Effects of Submaximal Performances on Critical Speed and Power: Uses of an Arbitrary-Unit Method with Different Protocols

The effects of submaximal performances on critical speed (SCrit) and critical power (PCrit) were studied in 3 protocols: a constant-speed protocol (protocol 1), a constant-time protocol (protocol 2) and a constant-distance protocol (protocol 3). The effects of submaximal performances on SCrit and PCrit were studied with the results of two theoretical maximal exercises multiplied by coefficients lower or equal to 1 (from 0.8 to 1 for protocol 1; from 0.95 to 1 for protocols 2 and 3): coefficient C1 for the shortest exercises and C2 for the longest exercises. Arbitrary units were used for exhaustion times (tlim), speeds (or power-output in cycling) and distances (or work in cycling). The submaximal-performance effects on SCrit and PCrit were computed from two ranges of tlim (1–4 and 1–7). These effects have been compared for a low-endurance athlete (exponent = 0.8 in the power-law model of Kennelly) and a high-endurance athlete (exponent = 0.95). Unexpectedly, the effects of submaximal performances on SCrit and PCrit are lower in protocol 1. For the 3 protocols, the effects of submaximal performances on SCrit, and PCrit, are low in many cases and are lower when the range of tlim is longer. The results of the present theoretical study confirm the possibility of the computation of SCrit and PCrit from several submaximal exercises performed in the same session.


Empirical Model of Running and Cycling Performances
Several empirical and descriptive models of performance have been proposed: the power-law model by Kennelly [1], asymptotic hyperbolic models by Hill and Scherrer [2,3], and, more recently, the logarithmic model of Péronnet and Thibault [4] and the 3-parameter asymptotic models by Hopkins [5] and Morton [6]. These empirical models are often used to estimate (i) the improvement in performance (ii) the future performances and running speeds over given distances (iii) the endurance capability, i.e., "the ability to sustain a high fractional utilization of maximal oxygen uptake for a prolonged period of time" [4].
In 1954, Scherrer et al. proposed a linear relationship [3] between the exhaustion time (t lim ) of a local exercise (flexions or extensions of the elbow or the knee) performed at different constant power outputs (P) and the total amount of work performed at exhaustion (W lim ) for t lim ranging between 3 and 30 min: W lim = a + bt lim = Pt lim Consequently, the relationship between P and t lim is hyperbolic: In 1981, the linear W lim -t lim relationship was adapted to exercises on a stationary cycle ergometer and it was demonstrated that slope b of the W lim -t lim relationship was correlated with the ventilatory threshold [10]. Therefore, slope b was proposed as an index of general endurance (P Crit ). Thereafter, Whipp et al. [11] proposed another linear model with S Crit (or P Crit ) and a: S = S Crit 1/t + a (1/t lim ) Running P = P Crit 1/t + a (1/t lim ) Cycling where S is running speed.
Actually, the hyperbolic model is often used as it is the simplest model that corresponds to a linear relationship between exhaustion time (t lim ) and distance (D lim ) in running and swimming or total work (W lim ) in cycling. For many exercise physiologists, S Crit and P Crit are considered as fatigue thresholds [12]. Moreover, the values of S Crit and P Crit become endurance indices when they are normalized to Maximal Aerobic Speed (S Crit /MAS) or Maximal Aerobic Power (P Crit /MAP). Parameter "a" is also called ADC (Anaerobic Distance Capacity) or ARC (Anaerobic Running Capacity) in running [7,9] and AWC (Anaerobic Work Capacity) in cycling [13,14].
However, the relationship between t lim and D lim is not perfectly linear as suggested by the power-law model by Kennelly: D lim = k t lim g S = D lim /t lim = (k t lim g )/t lim = k t lim g−1 where exponent g can be considered as an endurance index and parameter k is equal to the maximal speed corresponding to the unit of t lim [15].

Variability of the Performances of Exhausting Exercises.
Three protocols are used for the estimation of the performances of exhausting exercises: (1) to run as long as possible at a constant speed or to cycle as long as possible at a constant power output. This constant-speed protocol is often called Time-to-Exhaustion; (2) to run as much distance (or to produce as much work on a cycle-ergometer) as they can within a given time period (constant-time protocol); (3) to run a set distance (or to produce a set work on a cycle-ergometer) as fast as possible (constant-distance protocol).
The first protocol (Time-to-Exhaustion) is used for the estimation of S Crit on a treadmill or the estimation of P Crit on a cycle ergometer. The second protocol was used for prediction of one-hour running performance [16]. The third protocol was used in the studies on the modelling of running performances that were based on the world records [17][18][19][20] or performances in the Olympic games [21] or individual performance of elite endurance runners [15]. The reliability of performances in protocol 1 (constant speed) is low, whereas the reliability of the other protocols is higher [22][23][24][25][26][27]. For example in swimming, the Coefficient of Variation of constant-speed protocol (CV = 6.46 ± 6.24%) was significantly less reliable (p < 0.001) than those of constant-time protocol (CV = 0.63 ± 0.54%) and constant-distance protocol (CV = 0.56 ± 0.60%) [27].
In a recent study [28] critical speeds measured on a treadmill with a constant-speed protocol were compared with a Single-Visit Field Test of Critical Speed. The constant-speed runs to exhaustion on treadmill were performed with 3 running speeds during 3 separate sessions. Two single-visit field tests on separate days consisted to the measurement of maximal performances over 3600, 2400, and 1200 m (constant-distance protocol) with 30-min or 60-min recovery. Unexpectedly, there was no difference in S Crit measured with the treadmill and 30-min-and 60-min recovery field tests although the reliability of protocol 1 is lower than that of protocol 3. Thereafter, a Single-Visit Field Test of Critical Speed was tested in trained and untrained runners [29]: the reliability of S Crit was better in the trained runners.

Purpose of the Present Study
In few studies, the values of S Crit and P Crit are computed from the best maximal performances of several exhausting exercises of the same subjects [15] or world records [17][18][19][20] or performances in the Olympic games [21]. In most studies, it is not obvious that the data used in the computation of S Crit and P Crit are maximal. For example, the performance variability is important in protocol 1 that is mainly used in laboratories. Moreover, several exhausting exercises are often performed in the same session with protocols 1, 2 and 3, which could increase the performance variability because of fatigue. As suggested in a review [30], the purpose of the present study was to confirm the interest of S Crit and P Crit computed from exercises whose performances are submaximal.
In the Single-Visit Field Test of Critical Speed, there were trained and untrained runners whose reliability of S Crit was different [29]. Therefore, the effects of submaximal performance on S Crit and P Crit in the present study have been compared between a low-endurance athlete and a high-endurance athlete, i.e., athletes with low and high endurance indices (for example, exponent g or S Crit /MAS or P Crit /MAP). The values of exponent g were about 0.95 in the best elite endurance runners [15,31] as Gebrselassie whose ratio S Crit /MAS was equal to 0.945 (MAS corresponded to the maximal running speed during 7 min). In the low-endurance runners whose ratios S Crit /MAS were equal to 0.764 ± 0.078, exponent g was about 0.80 [31].
As S Crit and P Crit can be computed from two exhausting exercises, the effects of submaximal performances on these indices have been estimated by multiplying both theoretical maximal data by two coefficients: C 1 for the shortest performances and C 2 for the longest performances.
The effects of submaximal performances on S Crit and P Crit were estimated with arbitrary units for t lim , D lim and S (or P) in the present study. Indeed, there are many different cases: • the range of t lim can be different for each athlete in protocols 1 and 3; • the range of speeds and distances (or power-output and work in cycling) can be different for each athlete in protocol 2; • the same endurance indices can correspond to different maximal aerobic speed (MAS) or maximal aerobic power (MAP in cycling).
The effects of submaximal performances on endurance indices have been tested for values of t lim equal to 1 and 4 (arbitrary units), which corresponds to the usual range of t lim (3-12m in [28,29]) in many study. In some studies, the range of t lim is 2-15 min [12,32,33], Therefore, the effects of submaximal performances have also been tested for values of t lim equal to 1-7 (arbitrary unit).

Arbitrary Units
The values of t lim1 , D lim1 and S 1 in arbitrary units in protocols 1, 2 and 3 are equal to 1: The distances and running speeds corresponding to t lim2 and t lim3 was computed from the power-law model b Kennelly with arbitrary units of t lim : For the high-endurance athlete, exponent g of the power-law model is equal to 0.95. Therefore , the values of D lim2 , D lim3 , S 2 and S 3 in arbitrary units are equal to: For the low-endurance athlete, exponent g of the power-law model is equal to 0.80. Therefore, the values of D lim2 , D lim3 , S 2  The values of constant-distances (D lim ) in the present study were equal to the averages of the distances of low and high-endurance athletes, in protocols 1 and 2, i.e., 1 (D lim1 ), 3.3819 (D lim2 ) and 5.5471 (D lim3 ). The values of t lim2 and t lim3 corresponding to these distances were:

Coefficients C 1 and C 2
The ranges of coefficients C 1 and C 2 used in protocol 2 and 3 were from 0.95 to 1. Indeed, in protocols 2 and 3, the submaximal performances are the results of submaximal speeds (or submaximal power outputs in cycling). If the ratio C 1 /C 2 is lower than 0.9330, the speed corresponding to t lim2 would be higher than the speed at t lim1 in the high-endurance athlete. In protocol 1 (constant-speed protocol), the speed (or power) does not depend on C 1 or C 2 and the variability of performances are higher [27]. Therefore, the ranges of C 1 and C 2 were larger (from 0.8 to 1).

Computation in the Constant-Speed (or Constant Power Output) Protocol (Protocol 1)
The submaximal performances are the result of the submaximal values of t lim . The submaximal values of t lim (t lim1 sub and t lim2 sub ) are: Therefore, the submaximal values of D lim (D lim1 sub and D lim2 sub ) are equal to: For running: For cycling: The submaximal performances are the result of submaximal speeds.
For cycling, the submaximal performances correspond to lower powers.
For running: For cycling: The submaximal performances are the result of submaximal speeds.
S 1sub = C 1 S 1 and S 2sub = C 2 S 2 For cycling, the submaximal performances correspond to lower powers.

Results
The interest of the use of arbitrary units is demonstrated in Tables 1 and 2 for protocol 1. Athletes A, B, C, D, E and F who have the same ratio t lim1 /t lim2 (4) and the same ratio S 2 /S 1 (0.7579) have the same effects (S Critsub /S Crit ) corresponding to the same coefficients C 1 and C 2 (Table 1). Moreover, the use of the same arbitrary units can also been applied to cycling exercises (Table 2): the effects (P Critsub /P Crit ) of submaximal performances are the same when ratio t lim1 /t lim2 and ratio P 2 /P 1 are similar as in running exercises. Table 1. Athletes A, B and C have the same values of t lim1 and t lim2 but different running speeds (S 1 and S 2 ). Athletes D, E and F have the same values of running speed as athletes A, B and C, respectively. The values of t lim1 and t lim2 are higher in athletes D, E and F but ratio t lim2 /t lim1 is the same and equal to 4. C 1 = 0.9 and C 2 = 1 C 1 = 1 and C 2 = 0.9 The effects of submaximal performances on endurance indices are presented in Figure 1 Figure 1A. Empty circles correspond to C1 equal to C2.
In Figure 1A, the lowest and the highest ratios SCrit sub / SCrit are equal to 0.9567 and 1.0295, respectively.
When C1 is equal to C2 (empty circles), i.e., when the levels of submaximal performances are the same for both exhausting exercises, there is no effect of submaximal performances on ratio SCrit sub / SCrit (or ratio PCrit sub / PCrit) according to Equation (1):  (1-7). The specifications of the lines are presented in Figure 1A. Empty circles correspond to C 1 equal to C 2 .

Results for Protocol 1 (Constant Speed or Power Output Protocol)
For protocol 1, five curves of ratio S Crit sub /S Crit corresponding to five values of C 1 (0.80, 0.85, 0.90, 0.95 and 1.00) were computed from equation 1 with an increment of C 2 equal to 0.001.
The effects of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) are lower in the high-endurance athlete ( Figure 1B,D) than in the low-endurance athlete ( Figure 1A,C).
In Figure 1B,D, the effects of submaximal performances on ratio S Crit sub /S Crit are lower when the range of t lim is longer (1-7 instead of 1-4).
In Figure 1A, the lowest and the highest ratios S Crit sub /S Crit are equal to 0.9567 and 1.0295, respectively.
When C 1 is equal to C 2 (empty circles), i.e., when the levels of submaximal performances are the same for both exhausting exercises, there is no effect of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) according to Equation (1): In Figure 2A, the lowest and the highest ratios SCrit sub / SCrit are equal to 0.9254 and 1.0246, respectively.
When C1 is equal to C2 (empty circles in Figure 2A), i.e., when the levels of submaximal performances are the same for both exhausting exercises, the ratios SCrit sub/SCrit (or PCrit sub/PCrit) are equal to C2 (or C1) according to Equation (2):  Figures A and B correspond to the range t lim1 -t lim2 (1)(2)(3)(4) whereas Figures C and D correspond to the range t lim1 -t lim3 (1-7). The specification of the lines is presented in Figure A. Empty circles in Figure A correspond to C 1 equal to C 2 . In Figure 3B, the lowest and the highest ratios SCrit sub / SCrit are equal to 0.9321 and 1.0206, respectively.
When C1 is equal to C2 (empty circles in Figure 3B), i.e., when the levels of submaximal performances are the same for both exhausting exercises, the ratios SCrit sub/SCrit (or PCrit sub/PCrit) are equal to C2 (or C1) according to Equation (3):

Discussion
The effects of submaximal performances on SCrit 1/ t and PCrit 1/ t in the model proposed by Whipp et al. [11] are not presented in the present study. Indeed, SCrit 1/ t (or PCrit 1/ t) is equal to SCrit (or PCrit) when both indices are computed only from two exhausting exercises with constant-distance [15] or constant-power [34] protocols in running and cycling. Similarly, in the present study, the effects of submaximal performances were the same for SCrit 1/ t and SCrit (or PCrit 1/ t and PCrit) when they were computed from two submaximal exercises whatever the protocol. Consequently, the Figures about the effects of submaximal performances on SCrit 1/ t or PCrit 1/ t are not added in the present study.  Figure B are the same as in Figure 2) and C 2 . Figures A and B correspond to the range t lim1 -t lim2 whereas Figures C and D correspond to the range t lim1 -t lim3 . Empty circles in Figure B correspond to C 1 equal to C 2 .

Results for Protocol 2 (Constant-Time Protocol)
The values of coefficients C 1 and C 2 were limited from 0.95 to 1. Six curves of ratio S Crit sub /S Crit corresponding to six values of C 1 (0.95, 0.96, 0.97, 0.98, 0.99 and 1.00) were computed from Equation (2) with an increment of C 2 equal to 0.001.
As for protocol 1, the effects of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) are lower in the high-endurance athlete ( Figure 2B,D) than in the low-endurance athlete (Figure 2A,C).
In Figure 2C,D, the effects of submaximal performances on ratio S Crit sub /S Crit are lower when the range of t lim is longer (1-7 instead of 1-4).
In Figure 2A, the lowest and the highest ratios S Crit sub /S Crit are equal to 0.9254 and 1.0246, respectively.
When C 1 is equal to C 2 (empty circles in Figure 2A), i.e., when the levels of submaximal performances are the same for both exhausting exercises, the ratios S Crit sub /S Crit (or P Crit sub /P Crit ) are equal to C 2 (or C 1 ) according to Equation (2): S Crit sub /S Crit = (C 2 D lim2 − C 2 D lim1 )/(D lim2 − D lim1 ) = C 2 (or C 1 )

Results for Protocol 3 (Constant-Distance Protocol)
Six curves of ratio S Crit sub /S Crit corresponding to six values of C 1 (0.95, 0.96, 0.97, 0.98, 0.99 and 1.00) were computed from equation 3 with an increment of C 2 equal to 0.001.
In contrast with protocols 1 and 2, the effects of submaximal performance on ratio S Crit sub /S Crit (or P Crit sub /P Crit ) are more important in the high-endurance athlete. However, in the high-endurance athlete, the ranges of t lim1 -t lim2 (1-3.6059) and t lim1 -t lim3 (1-6.0705) is shorter than the ranges of t lim1 -t lim2 (1-4.5862) and t lim1 -t lim3 (1-8.5130) in the low-endurance athlete.
In Figure 3B, the lowest and the highest ratios S Crit sub /S Crit are equal to 0.9321 and 1.0206, respectively.
When C 1 is equal to C 2 (empty circles in Figure 3B), i.e., when the levels of submaximal performances are the same for both exhausting exercises, the ratios S Crit sub /S Crit (or P Crit sub /P Crit ) are equal to C 2 (or C 1 ) according to Equation (3):

Discussion
The effects of submaximal performances on S Crit 1/t and P Crit 1/t in the model proposed by Whipp et al. [11] are not presented in the present study. Indeed, S Crit 1/t (or P Crit 1/t ) is equal to S Crit (or P Crit ) when both indices are computed only from two exhausting exercises with constant-distance [15] or constant-power [34] protocols in running and cycling. Similarly, in the present study, the effects of submaximal performances were the same for S Crit 1/t and S Crit (or P Crit 1/t and P Crit ) when they were computed from two submaximal exercises whatever the protocol. Consequently, the Figures about the effects of submaximal performances on S Crit 1/t or P Crit 1/t are not added in the present study.
Previous experimental studies [22][23][24][25][26][27] showed that performance reliability with constant-speed protocol is significantly lower than those with the other protocols (constant-time or constant-distance protocols). However, for S Crit or P Crit in the present theoretical study, the effects of 20%-submaximal performances in protocol 1 are lower than the effects of 5%-submaximal performances in protocols 2 and 3. For example, the lowest ratio S Crit sub /S Crit in protocol 1 is equal to 0.9567 ( Figure 1A) whereas the lowest ratio in protocol 2 is equal to 0.9254 (Figure 2A).
Ratio S Cri tsub /S Crit (or ratio P Crit sub /P Crit ) is equal to 1 when the maximal and submaximal D lim -t lim relationships are parallel i.e., when the distances of both submaximal performances to the maximal D lim -t lim line are similar: -C 1 is equal to C 2 for protocol 1(empty circles in Figure 1); -D lim1 (1 − C 1 ) is equal to D lim2 (1 − C 2 ) for protocol 2 ; -t lim1 (1 − 1/C 1 ) is equal to t lim2 (1 − 1/C 2 ) for protocol 3.
In protocol 1, the submaximal performances correspond to similar decreases in D lim and t lim whereas, in protocol 2 and 3, they only correspond to a decrease in D lim or t lim (Figure 4). These simultaneous decreases in D lim and t lim limit the distance between the submaximal performance and the maximal D lim -t lim line, which explain the lower effects of submaximal performances on ratio S Crit sub /S Crit . possible low effects of submaximal performances on ratio SCrit sub / SCrit could explain that it is possible to compute SCrit from the values of tlim of 3 trials performed with protocol 3 in a same session with only 30 min of recovery between the trials as in a Single-Visit Field Test [28,29]. This low sensitivity of SCrit or PCrit to submaximal performances was previously suggested in a study on the comparison of critical speeds of continuous and intermittent running exercise on a track [35] and also in a review [30].  In protocols 2 and 3, when C 1 is equal to C 2 (empty circles in Figure 2A, Figure 3B) ratios S Crit sub /S Crit are equal to C 1 and are not very low (S Crit sub /S Crit ≥ 0.95). When C 2 is lower than C 1 , ratios S Crit sub /S Crit are lower and sometimes not negligible in protocols 2 and 3 (for example, in Figure 2A, S Crit sub /S Crit = 0.9254 for C 2 = 0.95 and C 1 = 1). However, although the range of C 1 -C 2 is much larger (0.8-1.0), ratios S Crit sub /S Crit are not very low (≥ 0.9567) in protocol 1, even when C 2 is lower than C 1 ( Figure 1A).
The effects of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) may be low (S Crit sub /S Crit ≥ 0.95) for both low-endurance and high-endurance athletes in the three protocols. These possible low effects of submaximal performances on ratio S Crit sub /S Crit could explain that it is possible to compute S Crit from the values of t lim of 3 trials performed with protocol 3 in a same session with only 30 min of recovery between the trials as in a Single-Visit Field Test [28,29]. This low sensitivity of S Crit or P Crit to submaximal performances was previously suggested in a study on the comparison of critical speeds of continuous and intermittent running exercise on a track [35] and also in a review [30].
The effects of submaximal performances on ratio S Crit sub /S Crit are lower in the high-endurance athlete for constant-speed and constant-time protocols (Figures 1 and 2). That said, the effects of submaximal performances on ratio S Crit sub /S Crit in constant-distance protocol (protocol 3) are higher in the high-endurance athlete ( Figure 3). However, the effects of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) are lower when the range of t lim is longer (for example, 1-7 instead of 1-4) as illustrated in Figures 1-3. Therefore, in protocol 3, the shorter ranges of t lim1 -t lim2 and t lim1 -t lim3 in the high-endurance athlete explain these computed higher effects of submaximal performances on ratio S Crit sub /S Crit . In contrast, the coefficient of variation of the Single-Visit Field Test that corresponds to this constant-distance protocol was lower in trained runners whose S Crit were faster than untrained runners [29]. It was likely that the reliability of S Crit in the trained runners was higher because of control of the maximal running speed corresponding to a given distance and better recovery.
In the Single-Visit Field Test [28], the running performances were submaximal because the values of D' (equivalent of ADC) were significantly lower in the 30-min (106.4-m) and 60-min-recovery (102.4-m) than in the 3-session treadmill test (249.7 m). However, S Crit in the Single-Visit Field Test was not different of S Crit in the 3-session treadmill test. These results were consistent with those of a previous experimental study on the effects of a 6-min exhausting exercise on S Crit in cycling [36].
In the present theoretical study, the submaximal performances have also effects on parameter "a" (ADC). For example, in protocol 1, parameter "a" decreases when C 1 and C 2 are equal and lower than 1 (empty circles in Figure 1) but increases when C 1 is equal to 1 and C 2 is lower than 1. These effects of submaximal performances on parameter "a" are not computed in the present study because this parameter is not an endurance index and its meaning is questionable [11].
The present method can also be used for the submaximal-performance effects on the exponent g of the power-law model by Kennelly [1] and the endurance index (EI) of the logarithmic model by Péronnet and Thibault [4] as they are 2-parameter models:

Conclusions
The results of the present theoretical study confirm the interest of S Crit and P Crit computed from exercises whose performances are submaximal and performed in the same session. Indeed, for the 3 protocols, the theoretical effects of submaximal performances on ratio S Crit sub /S Crit (or ratio P Crit sub /P Crit ) are low in many cases. The effects of submaximal performances are lower when the ratio t lim2 /t lim1 is larger. In protocol 3, it is likely that, in practice, the reliability of S Crit is better in trained runners due to the control of the maximal running speed corresponding to a given distance.