Functional muscle group- and sex-specific parameters for a three-compartment controller muscle fatigue model applied to isometric contractions

Abstract

The three-compartment controller with enhanced recovery (3CC-r) model of muscle fatigue has previously been validated separately for both sustained (SIC) and intermittent isometric contractions (IIC) using different objective functions, but its performance has not yet been tested against both contraction types simultaneously using a common objective function. Additionally, prior validation has been performed using common parameters at the joint level, whereas applications to many real-world tasks will require the model to be applied to agonistic and synergistic muscle groups. Lastly, parameters for the model have previously been derived for a mixed-sex cohort not considering the differece in fatigabilities between the sexes. In this work we validate the 3CC-r model using a comprehensive isometric contraction database drawn from 172 publications segregated by functional muscle group (FMG) and sex. We find that prediction errors are reduced by 19% on average when segregating the dataset by FMG alone, and by 34% when segregating by both sex and FMG. However, minimum prediction errors are found to be higher when validated against both SIC and IIC data together using torque decline as the outcome variable than when validated sequentially against hypothesized SIC intensity-endurance time curves with endurance time as the outcome variable and against raw IIC data with torque decline as the outcome variable.

Introduction

Muscle fatigue is an inevitable but reversible byproduct of physical exertion which manifests in a temporarily reduced force capacity for the muscles involved in the activity (Bigland-Ritchie et al., 1995). It is a complex biochemical process (Vøllestad, 1997) that affects all skeletal muscle for all activity types regardless of duration, but the extent to which each muscle is affected can depend on a multitude of physiological, task, and external factors (Chaffin, 1973). Since it is eminently desirable that tasks never demand more of the worker than their physical limits to prevent the incidence of injuries, it is immensely helpful to predict how a certain task affects muscle strength over time. Measuring strength declines over the course of a real-world task can be tedious, time-consuming, and invasive, and the results will likely be inapplicable to another person or situation. Mathematical models of fatigue overcome these difficulties with their predictive abilities. While each model may be suitable for a certain subset of tasks and may specify its own limitations, in general it can reasonably predict the course of strength decline for a particular joint in the human body (Rashedi and Nussbaum, 2015).

The three-compartment controller (3CC) family of models (Frey-Law et al., 2012, Looft et al., 2018, Xia and Frey Law, 2008) divide the constituent motor units (MU) of a muscle into three states or compartments—resting, active, and fatigued—and calculate the relative size of each to estimate the extent of fatigue. They specify rules (Eqs. (1), (2), (3), (4), (5), (6)) for determining the rate at which MUs transition from one state into another.

$$\frac{d{M}_R}{\mathit{dt}}=-C\left(t\right)+r\left(k,TL\right)\times R\times M_F$$

$$\frac{d{M}_A}{\mathit{dt}}=C\left(t\right)-F\times M_A$$

$$\frac{d{M}_F}{\mathit{dt}}=F\times M_A-r\left(k,TL\right)\times R\times M_F$$

$$\mathrm{If}{M}_A<TL\mathrm{and}{M}_R>TL-M_A,C\left(t\right)=L_D\times \left(TL-M_A\right)$$

$$\mathrm{If}{M}_A<TL\mathrm{and}{M}_R<TL-M_A,C\left(t\right)=L_D\times M_R$$

$$\mathrm{If}{M}_A\ge TL,C\left(t\right)=L_R\times \left(TL-M_A\right)$$

Here, $M_R$, $M_A$, and $M_F$ are the fraction of motor units in the muscle that are currently resting, active, and fatigued, respectively. $C\left(t\right)$ is a bidirectional activation-deactivation drive. F and R are the fatigue and recovery rate constants, respectively, and $r$ is a recovery multiplier that is a discontinuous function of target load (TL) and a unitless optimization parameter k. r is defined by Eq. (7). The model is relatively insensitive to the muscle force development ($L_D$) and relaxation ($L_R$) factors (Xia and Frey-Law, 2008), so a constant value of 10 is chosen for both.

$$r=\left\{\begin{array}{c}k\mathrm{if}\mathit{TL}=0\\ 1\mathrm{if}TL>0\end{array}\right.$$

The 3CC model, in its present form, was developed by first determining the values of F and R by running a 2-parameter sweep against sustained isometric contraction (SIC) data. Endurance time (ET) was chosen as the output variable for differing input task intensities (Frey-Law et al., 2012). This approach ensured the best possible F and R values for predicting ET for a given sustained task intensity, but not necessarily for predicting the precise course of fatigue during that period. k was then determined by running a 1-parameter sensitivity analysis using the predetermined F and R values against intermittent isometric contraction (IIC) data, but at this stage the input and output variables were reversed—torque declines were estimated for given input times (Looft et al., 2018). k—subsequently calculated—produced the best fit to IIC data for the given F and R values. Ideally, all three parameters would have been chosen independently for IIC data, but it is likely that the F and R values obtained thereby would have conflicted with those obtained from the SIC data since the two optimization steps had different variables as their objective functions. It is therefore of interest to analyze the performance of the model when SIC data is held to the same standard as IIC data, and to obtain parameter sets that represent all isometric tasks equally well.

Additionally, most real-world tasks asymmetrically and alternately load agonistic and antagonistic muscle groups around the active joints. Furthermore, since synergists assist the agonistic muscles, it is desirable to obtain parameters that represent both agonists and synergists together—hereon referred to as functional muscle groups (FMG)—for a given directional contraction. Also of interest are the differences in fatigability between men and women, which, if present, will manifest in different model parameters for the sexes and reduced prediction errors. In this study, we analyze the effects of segregating a large isometric contraction (IC) database variously on the basis of sex and FMG to refine the applicability of the 3CC-r model to specific individuals and situations. We treat both SIC and IIC data as belonging to a continuum of isometric contraction (IC) data bearing three common attributes—TL, duty cycle (DC) and cycle time (CT). The distinction between SICs and IICs are made by assigning DC = 100% for SICs while IICs inherently have DC < 100%. We segregate the data by active joint, by FMG, and then by both FMG and sex and run this data through the model to estimate the parameters that best represent each segregated dataset.