Research on a muscle fatigue quantification based on an improved critical power model_Journal of Biomedical Engineering

Authors：

WU Xiaoguang ¹ ,  SHANG Jintao ¹ , NIU Xiaochen ² , LIN Hongbin ¹ , DU Yihao ¹

1. Key Laboratory of Measurement Technology and Instrumentation of Hebei Province, Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, P. R. China;
2. Qinhuangdao Vocational and Technical College, Qinhuangdao, Hebei 066100, P. R. China;

Corresponding?author：

SHANG Jintao, Email: shangjintao2001@126.com

Keywords：

Critical power model; Muscle power output; Muscle residual energy; Muscle fatigue

DOI：

10.7507/1001-5515.202510061

Video：

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Abstract Full text Figures/Tables Video References Cited by

To address the challenge of accurately quantifying and predicting muscle fatigue, this study proposes a method for measuring muscle fatigue levels during dumbbell curl exercises. First, the Hill muscle model was used to calculate the biceps brachii power output during dumbbell curls for 12 male subjects. The true value of average muscle power output required under the current physiological state was fitted based on the average power output during non-fatigued cycles. The variation pattern between the curvature constant of the critical power (CP) model and the true muscle power output was investigated, leading to the construction of an improved CP model capable of adapting to physiological states. Next, residual muscle energy was calculated using a muscle energy expenditure and recovery model to quantify fatigue levels. Test results demonstrated that under varying exercise intensities, the mean root mean square error of the improved CP model in predicting the duration of dumbbell curls compared to actual measurement time is 8.01 seconds, significantly lower than the 19.79 seconds of the original model, validating the effectiveness of this approach. In summary, this methodology provides scientific basis for fitness enthusiasts to plan repetition counts within sets and rest intervals between sets during dumbbell curls. It plays a positive role in enhancing exercise efficiency and lays a foundation for further advancing the popularization of fitness activities.

Citation： WU Xiaoguang, SHANG Jintao, NIU Xiaochen, LIN Hongbin, DU Yihao. Research on a muscle fatigue quantification based on an improved critical power model. Journal of Biomedical Engineering, 2026, 43(2): 259-266. doi: 10.7507/1001-5515.202510061 Copy

1.	牛惠祺, 張弼, 劉麗剛, 等. 人體肌肉狀態疲勞監測及其在外骨骼交互控制中的應用. 生物醫學工程學雜志, 2023, 40(4): 654-662.
2.	Pethick J, Tallent J. The neuromuscular fatigue-induced loss of muscle force control. Sports, 2022, 10(11): 184.
3.	陀浩雯. 疲勞前后消防員裸足/穿鞋跑的生物力學特征研究. 桂林: 廣西師范大學, 2025.
4.	郝珊珊. 基于運動生物力學原理探索老年人運動損傷的防護對策. 當代體育科技, 2025, 15(25): 8-11.
5.	Bakhshinejad A J, Ramer D J, Dunsmore A K, et al. Effects of intensity and fatigue on the kinetics and kinematics of the barbell squat, bench press, and deadlift in experienced lifters: a systematic review. Sports Med Open, 2025, 11(1): 115.
6.	楊成, 劉鐵, 安鈺峰. 跑步速度對男子運動員下肢運動學、動力學特征及足底壓力分布的影響研究. 應用力學學報, 2025, 42(4): 928-939.
7.	高照, 段銳. 運動性疲勞的分子生物學機制及相關特異性基因靶點的研究進展. 生理科學進展, 2024, 55(1): 13-20.
8.	Schwiete C, Roth C, Mester J, et al. Overlaps of skeletal muscle fatigue and skeletal muscle damage: the muscle injury continuum. Sports Med Open, 2025, 11(1): 73.
9.	張娜娜. 不同姿勢—時間—場景下駕駛人員肌肉疲勞特性研究. 邯鄲: 河北工程大學, 2025.
10.	Wiens L, Losciale J M, Fliss M D, et al. Does high-intensity interval training increase muscle strength, muscle mass, and muscle endurance? a systematic review and meta-analysis. Sports, 2025, 13(9): 293.
11.	Hackett D A, Ghayomzadeh M, Farrell S N, et al. Influence of total repetitions per set on local muscular endurance: a systematic review with meta-analysis and meta-regression. Sci Sports, 2022, 37(5-6): 405-420.
12.	宋濤, 吳鈺祥, 徐國棟, 等. 不同程度血流限制結合低強度抗阻訓練的增肌效應研究進展. 中華物理醫學與康復雜志, 2023, 45(5): 473-476.
13.	Brenner J S, Watson A. Overuse injuries, overtraining, and burnout in young athletes. Pediatrics, 2024, 153(2): e2023065129.
14.	Fiala O, Hanzlova M, Borska L, et al. Beyond physical exhaustion: understanding overtraining syndrome through the lens of molecular mechanisms and clinical manifestation. Sports Med Health Sci, 2025, 7(4): 237-248.
15.	Bernard K. \begindocument$\dot \mathrmV\mathrmO_2 $\enddocument (non-)linear increase in ramp-incremental exercise vs \begindocument$\dot \mathrmV\mathrmO_2 $\enddocument slow component in constant-power exercise: underlying mechanisms. Respir Physiol Neurobiol, 2023, 311: 104023.
16.	嚴仕嘉, 楊曄, 易鵬. 基于實時肌肉疲勞特征融合的表面肌電手勢識別增強算法. 生物醫學工程學雜志, 2024, 41(5): 958-968.
17.	白嘯天, 朱瑤佳, 霍洪峰. 運動性疲勞對跑步足踝肌肉協同特征的影響. 中國運動醫學雜志, 2023, 42(3): 191-200.
18.	Xu K, Blazevich A J, Boullosa D, et al. Optimizing post-activation performance enhancement in athletic tasks: a systematic review with meta-analysis for prescription variables and research methods. Sports Med, 2025, 55(4): 977-1008.
19.	Lai K Y, Hsu C H, Lin Y C, et al. Effect of induced extrinsic and intrinsic hand and forearm muscular fatigue on the control of finger force during piano playing. Hum Mov Sci, 2025, 99: 103319.
20.	吳倩, 魏夢力, 曹偲佳, 等. 跑步疲勞進程中人體下肢著地沖擊時的肌肉激活變化特征. 中國康復理論與實踐, 2024, 30(10): 1215-1223.
21.	方博儒, 仇大偉, 白洋, 等. 表面肌電信號在肌肉疲勞研究中的應用綜述. 計算機科學與探索, 2024, 18(9): 2261-2275.
22.	Qiu X, Zhao H, Xu P, et al. Research on a calculation model of ankle-joint-torque-based sEMG. Sensors, 2024, 24(9): 2875.
23.	Tao Q, Li Z, Liu L, et al. Upper limb muscle strength prediction based on motion capture and sEMG data//International Conference on Automation and Computing. London: IEEE, 2019: 1-5.
24.	韓歡, 呂倩, 尹冠軍, 等. 融合肌電和超聲射頻信號特征的骨骼肌肌力評估研究. 中國生物醫學工程學報, 2024, 43(3): 306-314.
25.	陳誠. 氣動肌肉與彈性元件組合驅動的下肢外骨骼控制研究. 武漢: 華中科技大學, 2023.
26.	Brink K J, Wiles T M, Stergiou N, et al. Higher entropy in movements may promote enhanced upper limb motor precision. Hum Mov Sci, 2025, 102: 103368.
27.	Contessa P, Adam A, De Luca C J. Motor unit control and force fluctuation during fatigue. J Appl Physiol, 2009, 107(1): 235-243.
28.	Rohmert W. Ermittlung von erholungspausen für statische arbeit des menschen. Int Z Angew Physiol Einschl Arbeitsphysiol, 1960, 18: 123-164.
29.	Monod H, Scherrer J. The work capacity of a synergic muscular group. Ergonomics, 1965, 8(3): 329-338.
30.	Chorley A, Lamb K L. The application of critical power, the work capacity above critical power (W′), and its reconstitution: a narrative review of current evidence and implications for cycling training prescription. Sports, 2020, 8(9): 123.
31.	Hartman E M, Michael K, Kirsten T, et al. Sequential oxygen mismatch from skeletal muscle to prefrontal cortex underpins the rate of exhaustion during all-out exercise. Med Sci Sports Exerc, 2025. DOI: 10.1249/MSS.0000000000003771.
32.	魏夢力, 鐘亞平, 吳倩, 等. 跑步疲勞進程中人體下肢肌肉協同模式變化. 中國康復醫學雜志, 2024, 39(9): 1295-1303.
33.	管露露, 戚博特, 馮讀碩, 等. 神經遞質及其受體在運動中樞性疲勞中的作用機制. 生物化學與生物物理進展, 2025, 52(6): 1321-1336.
34.	Tiller N B, Millet G Y. Decoding ultramarathon: muscle damage as the main impediment to performance. Sports Med, 2025, 55(3): 535-543.
35.	Lee M J, Sung J Y, Kim J. Effect of low-intensity high-repetition versus high-intensity low-repetition elastic band resistance training on functional physical fitness and myokine levels in older adults. Appl Sci, 2025, 15(2): 757.
36.	耿治中, 王金昊, 曹國歡, 等. 高溫高濕與低氧環境對運動者能量代謝和骨骼肌氧合的差異. 中國組織工程研究, 2025, 29(32): 6866-6876.
37.	Frutos S, Sierra S A, Flor L, et al. Influence of a diaphragmatic fatigue protocol in healthy male population on muscle strength, respiratory function and exercise capacity: a randomized clinical trial. J Exerc Sci Fit, 2025, 23(4): 473-479.
38.	Afsharipour B, Soedirdjo S, Merletti R. Two-dimensional surface EMG: the effects of electrode size, interelectrode distance and image truncation. Biomed Signal Process Control, 2019, 49: 298-307.
39.	厲中山, 王春露, 劉潔, 等. 短期低頻脈沖磁場誘導經典瞬時感受器電位通道1對肱二頭肌最大自主收縮力與力量耐力的影響. 中國組織工程研究, 2023, 27(11): 1796-1804.
40.	Zajac F E. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit Rev Biomed Eng, 1989, 17(4): 359-411.
41.	楊俊超, 盧智慧, 李祥鑫, 等. 基于W’平衡模型的4分鐘低量高強度間歇運動能量代謝特征分析. 中國運動醫學雜志, 2025, 44(5): 358-364.
42.	畢志遠, 王欣欣, 顧正秋, 等. 臨界功率: 生理學基礎與實踐應用. 成都體育學院學報, 2026, 52(1): 1-12.
43.	Skiba P F, Chidnok W, Vanhatalo A, et al. Modeling the expenditure and reconstitution of work capacity above critical power. Med Sci Sports Exerc, 2012, 44(8): 1526-1532.
44.	Li Z, Wang X, Li Q, et al. Muscle fatigue identification and prediction in motion using wearable device with power and torque-based features. Wearable Electron, 2025, 2: 62-68.
45.	Xu H, Xue Y, Zhou Z, et al. Retentive capacity of power output and linear versus non-linear mapping of power loss in the isotonic muscular endurance test. Sci Rep, 2021, 11(1): 22677.
46.	Hill D W. The critical power concept: a review. Sports Med, 1993, 16(4): 237-254.
47.	Alias R S A, Andrade N A A, Castilla P A, et al. Determining critical power and W’ in running: accuracy of different two-point models using the power metric. Proc Inst Mech Eng P J Sports Eng Technol, 2025, 239(4): 734-740.

1. 牛惠祺, 張弼, 劉麗剛, 等. 人體肌肉狀態疲勞監測及其在外骨骼交互控制中的應用. 生物醫學工程學雜志, 2023, 40(4): 654-662.
2. Pethick J, Tallent J. The neuromuscular fatigue-induced loss of muscle force control. Sports, 2022, 10(11): 184.
3. 陀浩雯. 疲勞前后消防員裸足/穿鞋跑的生物力學特征研究. 桂林: 廣西師范大學, 2025.
4. 郝珊珊. 基于運動生物力學原理探索老年人運動損傷的防護對策. 當代體育科技, 2025, 15(25): 8-11.
5. Bakhshinejad A J, Ramer D J, Dunsmore A K, et al. Effects of intensity and fatigue on the kinetics and kinematics of the barbell squat, bench press, and deadlift in experienced lifters: a systematic review. Sports Med Open, 2025, 11(1): 115.
6. 楊成, 劉鐵, 安鈺峰. 跑步速度對男子運動員下肢運動學、動力學特征及足底壓力分布的影響研究. 應用力學學報, 2025, 42(4): 928-939.
7. 高照, 段銳. 運動性疲勞的分子生物學機制及相關特異性基因靶點的研究進展. 生理科學進展, 2024, 55(1): 13-20.
8. Schwiete C, Roth C, Mester J, et al. Overlaps of skeletal muscle fatigue and skeletal muscle damage: the muscle injury continuum. Sports Med Open, 2025, 11(1): 73.
9. 張娜娜. 不同姿勢—時間—場景下駕駛人員肌肉疲勞特性研究. 邯鄲: 河北工程大學, 2025.
10. Wiens L, Losciale J M, Fliss M D, et al. Does high-intensity interval training increase muscle strength, muscle mass, and muscle endurance? a systematic review and meta-analysis. Sports, 2025, 13(9): 293.
11. Hackett D A, Ghayomzadeh M, Farrell S N, et al. Influence of total repetitions per set on local muscular endurance: a systematic review with meta-analysis and meta-regression. Sci Sports, 2022, 37(5-6): 405-420.
12. 宋濤, 吳鈺祥, 徐國棟, 等. 不同程度血流限制結合低強度抗阻訓練的增肌效應研究進展. 中華物理醫學與康復雜志, 2023, 45(5): 473-476.
13. Brenner J S, Watson A. Overuse injuries, overtraining, and burnout in young athletes. Pediatrics, 2024, 153(2): e2023065129.
14. Fiala O, Hanzlova M, Borska L, et al. Beyond physical exhaustion: understanding overtraining syndrome through the lens of molecular mechanisms and clinical manifestation. Sports Med Health Sci, 2025, 7(4): 237-248.
15. Bernard K. \begindocument$\dot \mathrmV\mathrmO_2 $\enddocument (non-)linear increase in ramp-incremental exercise vs \begindocument$\dot \mathrmV\mathrmO_2 $\enddocument slow component in constant-power exercise: underlying mechanisms. Respir Physiol Neurobiol, 2023, 311: 104023.
16. 嚴仕嘉, 楊曄, 易鵬. 基于實時肌肉疲勞特征融合的表面肌電手勢識別增強算法. 生物醫學工程學雜志, 2024, 41(5): 958-968.
17. 白嘯天, 朱瑤佳, 霍洪峰. 運動性疲勞對跑步足踝肌肉協同特征的影響. 中國運動醫學雜志, 2023, 42(3): 191-200.
18. Xu K, Blazevich A J, Boullosa D, et al. Optimizing post-activation performance enhancement in athletic tasks: a systematic review with meta-analysis for prescription variables and research methods. Sports Med, 2025, 55(4): 977-1008.
19. Lai K Y, Hsu C H, Lin Y C, et al. Effect of induced extrinsic and intrinsic hand and forearm muscular fatigue on the control of finger force during piano playing. Hum Mov Sci, 2025, 99: 103319.
20. 吳倩, 魏夢力, 曹偲佳, 等. 跑步疲勞進程中人體下肢著地沖擊時的肌肉激活變化特征. 中國康復理論與實踐, 2024, 30(10): 1215-1223.
21. 方博儒, 仇大偉, 白洋, 等. 表面肌電信號在肌肉疲勞研究中的應用綜述. 計算機科學與探索, 2024, 18(9): 2261-2275.
22. Qiu X, Zhao H, Xu P, et al. Research on a calculation model of ankle-joint-torque-based sEMG. Sensors, 2024, 24(9): 2875.
23. Tao Q, Li Z, Liu L, et al. Upper limb muscle strength prediction based on motion capture and sEMG data//International Conference on Automation and Computing. London: IEEE, 2019: 1-5.
24. 韓歡, 呂倩, 尹冠軍, 等. 融合肌電和超聲射頻信號特征的骨骼肌肌力評估研究. 中國生物醫學工程學報, 2024, 43(3): 306-314.
25. 陳誠. 氣動肌肉與彈性元件組合驅動的下肢外骨骼控制研究. 武漢: 華中科技大學, 2023.
26. Brink K J, Wiles T M, Stergiou N, et al. Higher entropy in movements may promote enhanced upper limb motor precision. Hum Mov Sci, 2025, 102: 103368.
27. Contessa P, Adam A, De Luca C J. Motor unit control and force fluctuation during fatigue. J Appl Physiol, 2009, 107(1): 235-243.
28. Rohmert W. Ermittlung von erholungspausen für statische arbeit des menschen. Int Z Angew Physiol Einschl Arbeitsphysiol, 1960, 18: 123-164.
29. Monod H, Scherrer J. The work capacity of a synergic muscular group. Ergonomics, 1965, 8(3): 329-338.
30. Chorley A, Lamb K L. The application of critical power, the work capacity above critical power (W′), and its reconstitution: a narrative review of current evidence and implications for cycling training prescription. Sports, 2020, 8(9): 123.
31. Hartman E M, Michael K, Kirsten T, et al. Sequential oxygen mismatch from skeletal muscle to prefrontal cortex underpins the rate of exhaustion during all-out exercise. Med Sci Sports Exerc, 2025. DOI: 10.1249/MSS.0000000000003771.
32. 魏夢力, 鐘亞平, 吳倩, 等. 跑步疲勞進程中人體下肢肌肉協同模式變化. 中國康復醫學雜志, 2024, 39(9): 1295-1303.
33. 管露露, 戚博特, 馮讀碩, 等. 神經遞質及其受體在運動中樞性疲勞中的作用機制. 生物化學與生物物理進展, 2025, 52(6): 1321-1336.
34. Tiller N B, Millet G Y. Decoding ultramarathon: muscle damage as the main impediment to performance. Sports Med, 2025, 55(3): 535-543.
35. Lee M J, Sung J Y, Kim J. Effect of low-intensity high-repetition versus high-intensity low-repetition elastic band resistance training on functional physical fitness and myokine levels in older adults. Appl Sci, 2025, 15(2): 757.
36. 耿治中, 王金昊, 曹國歡, 等. 高溫高濕與低氧環境對運動者能量代謝和骨骼肌氧合的差異. 中國組織工程研究, 2025, 29(32): 6866-6876.
37. Frutos S, Sierra S A, Flor L, et al. Influence of a diaphragmatic fatigue protocol in healthy male population on muscle strength, respiratory function and exercise capacity: a randomized clinical trial. J Exerc Sci Fit, 2025, 23(4): 473-479.
38. Afsharipour B, Soedirdjo S, Merletti R. Two-dimensional surface EMG: the effects of electrode size, interelectrode distance and image truncation. Biomed Signal Process Control, 2019, 49: 298-307.
39. 厲中山, 王春露, 劉潔, 等. 短期低頻脈沖磁場誘導經典瞬時感受器電位通道1對肱二頭肌最大自主收縮力與力量耐力的影響. 中國組織工程研究, 2023, 27(11): 1796-1804.
40. Zajac F E. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit Rev Biomed Eng, 1989, 17(4): 359-411.
41. 楊俊超, 盧智慧, 李祥鑫, 等. 基于W’平衡模型的4分鐘低量高強度間歇運動能量代謝特征分析. 中國運動醫學雜志, 2025, 44(5): 358-364.
42. 畢志遠, 王欣欣, 顧正秋, 等. 臨界功率: 生理學基礎與實踐應用. 成都體育學院學報, 2026, 52(1): 1-12.
43. Skiba P F, Chidnok W, Vanhatalo A, et al. Modeling the expenditure and reconstitution of work capacity above critical power. Med Sci Sports Exerc, 2012, 44(8): 1526-1532.
44. Li Z, Wang X, Li Q, et al. Muscle fatigue identification and prediction in motion using wearable device with power and torque-based features. Wearable Electron, 2025, 2: 62-68.
45. Xu H, Xue Y, Zhou Z, et al. Retentive capacity of power output and linear versus non-linear mapping of power loss in the isotonic muscular endurance test. Sci Rep, 2021, 11(1): 22677.
46. Hill D W. The critical power concept: a review. Sports Med, 1993, 16(4): 237-254.
47. Alias R S A, Andrade N A A, Castilla P A, et al. Determining critical power and W’ in running: accuracy of different two-point models using the power metric. Proc Inst Mech Eng P J Sports Eng Technol, 2025, 239(4): 734-740.

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