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. Author manuscript; available in PMC: 2019 Aug 22.
Published in final edited form as: J Biomech. 2018 Jun 18;77:16–25. doi: 10.1016/j.jbiomech.2018.06.005

Modification of a Three-Compartment Muscle Fatigue Model to Predict Peak Torque Decline During Intermittent Tasks

John M Looft 1,2, Nicole Herkert 2, Laura Frey-Law 2
PMCID: PMC6092960  NIHMSID: NIHMS978936  PMID: 29960732

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 (MR), Active (MA), and Fatigued (MF). 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 – 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: Muscle Fatigue, Isometric Contraction, Meta-analysis, Mathematical Modeling, Ergonomics

Introduction

Musculoskeletal disorders (MSDs) account for 33% of all illnesses and injuries requiring days off work ( (Bureau of Labor Statistics, 2012). Accordingly MSDs continue to financially burden the U.S. with annual direct costs estimated at $50 billion per year over a decade ago (Silverstein, et al., 2002, Yelin, et al., 1995). Realistically, the total economic burden of MSDs is on the order of 100’s of billions of dollars per year (Buckle and Devereux, 2002, Yelin, et al., 1995) and MSDs contribute to reduced quality of life and well-being (Larsen, et al., 2017).

Muscle fatigue has been cited as a risk factor for injury in recreational activities (Small, et al., 2010) and in the workplace (Kinali, et al., 2016). However, whether muscle fatigue is a risk factor for MSDs remains unclear. Indirect survey methods (Rodgers, 1992) are often used to assess fatigue development during functional tasks. These subjective measures provide limited objective information, making it difficult to investigate associations between muscle fatigue and MSDs. The advancement of mathematical models may assist in quantifying muscle fatigue development.

Our analytical three-compartment controller (3CC) fatigue model (Figure 1A, Xia and Frey Law, 2008) can accurately predict endurance time for isometric tasks (Frey-Law, et al., 2012), and performs well when compared to other models (Rashedi and Nussbaum, 2015, Sonne and Potvin, 2016). However, under complex conditions (i.e., addition of rest intervals) the accuracy of the 3CC fatigue model declines, likely due to poorer representation of muscle recovery (Rashedi and Nussbaum, 2015, Sonne and Potvin 2015). This may be due to the recovery rate, R, being previously optimized using sustained isometric tasks to failure (Frey-Law, et al., 2012).

Figure 1.

Figure 1.

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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.

During muscle contraction, blood flow is reduced from increased intramuscular pressure on the vascular system (Degens, et al., 1998, Sejersted, et al., 1984). Muscle perfusion decreases with increasing muscle contraction (Humphreys and Lind, 1963, Robergs, et al., 1997), resulting in graded levels of ischemia. Intermittent rest periods allow for the occurrence of subsequent reactive vasodilation and post-contraction hyperemia (Humphreys and Lind, 1963, Nowak and Wennmalm, 1979, Robergs, et al., 1997, Wigmore, et al., 2004). This rebound perfusion can enhance force recovery following fatigue (Cerqueira, et al., 2017).

We propose a modification to the 3CC fatigue model to represent enhanced recovery resulting from reactive hyperemia. We hypothesized predictions of force decay during intermittent contractions would improve with the addition of a rest recovery multiplier, r, to augment recovery during rest intervals. Recognizing there may be additional factors contributing to model errors, the objective of the current study was to test this hypothesis and determine optimal r parameter values for this modified 3CC-r model relative to available data in the literature.

Methods

Modified Fatigue Model

The 3CC fatigue model (Frey-Law, et al., 2012, Xia and Frey Law, 2008) relies on three differential equations to dictate the “flow” between three muscle states: resting (Mr), active (MA), and fatigued (MF, Figure 1A). See Supplemental materials a brief summary with model equations. The proposed modified 3CC-r model (Figure 1B) only alters one equation, augmenting recovery from the fatigued state with a rest recovery multiplier, r:

When TL > 0 dMR/dt = -C(t) + (R)*MF (original 3CC)
When TL = 0 dMR/dt = -C(t) + (R*r)*MF (modified 3CC-r)
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This enhanced recovery occurs during rest intervals only (i.e. TL =0). For example, r=1 maintains the original 3CC model. An r=10 increases recovery 10-fold during rest intervals, leaving it unchanged during contractions.

Literature Review of Intermittent Task Fatigue

Because intermittent fatigue data has not been compiled previously, we performed a large-scale systematic review of available fatigue data for model prediction comparisons. As intermittent isometric fatiguing tasks are rarely maintained until endurance time, we extracted decline in peak torque (% torque decline) as the primary fatigue measure. That is, 0% torque decline represents no fatigue and 100% torque decline represents complete fatigue. Experimentally, participants perform maximal voluntary contractions (MVCs) occasionally throughout the fatigue protocol.

The systematic literature review consisted of two stages. The first stage involved database searches, including: PubMed, CINAHL, Web of Science, and Google Scholar thru December, 2012. A total of 17 search terms/keywords, alone and in multiple combinations, were used, such as: “intermittent”, “static fatigue”, “fatigue”, “isometric”, “endurance”, “muscle torque decline”. In addition, early searches found several studies evaluating fatigue with creatine supplementation compared to controls. Thus, “creatine supplementation” was added as a secondary search term (control group data only). The second stage involved searching the bibliographies of included studies.

Inclusion criteria included: healthy adults, 18–55 years old, voluntary contractions, intermittent isometric tasks at a known intensity and duty cycle, force/torque data reported at discrete time points, and published in English. Exclusion criteria included: dynamic contractions, simultaneous multi-joint functional testing (e.g. squat lifts), body/limb weight as primary resistance, interventional arms of cohort studies (i.e. creatine supplementation) and electrical stimulation. The primary outcome variable, % torque decline, was extracted at 30 second intervals as available. We also extracted sample size, joint region, intensity (%MVC), duty cycle, assessment time (min), author, date, and subject gender(s). A minimum of two authors reviewed potential studies to minimize recording errors.

The literature search produced 2392 potential publications. Applying the inclusion criteria and removing duplicates reduced the number of appropriate articles to 59. Four additional articles were found from bibliographies, for a total of 63 publications. Fatigue data was found for four joint regions (ankle, knee, elbow and hand/grip) but no shoulder or trunk studies met all inclusion/exclusion criteria. Data from each joint region were considered separately because a previous study found endurance times differed between joints for sustained isometric tasks (Frey Law and Avin, 2010). The hand/grip region included studies of grip, first dorsal interosseaus, abductor pollicis brevis, and adductor pollicis, similar to previous fatigue meta-analyses (Frey Law and Avin, 2010). A total of 351 data points were extracted (Table 1), ranging from 58 (knee) to 144 (ankle). Multiple intensity-duty cycle combinations were found for each joint region: ankle (14), knee (9), elbow (9), and hand/grip (17), with as few as 1 or as many as 20 time points reported for each combination. The majority of studies used task conditions at 100% MVC with 50% or higher duty cycle (52.1% of datapoints). Average sample size per datum was 11.9 participants (range 4 – 74). Most studies included males (M, 59.8%) or were a mixed cohort (MX, 26.5%), thus separate analyses by sex were not performed.

Table 1:

Studies included in the meta-analysis by author for the ankle, knee, elbow, and hand/grip regions.

Joint Author, Date Subjects Intensity
(%MVC)		 Duty Cycle
(%)		 Data
Points		 Time
(min)
Ankle (Alway, et al, 1987) 8M 100 50 1 5
(Bemben, et al., 1996) 74M 100 40 2 1
(Birtles, et al., 2002) 22MX 100 50 20 1 – 20
(Birtles, et al., 2003) 10MX 100 50 17 1 – 20
(Chung, et al., 2007) 12M 100 50 11 0.5–5.5
(Egana and Green, 2007) 7M 30, 40, 50,
60, 80, 90		 33 57 1 – 20
(Fimland, et al., 2010) 13M 100 83 6 1–3.5
(Finlayson, et al., 2008) 8F 80 80 1 5
(Kent-Braun, et al., 1994) 8MX 10 40 1 0.5
(Kent-Braun, et al., 2002) 20MX 10 40 1 0.5
(Lanza, et al., 2004) 9M 100 50 3 1–3
(Lanza, et al., 2006) 12MX 100 50 1 2
(Mademli and Arampatzis, 2008) 11M 65 40 2 3,6
(McNeil, et al., 2006) 10M 50 67 1 2
(Mitsukawa, et al., 2009) 7M 100 50 2 3,4
(Russ and Kent-Braun,2003) 8M,8F 100 50 8 1–4
(Russ, et al., 2008) 16MX 100 70 10 1–5
Knee (Armatas, et al., 2010) 13M 100 50 7 0.5–6
(Baker-Fulco, et al., 2006) 7M 32 50 1 35
(Bemben, et al., 2001) 8M 100 50 2 1
(Burnley, 2009) 8M 100 60 9 0.5 – 4.5
(Callahan, et al., 2009) 16MX 100 50 7 0.5 – 4
(Callahan and Kent-Braun, 2011) 11F 100 50 7 0.5 – 3.5
(Hamada, et al., 2003) 4M 100 63 3 0.5 – 2
(Homby, et al., 2009) 10MX 100 50 3 0.5 – 2.5
(Kalmar and Cafarelli, 2006) 8M 50 67 1 1
(Katayama, et al., 2006) 6M 62 50 1 1.5
(Morse, et al., 2008) 12M 100 50 1 1
(Mulder, et al., 2007) 10M 45 60 5 1–5
(Ordway, et al., 1977) 27M 100 50 5 1–5
(Saugen, et al., 1997) 8M 40 60 4 5–40
(Stackhouse, et al., 2001) 20MX 100 71 2 1.5,2
Elbow (Allman and Rice, 2003) 6M 60 60 1 1
(Bemben, et al., 2001) 8M 100 50 2 1
(Bilodeau, 2006) 8MX 100 86 6 0.5–3
(Hunter, et al., 2004) 10M, 10F 50 60 2 9, 23
(Jakobi, et al., 2000) 7M 50 60 1 5
(Jubeau, et al., 2012) 12M 100 21 10 1 – 15.5
(Lloyd, et al., 1991) 13M 30 60 8 5–40
(Mendez-Villanueva, et al., 2009) 9M 50 30, 60 5 4–33
(Muthalib, et al., 2010) 10M 100 21 10 1 – 15.5
(Ordway, et al., 1977) 27M 100 50 5 1–5
(Seghers and Spaepen,2004) 10MX 25, 50 50, 25 2 20
(Taylor, et al., 2000) 9MX 100 50 4 2–8
(Thomas and del Valle,2001) 4MX 50 60 5 3–12
Hand/Grip (Bemben, et al., 1996) 74M 100 40 2 1
(Benwell, et al., 2007b) 12MX 30 60 5 2–5
(Ditor and Hicks, 2000) 12M, 12F 100 71 12 0.5–3
(Duchateau, et al., 2002) 13MX 25 60 1 8
(Fujimoto and Nishizono, 1993) 8M, 6M 40 60 7 2–8
(Fulco, et al., 1994) 8M 50 50 3 1–3
(Fulco, et al., 2001) 12M, 21F 50 50 6 1–3
(Gonzales and Scheuermann, 2007) 11M, 11F 50 50 4 2–13
(Hunter, et al., 2009) 20M, 20F 50 60 2 5,7
(Jaskolska and Jaskolski, 1997) 22M 100 50 1 6
(Liu, et al., 2005) 14MX 100 67 4 0.5 – 2
(Newham and Cady, 1990) 6MX 25, 50, 100 50 12 1 – 10
(Pitcher and Miles, 1997) 9M 80 60 1 15
(Quaine, et al., 2003) 10M 80 50 1 2
(Saito, et al., 2008) 8M, 8F 100 50 12 0.5 – 4
(Thickbroom, et al., 2006) 15MX 40 70 1 20
(Vigouroux and Quaine, 2006) 10M 80 50 1 3
(Wood, et al., 1997) 20F 16, 32, 48 63, 31,21 12 9 – 36
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Original Model Predictions

We assessed the accuracy of the original 3CC fatigue model (e.g., r=1) to predict torque decline for intermittent tasks. This was accomplished by running the model for each joint and set of task conditions extracted from the literature. Because F and R parameter values previously were found to differ between joint regions, we used joint-specific F and R values when assessing the model (Frey-Law, et al., 2012). That is, the F and R values previously optimized for sustained isometric tasks were used to simulate muscle fatigue for intermittent tasks. Model errors between predicted and observed fatigue were assessed in two ways. Directional errors, where positive errors indicated over-estimated fatigue, as well as absolute errors (absolute values) were assessed. Due to varying assessment times reported in the literature, model errors were also adjusted to account for the additive nature of fatigue error over time by dividing each error value by the square root of the assessment time (minutes). That is, at one minute there was no change (divided by one), but as time increased there was a nonlinear reduction in total error, resulting in relatively constant error levels across time (Supplemental Materials, Figure S1).

Modified Model Predictions

A large sensitivity analysis of the new rest recovery parameter, r, was performed. Values from 2–100 were used to predict fatigue (% torque decline) using the 3CC-r fatigue model. That is, for each joint, % torque decline was estimated for all intensity-duty cycle permutations found in the literature at time points from 30 seconds to 40 minutes at 30 second intervals for each r value. Errors between predicted and observed % torque decline were calculated as described above for each value of r. A range of potential optimal r values for each joint region were identified as those producing the smallest marginal mean errors. Ranges for ropt, producing errors within ± 1 % torque decline of these minima, were assessed in lieu of confidence intervals as a measure of the sensitivity of r. Four error determinations were considered: directional and absolute errors, with and without time adjustment, to ensure robust results.

Statistical Analyses

To test our hypothesis, mixed repeated measures analysis of covariance (RM ANCOVA) assessments were performed (SPSS V. 24.0; Chicago, IL) to determine if directional or absolute model errors differed across values of r, for each anatomical joint region. Both unadjusted and time-adjusted errors (see above) were considered. For each analysis, the dependent variable was observed error and r was the repeated independent variable. Intensity (%MVC) and duty cycle were included as covariates in all models, with time only included as a covariate in the unadjusted error models. Sample size was used as a weighting factor. Each joint region was assessed separately as observed fatigue behavior (Frey Law and Avin, 2010) and F and R model parameters differ by joint (Frey-Law, et al., 2012). When r produced significant differences in error, simple contrasts (i.e., using general linear model matrices) were performed to compare the resulting mean errors for each r ≥ 2 to the original 3CC model (r=1). We then considered whether these improved errors met an operationally defined minimally important difference (MID) of at least 50% improvement. Finally, Akaike’s Information Criterion (AIC) was calculated for the original 3CC and modified 3CC-r models, using potential optimal r values, to determine which model produced the best fit for each joint region. Secondary analyses were performed to explore whether task condition (100% MVC vs. < 100% MVC) and/or duty cycle (≥ 50% vs. < 50%) influenced optimal r values. Significance was set at p ≤ 0.05 for all analyses.

Results

Model Comparisons

The original 3CC fatigue model consistently over-predicted fatigue by 23.3% torque decline on average across all joints, task conditions, and times assessed. When predicted versus observed fatigue was plotted (Figure 2), the errors (i.e., deviations from the identity line) tended to increase with greater fatigue (and time Supplemental Figure S1).

Figure 2.

Figure 2.

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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).

The addition of a rest recovery multiplier improved fatigue predictions for intermittent tasks across all joints (p < 0.001; see Supplemental Materials Table S1 for RM ANCOVA results). Contrasts revealed all r values from 2 to 100 resulted in significant improvements in model predicitons (p < 0.05) compared to the original model (r = 1) for ankle and elbow data, but only r = 2 to 37 and r = 2 to 66 for the knee and hand/grip data, respectively (Table S1). Similar results were found for directional errors and after adjusting for assessment time (Supplemental Materials Figure S2).

Optimal Rest Recovery Multiplier

The regions of minimal absolute error (±1% torque decline) overlapped for each of the joint regions (Figure 3). The knee, ankle, and elbow regions had their lowest prediction errors for r values ranging from 5 to 26, while the hand/grip region had lowest error with r values ranging from 14 to 69 (Table 2). Fatigue predictions were more centered on the identity line (Figure 4) using the modified compared to the original fatigue model. The AIC analysis confirmed the best fits were with the modified 3CC-r model, even accounting for an additional parameter. The potential ropt values producing the lowest AICs for the knee, elbow, and ankle were not different from each other (ropt = 15). While the absolute minimum error for the elbow occurred at r = 8 to 10, the AICs for r of 8 versus 15 were not different. Conversely, the lowest AIC for hand/grip occurred for ropt = 30, which was an improvement over r =15. Thus, 15 was chosen as the optimal value for all regions except for hand/grip muscles.

Figure 3.

Figure 3.

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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 (ropt) are displayed as solid vertical lines, representing the best estimates for ropt. 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.

Table 2:

Range of optimal rest recovery multiplier (ropt) values [range ± 1 % torque decline] resulting in minimum fatigue prediction errors relative to observed fatigue, for the modified three-compartment controller (3CC) model based on absolute and directional model errors, with and without adjustment for time. All net error estimates were adjusted for MVC and duty cycle.

Joint
region		 ropt Candidate Values
 Final
ropt
Absolute
Error
(unadj)		 Absolute
Error
(time adj *)		 Directional
Error
(unadj)		 Directional
Error
(time adj*)
Ankle 15
[12 – 19]		 15
[9 – 26]		 16
[14 – 18]		 17
[13 – 22]		 15
Knee 12
[8 – 17]		 13
[7 – 21]		 13
[12 – 14]		 15
[13 – 17]		 15
Elbow 8
[7 – 15]		 8
[5 – 17]		 10
[9 – 11]		 10
[8 – 12]		 15
Hand/Grip 27
[18 – 42]		 28
[14 – 69]		 30
[27 – 34]		 37
[31 – 45]		 30
General (All) 15
[11 – 22]		 15
[8 – 29]		 17
[15 – 181		 19
[15 – 23]		 15
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*

errors divided by sqrt(time) in minutes to adjust for variations in time of assessment between studies and conditions, as prediction errors increase over time.

Figure 4.

Figure 4.

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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).

The resulting 3CC-r models over-predicted fatigue by 1.2% torque decline (directional error), with an absolute error of 8.2% torque decline overall (Table 3). The modified recovery to fatigue ratios (R*r/F) during rests using the ropt values (Table 4) varied from 1.41–1.96 (i.e. increase in recovery to fatigue ratio during rest intervals).

Table 3:

Mean absolute and directional model errors for the original three-compartment controller (3CC) model (r = 1) and the modified 3CC-r model at the optimal rest multiplier (ropt) for each joint region.

Joint
region		 ropt* Absolute Error
(% torque decline)		 Directional Error
(% torque decline)† 		Change in AIC**
r = 1 ropt r = 1 ropt AICopt–AIC1
Ankle 15 20.7 5.7 19.3 0.5 −68.2
Knee 15 21.0 8.6 21.0 −2.1 −54.8
Elbow 15 25.6 9.9 25.5 5.8 −43.8
Hand/Grip 30 29.2 8.7 28.9 0.1 −38.3
General (All) 15 23.9 8.2 23.3 1.2 −442.3
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†

A positive error indicates over-prediction of fatigue, across all task conditions adjusting for intensity, duty cycle, and time.

*

ropt values are based on the four error methods and the Akaike’s Information Criterion (AIC).

**

Negative values indicate superior fit of modified 3CC-r model despite addition of r parameter over original 3CC model.

Table 4:

Optimal fatigue (F), recovery (R), and rest multiplier (r) parameters by joint region.

Joint
Region		 3CC and 3CCr model
parameters		 During
contraction		 During rest
intervals
F* R* r R*r F:R
Ratio* 		R:F
Ratio* 		F:(R*r)
Ratio		 (R*r):F
Ratio
Ankle 0.00589 0.00058 15 0.0087 10.2 0.098 0.68 1.47
Knee 0.01500 0.00149 15 0.02235 10.1 0.099 0.67 1.49
Elbow 0.00912 0.00094 15 0.0141 9.70 0.103 0.65 1.55
Grip 0.00980 0.00064 30 0.0192 15.3 0.065 0.51 1.96
General 0.00970 0.00091 15 0.01365 10.1 0.094 0.67 1.41
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*

Optimal parameters reported by (Frey-Law et al., 2012); controller parameters set at LD and LR = 10

Intensity, duty cycle and time all had significant influences on the original model (r=1) error estimates (Supplemental Materials, Table S1, Figure 5). Overall, higher errors (~27% torque decline) occurred for lower intensity tasks (MVC < 100%) with longer intermittent rest intervals (duty cycle < 50%) compared to 20% error for high intensity tasks (MVC = 100%) with shorter rest intervals (duty cycle ≥ 50%). This trend was observed for all but the ankle regions (see r=1 values, Figure 5). However, the ranges of optimal r-values for the 3CC-r model were not consistently different between task conditions when stratifying by high versus low intensity and duty cycle (Figure 5).

Figure 5.

Figure 5.

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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 ropt values (vertical gray lines).

Discussion

The primary finding of this study was that our initial hypothesis was supported. That is, the addition of a rest recovery parameter to augment recovery during rest intervals improved fatigue model predictions of force decay during intermittent contractions. For all but the hand/grip muscles, a 15-fold increase in recovery rate was optimal. In addition, a large meta-analysis was performed assimilating intermittent fatigue data from the literature. Fatigue behavior had not previously been compiled in this manner for intermittent tasks. This systematic review compiled a large number of available data, while also identifying gaps in the literature.

Muscle fatigue development is a complex, nonlinear phenomenon. For sustained static contractions, simple statistical models have been applied, often referred to as Rohmert’s curves (Rohmert, 1960). Unfortunately, there is no simple counterpart to describe normative fatigue behavior for tasks involving rests. This is the first study to compile such a large set of fatigue data for intermittent tasks. However, despite including 351 data points from 63 studies, the distribution of task conditions remained unequal. While many functional tasks occur at low intensities or duty cycles, only 14% of the data involved <50% MVC or duty cycle: none for elbow or knee, 9% for hand/grip and 29% for ankle. Thus, these results do not fully represent all possible task conditions. Further, comparisons with raw data points result in inherently greater measurement noise than summary models of fatigue. Indeed, heterogeneous estimates of % torque decline occurred even for similar task conditions across publications. Despite this obstacle, the fatigue model errors were reduced from approximately 20% (3CC) to only 5–10% (3CC-r). Variance between studies reaffirms the importance of using larger datasets when possible for fitting parameters or evaluating model behavior.

Rashedi & Nussbaum (2015) previously suggested improvements in representing muscle recovery were strategic areas for fatigue model advancement. Our current study targeted one aspect of recovery, namely augmenting recovery during rest intervals to represent reactive hyperemia. Decreases in blood flow are well documented during sustained contraction due to increased intramuscular pressure on the vascular system (Degens, et al., 1998, Humphreys and Lind, 1963, Sejersted, et al., 1984, Wigmore, et al., 2004). However, increases in blood flow during low level contractions have also been observed (Bystrom and Sjogaard, 1991, Larsson, et al., 1995), possibly due to metabolite accumulation triggering vasodilation. Immediately following contraction, however, reactive vasodilation and resultant hyperemia occurs (Humphreys and Lind, 1963, Robergs, et al., 1997, Sejersted, et al., 1984, Wigmore, et al., 2004). This subsequently increases oxidative adenosine triphosphate (ATP) synthesis, thereby increasing muscle recovery (Layec, et al., 2013).

Similar results have been seen following cuff occlusion (Birtles, et al., 2003, Chung, et al., 2007) to manipulate ischemia, where blood flow rebounds once occlusion is released (Larsson, et al., 1995). Muscle recovery, as measured by phosphocreatine levels, also rebounds after blood flow is restored (Blei, et al., 1993, Greiner, et al., 2007). Similarly, greater force recovery occurs following release of cuff occlusion compared to recovery in the non-occluded state (Cerqueira, et al., 2017). Lastly, psychophysical studies show disproportionately longer endurance times for intermittent versus sustained isometric contractions of the same intensity (Iridiastadi and Nussbaum, 2006, Iridiastadi and Nussbaum, 2006b, Wood, et al., 1997). Collectively, these physiologic studies are consistent with our modeled augmented muscle recovery rates during postcontraction rests. Yet, this physiologically driven modification does not alter the original 3CC model’s fatigue predictions of sustained isometric contractions as it is only applied at rest. Additionally, this modification does not preclude the influence of other mechanisms on fatigue recovery that may result in future advancements.

While F and R parameters of the 3CC model are joint-specific (Frey-Law, et al., 2012), consistent with observed fatigue behavior (Frey Law and Avin, 2010), a common optimal r value for all but hand/grip muscles was observed. This different response for hand/grip may be due to methodology, i.e., heterogeneity of muscles or task conditions tested. For example, unlike the other regions, hand/grip data included isolated small finger muscles (67% cases) and larger muscle grip contractions (33%). Further the response surface for r was flatter for the hand/grip than the other regions. This suggests hand/grip data appears less sensitive to change in r. Another possible explanation is that a single optimal r is an oversimplification. Rashedi and Nussbaum (2016) found more frequent, shorter rests induced less fatigue than equivalent workloads with less frequent, longer rest periods, suggesting recovery augmentation may be cycle time dependent. Thus, these preliminary results support a common r value for all but hand/grip muscles, recognizing additional insights may be uncovered in the future.

There are several limitations worth noting. The first is our model does not take into account muscle fibertype composition. This may contribute to observed recovery errors. It is possible varying muscle fiber distributions may contribute to differences in fatigue errors and reperfusion responses between anatomical regions. Yet, differences in fatigue to recovery ratios between joint regions were less for the modified than the original model. Future studies may be useful to determine whether accounting for different muscle fiber concentrations would provide yet another improvement to fatigue estimates. A second limitation is the lack of data available for shoulder or trunk regions. To mitigate this, we fit a general 3CC-r fatigue model to all data combined. The common ropt value for all but the hand/grip region suggests a rest recovery value of 15 is a reasonable initial estimate until more data become available. A third limitation is the improvements noted here may not translate to dynamic tasks, where muscle reperfusion and hyperemia can also occur. Thus, future studies are needed to assess whether this model modification also improves fatigue estimates for more complex, dynamic tasks. The improvements noted with the addition of a rest recovery parameter to represent muscle reperfusion does not address all potential sources of model error. For example, cycle time may be an important consideration in fatigue development and recovery (Rashedi and Nussbaum, 2016), which our model cannot account for explicitly. Thus, the remaining 5–10% error may be a result of multiple additional sources of nonlinear muscle behavior. Finally, the use of ANOVA for the comparison of multiple r values may increase the potential for Type I errors. The lack of robust between-joint statistical analyses limits our ability to clearly discern whether the rest recovery parameter varies between joint regions. Future studies evaluating the Jacobian of the equations may be useful to assess for potential parameter interactions.

In summary, this study found the addition of a physiologically based rest recovery multiplier reduced fatigue model errors for intermittent contractions by over 50%. Because the additional parameter only alters model recovery during rest intervals, this modification maintains the heuristic nature of the original model.

Supplementary Material

Acknowledgements:

The authors were funded in part by the United States Council for Automotive research (USCAR), the National Institutes for Health, K01AR056134, and the University of Iowa Heartland Center (graduate student stipend). However these sponsors had no involvement in the study design, implementation, data analysis or writing of this publication.

Footnotes

Conflict of Interest Statement:

None of the authors have any financial or personal conflicts of interest to report. The authors were funded in part by the United States Council for Automotive Research (USCAR), the National Institutes for Health, K01AR056134, and the University of Iowa Heartland Center (graduate student stipend).

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