A novel approach for determining fatigue resistances of different muscle groups in static cases

Abstract: In ergonomics and biomechanics, muscle fatigue models based on maximum endurance time (MET) models are often used to integrate fatigue effect into ergonomic and biomechanical application. However, due to the empirical principle of those MET models, the disadvantages of this method are: 1) the MET models cannot reveal the muscle physiology background very well; 2) there is no general formation for those MET models to predict MET. In this paper, a theoretical MET model is extended from a simple muscle fatigue model with consideration of the external load and maximum voluntary contraction in passive static exertion cases. The universal availability of the extended MET model is analyzed in comparison to 24 existing empirical MET models. Using mathematical regression method, 21 of the 24 MET models have intraclass correlations over 0.9, which means the extended MET model could replace the existing MET models in a general and computationally efficient way. In addition, an important parameter, fatigability (or fatigue resistance) of different muscle groups, could be calculated via the mathematical regression approach. Its mean value and its standard deviation are useful for predicting MET values of a given population during static operations. The possible reasons influencing the fatigue resistance were classified and discussed, and it is still a very challenging work to find out the quantitative relationship between the fatigue resistance and the influencing factors.

• The paper introduces a novel theoretical model that derives a Maximum Endurance Time equation by extending a dynamic muscle fatigue model to quantify fatigue resistance.
• The paper uses regression analysis to optimize fatigue resistance values across muscle groups, achieving an ICC over 0.90 in most cases.
• The paper demonstrates practical applications by offering a unified framework for ergonomic assessments and virtual human simulations in static exertion scenarios.

This paper introduces a novel, theoretically grounded approach to model muscle fatigue during static exertions, specifically focusing on determining Maximum Endurance Time (MET) and quantifying fatigue resistance ($m$) across different muscle groups. It addresses limitations of existing empirical MET models, which often lack physiological basis and a universal formulation.

The core of the approach is an extension of a simple dynamic muscle fatigue model previously developed by the authors [ma2008nsd]. This dynamic model describes the decrease in a muscle's current maximum exertable force ($F_{cem}(t)$) based on the current load ($F_{load}(t)$), the muscle's maximum voluntary contraction ($MVC$), and a fatigue ratio parameter ($k$). The model is represented by the differential equation:

$$\frac{dF_{cem}(t)}{dt} = -k \frac{F_{cem}(t)}{MVC}F_{load}(t)$$

For static cases where the load $F_{load}$ is constant and expressed as a fraction of MVC ($f_{MVC} = F_{load}/MVC$), this model is integrated to derive an equation for MET, the time until $F_{cem}(t)$ drops to the level of the required $F_{load}$:

$MET = -\dfrac{ ln(f_{MVC})}{k\, f_{MVC}}$

This equation represents the extended MET model. Unlike empirical models, it is derived from a theoretical basis linked to motor unit recruitment. A key feature is the parameter $k$, representing the muscle group's fatigability. Its reciprocal, $m = 1/k$, is defined as the fatigue resistance.

To validate this extended MET model and determine typical fatigue resistance values, the authors performed a mathematical regression analysis comparing it against 24 existing empirical MET models from the literature, covering general tasks, shoulder, elbow, hand, and hip/back muscle groups.

The regression aimed to find the optimal fatigue resistance parameter $m$ for each empirical model $f(x)$ by fitting it to the extended MET model $p(x)$ using a linear relationship $f(x) = m \cdot p(x)$ (assuming zero intercept). The parameter $m$ was calculated by minimizing the sum of squared differences:

$$m = \dfrac{\sum\limits_{i=1}^{N}p(x_i)f(x_i)}{\sum\limits_{i=1}^{N}p(x_i)^2}$$

The goodness-of-fit was evaluated using the Intraclass Correlation coefficient (ICC) before ($ICC_1$) and after ($ICC_2$) the regression.

Key Findings:

1. Generalizability: The regression significantly improved the fit for most models. 21 out of the 24 empirical models achieved an $ICC_2 > 0.90$ after regression, demonstrating that the extended MET model, by adjusting the fatigue resistance parameter $m$, can effectively represent the fatigue behavior described by diverse empirical models across different muscle groups.
2. Fatigue Resistance Quantification: The regression yielded specific fatigue resistance values ($m$) for each empirical model. These were grouped by body part, and mean ($\bar{m}$) and standard deviation ($\sigma_m$) were calculated:
   • General models: $\bar{m} = 0.8135, \sigma_m = 0.2320$
   • Shoulder models: $\bar{m} = 0.7562, \sigma_m = 0.4347$
   • Elbow models: $\bar{m} = 0.8609, \sigma_m = 0.4079$
   • Hand model: $m = 0.8907$
   • Hip/Back models: $\bar{m} = 1.9701, \sigma_m = 1.1476$
   • Hip/back muscles showed the highest average fatigue resistance but also the greatest variability.
3. Population Prediction: Plotting the extended MET model using $\bar{m} \pm \sigma_m$ showed it could encompass the predictions of most existing MET models, particularly within the common 15% to 80% MVC range, suggesting its utility for predicting endurance for a population.
4. Influencing Factors: The paper discusses factors contributing to variations in fatigue resistance ($m$), including systematic experimental bias, inter-individual differences, intra-muscle group variations (due to gender, age, fiber composition, posture), and inter-muscle group differences (linked to muscle fiber type composition and load-sharing mechanisms).

Practical Implications:

• Unified Fatigue Modeling: The extended MET model offers a simpler, theoretically grounded, and computationally efficient alternative to using multiple empirical MET models.
• Ergonomic Assessment: It allows quantifying muscle-group-specific fatigue resistance, which can be determined via regression from experimental data for a target population. This parameter ($m$ or $k$) can then be used in the extended MET model for fatigue prediction in ergonomic analyses.
• Virtual Human Simulation: The model is suitable for integration into digital human modeling tools to simulate fatigue effects during static or quasi-static tasks, aiding in proactive ergonomic design.

In conclusion, the paper presents a valuable theoretical framework for understanding and predicting muscle fatigue in static exertions, offering a generalizable model and a method to quantify fatigue resistance across different muscle groups and populations.