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.	常鳳, 趙文艷, 魏亞茹, 等. LIPC-514C/T多態性與運動干預PDM人群血糖及健康體適能療效的關聯研究. 中國體育科技, 2023, 59(12): 62-69.
2.	中國老年護理聯盟, 中南大學湘雅護理學院(中南大學湘雅泛海健康管理研究院), 中南大學湘雅醫院(國家老年疾病臨床醫學研究中心), 等. 認知衰退老年人非藥物干預臨床實踐指南: 身體活動. 中國全科醫學, 2023, 26(16): 1927-1937,1971.
3.	季謀芳, 李若冰, 胡廷進, 等. 不同非藥物干預方式對產后抑郁癥患者影響的網狀meta分析. 中華護理雜志, 2024, 59(2): 228-235.
4.	邢雙濤, 馮世杰, 關朝陽. 不同運動對老年人骨密度影響的網狀meta分析. 中國骨質疏松雜志, 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, Kyr?l?inen 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.	曹宏慶, 康立山, 陳毓屏. 高階常微分方程的動態演化建模. 武漢大學學報(自然科學版), 2000, 46(1): 19-23.
17.	曹宏慶, 康立山, 陳毓屏. 高階常微分方程的演化建模用于時間序列的分析. 小型微型計算機系統, 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, 虞麗娟, 吳衛兵. 訓練負荷—體能狀態數學模型的建立及參數估計. 上海體育學院學報, 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.	趙海燕, 王林霞, 趙德峰, 等. 心率變異性指標RMSSD和TLHRV在持續性運動訓練負荷監控中的有效性研究. 中國運動醫學雜志, 2018, 37(6): 461-467.
22.	Busso T, H?kkinen 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. 運動員競技能力非線性系統理論構建研究. 天津體育學院學報, 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. 競技能力增長理論模型及其演進. 體育科學, 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.	陳小平. 運動訓練的基石——“超量恢復”學說受到質疑. 首都體育學院學報, 2004, 16(4): 3-7.

1. 常鳳, 趙文艷, 魏亞茹, 等. LIPC-514C/T多態性與運動干預PDM人群血糖及健康體適能療效的關聯研究. 中國體育科技, 2023, 59(12): 62-69.
2. 中國老年護理聯盟, 中南大學湘雅護理學院(中南大學湘雅泛海健康管理研究院), 中南大學湘雅醫院(國家老年疾病臨床醫學研究中心), 等. 認知衰退老年人非藥物干預臨床實踐指南: 身體活動. 中國全科醫學, 2023, 26(16): 1927-1937,1971.
3. 季謀芳, 李若冰, 胡廷進, 等. 不同非藥物干預方式對產后抑郁癥患者影響的網狀meta分析. 中華護理雜志, 2024, 59(2): 228-235.
4. 邢雙濤, 馮世杰, 關朝陽. 不同運動對老年人骨密度影響的網狀meta分析. 中國骨質疏松雜志, 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, Kyr?l?inen 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. 曹宏慶, 康立山, 陳毓屏. 高階常微分方程的動態演化建模. 武漢大學學報(自然科學版), 2000, 46(1): 19-23.
17. 曹宏慶, 康立山, 陳毓屏. 高階常微分方程的演化建模用于時間序列的分析. 小型微型計算機系統, 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, 虞麗娟, 吳衛兵. 訓練負荷—體能狀態數學模型的建立及參數估計. 上海體育學院學報, 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. 趙海燕, 王林霞, 趙德峰, 等. 心率變異性指標RMSSD和TLHRV在持續性運動訓練負荷監控中的有效性研究. 中國運動醫學雜志, 2018, 37(6): 461-467.
22. Busso T, H?kkinen 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. 運動員競技能力非線性系統理論構建研究. 天津體育學院學報, 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. 競技能力增長理論模型及其演進. 體育科學, 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. 陳小平. 運動訓練的基石——“超量恢復”學說受到質疑. 首都體育學院學報, 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

Abstract Full text Figures/Tables Video References Cited by

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