# Applications of the Critical Power Model to Dynamic Constant External Resistance Exercise: A Brief Review of the Critical Load Test

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## Abstract

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## 1. Introduction

#### Historical Perspective: The Influence of Dr. Herbert A. deVries

## 2. The Modeling of Human Performance

^{−1}) for world records versus the time in seconds to complete the race [12]. This curvilinear, asymptotic relationship was further examined for dynamic, continuous isometric, and intermittent isometric exercise of local muscle actions (<1/3 total muscle mass), and a linear model [6] was developed from measures of the total work performed or limit work (W

_{Lim}) and time to exhaustion or limit time (T

_{Lim}) (Figure 1). Together, these parameters formed the linear equation W

_{Lim}= y-intercept + slope × T

_{Lim}, where the slope was termed CP, which corresponded to the “maximum rate [a muscle or muscle group] can keep up for a very long time without fatigue” [6] (p. 329). The y-intercept was defined as an “energetic reserve” that is used during exercise above CP [6], which later investigators called the anaerobic work capacity (AWC) [7,8] or curvature constant (W′) [13,14,15]. Thus, Monod and Scherrer [6] expanded the early observations of A. V. Hill [12] of the curvilinear relationship between average speed and world record time and described the linear relationship between the time to exhaustion and the work performed from multiple work bouts, to define individual performance capabilities.

## 3. The Critical Power Test

_{lim}) (power output × time to exhaustion (T

_{lim})) is then calculated for each work bout and plotted against T

_{lim}. A simple linear regression analysis of the W

_{Lim}versus T

_{Lim}relationship gives a slope, defined as CP, and the y-intercept, previously described as an energy reserve, defined as the W′ (Figure 2a). Therefore, Moritani et al. [7] expanded on the previous work of Monod and Scherrer [6] in describing the linear relationship between time to exhaustion and work performed during whole-body cycle ergometry.

## 4. Critical Power Test Parameters

_{2}and blood lactate reach steady state responses. This was supported through the demonstration that CP was correlated with the anaerobic threshold and was dependent on oxygen supply [7]. Additionally, it was demonstrated that $\dot{\mathrm{V}}$O

_{2}and blood lactate reached a delayed steady state for exercise at or below CP, but both physiological markers increased until exhaustion for exercise performed above CP [14]. Based on these responses, it has been suggested [16] that CP represents the demarcation, or separation, between the heavy and severe intensity domains. Generally, for untrained or recreationally trained individuals, this model overestimates the power output that meets this definition [17], but more closely approximates this power output for elite athletes due to differences in the presence of the $\dot{\mathrm{V}}$O

_{2}slow component phenomenon [16,18]. Thus, in actuality, CP is not a power output that can be maintained indefinitely without exhaustion but may reflect a power output that demarcates (with some error) differences in physiological responses and/or a transition phase between the heavy and severe intensity domains [19].

_{Lim}for a power output above CP. The W′ reflects the total amount of work that can be performed above CP using only energy stored within the muscle (i.e., ATP bound to myosin, phosphocreatine, glycogen, and oxygen bound to myoglobin) before it is limited by exhaustion [7,9]. Based on the linear, 2-paramter regression equation from Moritani et al. [7] T

_{Lim}= W′/(P − CP), where T

_{Lim}equals time to exhaustion, W′ is the anaerobic work capacity, P is the imposed power output above CP, and CP is the derived critical power, a coach or practitioner can theoretically predict the time to exhaustion at a given power output above the CP due to intramuscular energy stores being used at a “predictable rate based on the magnitude of the difference between the imposed power loading (P) and CP” [17] (pp. 1001–1002).

## 5. Methodological Considerations

_{max}, maximal instantaneous power, which lowers the CP estimate [20,21]. Thus, the CP estimates from the linear, 2-parameter models may be more accurate for highly trained athletes, while the nonlinear, 3-parameter model may provide more accurate estimates for untrained or recreationally trained individuals. However, more research is needed on the nonlinear, 3-parameter model, which has been limited thus far by the complexity of the modeling and the challenges in elucidating the differences between physiology and mathematics.

## 6. Progression of the Modeling across Exercise Modalities: Applications of the Critical Power Model to DCER Exercises

^{2}) values nor estimates of the CL were provided for the linear, 2-parameter total work model [34]. The nonlinearity appeared to be driven primarily by the total work performed at the lowest load (~34% 1RM, repetitions > 41) [24] (p. 156). Based on the observed nonlinear response for the linear, 2-parameter total work model, the authors [34] further examined the bench press performance with the nonlinear, 3-parameter model (Figure 4a). For this model, the asymptote of the repetition versus load relationship was defined as the CL. This modeling resulted in goodness of fit (r

^{2}) values that ranged from 0.6698–0.9999. Interestingly, the asymptote (i.e., CL) was reported to be 0 kg for most subjects (12 of 16) [34]. However, the load selections for the modeling in this study may have also contributed to these zero estimations for the CL from the nonlinear, 3-parameter model. Specifically, it is likely that the lowest load (~34% 1RM, repetitions > 41) was peri-asymptotic and likely lowered the estimates of the CL from this model. This initial application highlighted the importance of load selection in the estimation of the CL.

^{2}values were higher for the three loads (range = 0.9512–0.9988) compared to the four loads (range = 0.7799–0.8909). The decreased linearity for the model utilizing all four loads may be explained by the use of a load (30% 1RM) below the estimated CL (38% 1RM). Like CP, the mathematical modeling of the CL relies on the assumption that loads are selected above the CL for its estimation. Peri-asymptotic loads will decrease the linearity and accuracy of the estimation of the CL. Although loads between 30% and 80% maximal voluntary contraction (MVC) force may be appropriate for isometric exercise and the determination of critical torque or critical force [6,39], 30% of 1RM appears to be too low in these initial DCER applications, as evidenced by CL estimates that range from ~25% and 40% 1RM [35,36,37]. This may be related to the intermittent nature of DCER exercise that allows for some restoration of blood flow during the eccentric phase and/or between repetitions (depending on the cadence).

^{2}values ranged from 0.864–0.989. Of particular importance was the fact that the lowest load was above the asymptote, which corresponded to 40% 1RM for the deadlift [36] and 26% 1RM for the leg extension [37]. The high linearity between total work (load (kg) × repetitions) and repetitions completed for both DCER exercises highlighted the necessity for choosing loads that are above the asymptote for the derivation of the CL. Thus, there are methodological considerations for the mathematical modeling of DCER exercise that have been identified in these initial applications [34,35,36,37,38] that underlie the importance of selecting loads that are neither too high nor too low (peri-asymptotic), which can result in the loss of linearity in the total work versus repetition relationship and decrease the validity of the CL estimation.

## 7. Additional Methodological Considerations for the Determination of the CL Test Parameters

## 8. Test Parameters: Critical Load and the y-Intercept (L′)

## 9. Research and Training Applications of the Critical Load Model

## 10. Recommendations for the Determination of the Critical Load

- At least four loads are recommended for the determination of the CL and L′, and each load used in the mathematical model should be greater than the CL. At this time, 50% 1RM or greater is recommended for the lowest load, and under most conditions, increases in loads should be made at increments of 10% (i.e., 50%, 60%, 70%, 80% 1RM).
- A cadence should be selected specific to each movement and standardized across subjects. This cadence should allow for successful completion of repetitions through the full range of motion for the lowest and highest loads.
- For subjects unfamiliar with performing repetitions to failure, a familiarization session at a submaximal load around 50–60% may improve the accuracy of the modeling.
- The model should be examined for each subject, and the r
^{2}of the total work versus repetition relationship should be at least 0.75 or greater. - If an r
^{2}is lower than 0.75 or the lowest load used in the model is lower than the CL for an individual subject, that load should be eliminated and an additional load setting greater than 50% 1RM should be performed and used in the analyses. - The CL and L′ can be estimated using the linear, 2-parameter total work (load (kg) × repetitions) versus duration relationship, the linear, 2-parameter load versus the inverse duration, or the nonlinear, 3-parameter model, and the duration should be expressed as repetitions.
- The mean and range of r
^{2}and standard error of the estimate (SEE) values from the regression model should be reported in all future works.

## 11. Future Research on the Critical Load Model

- Load selections—A wider range (e.g., 35–40% 1RM to 95% 1RM) of relative load settings should be examined across whole-body, upper-body, and lower-body, unilateral and bilateral muscle actions to determine the effects of the load setting on the mathematical modeling.
- Number of loads—The effects of using two loads versus three, four, or five loads on the parameter estimates CL and L′ from the linear and non-linear mathematical models should be examined.
- Effects of cadence—The effect of various cadences, including a self-selected cadence, on the estimation of the CL and L′ should be examined.
- Reliability—Future studies should examine the reliability of the CL and L′ for various DCER exercises.
- Muscle specific thresholds—The CL model should be examined for agonist versus antagonist muscle actions, bilateral versus unilateral muscle actions, and upper- versus lower-body muscle groups to determine if the mathematical model is sensitive to detect muscle group-specific fatigue characteristics.
- Mode-specific thresholds—Studies should compare the parameter estimates for isometric versus DCER movements in the same muscle group.
- Physiological underpinnings—Further investigation is warranted to examine the potential metabolic and circulatory factors underlying the determination of the CL and L′ as well as the prediction of performance using the CL model.
- Training studies—Training adaptations for strength and hypertrophy should be examined for loads prescribed above and below the CL for each individual.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Theoretical representation of the linearization of the power output versus duration curve to derive the parameters of the critical power (CP) test. The upper panel demonstrates the negative, curvilinear relationship between power output (P) and time to exhaustion (T

_{Lim}) that can be used to estimate the T

_{Lim}for any P that is greater than CP using the equation T

_{Lim}= W′/(P − CP). The lower panel demonstrates the linear relationship between the total work completed (W

_{Lim}) and T

_{Lim}. Each work bout at a constant power output can only be maintained for a finite amount of time, which results in the completion of a finite amount of work. When the W

_{Lim}and T

_{Lim}are plotted against each other (lower panel), there is a linear relationship that can used to derive the parameters of the CP test. The slope is CP and the y-intercept is the W′ (W

_{Lim}= CP × (T

_{Lim}) + W′).

**Figure 2.**A schematic representation of the five mathematical models that have been used to estimate the parameters of the critical power (CP) test, CP and W′; (

**a**) linear model based on the regression analysis of the total work (TW) versus time to exhaustion, where CP is the slope and W′ is the y-intercept; (

**b**) linear model based on the regression analysis of the power output versus the inverse of time to exhaustion, where W′ is the slope and CP is the y-intercept; (

**c**) nonlinear, 2-parameter regression model of time versus power output, where CP is the asymptote and W′ is the curvature constant; (

**d**) nonlinear, 3-parameter regression model of time versus power output, where CP is the asymptote, W′ is the curvature constant, and maximal instantaneous power (Pmax) is the x-intercept; and (

**e**) exponential regression model of time versus power output, where CP is the asymptote and Pmax is the x-intercept.

**Figure 4.**The mathematical models used to determine the critical load (CL) and L′ for dynamic constant external resistance exercise; (

**a**) non-linear, 3-parameter regression model of repetitions completed versus load (% one repetition maximum [1RM]), where CL is the asymptote, L′ is the curvature constant, and maximal instantaneous lift (L

_{max}) is the x-intercept; (

**b**) linear regression model of the load versus inverse of time, where L′ is the slope and CL is the y-intercept; (

**c**) linear regression model of the total work versus distance traveled, where CL is the slope and L′ is the y-intercept; (

**d**) linear regression model of the total work versus total repetitions completed, where CL is the slope and L′ is the y-intercept.

**Figure 5.**A representative example of the mathematical model used to derive the critical load (CL) and L′ for a male and female subject for deadlift and leg extension exercises. The CL is slope and L′ is the y-intercept; (

**a**) leg extension responses for a male subject; (

**b**) deadlift responses for a male subject; (

**c**) leg extension responses for a female subject; (

**d**) deadlift responses for a female subject.

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Bergstrom, H.C.; Dinyer, T.K.; Succi, P.J.; Voskuil, C.C.; Housh, T.J.
Applications of the Critical Power Model to Dynamic Constant External Resistance Exercise: A Brief Review of the Critical Load Test. *Sports* **2021**, *9*, 15.
https://doi.org/10.3390/sports9020015

**AMA Style**

Bergstrom HC, Dinyer TK, Succi PJ, Voskuil CC, Housh TJ.
Applications of the Critical Power Model to Dynamic Constant External Resistance Exercise: A Brief Review of the Critical Load Test. *Sports*. 2021; 9(2):15.
https://doi.org/10.3390/sports9020015

**Chicago/Turabian Style**

Bergstrom, Haley C., Taylor K. Dinyer, Pasquale J. Succi, Caleb C. Voskuil, and Terry J. Housh.
2021. "Applications of the Critical Power Model to Dynamic Constant External Resistance Exercise: A Brief Review of the Critical Load Test" *Sports* 9, no. 2: 15.
https://doi.org/10.3390/sports9020015