The parameters of the hyperbolic relationship between power and time to exhaustion, i.e., critical power (CP) and curvature constant (W′ ), have been used to describe energy exertion in the severe exercise intensity domain. CP, the power asymptote of the relationship, represents a power output above which muscle metabolic homeostasis cannot be attained (^{1–3} ), and W′ represents the finite amount of work that can be done above CP (^{2,3} ). The hyperbolic relationship between power and time to exhaustion is given by

$\mathrm{P}=\mathrm{CP}+\frac{W\text{'}}{{t}_{\mathrm{lim}}}$

where P is the power in watts, CP is critical power in watts, W′ is the curvature constant in joules, and t _{lim} is the time to exhaustion in seconds.

Equation 1, popularly known as the two-parameter model, originated from Monod and Scherrer’s (^{4} ) work on muscle groups and has since been applied to other exercise forms such as cycling (^{5} ), running (^{6} ), swimming (^{7} ), and rowing (^{8} ) with appropriate modifications to the parameters. The two-parameter model describes W′ expenditure in the severe-intensity domain (>CP) and not its recovery in the moderate to heavy-intensity domain (<CP) (^{9,10} ). Attempts have been made to develop a unified expenditure–recovery model (^{10–13} ) to aid in planning race strategies, performance optimization , and understanding the effect of physical exercise on overall health (^{10,14,15} ).

Morton and Billat (^{11} ) were the first to consider the recovery of W′ while deriving a model for intermittent exercise that was based on the two-parameter model. They assumed that the recovery of W′ occurs at the same rate as its expenditure. However, Ferguson and colleagues (^{12} ) illustrated that the recovery of W′ was not proportional to its expenditure but “curvilinear.” Skiba and colleagues (^{10,13,16,17} ) proposed an exponential model acknowledging this nonlinear behavior of recovery of W′ given by

$W{\text{'}}_{\mathrm{bal}}=W\text{'}-\underset{0}{\overset{t}{\int}}W{\text{'}}_{\mathrm{exp}}{e}^{\frac{-\left(t-u\right)}{{\tau}_{W\text{'}}}}\mathit{du}$

where W′ _{bal} is the W′ balance at any time during exercise (J), W′ _{exp} is the amount of W′ expended (J), (t − u ) is the duration of the recovery interval (s), and τ _{W′} is the time constant of reconstitution of W′ (s) given by

${\tau}_{W\text{'}}=546{e}^{\left(-0.01{D}_{\mathrm{CP}}\right)}+316$

where D _{CP} is the difference between CP and average power output during all intervals below CP. Equation 3 is a nonlinear regression obtained by plotting τ _{W′} values (calculated by setting the W′ _{bal} equal to 0 in equation 2 at the termination of exercise) against respective D _{CP} values.

Although the W′ _{bal} model in equation 2, referred to as SK1 henceforth, was validated by Skiba et al. (^{16} ), it has been shown to underestimate the actual W′ balance by Caen and colleagues (^{18} ). Furthermore, Skiba et al. (^{19} ), (p. 78) suggests that there is some recovery of W′ above CP, which violates the assumption of Skiba and colleagues (^{10} ) about W′ recovery occurring the instant a subject’s power output falls below CP, unless there are two different time constants for expenditure and recovery. In addition, a τ _{W′} in equation 3 cannot be determined for an exertion interval as a D _{CP} value is needed, which is not available until a subject goes into the recovery interval. From a mathematical sense, SK1 assumes an exponential decay of W′ above CP instead of the linear decay described by equation 1. Moreover, a closer investigation of equation 2 reveals an imbalance of units on the right-hand side (RHS). The second term on the RHS, upon integration, yields the units of joules-second, whereas the first term on the RHS has the units of joules. Furthermore, two versions of SK1 have been reported by Skiba and colleagues (^{10,17} ) with two different differential terms du and dt in equation 2, which yield different results upon integration. Some of these concerns were addressed by Skiba and colleagues (^{13} ) in their biconditional model for exertion and recovery (Appendix 1 of Skiba et al. [^{13} ]) given by

$\text{if}\phantom{\rule{0.12em}{0ex}}\mathit{P}\ge \mathrm{CP},W{\text{'}}_{\mathrm{bal}}=W{\text{'}}_{0}-\left[\left(\mathit{P}-\mathrm{CP}\right)t\right]$

$\text{if}\phantom{\rule{0.12em}{0ex}}{P}<\mathrm{CP},W{\text{'}}_{\mathrm{bal}}=W{\text{'}}_{0}-W{\text{'}}_{\mathrm{exp}}{e}^{\frac{-t}{\tau}}$

where W′ _{0} is W′ at time t = 0 in joules, t is the duration of the exertion/recovery interval in seconds (recovery if P < CP), W′ _{exp} is the amount of W′ expended before the recovery interval in joules, and τ = W′ _{0} /D _{CP} is the time constant of W′ recovery in seconds. Equation 4 will be referred to as SK2 henceforth. Supplemental Digital Content 1 provides the solutions of different forms of SK1 illustrating the imbalance of units and a detailed derivation of the P < CP case of SK2 from first principles, https://links.lww.com/MSS/C30 .

Bartram and colleagues (^{20} ) illustrated that SK2 underestimates the recovery of W′ in elite athletes and proposed that τ be modified as

$\tau =2287.2{D}_{\mathrm{CP}}^{-0.688}$

They also suggested deriving group/athlete-specific time constants to accurately estimate W′ recovery. The model using the τ from equation 5 will henceforth be referred to as BAR.

The available recovery models provide a good starting point to further investigate the underpinnings of W′ recovery. Studies that have investigated the recovery of W′ either have their subjects completely expend their W′ in the prerecovery interval (^{12,18} ) or have shorter exertion intervals in the range of 30 s (^{10,20} ). Thus, the goals of this study were to investigate the following: (i) the effect of recovery power and duration on the recovery of W′ after a semiexhaustive exertion interval, (ii) if the W′ _{bal} calculated from SK2 and BAR models accurately predict the actual W′ recovered, and (iii) real-time performance optimization using an individual’s specific recovery data. To address the first objective, we hypothesized that W′ recovery is affected by both recovery power and duration. To address the second objective, we hypothesized that the SK2 and BAR models overestimate the actual W′ recovery. Finally, to optimize performance, as a test case, we applied optimal control methodology and dynamic programming to investigate if the athlete-specific recovery model results in a reduction in time required to complete a known course.

METHODS
Subjects
Seven recreational cyclists (4 males and 3 females, age = 36 ± 11 yr, weight = 73 ± 14 kg, height = 1.76 ± 0.08 m) volunteered to participate in the study. The number of subjects is similar to that reported in similar studies (^{10,16,21} ). The subjects were recruited using a survey on their activity levels. All subjects trained 3–5 d·wk^{−1} with the training load ranging 100–200 km·wk^{−1} . Each subject signed an informed consent approved by university’s institutional review board. Each testing day was approximately 2 h long. Subjects were given instructions to (i) refrain from any strenuous physical activity for at least 24 h before each test, (ii) avoid alcohol consumption for at least 24 h before each test, (iii) avoid any caffeinated drink for at least 3 h before each test, and (iv) consume a carbohydrate-rich meal 90 min to 2 h before each test. The scheduling ensured at least 24 h of rest between two tests. All tests were conducted in the same time window (±2 h) of the day and the total duration to complete all tests was 4–7 wk per subject.

Equipment and Experimental Overview
All tests were conducted on the subjects’ own bicycles by mounting them onto a RacerMate CompuTrainer (CompuTrainer; RacerMate, Seattle, WA). The CompuTrainer was calibrated per the manufacturer’s guidelines before each test. PerfPRO Studio software (Hartware Technologies, Rockford, MI) was used to program the protocols on the trainer. The bicycle computer of the trainer showed only the cadence to the subjects during the tests. To account for differences in each subject’s gear combinations, a gear-inches range of 52–57 inches was chosen. Subjects were not allowed to change gears during testing. Heart rate data were collected using subjects’ own chest-strap heart rate monitors. The COSMED Quark CPET apparatus (COSMED, Rome, Italy) was used during the ramp test to measure oxygen uptake (V˙O_{2} ). Each subject visited the laboratory 14 to 16 times. The first day consisted of a ramp test and a 3-min all-out familiarization test. In the next two or four visits, the subjects performed 3-min all-out tests (3MT) to determine their CP and W′ (subjects 1 and 2 were able to perform only two 3MT, the rest performed four). The intermittent test familiarization was carried out either after the last 3MT or on the next visit based on the subject’s discretion. In the next two visits, subjects repeated their first intermittent test. In the next eight visits, the remaining intermittent tests were conducted. The order of the intermittent cycling tests was randomized.

Experimental Protocols
Incremental ramp test
The incremental ramp test was conducted to determine the peak oxygen uptake (V˙O_{2peak} ), the gas exchange threshold (GET), the maximum power observed in the ramp test (P_{max} ), and the power at GET (P_{GET} ). V˙O_{2peak} is the highest V˙O_{2} value attained during an incremental/severe-intensity test designed to elicit a subject’s limit of exercise tolerance (^{22} ). The subjects were instructed to pedal at 80 rpm throughout the ramp test. The warm-up included (i) a 5-min interval at 100 W and 80 rpm, (ii) a 5-min rest interval, and (iii) a 3-min unloaded pedaling at 80 rpm. The ramp interval immediately followed the warm-up at 100 W with an increase of 0.5 W·s^{−1} . The termination of the test was determined by a 5-rpm reduction in the cadence for more than 10 s (^{23} ). Strong verbal encouragement was given to the subjects by repeatedly instructing them to hold 80 rpm.

Three-minute all-out test
The 3MT was conducted on a CompuTrainer. The CompuTrainer has been validated to conduct 3MT by Clark and colleagues (^{24} ). They reported a coefficient of variation of 2.8% for CP and 24.4% for W′ between the 3MT and the constant work-rate protocol. However, 7 of 10 subjects had an average absolute difference of 0.76 ± 0.49 kJ (range, 0.08 to 1.5 kJ) between the W′ determined from the two protocols.

The warm-up for the 3MT was identical with that of the ramp test apart from the last 5 s of the unloaded interval where the subjects were instructed to increase their cadence to at least 110 rpm. The all-out effort followed the warm-up with the subjects given strong verbal encouragement throughout the test.

Intermittent cycling tests
The intermittent cycling test protocol shown in Figure 1 was developed based on the following assumptions:

FIGURE 1: The intermittent cycling test protocol. The warm-up is the same as that of the ramp test. L, M, and H refer to low, medium, and high levels of the recovery power, P_{rec} . Both P_{rec} and t _{rec} were manipulated to result in a full factorial design of nine tests. The amount of W′ recovered (E _{rec} ) in the recovery interval was calculated by subtracting W′ from the sum of the areas A_{1} and A_{3} . The actual W′ balance at the end of the recovery interval would be equal to A_{3} . ^{1} P_{GET} is the power at which GET occurs.

The 3MT accurately estimates CP and W′ (^{25–27} ).
Exercise above CP results only in the expenditure of W′ , not its recovery (^{13} ).
Exercise below CP results in recovery of W′ , thus increasing the W′ balance (^{10–13} ).
The recovery of W′ is a function of the level of power below CP and the recovery duration (^{10–12} ).
The power held during the recovery interval is constant. The behavior of power versus time below CP is unknown and, therefore, needs to be constant to mathematically model the recovery of W′ .
The warm-up for the intermittent test was also identical with that of the ramp test. The test protocol was composed of (i) a 2-min exertion interval at CP4, the power at which a subject would exhaust all of their W′ in 4 min (calculated using equation 1); (ii) a recovery interval at three recovery powers, P_{rec} (low [L], 20 W; medium [M], 0.9 P_{GET} ; and high [H], P_{GET} + 0.5 (CP − P_{GET} ]), and three recovery durations, t _{rec} (2, 6, and 15 min); and (iii) a 3-min all-out interval. The exertion interval was designed to expend ~50% of the subject’s W′ . For the recovery interval, the powers were chosen for comparison purposes to previously published studies by Skiba and colleagues (^{10} ) and Chidnok and colleagues (^{9} ), whereas the recovery durations were chosen from Ferguson and colleagues (^{12} ).

The subjects were instructed to maintain 80 rpm in the warm-up, CP4, and recovery intervals. In the last 5 s of the recovery interval, the subjects were instructed to ramp up to at least 110 rpm. To ensure an all-out effort, the subjects were instructed to pedal as hard as possible in the 3-min all-out interval. Strong verbal encouragement was given throughout the test. A cooldown at 20 W immediately followed the all-out interval.

Optimization tests
One subject participated in the optimization tests, which involved (i) a self-strategy test and (ii) an optimal strategy test. The subject chose an 18-km long out-and-back course, which was simulated on the CompuTrainer. The warm-up (left to the discretion of the subject) lasted ~15 min and was identical across the two tests. The subject was allowed to change gears during both the tests. For the self-strategy test, the subject was advised to use their own strategy to complete the course as quickly as possible. The subject was shown their power output, time lapsed, speed, cadence, heart rate, and distance covered during the test. For the optimal strategy test, the entire distance of the course was discretized into 100-m segments. The optimal power for each segment was determined using the subject’s individual fatigue and recovery models derived from the intermittent tests as illustrated in our previous work (^{28} ). The subject was shown both the target power and their real-time power during the test and was instructed to try and match the target power. No other parameters were shown to the subject during the optimal strategy test. Strong verbal encouragement was provided for both the tests.

Data Analyses
Incremental ramp test
The V˙O_{2peak} was computed as the highest 30-s average of oxygen uptake observed during the test (^{25} ). The v-slope method (^{29} ) was used to determine the GET. The max power during the ramp test (P_{max} ) and the power at GET (P_{GET} ) were noted from the power data file. P_{max} and P_{GET} were used to determine a gradient for the 3MT that would result in an end power halfway between the two powers at 80 rpm. This gradient was used to set up the 3MT on the CompuTrainer. The gradient–cadence relationship was established by trial and error during pilot testing to mimic similar power–cadence relationships published using cycling ergometer modes (^{25,30} ).

3MT
The maximal effort in the 3MT was confirmed by comparing the maximum heart rate to that observed during the ramp test. CP was computed as the average power of the last 30 s of each 3MT, and W′ was computed by numerical integration of power values above CP (i.e., P − CP) over time (^{25} ). The total work done (TWD) was calculated by numerical integration of power values over the test duration. In addition, the peak power (Pp) observed during the test was noted.

Intermittent cycling tests
The amount of W′ expended in the CP4 and the 3-min all-out intervals, A_{1} and A_{3} (Fig. 1 ), were calculated by numerical integration of power values above CP. The area of the recovery interval is larger than the amount of W′ recovered. Hence, the amount of W′ recovered in the recovery interval was expressed as a percentage of W′ and was computed using the formula

${E}_{\mathrm{rec}}=\frac{{A}_{1}+{A}_{3}-W\text{'}}{W\text{'}}.$

Statistical analyses
The repeatability of the four 3MT and the two intermittent tests were separately evaluated using intraclass correlation coefficient (ICC), typical error (TE), and coefficient of variation (CV) (^{31} ). CP, W′ , Pp, and TWD from the 3MT, as well as Pp and TWD during the 3-min all-out interval of the two repeated trials of the intermittent test, were compared for repeatability. A two-way repeated-measures ANOVA was conducted to investigate the effects of P_{rec} and t _{rec} on E_{rec} . Subject 7 was excluded from the analysis as their CP had an increasing trend from trial 1 to trial 4. Subject 7’s CP values across the four 3MT trials were 295, 305, 319, and 327 W. In addition, there was an increase in TWD from trial 1 (65.44 kJ) to trial 4 (70 kJ). This caused an inaccurate estimation of CP4, which resulted in less W′ being expended in the CP4 interval. This in turn resulted in similar E _{rec} for all intermittent tests. Post hoc tests were conducted using the Bonferroni correction (^{3,30,32} ). To investigate the influence of the intermittent test on CP, fresh CP from the four 3MT (CP_{fr} ) and fatigued CP from the intermittent tests (CP_{ft} ) were compared at both group and subject levels using independent-sample t -tests as the sample sizes were not equal. Mann–Whitney U tests were conducted in case of a violation of the normality assumption. Similarly, the actual W′ balance at the end of recovery interval (given by A_{3} ) was compared with W′ _{bal} predicted from SK2 and BAR models. Effect sizes are reported as partial η ^{2} (small = 0.01, medium = 0.06, and large = 0.14) and Cohen’s d (small = 0.2, medium = 0.5, and large = 0.8) wherever appropriate (^{33} ). The violations to assumptions of normality, sphericity, and homogeneity of variance were checked using Shapiro–Wilk’s, Mauchly’s, and Levene’s tests respectively. The data are represented as mean ± SD. All statistical analyses were conducted in SPSS Statistics 25 (IBM Corp., Armonk, NY), and the level of significance was 0.05.

RESULTS
A summary of subject-level V˙O_{2peak} , GET, P_{max} , and P_{GET} is reported in Supplemental Digital Content 2, https://links.lww.com/MSS/C31 . The average relative V˙O_{2peak} was 51.91 ± 6.27 mL·kg^{−1} ·min^{−1} , and the average relative GET was 36.25 ± 4.51 mL·kg^{−1} ·min^{−1} . CP, W′ , TWD, and Pp from all the 3MT were averaged and are reported in Table 1 . The peak power during the 3MT occurred at 3.82 ± 1.23 s (range, 3–7 s) from the start of the test. P_{GET} and P_{max} from the ramp test were 0.81 CP and 1.33 CP, respectively. The maximum heart rate during all the 3-min all-out efforts (including the intermittent cycling tests) occurred within the first minute of the interval and was comparable (~±5 bpm) with that observed in the ramp test for all subjects. The average subject-level coefficient of variance (calculated from Table 1 ) for CP was 2.09% (range, 0.86% to 4.85%) and for W′ was 7.54% (range, 3.27% to 13.79%).

TABLE 1 -
Summary of the parameters from the four trials of the 3MT for all subjects.

Subject
No. of 3MT
CP (W)
W′ (kJ)
TWD (kJ)
Pp (W)
1
2
269 ± 3
12.03 ± 0.58
60.08 ± 0.22
766 ± 6
2
2
233 ± 2
10.10 ± 0.33
51.69 ± 0.63
714 ± 8
3
4
335 ± 7
15.09 ± 1.00
75.09 ± 0.84
1043 ± 44
4
4
217 ± 6
5.64 ± 0.56
43.94 ± 1.26
438 ± 15
5
4
242 ± 4
7.84 ± 0.34
51.00 ± 0.67
549 ± 28
6
4
206 ± 10
9.14 ± 1.26
46.11 ± 0.86
421 ± 27
7
4
311 ± 14
12.65 ± 1.27
68.32 ± 2.00
838 ± 47

Data are presented as mean ± SD.

Pp, peak power observed during the 3MT; TWD, total work done during the 3MT; W′ , curvature constant (work capacity).

Repeatability of 3MT and the intermittent test
The repeatability statistics for the 3MT and the repeated intermittent tests are available in Supplemental Digital Content 3, https://links.lww.com/MSS/C32 . ICC, TE, and CV for all the parameters of the 3MT and the intermittent cycling tests indicate excellent agreement between the trials. Subject 5 performed only one trial of their first intermittent test as the second trial was unsuccessful due to their shoes coming unclipped from the pedal, and the test was not repeated to prevent delays on the schedule.

Effect of P_{rec} and t _{rec} on E _{rec}
The repeated-measures ANOVA showed a statistically significant two-way interaction effect between P_{rec} and t _{rec} on E_{rec} (P = 0.004, η ^{2} = 0.519), which is also illustrated in Figure 2 A. The assumption of sphericity was not violated as indicated by Mauchly’s test for the two-way interaction (χ ^{2} _{9} = 5.547, P = 0.812). A statistical power (post hoc ) of 0.929 was observed for the two-way interaction.

FIGURE 2: Interaction plots. E _{rec} vs P_{rec} (A) and E _{rec} vs t _{rec} (B). As seen in panel A, the crossing of the dotted line and the gray line indicates the interaction effect of P_{rec} × t _{rec} on E _{rec} . Simple main effects were present only with respect to P_{rec} at each t _{rec} . E _{rec} is the amount of W′ recovered in the recovery interval normalized with respect to W′ , P_{rec} is the power held in the recovery interval, and t _{rec} is the duration of the recovery interval.

Simple main effects analyses were conducted because of the presence of interaction effects between P_{rec} and t _{rec} . There was a statistically significant difference in mean E_{rec} at t _{rec} = 2 min (P = 0.001, η ^{2} = 0.747), at t _{rec} = 6 min (P = 0.006, η ^{2} = 0.640), and at t _{rec} = 15 min (P < 0.001, η ^{2} = 0.914). Table 2 shows the summary of the simple main effects analysis. The post hoc powers observed for simple main effects of P_{rec} on E _{rec} were >0.9.

TABLE 2 -
Mean

E _{rec} at different

t _{rec} and P

_{rec} with summary of simple main effects of P

_{rec} at each

t _{rec} .

t
_{rec} (Min)
Mean E
_{rec} (%)
P_{rec-L}
P_{rec-M}
P_{rec-H}
2
33.7% ± 10.1%^{
a,b
}
18.95% ± 9.42%
3.31% ± 21.84%
6
40.62% ± 12.3%
^{c}
31.51% ± 13.97%
6.47% ± 24.57%
15
39.01% ± 14.12%^{
d,e
}
19.20% ± 16.77%
^{f}
−15.53% ± 23.58%

^{a} Statistically significantly different from P_{rec-M} (mean difference = 14.75%, 95% CI [4.84–24.66], P = 0.01).

^{b} Statistically significantly different from P_{rec-H} (mean difference = 30.39%, 95% CI [8.06–52.72], P = 0.015).

^{c} Statistically significantly different from P_{rec-H} (mean difference = 34.15%, 95% CI [4.53–63.77], P = 0.029).

^{d} Statistically significantly different from P_{rec-M} (mean difference = 19.81%, 95% CI [2.63–36.99], P = 0.029).

^{e} Statistically significantly different from P_{rec-H} (mean difference = 54.54%, 95% CI [34.05–75.02], P = 0.0007).

^{f} Statistically significantly different from P_{rec-H} (mean difference = 34.72%, 95% CI [15.87–53.57], P = 0.004).

E _{rec} , energy recovered in the recovery interval as a percentage of W′ ; P_{rec} , power held during the recovery interval; L, low; M, medium; H, high; t _{rec} , duration of the recovery interval.

The negative mean E _{rec} seen at P_{rec-H} indicates that the subjects did not recover any W′ but depleted it in the recovery interval, suggesting that they were functioning above CP. This indicates a variability associated with CP either within a trial or between trials.

With regard to simple main effects of t _{rec} at the three recovery powers, there was no statistically significant difference in mean E _{rec} at the different t _{rec} for P_{rec-L} (P = 0.303, η ^{2} = 0.213). Similar results were observed at P_{rec-M} (P = 0.094, η ^{2} = 0.376) and at P_{rec-H} (P = 0.053, η ^{2} = 0.536) (Greenhouse–Geisser correction was applied for P_{rec-H} as epsilons for Greenhouse–Geisser and Huynh–Feldt corrections were 0.570 and 0.628, respectively). The observed post hoc powers for the simple main effects of t _{rec} on E _{rec} were 0.227, 0.456, and 0.537 for P_{rec-L} , P_{rec-M} , and P_{rec-H} , respectively. Data and additional details of statistical analyses pertaining to this section are available in Supplemental Digital Content 4, https://links.lww.com/MSS/C33 .

Comparison of actual W′ balance (A_{3} ) and W′ _{bal} predicted by SK2 and BAR models
Mann–Whitney U tests were conducted to determine whether there were differences between actual W′ balance (A_{3} ) and W′ _{bal} predicted by SK2 (W′ _{SK2} ) and BAR (W′ _{BAR} ) due to a violation of the normality assumption. Comparing A_{3} and W′ _{SK2} , A_{3} was statistically significantly less than W′ _{SK2} (P = 0.035, η ^{2} = 0.083). Similarly, A_{3} was statistically significantly less than W′ _{BAR} (P = 0.015, η ^{2} = 0.109). Data and additional details of statistical analyses pertaining to this section are available in Supplemental Digital Content 5, https://links.lww.com/MSS/C34 .

Influence of intermittent test on CP
The influence of the intermittent test on CP was analyzed by comparing CP_{fr} and CP_{ft} at both group and subject levels (excluding subject 7). Data from subjects 1 and 2 were not analyzed at the subject level as there were only two data points for CP_{fr} . At the group level, a Mann–Whitney U test was conducted due to a violation of the normality assumption. The mean CP_{fr} was not statistically different from that of CP_{ft} , P = 0.327.

At the subject level, there were no violations of assumptions as assessed by Shapiro–Wilk’s and Levene’s tests for each subject. Independent-sample t -test indicated no statistically significant difference between the mean CP_{fr} and the mean CP_{ft} for subject 4 (P = 0.166, d = 0.89) and subject 6 (P = 0.517, d = 0.40). However, mean CP_{ft} was statistically significantly higher than mean CP_{fr} for subject 3 (P = 0.025, d = 1.56) and subject 5 (P = 0.032, d = 1.48). The P values along with the effect sizes for subjects 3 and 5 indicate the within-subject variability of CP. Data and additional details of statistical analyses pertaining to this section are available in Supplemental Digital Content 6, https://links.lww.com/MSS/C35 .

Optimization tests
The goal of the optimization was to minimize time by managing the W′ balance. Subject 5 volunteered to participate in the optimization tests and chose the out-and-back 18-km course that was simulated on the CompuTrainer. The subject’s individual data were used to arrive at a recovery model and to determine the optimal power profile for the test (for methodology, refer to our previous work (^{28} )). Table 3 summarizes the results from both the tests. Figures 3 A and 3 B show the power versus distance profiles for both the tests, whereas Figure 3 C shows the W′ balance plotted against the distance.

TABLE 3 -
Comparison of results between self-strategy and optimal strategy tests.

Parameter
Self-Strategy Test
Optimal Strategy Test
Time (min:s)
34:08
33:13
Average power (W)
212
219
Max power (W)
429
343
Average velocity (mph)
31.64
32.51
Average heart rate (bpm)
148
146

FIGURE 3: Results from the optimization tests. Power vs distance profile for self-strategy (A), power vs distance profile for optimal strategy (B), and W′ balance vs distance for both the tests (C). The subject’s W′ is back to the initial value in the self-strategy test as the subject was mostly pedaling below CP after approximately 5 km. The subject expended all their W′ toward the end in the optimal strategy trial. The course grade is shown below the power profile and is plotted on the secondary axis.

The optimal strategy test shows an improvement of 55 s from the subject’s self-strategy. In the self-strategy test (Fig. 3 A), the subject began the test at higher powers and then settled below their CP around the 5-km mark. By contrast, in the optimal strategy test (Fig. 3 B), the subject pedaled above and below CP giving them ample recovery to finish the test faster. Figure 3 C shows that the W′ _{bal} at the end of the self-strategy trial was equal to the subject’s W′ , whereas W′ _{bal} was equal to 0 toward the end of the optimal strategy trial although not exactly at the finish line. This can be attributed to (i) the variability of subject 5’s W′ and (ii) the constraint that the maximum power which can be generated at W′ _{bal} = 0 is CP. Overcoming the uphill section at the end of the course at CP results in an increased race completion time. Hence, it is optimal to recover before the uphill section to go up the hill faster.

DISCUSSION
The objectives of this study were to investigate (i) the effect of P_{rec} and t _{rec} on E _{rec} after a semiexhaustive interval above CP, (ii) if W′ _{bal} as calculated from SK2 and BAR models accurately predict the actual W′ remaining, and (iii) real-time performance optimization using subject-specific recovery data. A significant result of this study was the two-way interaction effect between recovery parameters, P_{rec} and t _{rec} , followed by the simple main effects of P_{rec} on E _{rec} (see Fig. 2 A and the figure in Supplemental Digital Content 4, https://links.lww.com/MSS/C33 ). This illustrates that recovery power has a greater influence on the recovery of W′ in comparison with recovery duration. Furthermore, the overestimation of W′ balance at the end of the recovery interval by both SK2 and BAR models illustrates the need to establish athlete-specific recovery parameters or models echoing the conclusions from Bartram and colleagues (^{20} ). Moreover, the result from the optimal strategy test performed with one subject shows the potential of athlete-specific models in performance optimization .

The assumptions of this study were that the 3MT estimates CP and W′ reliably, and equation 1 accurately describes the expenditure of W′ in the severe-intensity domain. A limitation of the study was that the actual power output in 20 W recovery interval was 75–90 W for all subjects. It was not possible to generate a power output of 20 W at 80 rpm because of the rolling resistance calibration recommendations. Therefore, the average of the actual recovery powers observed in the P_{rec-L} interval was used to derive the individualized recovery models.

From the 3MT, the P_{GET} occurred at 0.81 CP, which is higher than that reported by Vanhatalo and colleagues (^{25} ) of ~0.625 CP. This may be due to the unaccounted mean response time of V˙O_{2} at the start of the ramp interval (i.e., the delay in V˙O_{2} response). Iannetta and colleagues (^{34} ) used a 42-min step transition before the ramp interval and reported an average mean response time of <30 s for athletes similar to those involved in this study. Consequently, P_{GET} may have been overestimated by ~15 W (30 W·min^{−1} ramp for 30 s), resulting in an increased gradient for the 3MT. However, the average cadence of the last 30 s of the 3MT for all subjects, except subject 7, was ~80 rpm (range, 75–90 rpm). This is similar to Vanhatalo and colleagues (^{26} ), suggesting that the increased gradient had minimal influence on the 3MT outcomes.

The repeatability statistics reported in Supplemental Digital Content 3, https://links.lww.com/MSS/C32 for CP (ICC = 0.996, TE = 8 W, and CV = 3.04%) were stronger than those by Johnson and colleagues (^{27} ) (ICC = 0.93, TE = 15 W, and CV = 6.7%) and similar to Burnley and colleagues (^{23} ) (ICC = 0.99, TE = 7 W, CV = 3%). Similarly, repeatability statistics for W′ (ICC = 0.988, TE = 0.956 kJ, and CV = 10.14%) were stronger than Johnson and colleagues (^{27} ) (ICC = 0.87, TE = 1.456 kJ, and CV = 20.7%). Burnley and colleagues (^{23} ) did not compute repeatability statistics for W′ , and therefore a comparison is not possible. The stronger repeatability along with the average subject-level coefficient of variance of 2.09% for CP and 7.54% for W′ gives us reason to believe that the higher variability of W′ (compared with CP) did not substantially influence the outcomes of the study.

The statistical powers observed for the two-way interaction and the simple main effects of P_{rec} (>0.9) illustrate that these analyses were appropriately powered. However, the low statistical power observed on simple main effects of t _{rec} could be due to the low sample size used in the study. The two-way interaction effect between P_{rec} and t _{rec} on E _{rec} was not observed by Caen and colleagues (^{18} ). The simple main effects of lower P_{rec} resulting in greater W′ recovery has also been illustrated by Bickford and colleagues (^{35} ), with P_{rec} having a greater influence on W′ recovered than t _{rec} . Furthermore, the lack of simple main effects with respect to t _{rec} contrasts the results from Caen and colleagues (^{18} ). They reported main effects with respect to recovery duration with more energy recovered at 6 min (59.4% ± 4.1%) when compared with 2 min (46% ± 2.1%). Similar results to Caen and colleagues (^{18} ) were observed by Ferguson and colleagues (^{12} ) with greater W′ recovery as recovery duration increased. This difference in results could be attributed to the equipment used or the CP4 exertion interval. It has been shown that the CompuTrainer values for Peak power output and time to exhaustion during a ramp test are lower in comparison with those from a Lode ergometer (^{36} ). However, the V˙O_{2peak} and the peak heart rate were not statistically different from each other across the two ergometers indicating that the effort levels were similar. Hence, it is reasonable to believe that similar results would be observed if the current study were to be conducted on a Lode ergometer. Therefore, the difference in results can be attributed to the CP4 exertion interval, where ~50% of W′ (linear depletion as per Skiba and colleagues (^{13} )) was expended while, in the aforementioned studies, all of W′ was expended before the recovery interval. The rate of W′ recovery could be different for recovery intervals that follow exhaustive intervals versus semiexhaustive intervals, like this study.

Another significant result from this study was that SK2 and BAR models overestimated the actual W′ balance at the end of the recovery interval. This finding is in contrast to the results found by Bartram and colleagues (^{20} ), where SK2 underestimated W′ balance. The reason for this could be that the athletes who participated in this study were competitive amateur cyclists as compared with elite cyclists in the study of Bartram and colleagues (^{20} ). In addition, SK1, SK2, and BAR assume the recovery of W′ to be exponential with respect to time. The results from this study did not find any such trends (Fig. 2 ). There was an increase in W′ recovered between 2 and 6 min, whereas a negative trend was seen in one case for two subjects, which could be attributed to the intraindividual variability. This was the reason for comparing TWD and Pp to establish the repeatability of the intermittent testing protocol, which showed less variability as indicated by the ICC, TE, and CV (refer to Supplementary Digital Content 3, https://links.lww.com/MSS/C32 ). Furthermore, the medium effect sizes for both cases (0.083 and 0.109) provide merit to the conclusion that the recovery of W′ may not be exponential for all athletes and all recovery durations. This is further illustrated by the average prediction errors in W′ balance of −1.312 ± 1.836 kJ (A3-SK2) and −1.596 ± 1.966 kJ (A3-BAR).

None of the existing literature accounts for the variability of CP and W′ at the subject level as it opposes the assumptions of these parameters being discrete and constant throughout the experiments. In the present study, in some cases, depletion of W′ was observed at the highest recovery power P_{rec-H} , indicated by the negative E _{rec} . This can be attributed to the variability of V˙O_{2} data around GET (^{29} ), which resulted in P_{GET} being in the range of ~0.9 CP. Regardless, the subjects were pedaling above CP (i.e., their actual CP on that day or during that exercise bout) although it was meant to be a recovery interval. This suggests that there is a variability associated with CP at the individual level, which is illustrated by the increase in CP seen in two of four subjects (excluding subject 7) who repeated the fresh 3MT four times. Similar results were reported by Miura and colleagues (^{37} ), where prior exercise in the heavy-intensity domain resulting in increased CP estimates. Furthermore, prior heavy-intensity exercise has also shown to increase W′ (^{38,39} ). However, these studies used the constant work-rate protocol to determine CP and W′ as opposed to the 3MT used in this study. The heavy-intensity exercise at P_{rec-M} and P_{rec-H} may have acted as an additional warm-up (^{38,39} ). These results indicate that CP and W′ have an associated variability that could be a trial-to-trial phenomenon or an intratrial phenomenon, thus pointing toward individualized time constants or models as suggested in several studies (^{17,18,20} ).

Another contribution of this study is the real-time performance optimization performed with one subject; however, there are a few limitations to the optimal power profile calculation. First, the recovery of W′ was assumed to depend only on recovery power for dynamic programming. Second, the subject would hover above and below the suggested optimal power and was unable to match it exactly at each instant within the 100-m interval. Third, the effects of drag were disregarded while determining the optimal strategy. Fourth, the trainer was unable to simulate the effect of coasting in the downhill sections, and hence the subject would keep pedaling in these segments. Fifth, there was more than a 4-wk gap between the completion of the intermittent tests and the optimization tests that could have resulted in changes in the subject’s CP, W′ , and recovery mechanisms. The optimization tests with elite cyclists instead of one competitive amateur may yield a better comparison between self-strategy and optimal strategy. Furthermore, the improved performance may be due to other factors such as the subject’s psychological aspects and the novelty of the test. Hence, similar studies in the future should conduct both the tests with more subjects, ample familiarization trials to develop self-strategies, and at least two optimal strategy trials to determine their repeatability and mitigate some of the limitations.

Considering all these limitations, the improvement of 55 s provides encouraging signs for future studies investigating W′ recovery to be used in real-time in situ performance optimization . The recovery of W′ may not be exponential in the range of 2 to 15 min when a semiexhaustive exertion interval precedes the recovery interval. We believe that a semiexhaustive exertion interval is a more realistic representation of a race/interval training scenario. The optimal power profile suggested changes the target power every 100 m, which at a speed of 15 mph is covered in ~15 s. This approach is similar to the microinterval training, which as suggested by Skiba and colleagues (^{17} ) is a common coaching practice.

In conclusion, this study illustrated the interaction effect between recovery characteristics (i.e., recovery power and duration) on recovery of W′ . This study showed that recovery power has a greater influence on the recovery of W′ in comparison with recovery duration. Moreover, in some cases, depletion of W′ was observed when the recovery power was in the vicinity of 0.9 CP, indicating the variability associated with CP. In addition, the present study showed the limitations of SK2 and BAR models with the overestimation of W′ balance at the end of the recovery intervals, thus highlighting the need to further investigate athlete-specific recovery parameters to tune existing models or developing new models for exercise below CP. Furthermore, the results of the optimal strategy test show promising signs for in situ real-time performance optimization using the CP concept.

The authors thank Lee Shearer, Nicholas Hayden, Frank Lara, Mason Coppi, Jake Ogden, and Brendan Rhim for their assistance in data collection.

No funding was received to assist in the preparation of this article.

The authors declare that they have no conflict of interest with regard to the content of this article.

The results of this study are presented clearly, honestly, and without fabrication, falsification, or inappropriate data manipulation. Results of this study do not constitute endorsement by the American College of Sports Medicine.

REFERENCES
1. Poole DC, Ward SA, Gardner GW, Whipp BJ. Metabolic and respiratory profile of the upper limit for prolonged exercise in man.

Ergonomics . 1988;31(19):1265–79.

2. Poole DC, Ward SA, Whipp BJ. The effects of training on the metabolic and respiratory profile of high-intensity cycle ergometer exercise.

Eur J Appl Physiol Occup Physiol . 1990;59(6):421–9.

3. Jones AM, Wilkerson DP, DiMenna F, Fulford J, Poole DC. Muscle metabolic responses to exercise above and below the “critical power” assessed using 31P-MRS.

Am J Physiol - Regul Integr Comp Physiol . 2008;294(2):585–93.

4. Monod H, Scherrer J. The work capacity of a synergic muscular group.

Ergonomics . 1965;8(3):329–38.

5. Moritani T, Nagata A, Devries HA, Muro M. Critical power as a measure of physical work capacity and anaerobic threshold.

Ergonomics . 1981;24(5):339–50.

6. Hughson RL, Orok CJ, Staudt LE. A high velocity treadmill running test to assess endurance running potential.

Int J Sports Med . 1984;5(1):23–5.

7. Wakayoshi K, Ikuta K, Yoshida T, et al. Determination and validity of critical velocity as an index of swimming performance in the competitive swimmer.

Eur J Appl Physiol Occup Physiol . 1992;64(2):153–7.

8. Kennedy MD, Bell GJ. A comparison of critical velocity estimates to actual velocities in predicting simulated rowing performance.

Can J Appl Physiol . 2000;25(4):223–35.

9. Chidnok W, Dimenna FJ, Bailey SJ, et al. Exercise tolerance in intermittent cycling: application of the critical power concept.

Med Sci Sports Exerc . 2012;44(5):966–76.

10. Skiba PF, Chidnok W, Vanhatalo A, Jones AM. Modeling the expenditure and reconstitution of work capacity above critical power.

Med Sci Sports Exerc . 2012;44(8):1526–32.

11. Morton RH, Billat LV. The critical power model for intermittent exercise.

Eur J Appl Physiol . 2004;91(2–3):303–7.

12. Ferguson C, Rossiter HB, Whipp BJ, Cathcart AJ, Murgatroyd SR, Ward SA. Effect of recovery duration from prior exhaustive exercise on the parameters of the power–duration relationship.

J Appl Physiol . 2010;108(4):866–74.

13. Skiba PF, Fulford J, Clarke DC, Vanhatalo A, Jones AM. Intramuscular determinants of the ability to recover work capacity above critical power.

Eur J Appl Physiol . 2015;115(4):703–13.

14. Fukuba Y, Whipp BJ. A metabolic limit on the ability to make up for lost time in endurance events.

J Appl Physiol . 1999;87(2):853–61.

15. Poole DC, Burnley M, Vanhatalo A, Rossiter HB, Jones AM. Critical power: an important fatigue threshold in exercise physiology.

Med Sci Sport Exerc . 2016;48(11):2320–34.

16. Skiba PF, Clarke D, Vanhatalo A, Jones AM. Validation of a novel intermittent W′ model for cycling using field data.

Int J Sports Physiol Perform . 2014;9(6):900–4.

17. Skiba PF, Jackman S, Clarke D, Vanhatalo A, Jones AM. Effect of work and recovery durations on

W′ reconstitution during intermittent exercise.

Med Sci Sports Exerc . 2014;46(7):1433–40.

18. Caen K, Bourgois JG, Bourgois G, Van der Stede T, Vermeire K, Boone J. The reconstitution of

W′ depends on both work and recovery characteristics.

Med Sci Sports Exerc . 2019;51(8):1745–51.

19. Skiba PF.

The Kinetics of the Work Capacity Above Critical Power . Dissertation, University of Exeter; 2014. p. 190.

20. Bartram JC, Thewlis D, Martin DT, Norton KI. Accuracy of

W′ recovery kinetics in high performance cyclists—modelling intermittent work capacity.

Int J Sports Physiol Perform . 2018;13(6):724–8.

21. Bartram JC, Thewlis D, Martin DT, Norton KI. Predicting critical power in elite cyclists: questioning the validity of the 3-minute all-out test.

Int J Sports Physiol Perform . 2017;12(6):783–7.

22. Cade WT, Bohnert KL, Reeds DN, et al. Peak oxygen uptake (V˙O

_{2peak} ) across childhood, adolescence and young adulthood in Barth syndrome: data from cross-sectional and longitudinal studies.

PLoS One . 2018;13(5):1–12.

23. Burnley M, Doust JH, Vanhatalo A. A 3-min all-out test to determine peak oxygen uptake and the maximal steady state.

Med Sci Sports Exerc . 2006;38(11):1995–2003.

24. Clark IE, Gartner HE, Williams JL, Pettitt RW. Validity of the 3-minute all-out exercise test on the CompuTrainer.

J Strength Cond Res . 2016;30(3):825–9.

25. Vanhatalo A, Doust JH, Burnley M. Determination of critical power using a 3-min all-out cycling test.

Med Sci Sports Exerc . 2007;39(3):548–55.

26. Vanhatalo A, Doust JH, Burnley M. Robustness of a 3 min all-out cycling test to manipulations of power profile and cadence in humans.

Exp Physiol . 2008;93(3):383–90.

27. Johnson TM, Sexton PJ, Placek AM, Murray SR, Pettitt RW. Reliability analysis of the 3-min all-out exercise test for cycle ergometry.

Med Sci Sports Exerc . 2011;43(12):2375–80.

28. Ashtiani F, Sreedhara VSM, Vahidi A, Mocko G, Hutchison R. Experimental modeling of cyclists fatigue and recovery dynamics enabling optimal pacing in a time trial. In:

2019 American Control Conference (ACC) . American Automatic Control Council; 2019. pp. 5083–8.

29. Beaver WL, Wasserman K, Whipp BJ. A new method for detecting anaerobic threshold by gas exchange.

J Appl Physiol . 1986;60(6):2020–7.

30. Clark IE, Murray SR, Pettitt RW. Alternative procedures for the three-minute all-out exercise test.

J Strength Cond Res . 2013;27(8):2104–12.

31. Hopkins WG. Measures of reliability in sports medicine and science.

Sport Med . 2000;30(1):1–15.

32. Constantini K, Sabapathy S, Cross TJ. A single-session testing protocol to determine critical power and W.

Eur J Appl Physiol . 2014;114(6):1153–61.

33. Cohen J.

Statistical Power Analysis for the Behavioral Sciences . 2nd ed. Hillsdale (NJ): Lawrence Erlbaum Associates; 1988. p. 567.

34. Iannetta D, Murias JM, Keir DA. A simple method to quantify the V˙O

_{2} mean response time of ramp-incremental exercise.

Med Sci Sports Exerc . 2019;51(5):1080–6.

35. Bickford P, Sreedhara VSM, Mocko GM, Vahidi A, Hutchison RE. Modeling the expenditure and recovery of anaerobic work capacity in cycling. In:

Proceedings of the 12th Conference of the International Sports Engineering Association ; 2018 Mar 26–29: Brisbane (Australia). 2018, 2, 219 (article number: 219.

https://doi.org/10.3390/proceedings2060219 ).

36. Earnest CP, Wharton RP, Church TS, Lucia A. Reliability of the Lode Excalibur Sport Ergometer and applicability to CompuTrainer electromagnetically braked cycling training device.

J Strength Cond Res . 2005;19(2):344–8.

37. Miura A, Shiragiku C, Hirotoshi Y, et al. The effect of prior heavy exercise on the parameters of the power–duration curve for cycle ergometry.

Appl Physiol Nutr Metab . 2009;34(6):1001–7.

38. Jones AM, Wilkerson DP, Burnley M, Koppo K. Prior heavy exercise enhances performance during subsequent perimaximal exercise.

Med Sci Sports Exerc . 2003;35(12):2085–92.

39. Burnley M, Davison G, Baker JR. Effects of priming exercise on V˙O

_{2} kinetics and the power–duration relationship.

Med Sci Sports Exerc . 2011;43(11):2171–9.