Abstract
Purpose
The time course of maximal voluntary isometric contraction (MVIC) force is of particular interest whenever force capacities are a limiting factor, e.g., during heavy manual work or resistance training (RT) sessions. The objective of this work was to develop a mathematical model of this time course that is suitable for optimization of complex loading schemes.
Materials and methods
We compiled a literature overview of existing models and justified the need for a new model. We then constructed a phenomenological ordinary differential equation model to describe the time course of MVIC force during voluntary isometric contractions and at rest. We validated the model with a comprehensive set of published data from the elbow flexors. For this, we estimated parameters from a subset of the available data and used those estimates to predict the remaining data. Afterwards, we illustrated the benefits of our model using the calibrated model to (1) analyze fatigue and recovery patterns observed in the literature (2) compute a work–rest schedule that minimizes fatigue (3) determine an isometric RT session that maximizes training volume.
Results
We demonstrated that our model (1) is able to describe MVIC force under complex loading schemes (2) can be used to analyze fatigue and recovery patterns observed in the literature (3) can be used to optimize complex loading schemes.
Conclusions
We developed a mathematical model of the time course of MVIC force that can be efficiently employed to optimize complex loading schemes. This enables an optimal use of MVIC force capacities.
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Abbreviations
- FTI:
-
Force-time integral
- MAE:
-
Mean absolute error
- MVIC:
-
Maximal voluntary isometric contraction
- ODE:
-
Ordinary differential equation
- SD:
-
Standard deviation
- RT:
-
Resistance training
- WRSS:
-
Weighted residual sum of squares
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Acknowledgements
We gratefully thank Dr. Janet L. Taylor of Neuroscience Research Australia, Sydney, Australia for providing and explaining the experimental data of the studies Taylor et al. (1999, 2000), Søgaard et al. (2006), and Smith et al. (2007). We furthermore would like to thank the anonymous reviewers whose comments helped to improve this manuscript significantly.
Funding
JLH acknowledges support from the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (Graduate School 220), funded by the Deutsche Forschungsgemeinschaft (DFG) within the German Excellence Initiative.
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JLH and CK conceived the idea for this work. JLH conducted the literature research, developed the model, performed the numerical experiments, and drafted the manuscript. JLH, CK, and JPS discussed and edited the draft. JLH, CK, and JPS revised the manuscript. JLH, CK, and JPS approved the final version of the manuscript.
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A preprint of this work is available on bioRxiv.org. URL https://www.biorxiv.org/content/early/2018/01/30/256578, DOI https://doi.org/10.1101/256578.
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Communicated by Jean-René Lacour.
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Herold, J.L., Kirches, C. & Schlöder, J.P. A phenomenological model of the time course of maximal voluntary isometric contraction force for optimization of complex loading schemes. Eur J Appl Physiol 118, 2587–2605 (2018). https://doi.org/10.1007/s00421-018-3983-z
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DOI: https://doi.org/10.1007/s00421-018-3983-z