benjamini Y, Hochberg Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B. 1995;57:289–300. Kanda Y. Investigation of the freely available easy-to-use software 'EZR' for medical statistics. Bone Marrow Transpl. 2013;48:452–8. 10.1038/bmt.2012.244 . Wang X, Gao L, Xiong J, Cheng H, Liu L, Dong H, et al. The life-course changes in muscle mass using dual-energy x-ray absorptiometry: the China BCL study and the US NHANES study. J Cachexia Sarcopenia Muscle. 2024;15:1687–95. 10.1002/jcsm.13522 . Belzunce MA, Henckel J, Di Laura A, Horga LM, Hart AJ. Gender similarities and differences in skeletal muscle and body composition: an MRI study of recreational cyclists. BMJ Open Sport Exerc Med. 2023;9:e001672. 10.1136/bmjsem-2023-001672 . Nikander R, Sievänen H, Heinonen A, Daly RM, Uusi-Rasi K, Kannus P. Femoral neck structure in adult female athletes subjected to different loading modalities. J Bone Min Res. 2005;20:520–8. 10.1359/JBMR.041130 . Rauch F, Schoenau E. The developing bone: slave or master of its cells and molecules? Pediatr Res. 2001;50:309–14. 10.1203/00006450-200109000-00003 . Miao T, Li X, Zhang W, Yang F, Wang X. Effects of high-impact jumping versus resistance exercise on bone mineral content in children and adolescents: a systematic review and meta-analysis. PeerJ. 2025;13:e19616. 10.7717/peerj.19616 . Tenforde AS, Fredericson M. Influence of sports participation on bone health in the young athlete: a review of the literature. PM R. 2011;3:861–7. 10.1016/j.pmrj.2011.05.019 . Hoenig T, Hollander K, Popp KL, Fredericson M, Kraus EA, Warden SJ, et al. International Delphi consensus on bone stress injuries in athletes. Br J Sports Med. 2025;59:78–90. 10.1136/bjsports-2024-108616 . Moayyeri A, Adams JE, Adler RA, Krieg MA, Hans D, Compston J, et al. Quantitative ultrasound of the heel and fracture risk assessment: an updated meta-analysis. Osteoporos Int. 2012;23:143–53. 10.1007/s00198-011-1817-5 . Mirwald RL, Baxter-Jones AD, Bailey DA, Beunen GP. An assessment of maturity from anthropometric measurements. Med Sci Sports Exerc. 2002;34:689–94. 10.1097/00005768-200204000-00020 . Moore SA, McKay HA, Macdonald H, Nettlefold L, Baxter-Jones AD, Cameron N, et al. Enhancing a somatic maturity prediction model. Med Sci Sports Exerc. 2015;47:1755–64. Additional Declarations No competing interests reported. Supplementary Files FigureS1.docx Supplementary Figure S1. Correlation profile r{sf_OSI(b), In(LNM)} Correlation profile between sf_OSI(b) and ln(LBM) across candidate scaling exponents ( b ). The optimal b value corresponded to the point at which the correlation approached zero (zero-correlation method, b = 0.199). For comparison, the slopes estimated by ordinary least squares (OLS) and Deming regressions (error ratios = 0.5, 1.0, and 2.0) are also shown. All methods yielded similar conclusions, supporting the idea that sf_OSI is independent of body size. Supplementarytable.docx Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 07 Jan, 2026 Editor invited by journal 10 Dec, 2025 Editor assigned by journal 08 Dec, 2025 Submission checks completed at journal 08 Dec, 2025 First submitted to journal 05 Dec, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8285227","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":571083315,"identity":"fcc070bc-90ca-4c6f-9a4d-0243e9a929bd","order_by":0,"name":"Rei Nemoto","email":"","orcid":"","institution":"Wakayama Medical University","correspondingAuthor":false,"prefix":"","firstName":"Rei","middleName":"","lastName":"Nemoto","suffix":""},{"id":571083319,"identity":"63677907-89fd-45f9-bef5-168d1169771d","order_by":1,"name":"Kenji Goto","email":"","orcid":"","institution":"Wakayama Medical 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16:01:25","extension":"html","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":133504,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/7b81a1b48af2028a3aa9a18f.html"},{"id":99824962,"identity":"1a60004b-ce74-47ea-9076-18ac96aad91a","added_by":"auto","created_at":"2026-01-08 16:01:25","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":226755,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRelationship between OSI and LBM before and after allometric adjustment\u003c/strong\u003e\u003cbr\u003e\n \u003cstrong\u003e(A)\u003c/strong\u003e Scatter plot of raw OSI versus LBM by sex. A clear body size dependence and differences in regression slopes between sexes were observed. In females, OSI showed a moderate and significant correlation with LBM (\u003cem\u003er\u003c/em\u003e = 0.484, 95% CI: 0.347–0.601, \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001), and a similar moderate and significant correlation was also observed in males (\u003cem\u003er\u003c/em\u003e = 0.543, 95% CI: 0.420–0.646, \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e(B)\u003c/strong\u003e After allometric adjustment (sf_OSI), the correlation with ln(LBM) disappeared (Pearson’s \u003cem\u003er\u003c/em\u003e = −0.0002, 95% CI: −0.114 to 0.114, \u003cem\u003ep\u003c/em\u003e = 0.998), confirming the effective removal of body size dependence.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/b1396fcfe18c57c3a9cd5196.png"},{"id":100356764,"identity":"03c8e8a3-d09e-4dd2-88b6-e5b980e739e2","added_by":"auto","created_at":"2026-01-16 07:17:24","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":60524,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDistribution of size-free OSI by gender and age group\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDistribution of sf_OSI across four groups defined by sex and age (female teens, male teens, female adults, and male adults). Yellow diamonds indicate group means. Horizontal bars denote significant between-group differences identified in post hoc comparisons (Benjamini–Hochberg FDR correction). (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.05, \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.01, \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001)\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/192719a8294c9705a443796e.png"},{"id":99824985,"identity":"88e17592-c7cc-4d93-a36a-bf5e53ff2393","added_by":"auto","created_at":"2026-01-08 16:01:27","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":66993,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eGender contrast in size-free OSI by sport type\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eForest plot showing sex differences in sf_OSI (\u003cem\u003eb\u003c/em\u003e= 0.199) across sports categories.\u003cbr\u003e\nValues represent least-squares mean differences (female − male) adjusted for age and measurement mode, with corresponding 95% confidence intervals.\u003c/p\u003e\n\u003cp\u003ePercentage differences were calculated as 100 × (exp(Estimate) − 1).\u003c/p\u003e\n\u003cp\u003eBecause the sex × sport interaction was not significant (\u003cem\u003ep\u003c/em\u003e = 0.642), sports-specific comparisons were presented as exploratory descriptive results.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/3defcbc5c9d2a0e7cbb84cc4.png"},{"id":100356817,"identity":"4f5de76c-8786-4c16-9659-1cee994f3964","added_by":"auto","created_at":"2026-01-16 07:17:37","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":79098,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eChange in ΔR² after adding muscle strength\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eChanges in model fit (ΔR²) after inclusion of muscle strength, stratified by sport category.\u003cbr\u003e\nDots indicate estimated values, and horizontal lines represent 95% confidence intervals.\u003c/p\u003e\n\u003cp\u003ePositive values (\u0026gt;0) indicate improvement in model explanatory power.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/36b1f932b92eee117a00d179.png"},{"id":100376810,"identity":"b7763fe0-136b-4215-a5b0-456bf0615823","added_by":"auto","created_at":"2026-01-16 08:45:55","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1283511,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/7595aa8d-a056-441f-a6bc-61a9a137bffa.pdf"},{"id":100356931,"identity":"6bebfb63-d019-4c7e-92a7-70d6b1f914b4","added_by":"auto","created_at":"2026-01-16 07:18:00","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":172534,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSupplementary Figure S1. Correlation profile r{sf_OSI(b), In(LNM)}\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCorrelation profile between sf_OSI(b) and ln(LBM) across candidate scaling exponents (\u003cem\u003eb\u003c/em\u003e).\u003c/p\u003e\n\u003cp\u003eThe optimal \u003cem\u003eb\u003c/em\u003evalue corresponded to the point at which the correlation approached zero (zero-correlation method, \u003cem\u003eb\u003c/em\u003e = 0.199).\u003c/p\u003e\n\u003cp\u003eFor comparison, the slopes estimated by ordinary least squares (OLS) and Deming regressions (error ratios = 0.5, 1.0, and 2.0) are also shown.\u003c/p\u003e\n\u003cp\u003eAll methods yielded similar conclusions, supporting the idea that sf_OSI is independent of body size.\u003c/p\u003e","description":"","filename":"FigureS1.docx","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/ddb0bc29da6a08cd04da6650.docx"},{"id":100356784,"identity":"d53e1796-687b-4ee7-9a88-94588cb78530","added_by":"auto","created_at":"2026-01-16 07:17:27","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":18447,"visible":true,"origin":"","legend":"","description":"","filename":"Supplementarytable.docx","url":"https://assets-eu.researchsquare.com/files/rs-8285227/v1/ef676f0cd4a60f5eeba00abe.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Allometric Scaling of Calcaneal Bone Stiffness in Young Athletes: An Exploratory Study of a Size-Free Index","fulltext":[{"header":"Introduction","content":"\u003cp\u003eBone health is essential for maintaining athletic performance and preventing stress fractures. During growth periods and among female athletes, maintaining an appropriate balance between bone mass development and energy availability is crucial.\u003c/p\u003e \u003cp\u003eA recent International Olympic Committee (IOC) consensus statement identified \u003cem\u003eRelative Energy Deficiency in Sport\u003c/em\u003e (REDs) as a major factor contributing to impaired bone development and reduced bone strength, emphasizing the need for standardized bone assessment using dual-energy X-ray absorptiometry (DXA) [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAlthough allometric scaling has gained increasing attention in physiology and biomechanics, its application to bone ultrasound metrics remains limited. However, DXA-based measurement of bone mineral density (BMD) is limited by high cost and radiation exposure, limiting its suitability for routine field screening. Quantitative ultrasound (QUS) is a simple, noninvasive technique widely used for bone assessment [\u003cspan additionalcitationids=\"CR4 CR5\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. The \u003cem\u003eOsteo Sono-Assessment Index\u003c/em\u003e (OSI) obtained from QUS reflects bone mechanical properties and has been applied to assess bone development during adolescence and early adulthood, and to predict fracture risk.\u003c/p\u003e \u003cp\u003eA comprehensive understanding of bone health requires quantifying bone mass and evaluating the efficiency of bone adaptation to mechanical loading and strength development [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Based on this concept, allometric adjustment using lean body mass (LBM) has been proposed to derive body size\u0026ndash;independent indices [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. This approach minimizes the confounding effects of body size, enabling more accurate comparisons between sexes and sports. Moreover, previous studies have reported associations between habitual mechanical loading or impact frequency and calcaneal QUS parameters [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], suggesting that bone strength development depends on daily physical loading.\u003c/p\u003e \u003cp\u003eNevertheless, most previous studies have focused on specific sports or age groups, and few have compared sex- and sports-specific characteristics in a cross-sectional design. Furthermore, residual sex differences and sports-specific bone adaptations that persist after adjusting for body size and muscle strength remain sufficiently elucidated. Recent evidence indicates that athletes participating in high-impact, multidirectional sports have 5\u0026ndash;30% greater bone mass or strength than those in low-impact sports [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]; however, the extent to which these effects relate to sex or sport characteristics remains unclear.\u003c/p\u003e \u003cp\u003eThis study aimed to investigate sex- and sports-related differences in relative calcaneal stiffness among young athletes using the size-free OSI (sf_OSI) derived from allometric adjustment. Additionally, the study explored the extent to which these sex differences could be explained by body composition\u0026ndash;related factors, such as muscle strength, thereby providing insight into the structural and functional basis of bone adaptation.\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eParticipants and data collection\u003c/h2\u003e \u003cp\u003eThis study used data obtained from routine medical checkups conducted among representative athletes participating in the \u003cem\u003eNational Sports Festival\u003c/em\u003e (a nationwide multi-sport competition in Japan) in Wakayama Prefecture.\u003c/p\u003e \u003cp\u003eA total of 980 athletes (414 males and 566 females, aged 12–66 years) who underwent medical evaluations between January 2012 and February 2023 were initially enrolled. For individuals with multiple entries, only the first record was retained. Participants aged 31 years were excluded, resulting in 666 eligible participants (303 males and 363 females). After excluding those with missing data for OSI, height, weight, LBM, or age, 446 participants (213 males and 233 females) remained. Subsequently, athletes who participated in predefined high-impact sports were selected (n = 315; 163 males and 152 females). After excluding unclassified sports, 295 athletes (153 males and 142 females) were included in the final analysis.\u003c/p\u003e \u003cp\u003e The study protocol was approved by the Ethics Committee of Wakayama Medical University (Approval No. 4042). Written informed consent was obtained from all participants. Written consent was obtained from both the athletes and their parents or guardians for those younger than 18 years of age. All data were anonymized prior to analysis.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eMeasurements\u003c/h3\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eBasic attributes:\u003c/h2\u003e \u003cp\u003eSex, date of birth, and sporting discipline were recorded.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eSport classification:\u003c/h3\u003e\n\u003cp\u003eBased on previous studies [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], high-impact sports were categorized into four groups.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cul\u003e \u003cli\u003e \u003cp\u003e\u003cb\u003eCourt-Jump\u003c/b\u003e: basketball, handball, and volleyball\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e\u003cb\u003eField-Sprint\u003c/b\u003e: track and field (short- and middle-distance), soccer, and rugby\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e\u003cb\u003eRacket-Strike\u003c/b\u003e: fencing, tennis, soft tennis, baseball, and badminton\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eCombat\u003c/strong\u003e \u003c/p\u003e\u003cp\u003ekendo, judo, boxing, wrestling, karate, sumo, and taekwondo\u003c/p\u003e \u003cp\u003e\u003c/p\u003e\n\u003ch3\u003eBody composition:\u003c/h3\u003e\n\u003cp\u003eLBM (kg) was measured using a multifrequency bioelectrical impedance analyzer (TANITA BC-118E, Tanita Corp., Japan). The \u003cem\u003eStandard\u003c/em\u003e mode was used for teenage athletes, and the \u003cem\u003eAthlete\u003c/em\u003e mode was used for athletes in their twenties. All measurements were performed in the morning under both fasting and resting conditions.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eMuscle strength:\u003c/h2\u003e \u003cp\u003eThe isokinetic knee extension peak torque (60°/s) of the right leg was measured using the BIODEX System 4 (Biodex Medical Systems, USA). The maximum torque was divided by the body mass to obtain relative muscle strength (Nm/kg).\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eBone strength:\u003c/h3\u003e\n\u003cp\u003eCalcaneal bone stiffness was assessed using an ultrasound bone densitometer (AOS-100SA; Aloka, Japan). The right heel was used to eliminate the influence of leg dominance. Measurements were performed at approximately the same time of day whenever possible to minimize diurnal variation [\u003cspan additionalcitationids=\"CR5\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e–\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003eStatistical analysis\u003c/h2\u003e \u003cp\u003eContinuous variables are presented as the mean ± standard deviation (SD). Sex differences were examined using Welch’s t-test, and effect sizes were expressed as Hedges’ \u003cem\u003eg\u003c/em\u003e [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFor allometric adjustment, the scaling exponent \u003cem\u003eb\u003c/em\u003e = 0.199 (95% CI: 0.132–0.266), derived from a log–log ordinary least squares (OLS) regression between the OSI and LBM, was used [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe sf_OSI was calculated as: \u003cb\u003esf_OSI = ln(OSI) − 0.199 × ln(LBM).\u003c/b\u003e\u003c/p\u003e \u003cp\u003eSensitivity analyses were conducted using Deming regression (error ratio = 1) [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] and the zero-correlation method, both yielding comparable results (Supplementary Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e). The effect of the LBM measurement mode (standards/athletes) was not significant and had minimal influence on the main analysis.\u003c/p\u003e \u003cp\u003ePearson’s product–moment correlation coefficients were calculated for the relationships between raw OSI and LBM and between sf_OSI and ln(LBM).\u003c/p\u003e \u003cp\u003eTwo-way analysis of variance (ANOVA) was used to examine the effects of sex and age group (teen = ≤ 19 years, adult = ≥ 20 years) on sf_OSI, with partial η² representing effect size.\u003c/p\u003e \u003cp\u003eDifferences among sports categories were analyzed using linear models adjusted for age and measurement mode, and least-squares means were computed. Multiple comparisons were corrected using the Benjamini–Hochberg procedure to control the false discovery rate (FDR) [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFurthermore, knee extension torque (absolute, body mass–normalized, and LBM-normalized values) was sequentially added to the models to evaluate the attenuation of sex differences, changes in Akaike’s Information Criterion (ΔAIC), and adjusted determination coefficients (ΔR²).\u003c/p\u003e \u003cp\u003eAll statistical analyses were performed using EZR ( graphical interface for R version 4.2.2) [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], with a two-tailed significance level set at \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.05.\u003c/p\u003e"},{"header":"Result","content":"\u003cp\u003e \u003cb\u003e1. Descriptive characteristics\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe participants’ characteristics are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"±\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"±\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParticipant characteristics by gender\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVariable\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemale (n = 142)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMale (n = 153)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eHedges’ g\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAge (years)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e17.2 ± 4.5\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e17.5 ± 5.0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.639\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHeight (cm)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e160.9 ± 6.4\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e170.9 ± 7.7\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBody mass (kg)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e52.3 ± 8.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e60.6 ± 13.0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBody mass index (kg/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e20.1 ± 2.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e21.1 ± 3.0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBody fat (%)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e22.3 ± 4.6\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e13.0 ± 4.0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLean body mass (kg)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e40.4 ± 5.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e52.4 ± 10.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOSI\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e3.31 ± 0.42\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e3.18 ± 0.42\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.009\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.31\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSize-free OSI (sf_OSI)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e0.45 ± 0.12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e0.36 ± 0.11\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.79\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKnee ext torque (Nm)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e127.0 ± 26.8\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e172.9 ± 50.0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKnee torque/body mass (Nm/kg)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e2.43 ± 0.33\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e2.84 ± 0.46\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKnee torque/lean body mass (Nm/kg)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c2\"\u003e \u003cp\u003e3.13 ± 0.43\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\"±\" colname=\"c3\"\u003e \u003cp\u003e3.26 ± 0.54\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.022\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e—\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003cstrong\u003eAbbreviations:\u003c/strong\u003e OSI, osteo-sono-assessment index; LBM, lean body mass.\u003c/p\u003e\n\u003cp\u003eNote\u003cstrong\u003e:\u003c/strong\u003e Values are presented as mean \u0026plusmn; standard deviation.\u003c/p\u003e\n\u003cp\u003eSex differences were examined using Welch\u0026rsquo;s t-test, and Hedges\u0026rsquo; \u003cem\u003eg\u003c/em\u003e was calculated for the OSI and sf_OSI.\u003c/p\u003e\n\u003cp\u003esf_OSI was calculated as ln(OSI) \u0026minus; 0.199 \u0026times; ln(LBM).\u003c/p\u003e\u003cp\u003eMales had significantly greater height, body mass, lean body mass (LBM), and knee extension torque than females, whereas females had a higher body fat percentage.\u003c/p\u003e\u003cp\u003eThe OSI values were significantly higher in females than in males (\u003cem\u003ep\u003c/em\u003e = 0.009), and the sex difference persisted after adjusting for body size using the sf_OSI (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001; Hedges’ \u003cem\u003eg\u003c/em\u003e = 0.79).\u003c/p\u003e\u003cp\u003e \u003cb\u003e2. Scaling validation\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe scaling exponent obtained from the log–log ordinary least squares (OLS) regression between OSI and LBM was b = 0.199 (95% confidence interval [CI]: 0.132–0.266), demonstrating a positive allometric relationship (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e1\u003c/span\u003eA). After allometric adjustment, no significant correlation was observed between sf_OSI and ln(LBM), indicating that the dependence on body size was effectively removed (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e1\u003c/span\u003eB). Sensitivity analyses using Deming regression (b = 0.292) and the zero-correlation method (b = 0.199) yielded comparable estimates, and the main conclusions were consistent across methods (Supplementary Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e). The correlation profile used to determine the optimal scaling exponent based on the zero-correlation method is shown in Supplementary Figure \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e \u003cb\u003e3. Sex/age differences\u003c/b\u003e \u003c/p\u003e\u003cp\u003eTwo-way ANOVA revealed significant effects of sex (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.01, Hedges’ \u003cem\u003eg\u003c/em\u003e = 0.79) and age (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001) on sf_OSI, whereas the sex × age interaction was not significant (\u003cem\u003ep\u003c/em\u003e = 0.22) (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e3\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eRegression models for size-free OSI (sf_OSI)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"8\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCovariates\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eβSexF (95% CI, p)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eβStrength (95% CI, p)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eβln(LBM) (95% CI, p)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAdj. R²\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eΔR²\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eAIC\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGender\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.090 (0.064–0.116), \u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.134\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e−445.846\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e+Age (cubic polynomial)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.245\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.111\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e−483.404\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e+Strength (Knee ext torque)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.103 (0.073–0.133), \u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.00035 (− 0.00002–0.00072), 0.064\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.252\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.006\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e−484.905\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM2 + LBM\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e+ln(LBM) (sensitivity)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.095 (0.059–0.132), \u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.00047 (− 0.00004–0.00098), 0.069\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e−0.054 (− 0.204–0.097), 0.482\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.250\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e−0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e–\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003ctfoot\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eDependent variable: size-free OSI (sf_OSI).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003esf_OSI was defined as ln(OSI) − \u003cem\u003eb\u003c/em\u003e·ln(LBM), with \u003cem\u003eb\u003c/em\u003e = 0.199 (95% CI: 0.132–0.266).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eM0–M2 represent the primary models that by design do not include ln(LBM).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eThe M2 + LBM model was a sensitivity analysis; inclusion of ln(LBM) slightly reduced the adjusted R² (Δ = −0.001), and ln(LBM) was not significant (\u003cem\u003ep\u003c/em\u003e = 0.482).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eFemales consistently showed higher sf_OSI values (β ≈ 0.09–0.10, \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eKnee extension strength exhibited a weak positive trend (\u003cem\u003ep\u003c/em\u003e ≈ 0.06–0.07).\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eThese results confirm that sf_OSI was adequately adjusted for body size, allowing robust inference across models.\u003c/td\u003e\u003c/tr\u003e\u003c/tfoot\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003ePost hoc comparisons showed that female teenagers had significantly higher sf_OSI than male teenagers, and female adults also exhibited higher sf_OSI than male adults (both FDR-adjusted \u003cem\u003ep\u003c/em\u003e \u0026lt; 0.05).\u003c/p\u003e\u003cp\u003eThe effect size for age (partial η² = 0.152) was larger than that for sex (partial η² = 0.036).\u003c/p\u003e\u003cp\u003e \u003cb\u003e4. Sport differences\u003c/b\u003e \u003c/p\u003e\u003cp\u003eAcross all sports categories, females exhibited higher sf_OSI values than males (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e2\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLS-means of size-free OSI (sf_OSI, b = 0.199) by sport and gender before adjustment for muscle strength\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"8\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSportstype\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eContrast\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEstimate\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e95% CI (Lower, Upper)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e% Difference\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003ep_adj (BH)\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCombat\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemale − Male\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.067\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.042\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e−0.015, 0.149\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e+ 6.9%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.109\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.109\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCourt-Jump\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemale − Male\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.098\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.024\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.051, 0.146\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e+ 10.3%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eField-Sprint\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemale − Male\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.090\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.025\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.041, 0.140\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e+ 9.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e\u0026lt; 0.001\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRacket-Strike\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemale − Male\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.062\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.020\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.022, 0.102\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e+ 6.4%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003ctfoot\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003eNote: Values represent least-squares mean differences (female–male) in size-free OSI (sf_OSI, \u003cem\u003eb\u003c/em\u003e = 0.199) adjusted for age and Tanita measurement mode.\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd colspan=\"8\"\u003ePositive values indicate higher sf_OSI in females.Percentage differences were calculated as 100 × (exp(Estimate) − 1).\u003cem\u003ep\u003c/em\u003e-values were corrected for multiple comparisons using the Benjamini–Hochberg false discovery rate (FDR) method, and the gender × sport interaction was not significant (Type II ANOVA, \u003cem\u003ep\u003c/em\u003e = 0.642); therefore, sport-specific comparisons are presented as descriptive results and should be interpreted with caution, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003e (forest plot), in which effect sizes and 95% confidence intervals are illustrated.\u003c/td\u003e\u003c/tr\u003e\u003c/tfoot\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eSex differences were most pronounced in Court-Jump (+ 10.3%) and Field-Sprint (+ 9.5%) sports (both \u003cem\u003ep\u003c/em\u003e(adj) \u0026lt; 0.001), whereas no significant difference was observed in Combat sports (\u003cem\u003ep\u003c/em\u003e(adj) = 0.109).\u003c/p\u003e\u003cp\u003eLinear model analysis identified significant main effects of sex (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001), sports category (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001), and age (\u003cem\u003ep\u003c/em\u003e \u0026lt; 0.001), whereas the measurement mode was not significant (\u003cem\u003ep\u003c/em\u003e = 0.88).\u003c/p\u003e\u003cp\u003eThe sex × sport interaction was also non-significant (\u003cem\u003ep\u003c/em\u003e = 0.65), indicating no statistically meaningful differences in the magnitude of sex effects across sports.\u003c/p\u003e\u003cp\u003eFor reference, the sports- and sex-specific percentiles of sf_OSI are presented in Supplementary Table S4.\u003c/p\u003e\u003cp\u003e \u003cb\u003e5. Muscle strength adjustment\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe inclusion of knee extension peak torque at 60°/s in the regression model attenuated the sex difference in sf_OSI by up to 13%, with only a minimal increase in explanatory power (ΔR² ≤ 0.04). These findings indicate that the contribution of muscle strength to the observed sex differences in sf_OSI is limited. The change in explanatory power (ΔR²) after adding muscle strength to the sf_OSI models across sport categories is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThis study is among the first to examine sex- and sports-specific differences in young athletes using a relative calcaneal stiffness index adjusted for body size (sf_OSI). The key findings are summarized as follows:\u003c/p\u003e \u003cp\u003eFirst, females consistently demonstrated higher sf_OSI values than males.\u003c/p\u003e \u003cp\u003eSecond, this sex difference was particularly pronounced in Court-Jump and Field-Sprint sports, whereas it was smaller in combat sports.\u003c/p\u003e \u003cp\u003eThird, the sex difference persisted even after adjustment for knee extension muscle strength.\u003c/p\u003e \u003cp\u003eConsistent with established knowledge on body composition differences, female athletes in this study exhibited lower LBM than males, consistent with previous research showing greater skeletal muscle mass and LBM in men in both the general and athletic populations [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Nevertheless, the size-adjusted sf_OSI revealed a higher relative calcaneal stiffness in females. This finding indicates that sex differences in bone strength cannot be fully explained by muscle mass alone, suggesting that structural adaptations driven by sports-specific mechanical loading\u0026mdash;such as frequent and multidirectional impacts\u0026mdash;may independently contribute to bone strength [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIndeed, peripheral quantitative computed tomography (pQCT) studies in female athletes have shown that muscle-derived joint moments and loading modalities, categorized as impact or odd-impact, are associated with differences in bone geometry [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. These findings are consistent with our observations.\u003c/p\u003e \u003cp\u003eFrom a methodological perspective, this study demonstrated the utility of allometric adjustment using a scaling exponent derived from log\u0026ndash;log regression to minimize body size dependence inherent in traditional BMD or unadjusted QUS indices. The near-zero correlation between sf_OSI and ln(LBM) confirmed the effective removal of the body size bias. Furthermore, the sequential inclusion of knee extension torque reduced the sex difference by up to 13%, with minimal improvement in model fit. These findings indicate that neither muscle mass nor simple torque measurements sufficiently explain the higher relative bone stiffness observed in females.\u003c/p\u003e \u003cp\u003eThis pattern also aligns with mechanostat theory [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], which posits that impact and multidirectional loading can stimulate structural bone adaptation beyond the influence of muscle-derived forces. Additionally, the hormonal milieu, bone geometry, and movement characteristics specific to adolescence and young adulthood, such as landing and cutting maneuvers, may have contributed to these sex differences. Future studies should incorporate direct measures of mechanical loading, such as ground reaction forces, jump tests, and accelerometer-based impact frequency analyses, to elucidate these mechanisms more clearly.\u003c/p\u003e \u003cp\u003eRegarding sport specificity, the relative sex differences were largest in Court-Jump and Field-Sprint sports and smallest in combat sports. Although the interaction was not statistically significant and should therefore be interpreted cautiously, the trend toward higher relative bone stiffness in sports involving jumping and multidirectional impact is consistent with findings from intervention studies and meta-analyses [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. In contrast, combat sports often involve endurance-based training and rapid weight reduction, which increases the risk of stress fractures, particularly in female athletes [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. Thus, future studies should consider risk stratification that integrates sport-specific loading characteristics with nutritional and weight-management factors.\u003c/p\u003e \u003cp\u003eThe sf_OSI is a useful indicator for visualizing sex- and sports-specific differences in bone properties. QUS is noninvasive, radiation-free, inexpensive, and portable, making it easily applicable in school-based health examinations and athletic training environments.\u003c/p\u003e \u003cp\u003eFrom a clinical perspective, combining QUS with body composition analysis enables straightforward onsite assessment of both bone and muscle status, facilitating early screening for stress fracture risk and relative energy deficiency in sports (REDs) [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e] among adolescent athletes. The sf_OSI\u0026rsquo;s independence from body size enables individualized risk assessment that considers personal physique- and sports-specific characteristics, thereby facilitating tailored preventive interventions such as nutritional guidance and training adjustments.\u003c/p\u003e \u003cp\u003eThus, QUS-based sf_OSI evaluation represents a safe and practical alternative to radiation-based techniques such as DXA or pQCT, offering a reliable means of repeated bone health monitoring in field settings. However, because this study was cross-sectional, the causal relationship between sf_OSI and fracture risk could not be determined. Future longitudinal studies are warranted to verify the predictive validity of the sf_OSI and establish its clinical and practical utility.\u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eLimitation\u003c/h2\u003e \u003cp\u003eThis study has some limitations.\u003c/p\u003e \u003cp\u003eFirst, due to its cross-sectional design, causal inferences cannot be made.\u003c/p\u003e \u003cp\u003eSecond, the nutritional status, hormonal environment, and biological maturity (e.g., Mirwald [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] and Moore [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] methods) were not directly assessed.\u003c/p\u003e \u003cp\u003eThird, muscle strength was measured only unilaterally, which may not have fully captured sport-specific asymmetry.\u003c/p\u003e \u003cp\u003eFourth, the integration of multiple sports categories might have introduced within-group heterogeneity.\u003c/p\u003e \u003cp\u003eAdditionally, standard imaging techniques such as DXA or pQCT were not employed for validation.\u003c/p\u003e \u003cp\u003eFuture longitudinal and multifaceted studies are warranted to verify the validity of the sf_OSI and establish its clinical and practical utility in both research and field settings.\u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThis study demonstrated that the size-free Osteo Sono-Assessment Index (sf_OSI), derived through allometric adjustment, can serve as an exploratory indicator of relative calcaneal stiffness, independent of body size. Females consistently exhibited higher sf_OSI values with sport-specific patterns observed; however, these sex differences could not be fully explained by muscle mass or knee extension strength.\u003c/p\u003e \u003cp\u003eThe sf_OSI offers a novel framework for assessing the qualitative aspects of bone adaptation and may be applicable to bone health monitoring and stress fracture risk assessment in young athletes.\u003c/p\u003e \u003cp\u003eNevertheless, because this study was cross-sectional, causal relationships remain uncertain, and maturity status and mechanical loading exposure were not sufficiently evaluated, leaving the possibility of residual confounding. The sf_OSI provides a new framework for evaluating relative bone stiffness in athletes. Future prospective studies should determine its predictive value for stress fractures and its usability in sports rehabilitation programs.\u003c/p\u003e"},{"header":"List of abbreviations","content":"\u003cp\u003eAIC, Akaike\u0026rsquo;s Information Criterion;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eANOVA, analysis of variance;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBMD, bone mineral density;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eCI, confidence interval;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDXA, dual-energy X-ray absorptiometry;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFDR, false discovery rate;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eLBM, lean body mass;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eOLS, ordinary least squares;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eOSI, Osteo Sono-Assessment Index;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003epQCT, peripheral quantitative computed tomography;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eQUS, quantitative ultrasound;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eREDs, Relative Energy Deficiency in Sport;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSD, standard deviation;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003esf_OSI, size-free Osteo Sono-Assessment Index.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe study protocol was approved by the Ethics Committee of the Wakayama Medical University (Approval No. 4042) and was conducted in accordance with the principles of the Declaration of Helsinki. Written informed consent was obtained from all participants, and for those under 18 years of age, consent was also obtained from their parents or legal guardians.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets generated and/or analyzed during the current study contain personal information and are therefore not publicly available. The data are securely stored at Wakayama Medical University and are available from the corresponding author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor’s contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eRN conceived and designed the study, analyzed and interpreted the data, and drafted the manuscript. KG contributed to data interpretation and critically revised the manuscript. MB and KK supervised the study and provided critical feedback on the manuscript. All authors read and approved the final version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe thank the staff of the Genki Development Institute and the Satellite Clinic and the athletes for their cooperation in this study.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eMountjoy M, Ackerman KE, Bailey DM, Burke LM, Constantini N, Hackney AC, et al. 2023 International Olympic Committee\u0026rsquo;s (IOC) consensus statement on Relative Energy Deficiency in Sport (REDs). Br J Sports Med. 2023;57:1073\u0026ndash;97. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1136/bjsports-2023-106994\u003c/span\u003e\u003cspan address=\"10.1136/bjsports-2023-106994\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDallman J, Herda A, Cleary CJ, Morey T, Diederich A, Vopat BG, et al. A brief review of the literature for published dual-energy x-ray absorptiometry protocols for athletes. Sports Health. 2024;16:735\u0026ndash;43. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1177/19417381231208204\u003c/span\u003e\u003cspan address=\"10.1177/19417381231208204\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDe Souza MJ, Nattiv A, Joy E, Misra M, Williams NI, Mallinson RJ, et al. Female athlete triad coalition consensus statement expert panel. 2014 female athlete triad coalition consensus statement on treatment and return to play of the female athlete triad. Br J Sports Med. 2014;48:289. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1136/bjsports-2013-093218\u003c/span\u003e\u003cspan address=\"10.1136/bjsports-2013-093218\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAlwis G, Rosengren B, Nilsson JA, Stenevi-Lundgren S, Sundberg M, Sernbo I, et al. Normative calcaneal quantitative ultrasound data as an estimation of skeletal development in Swedish children and adolescents. Calcif Tissue Int. 2010;87:493\u0026ndash;506.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLi C, Sun J, Yu L. Diagnostic value of calcaneal quantitative ultrasound in the evaluation of osteoporosis in middle-aged and elderly patients. Med (Baltim). 2022;101:e28325. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1097/MD.0000000000028325\u003c/span\u003e\u003cspan address=\"10.1097/MD.0000000000028325\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eM\u0026eacute;trailler A, Hans D, Lamy O, Gonzalez Rodriguez E, Shevroja E. Heel quantitative ultrasound (QUS) predicts incident fractures independently of trabecular bone score (TBS), bone mineral density (BMD), and FRAX: the OsteoLaus Study. Osteoporos Int. 2023;34:1401\u0026ndash;9. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/s00198-023-06728-4\u003c/span\u003e\u003cspan address=\"10.1007/s00198-023-06728-4\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDaly RM. The effect of exercise on bone mass and structural geometry during growth. Med Sport Sci. 2007;51:33\u0026ndash;49. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1159/000103003\u003c/span\u003e\u003cspan address=\"10.1159/000103003\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWarton DI, Wright IJ, Falster DS, Westoby M. Bivariate line-fitting methods for allometry. Biol Rev Camb Philos Soc. 2006;81:259\u0026ndash;91. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1017/S1464793106007007\u003c/span\u003e\u003cspan address=\"10.1017/S1464793106007007\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePackard GC. On the use of logarithmic transformations in allometric analyses. J Theor Biol. 2009;257:515\u0026ndash;8. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/j.jtbi.2008.10.016\u003c/span\u003e\u003cspan address=\"10.1016/j.jtbi.2008.10.016\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. discussion 519\u0026thinsp;\u0026ndash;\u0026thinsp;21.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePatel H, Woods L, Teesdale-Spittle P, Dennison E. A cross-sectional study of the relationship between recreational sporting activity and calcaneal bone density in adolescents and young adults. Calcif Tissue Int. 2021;109:218\u0026ndash;26. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1080/00913847.2021.1903819\u003c/span\u003e\u003cspan address=\"10.1080/00913847.2021.1903819\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJonvik KL, Torstveit MK, Sundgot-Borgen J, et al. Do we need to change the guideline values for determining low bone mineral density in athletes? J Appl Physiol (1985). 2022;132:1320\u0026ndash;2. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1152/japplphysiol.00851.2021\u003c/span\u003e\u003cspan address=\"10.1152/japplphysiol.00851.2021\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNikander R, Kannus P, Rantalainen T, Uusi-Rasi K, Heinonen A, Siev\u0026auml;nen H. Cross-sectional geometry of weight-bearing tibia in female athletes subjected to different exercise loadings. Osteoporos Int. 2010;21:1687\u0026ndash;94. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/s00198-009-1101-0\u003c/span\u003e\u003cspan address=\"10.1007/s00198-009-1101-0\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNikander R, Siev\u0026auml;nen H, Uusi-Rasi K, Heinonen A, Kannus P. Loading modalities and bone structures at nonweight-bearing upper extremity and weight-bearing lower extremity: a pQCT study of adult female athletes. Bone. 2006;39:886\u0026ndash;94. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/j.bone.2006.04.005\u003c/span\u003e\u003cspan address=\"10.1016/j.bone.2006.04.005\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHedges LV, Olkin I. Statistical methods for meta-analysis. Orlando, FL: Academic; 1985. p. 369.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLinnet K. Estimation of the linear relationship between the measurements of two methods with proportional errors. Stat Med. 1990;9:1463\u0026ndash;73. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1002/sim.4780091210\u003c/span\u003e\u003cspan address=\"10.1002/sim.4780091210\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBenjamini Y, Hochberg Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B. 1995;57:289\u0026ndash;300.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKanda Y. Investigation of the freely available easy-to-use software 'EZR' for medical statistics. Bone Marrow Transpl. 2013;48:452\u0026ndash;8. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1038/bmt.2012.244\u003c/span\u003e\u003cspan address=\"10.1038/bmt.2012.244\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWang X, Gao L, Xiong J, Cheng H, Liu L, Dong H, et al. The life-course changes in muscle mass using dual-energy x-ray absorptiometry: the China BCL study and the US NHANES study. J Cachexia Sarcopenia Muscle. 2024;15:1687\u0026ndash;95. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1002/jcsm.13522\u003c/span\u003e\u003cspan address=\"10.1002/jcsm.13522\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBelzunce MA, Henckel J, Di Laura A, Horga LM, Hart AJ. Gender similarities and differences in skeletal muscle and body composition: an MRI study of recreational cyclists. BMJ Open Sport Exerc Med. 2023;9:e001672. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1136/bmjsem-2023-001672\u003c/span\u003e\u003cspan address=\"10.1136/bmjsem-2023-001672\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNikander R, Siev\u0026auml;nen H, Heinonen A, Daly RM, Uusi-Rasi K, Kannus P. Femoral neck structure in adult female athletes subjected to different loading modalities. J Bone Min Res. 2005;20:520\u0026ndash;8. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1359/JBMR.041130\u003c/span\u003e\u003cspan address=\"10.1359/JBMR.041130\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRauch F, Schoenau E. The developing bone: slave or master of its cells and molecules? Pediatr Res. 2001;50:309\u0026ndash;14. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1203/00006450-200109000-00003\u003c/span\u003e\u003cspan address=\"10.1203/00006450-200109000-00003\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMiao T, Li X, Zhang W, Yang F, Wang X. Effects of high-impact jumping versus resistance exercise on bone mineral content in children and adolescents: a systematic review and meta-analysis. PeerJ. 2025;13:e19616. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.7717/peerj.19616\u003c/span\u003e\u003cspan address=\"10.7717/peerj.19616\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTenforde AS, Fredericson M. Influence of sports participation on bone health in the young athlete: a review of the literature. PM R. 2011;3:861\u0026ndash;7. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/j.pmrj.2011.05.019\u003c/span\u003e\u003cspan address=\"10.1016/j.pmrj.2011.05.019\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHoenig T, Hollander K, Popp KL, Fredericson M, Kraus EA, Warden SJ, et al. International Delphi consensus on bone stress injuries in athletes. Br J Sports Med. 2025;59:78\u0026ndash;90. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1136/bjsports-2024-108616\u003c/span\u003e\u003cspan address=\"10.1136/bjsports-2024-108616\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMoayyeri A, Adams JE, Adler RA, Krieg MA, Hans D, Compston J, et al. Quantitative ultrasound of the heel and fracture risk assessment: an updated meta-analysis. Osteoporos Int. 2012;23:143\u0026ndash;53. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/s00198-011-1817-5\u003c/span\u003e\u003cspan address=\"10.1007/s00198-011-1817-5\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMirwald RL, Baxter-Jones AD, Bailey DA, Beunen GP. An assessment of maturity from anthropometric measurements. Med Sci Sports Exerc. 2002;34:689\u0026ndash;94. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1097/00005768-200204000-00020\u003c/span\u003e\u003cspan address=\"10.1097/00005768-200204000-00020\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMoore SA, McKay HA, Macdonald H, Nettlefold L, Baxter-Jones AD, Cameron N, et al. Enhancing a somatic maturity prediction model. Med Sci Sports Exerc. 2015;47:1755\u0026ndash;64.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"bmc-sports-science-medicine-and-rehabilitation","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ssmr","sideBox":"Learn more about [BMC Sports Science, Medicine and Rehabilitation](http://bmcsportsscimedrehabil.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/ssmr/default.aspx","title":"BMC Sports Science, Medicine and Rehabilitation","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"OSI, bone strength, allometry, athletes, sex difference","lastPublishedDoi":"10.21203/rs.3.rs-8285227/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8285227/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eBackground: \u003c/strong\u003eBone strength is essential for maintaining athletic performance and preventing stress fractures. However, conventional indicators such as Bone Mineral Density (BMD) and the Osteo Sono-Assessment Index (OSI) are influenced by body size, limiting comparisons between sexes and sports disciplines. This study aimed to develop and validate a body size–independent indicator of calcaneal stiffness using allometric scaling in young athletes.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods: \u003c/strong\u003eThis cross-sectional study included 295 athletes aged 10–30 years (153 males, 142 females). OSI was measured using quantitative ultrasound, and lean body mass (LBM) was assessed by bioelectrical impedance analysis. Using the scaling exponent (b = 0.199) derived from the log–log regression between OSI and LBM, the size-free OSI (sf_OSI) was calculated as sf_OSI = ln(OSI) − 0.199 × ln(LBM).\u003cbr\u003e\nSex and sport differences were examined using Welch’s t-test and linear models, with muscle strength evaluated by including knee extension torque as a covariate.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults: \u003c/strong\u003eFemales showed significantly higher OSI values than males (3.31 ± 0.42 vs. 3.18 ± 0.42, \u003cem\u003ep\u003c/em\u003e = 0.009), with an even difference for sf_OSI (0.45 ± 0.12 vs. 0.36 ± 0.11, \u003cem\u003ep\u003c/em\u003e\u0026lt; 0.001, Hedges’ \u003cem\u003eg\u003c/em\u003e = 0.79). Sex (η² = 0.036) and age group (η² = 0.152) exerted independent effects. Sport-specific analysis showed greater female predominance in Court-Jump (+10.3%) and Field-Sprint (+9.5%) sports, with no significant difference in combat sports. Adjustment for knee extension torque reduced the sex difference by up to 13% (ΔR² ≤ 0.04).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusions:\u003c/strong\u003e sf_OSI appears to be a body size–adjusted indicator of calcaneal stiffness that clearly delineates sex- and sport-specific patterns. Although it demonstrated exploratory utility as a monitoring metric, direct causal or fracture risk associations were not established. sf_OSI may serve as a practical, noninvasive tool for monitoring sport-specific bone adaptation and stress fracture risk in field settings. In this study, females consistently exhibited higher sf_OSI values, particularly in Court-Jump and Field-Sprint sports. These sex differences were not fully explained by muscle strength, suggesting that sf_OSI is a useful parameter for evaluating sport-specific bone adaptation and stress fracture risk in young athletes.\u003c/p\u003e","manuscriptTitle":"Allometric Scaling of Calcaneal Bone Stiffness in Young Athletes: An Exploratory Study of a Size-Free Index","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-08 16:01:20","doi":"10.21203/rs.3.rs-8285227/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewersInvited","content":"","date":"2026-01-07T06:19:05+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-12-10T10:32:32+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-12-09T01:59:49+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-12-09T01:59:31+00:00","index":"","fulltext":""},{"type":"submitted","content":"BMC Sports Science, Medicine and Rehabilitation","date":"2025-12-05T08:00:12+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"bmc-sports-science-medicine-and-rehabilitation","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ssmr","sideBox":"Learn more about [BMC Sports Science, Medicine and Rehabilitation](http://bmcsportsscimedrehabil.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/ssmr/default.aspx","title":"BMC Sports Science, Medicine and Rehabilitation","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e4ed2234-92c1-44ef-b83a-9a58f58cb909","owner":[],"postedDate":"January 8th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-01-08T16:01:20+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-08 16:01:20","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8285227","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8285227","identity":"rs-8285227","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}