Modification of a three-compartment muscle fatigue model to predict peak torque decline during intermittent tasks

Abstract

This study aimed to test whether adding a rest recovery parameter, r, to the analytical three-compartment controller (3CC) fatigue model (Xia and Frey Law, 2008) will improve fatigue estimates during intermittent contractions. The 3CC muscle fatigue model uses differential equations to predict the flow of muscle between three muscle states: Resting (M_R), Active (M_A), and Fatigued (M_F). This model uses a feedback controller to match the active state to target loads and two joint-specific parameters: F, fatigue rate controlling flow from active to fatigued compartments) and R, the recovery rate controlling flow from the fatigued to the resting compartments. This model does well to predict intensity-endurance time curves for sustained isometric tasks. However, previous studies find when rest intervals are present that the model over predicts fatigue. Intermittent rest periods would allow for the occurrence of subsequent reactive vasodilation and post-contraction hyperemia. We hypothesize a modified 3CC-r fatigue model will improve predictions of force decay during intermittent contractions with the addition of a rest recovery parameter, r, to augment recovery during rest intervals, representing muscle re-perfusion. A meta-analysis compiling intermittent fatigue data from 63 publications reporting decline in peak torque (% torque decline) were used for comparison. The original model over-predicted fatigue development from 19 to 29% torque decline; the addition of a rest multiplier significantly improved fatigue estimates to 6-10% torque decline. We conclude the addition of a rest multiplier to the three-compartment controller fatigue model provides a physiologically consistent modification for tasks involving rest intervals, resulting in improved estimates of muscle fatigue.

Keywords: Ergonomics; Isometric contraction; Mathematical modeling; Meta-analysis; Muscle fatigue.

Figure 1.

A) Visual representation of the three-compartment controller (3CC) fatigue model proposed by Xia and Frey Law (2008) and B) the modified (3CCr) fatigue model with the addition of a rest multiplier, r, to augment recovery during rest intervals, when the target load, TL, equals 0 in the simulated task.

Figure 2.

Scatterplot of observed versus predicted fatigue (relative decline in peak torque, % torque decline) by joint region for the original 3CC fatigue model (r = 1): A) ankle (blue circles); B) knee (red squares); C) elbow (green diamonds); and D) hand/grip (cyan triangles). The identity line is shown to indicate 100% agreement. Data above the identity line reflect over-predictions of muscle fatigue whereas data below the line reflect under-predictions (i.e., directional errors).

Figure 3.

Absolute errors, unadjusted for time, between modeled and observed percent torque decline (%TD) across values of r for muscles about each joint region: A) ankle, B) knee, C) elbow, D) hand/grip, and E) all muscles. The original three-compartment controller (3CC) fatigue model error is shown as the red square (r = 1) and the modified 3CC-r fatigue model errors are shown as solid lines across varying r values (r ≥ 2). Optimal r values (r_opt) are displayed as solid vertical lines, representing the best estimates for r_opt. Errors resulting in ± 1 %TD error from the lowest possible error represent the range of potential optimal r values (vertical dashed lines). Minimally important differences (MID) of 50% reduction in original model errors (r = 1) are shown as the horizontal dashed line.

Figure 4.

Scatterplot of observed versus predicted fatigue (relative decline in peak torque, % torque decline) by joint region for the modified 3CCr fatigue model using the optimal r value: A) ankle (blue circles); B) knee (red squares); C) elbow (green diamonds); and D) hand/grip (cyan triangles). The identity line is shown to indicate where 100% agreement is located. Note that data are distributed more evenly above and below the identity lines than when r = 1 (see Figure 2).

Figure 5.

Absolute errors for each joint region: A) ankle; B) knee; C) elbow; and D) hand/grip, as a function of r (x-axis). The errors are stratified by 2 task conditions: 100% MVC and duty cycle (duty cycle) ≥ 50% (dark red solid line) or MVC < 100% and duty cycle < 50%. Note the largest differences in error (% torque decline) between task conditions are seen at r=1 (original 3CC model), but are less for the modified model in the region of r_opt values (vertical gray lines).