Optimization and validation of a mathematical model for precise assessment of personalized exercise load based on wearable devices_Journal of Biomedical Engineering

Authors：

WANG Wenxing , ZHAO Yuanhui , YU Wenlang ,  REN Hong

Physical Fitness and Health Research and Teaching Department, College of Sports Human Sciences, Beijing Sport University, Beijing 100084, P. R. China;

Corresponding author：

REN Hong, Email: renhong@bsu.edu.cn

Keywords：

Accurate assessment of exercise load; Training monitoring; Mathematical model; Chronic disease prevention and control; Non-pharmacological intervention

DOI：

10.7507/1001-5515.202406043

Video：

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Exercise intervention is an important non-pharmacological intervention for various diseases, and establishing precise exercise load assessment techniques can improve the quality of exercise intervention and the efficiency of disease prevention and control. Based on data collection from wearable devices, this study conducts nonlinear optimization and empirical verification of the original "Fitness-Fatigue Model". By constructing a time-varying attenuation function and specific coefficients, this study develops an optimized mathematical model that reflects the nonlinear characteristics of training responses. Thirteen participants underwent 12 weeks of moderate-intensity continuous cycling, three times per week. For each training session, external load (actual work done) and internal load (heart rate variability index) data were collected for each individual to conduct a performance comparison between the optimized model and the original model. The results show that the optimized model demonstrates a significantly improved overall goodness of fit and superior predictive ability. In summary, the findings of this study can support dynamic adjustments to participants' training programs and aid in the prevention and control of chronic diseases.

Citation： WANG Wenxing, ZHAO Yuanhui, YU Wenlang, REN Hong. Optimization and validation of a mathematical model for precise assessment of personalized exercise load based on wearable devices. Journal of Biomedical Engineering, 2025, 42(4): 739-747. doi: 10.7507/1001-5515.202406043 Copy

1.	常凤, 赵文艳, 魏亚茹, 等. LIPC-514C/T多态性与运动干预PDM人群血糖及健康体适能疗效的关联研究. 中国体育科技, 2023, 59(12): 62-69.
2.	中国老年护理联盟, 中南大学湘雅护理学院(中南大学湘雅泛海健康管理研究院), 中南大学湘雅医院(国家老年疾病临床医学研究中心), 等. 认知衰退老年人非药物干预临床实践指南: 身体活动. 中国全科医学, 2023, 26(16): 1927-1937,1971.
3.	季谋芳, 李若冰, 胡廷进, 等. 不同非药物干预方式对产后抑郁症患者影响的网状meta分析. 中华护理杂志, 2024, 59(2): 228-235.
4.	邢双涛, 冯世杰, 关朝阳. 不同运动对老年人骨密度影响的网状meta分析. 中国骨质疏松杂志, 2024, 30(3): 348-354.
5.	Plews D J, Laupsen P B, Stanley J, et al. Training adaptation and heart rate variability in elite endurance athletes: opening the door to effective monitoring. Sports Med, 2013, 43(9): 773-781.
6.	Mishica C, Kyröläinen H, Hynynen E, et al. Relationships between heart rate variability, sleep duration, cortisol and physical training in young athletes. J Sport Sci Med, 2021, 20(4): 778-788.
7.	Lechner S, Ammar A, Boukhris O, et al. Monitoring training load in youth soccer players: effects of a six-week preparatory training program and the association between external and internal loads. Biol Sport, 2023, 40(1): 63-75.
8.	Bellinger P. Functional overreaching in endurance athletes: a necessity or cause for concern?. Sports Med, 2020, 50(6): 1059-1073.
9.	Fitz-Clarke J R, Morton R H, Banister E W. Optimizing athletic performance by influence curves. J Appl Physiol, 1991, 71(3): 1151-1158.
10.	Imbach F, Sutton-Charani N, Montmain J, et al. The use of fitness-fatigue models for sport performance modelling: conceptual issues and contributions from machine-learning. Sports Med Open, 2022, 8(1): 29.
11.	Calvert T W, Banister E W, Savage M, et al. Systems model of the effects of training on physical performance. IEEE Trans Syst Man Cybernet, 1976, 6(12): 94-102.
12.	Banister E W, Hamilton C L. Variations in iron status with fatigue modelled from training in female distance runners. Eur J Appl Physiol Occup Physiol, 1985, 54(1): 16-23.
13.	Scott A L, Spaniol F J, Ocker L, et al. The application of a systems model to the training and performance of adolescent swimmers. J Strength Cond Res, 2011, 25: S27-S28.
14.	Wallace L K, Slattery K M, Coutts A J. A comparison of methods for quantifying training load: relationships between modelled and actual training responses. Eur J Appl Physiol, 2014, 114(1): 11-20.
15.	Stephens H B H, Burgess K E, Elyan E, et al. The effects of measurement error and testing frequency on the fitness-fatigue model applied to resistance training: a simulation approach. Int J Sports Sci Coach, 2020, 15(1): 60-71.
16.	曹宏庆, 康立山, 陈毓屏. 高阶常微分方程的动态演化建模. 武汉大学学报(自然科学版), 2000, 46(1): 19-23.
17.	曹宏庆, 康立山, 陈毓屏. 高阶常微分方程的演化建模用于时间序列的分析. 小型微型计算机系统, 2000, 21(4): 344-349.
18.	Busso T, Candau R, Lacour J R. Fatigue and fitness modelled from the effects of training on performance. Eur J Appl Physiol Occup Physiol, 1994, 69(1): 50-54.
19.	何勇, 虞丽娟, 吴卫兵. 训练负荷—体能状态数学模型的建立及参数估计. 上海体育学院学报, 2011, 35(2): 57-60.
20.	Saboul D, Balducci P, Millet G, et al. A pilot study on quantification of training load: the use of HRV in training practice. Eur J Sport Sci, 2016, 16(2): 172-181.
21.	赵海燕, 王林霞, 赵德峰, 等. 心率变异性指标RMSSD和TLHRV在持续性运动训练负荷监控中的有效性研究. 中国运动医学杂志, 2018, 37(6): 461-467.
22.	Busso T, Häkkinen K, Pakarinen A, et al. A systems model of training responses and its relationship to hormonal responses in elite weight-lifters. Eur J of Appl Physiol Occup Physiol, 1990, 61(1-2): 48-54.
23.	Vermeire K, Ghijs M, Bourgois J G, et al. The fitness-fatigue model: what's in the numbers?. Int J Sports Physiol Perform, 2022, 17(5): 810-813.
24.	仇乃民. 运动员竞技能力非线性系统理论构建研究. 天津体育学院学报, 2015, 30(1): 58-65.
25.	Chuckravanen D, Bulut S, Kürklü G B, et al. Review of exercise-induced physiological control models to explain the development of fatigue to improve sports performance and future trend. Sci Sports, 2019, 34(3): 131-140.
26.	Ludwig M, Asteroth A, Rasche C, et al. Including the past: performance modeling using a preload concept by means of the fitness-fatigue model. Int J Comput Sci Sport, 2019, 18(1): 115-134.
27.	胡海旭. 竞技能力增长理论模型及其演进. 体育科学, 2016, 36(2): 14-24, 40.
28.	Marrier B, Robineau J, Piscione J, et al. Supercompensation kinetics of physical qualities during a taper in team-sport athletes. Int J Sports Physiol Perform, 2017, 12(9): 1163-1169.
29.	Banister E W, Carter J B, Zarkadas P C. Training theory and taper: validation in triathlon athletes. Eur J Appl Physiol Occup Physiol, 1999, 79(2): 182-191.
30.	Busso T, Carasso C, Lacour J R. Adequacy of a systems structure in the modeling of training effects on performance. J Appl Physiol, 1991, 71(5): 2044-2049.
31.	Granero-Gallegos A, González-quílez A, Plews D, et al. HRV-based training for improving VO2max in endurance athletes. a systematic review with meta-analysis. Int J Environ Res Public Health, 2020, 17(21): 7999.
32.	da Silva D F, Ferraro Z M, Adamo K B, et al. Endurance running training individually guided by HRV in untrained women. J Strength Cond Res, 2019, 33(3): 736-746.
33.	Schaffarczyk M, Rogers B, Reer R, et al. Fractal correlation properties of HRV as a noninvasive biomarker to assess the physiological status of triathletes during simulated warm-up sessions at low exercise intensity: a pilot study. BMC Sports Sci Med Rehabil, 2022, 14(1): 203.
34.	Schüttler D, Hamm W, Bauer A, et al. Routine heart rate-based and novel ECG-based biomarkers of autonomic nervous system in sports medicine. Dtsch Z Sport Med, 2020, 71(6): 141-149.
35.	Buchheit M, Simon C, Charloux A, et al. Relationship between very high physical activity energy expenditure, heart rate variability and self-estimate of health status in middle-aged individuals. Int J Sports Med, 2006, 27(9): 697-701.
36.	陈小平. 运动训练的基石——“超量恢复”学说受到质疑. 首都体育学院学报, 2004, 16(4): 3-7.

1. 常凤, 赵文艳, 魏亚茹, 等. LIPC-514C/T多态性与运动干预PDM人群血糖及健康体适能疗效的关联研究. 中国体育科技, 2023, 59(12): 62-69.
2. 中国老年护理联盟, 中南大学湘雅护理学院(中南大学湘雅泛海健康管理研究院), 中南大学湘雅医院(国家老年疾病临床医学研究中心), 等. 认知衰退老年人非药物干预临床实践指南: 身体活动. 中国全科医学, 2023, 26(16): 1927-1937,1971.
3. 季谋芳, 李若冰, 胡廷进, 等. 不同非药物干预方式对产后抑郁症患者影响的网状meta分析. 中华护理杂志, 2024, 59(2): 228-235.
4. 邢双涛, 冯世杰, 关朝阳. 不同运动对老年人骨密度影响的网状meta分析. 中国骨质疏松杂志, 2024, 30(3): 348-354.
5. Plews D J, Laupsen P B, Stanley J, et al. Training adaptation and heart rate variability in elite endurance athletes: opening the door to effective monitoring. Sports Med, 2013, 43(9): 773-781.
6. Mishica C, Kyröläinen H, Hynynen E, et al. Relationships between heart rate variability, sleep duration, cortisol and physical training in young athletes. J Sport Sci Med, 2021, 20(4): 778-788.
7. Lechner S, Ammar A, Boukhris O, et al. Monitoring training load in youth soccer players: effects of a six-week preparatory training program and the association between external and internal loads. Biol Sport, 2023, 40(1): 63-75.
8. Bellinger P. Functional overreaching in endurance athletes: a necessity or cause for concern?. Sports Med, 2020, 50(6): 1059-1073.
9. Fitz-Clarke J R, Morton R H, Banister E W. Optimizing athletic performance by influence curves. J Appl Physiol, 1991, 71(3): 1151-1158.
10. Imbach F, Sutton-Charani N, Montmain J, et al. The use of fitness-fatigue models for sport performance modelling: conceptual issues and contributions from machine-learning. Sports Med Open, 2022, 8(1): 29.
11. Calvert T W, Banister E W, Savage M, et al. Systems model of the effects of training on physical performance. IEEE Trans Syst Man Cybernet, 1976, 6(12): 94-102.
12. Banister E W, Hamilton C L. Variations in iron status with fatigue modelled from training in female distance runners. Eur J Appl Physiol Occup Physiol, 1985, 54(1): 16-23.
13. Scott A L, Spaniol F J, Ocker L, et al. The application of a systems model to the training and performance of adolescent swimmers. J Strength Cond Res, 2011, 25: S27-S28.
14. Wallace L K, Slattery K M, Coutts A J. A comparison of methods for quantifying training load: relationships between modelled and actual training responses. Eur J Appl Physiol, 2014, 114(1): 11-20.
15. Stephens H B H, Burgess K E, Elyan E, et al. The effects of measurement error and testing frequency on the fitness-fatigue model applied to resistance training: a simulation approach. Int J Sports Sci Coach, 2020, 15(1): 60-71.
16. 曹宏庆, 康立山, 陈毓屏. 高阶常微分方程的动态演化建模. 武汉大学学报(自然科学版), 2000, 46(1): 19-23.
17. 曹宏庆, 康立山, 陈毓屏. 高阶常微分方程的演化建模用于时间序列的分析. 小型微型计算机系统, 2000, 21(4): 344-349.
18. Busso T, Candau R, Lacour J R. Fatigue and fitness modelled from the effects of training on performance. Eur J Appl Physiol Occup Physiol, 1994, 69(1): 50-54.
19. 何勇, 虞丽娟, 吴卫兵. 训练负荷—体能状态数学模型的建立及参数估计. 上海体育学院学报, 2011, 35(2): 57-60.
20. Saboul D, Balducci P, Millet G, et al. A pilot study on quantification of training load: the use of HRV in training practice. Eur J Sport Sci, 2016, 16(2): 172-181.
21. 赵海燕, 王林霞, 赵德峰, 等. 心率变异性指标RMSSD和TLHRV在持续性运动训练负荷监控中的有效性研究. 中国运动医学杂志, 2018, 37(6): 461-467.
22. Busso T, Häkkinen K, Pakarinen A, et al. A systems model of training responses and its relationship to hormonal responses in elite weight-lifters. Eur J of Appl Physiol Occup Physiol, 1990, 61(1-2): 48-54.
23. Vermeire K, Ghijs M, Bourgois J G, et al. The fitness-fatigue model: what's in the numbers?. Int J Sports Physiol Perform, 2022, 17(5): 810-813.
24. 仇乃民. 运动员竞技能力非线性系统理论构建研究. 天津体育学院学报, 2015, 30(1): 58-65.
25. Chuckravanen D, Bulut S, Kürklü G B, et al. Review of exercise-induced physiological control models to explain the development of fatigue to improve sports performance and future trend. Sci Sports, 2019, 34(3): 131-140.
26. Ludwig M, Asteroth A, Rasche C, et al. Including the past: performance modeling using a preload concept by means of the fitness-fatigue model. Int J Comput Sci Sport, 2019, 18(1): 115-134.
27. 胡海旭. 竞技能力增长理论模型及其演进. 体育科学, 2016, 36(2): 14-24, 40.
28. Marrier B, Robineau J, Piscione J, et al. Supercompensation kinetics of physical qualities during a taper in team-sport athletes. Int J Sports Physiol Perform, 2017, 12(9): 1163-1169.
29. Banister E W, Carter J B, Zarkadas P C. Training theory and taper: validation in triathlon athletes. Eur J Appl Physiol Occup Physiol, 1999, 79(2): 182-191.
30. Busso T, Carasso C, Lacour J R. Adequacy of a systems structure in the modeling of training effects on performance. J Appl Physiol, 1991, 71(5): 2044-2049.
31. Granero-Gallegos A, González-quílez A, Plews D, et al. HRV-based training for improving VO2max in endurance athletes. a systematic review with meta-analysis. Int J Environ Res Public Health, 2020, 17(21): 7999.
32. da Silva D F, Ferraro Z M, Adamo K B, et al. Endurance running training individually guided by HRV in untrained women. J Strength Cond Res, 2019, 33(3): 736-746.
33. Schaffarczyk M, Rogers B, Reer R, et al. Fractal correlation properties of HRV as a noninvasive biomarker to assess the physiological status of triathletes during simulated warm-up sessions at low exercise intensity: a pilot study. BMC Sports Sci Med Rehabil, 2022, 14(1): 203.
34. Schüttler D, Hamm W, Bauer A, et al. Routine heart rate-based and novel ECG-based biomarkers of autonomic nervous system in sports medicine. Dtsch Z Sport Med, 2020, 71(6): 141-149.
35. Buchheit M, Simon C, Charloux A, et al. Relationship between very high physical activity energy expenditure, heart rate variability and self-estimate of health status in middle-aged individuals. Int J Sports Med, 2006, 27(9): 697-701.
36. 陈小平. 运动训练的基石——“超量恢复”学说受到质疑. 首都体育学院学报, 2004, 16(4): 3-7.

Journal of Biomedical Engineering

Optimization and validation of a mathematical model for precise assessment of personalized exercise load based on wearable devices

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