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

## Abstract

**:**

_{Crit}) and critical power (P

_{Crit}) 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 S

_{Crit}and P

_{Crit}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 C

_{1}for the shortest exercises and C

_{2}for the longest exercises. Arbitrary units were used for exhaustion times (t

_{lim}), speeds (or power-output in cycling) and distances (or work in cycling). The submaximal-performance effects on S

_{Crit}and P

_{Crit}were computed from two ranges of t

_{lim}(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 S

_{Crit}and P

_{Crit}are lower in protocol 1. For the 3 protocols, the effects of submaximal performances on S

_{Crit}, and P

_{Crit}, are low in many cases and are lower when the range of t

_{lim}is longer. The results of the present theoretical study confirm the possibility of the computation of S

_{Crit}and P

_{Crit}from several submaximal exercises performed in the same session.

## 1. Introduction

#### 1.1. Empirical Model of Running and Cycling Performances

_{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:

_{lim}= a + bt

_{lim}= Pt

_{lim}

_{lim}is hyperbolic:

_{lim}− bt

_{lim}= (P − b) t

_{lim}

_{lim}= a/(P − b) = a/(P − P

_{Crit})

_{Crit}).

_{lim}) and world records (t

_{lim}) from 1500 to 10000 m:

_{lim}= a + b

_{tlim}

^{−1}.km

^{−1}) under 22 km.h

^{−1}[8]. Consequently, D

_{lim}and parameter “a” were equivalent to amounts of energy. Therefore, parameter “a” has been interpreted as equivalent to an energy store. Thereafter, parameter “a” was considered as an estimation of maximal Anaerobic Distance Capacity (ADC expressed in metres) for running exercises [7,9]. Slope b was considered as a critical velocity (S

_{Crit}).

_{lim}= ADC + S

_{Crit}t

_{lim}

_{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:

_{Crit 1/t}+ a (1/t

_{lim}) Running

_{Crit 1/t}+ a (1/t

_{lim}) Cycling

_{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].

_{lim}and D

_{lim}is not perfectly linear as suggested by the power-law model by Kennelly:

_{lim}= k t

_{lim}

^{g}

_{lim}/t

_{lim}= (k t

_{lim}

^{g})/t

_{lim}= k t

_{lim}

^{g−1}

_{lim}[15].

#### 1.2. Variability of the Performances of Exhausting Exercises.

_{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].

_{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.

#### 1.3. Purpose of the Present Study

_{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.

_{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].

_{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.

_{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).

_{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).

## 2. Methods

#### 2.1. Arbitrary Units

_{lim1}, D

_{lim1}and S

_{1}in arbitrary units in protocols 1, 2 and 3 are equal to 1:

_{lim1}= 1 and S

_{1}= 1 and D

_{lim1}= 1

#### 2.1.1. Arbitrary Units in Protocols 1 and 2

_{lim2}and t

_{lim3}in arbitrary units in protocols 1 and 2 are equal to 4 and 7, respectively.

_{lim2}= 4 and t

_{lim3}= 7

_{lim2}and t

_{lim3}was computed from the power-law model b Kennelly with arbitrary units of t

_{lim}:

_{lim1}= k t

_{lim1}

^{g}= k (1

^{g}) = 1 then k = 1 and D

_{lim}= t

_{lim}

^{g}and S = t

_{lim}

^{g−1}

_{,}the values of D

_{lim2}, D

_{lim3}, S

_{2}and S

_{3}in arbitrary units are equal to:

_{lim2}= t

_{lim2}

^{g}= 4

^{0.95}= 3.7321 and D

_{lim3}= t

_{lim3}

^{g}= 7

^{0.95}= 6.3510

_{2}= t

_{lim2}

^{g−1}= 4

^{−0.05}= 0.9330 and S

_{3}= t

_{lim3}

^{g−1}= 7

^{−0.05}= 0.9073

_{,}the values of D

_{lim2}, D

_{lim3}, S

_{2}and S

_{3}in arbitrary units are equal to:

_{lim2}= 4

^{0.80}= 3.0314 and D

_{lim3}= 7

^{0.80}= 4.7433

_{2}= 4

^{−0.2}= 0.7579 and S

_{3}= 7

^{−0.2}= 0.6776

#### 2.1.2. Arbitrary Units in Protocol 3

_{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:

_{lim}= t

_{lim}

^{g}

_{lim}

^{g})

^{1/g}= t

_{lim}= D

_{lim}

^{1/g}

_{lim1}= D

_{lim1}

^{1/g}= 1

^{1/g}= 1 and t

_{lim2}= D

_{lim2}

^{1/g}= 3.3819

^{1/g}

and t

_{lim3}= D

_{lim3}

^{1/g}= 5.5471

^{1/g}

_{lim2}= 3.3819

^{1/0.8}= 4.5862 and t

_{lim3}= 5.5471

^{1/0.80}= 8.5130

_{lim2}= 3.3819

^{1/0.95}= 3.6059 and t

_{lim3}= 5.5471

^{1/0.95}= 6. 0705

#### 2.2. Coefficients C_{1} and C_{2}

_{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).

#### 2.3. Computation of the Effects of Submaximal Performances on S_{Crit} and P_{Crit}

#### 2.3.1. Computation in the Constant-Speed (or Constant Power Output) Protocol (Protocol 1)

_{lim}. The submaximal values of t

_{lim}(t

_{lim1 sub}and t

_{lim2 sub}) are:

_{lim1 sub}= C

_{1}t

_{lim1}and t

_{lim2 sub}= C

_{2}t

_{lim2}

_{lim}(D

_{lim1 sub}and D

_{lim2 sub}) are equal to:

_{lim1 sub}= S

_{1}t

_{lim1 sub}= S

_{1}C

_{1}t

_{lim1}= C

_{1}D

_{lim1}

_{lim2 sub}= S

_{2}t

_{lim2 sub}= S

_{2}C

_{2}t

_{lim2}= C

_{2}t

_{lim2}

_{Crit}= (D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})

_{Crit sub}= (C

_{2}D

_{lim2}− C

_{1}D

_{lim1})/(C

_{2}t

_{lim2}− C

_{1}t

_{lim1})

_{Crit sub}/S

_{Crit}= [(C

_{2}D

_{lim2}− C

_{1}D

_{lim1})/(C

_{2}t

_{lim2}− C

_{1}t

_{lim1})]/[(D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})]

_{Crit sub}/P

_{Crit}= [(C

_{2}W

_{lim2}− C

_{1}W

_{lim1})/(C

_{2}t

_{lim2}− C

_{1}t

_{lim1})]/[(W

_{lim2}− W

_{lim1})/(t

_{lim2}− t

_{lim1})]

#### 2.3.2. Computation in the constant-time protocol (protocol 2)

_{1sub}= C

_{1}S

_{1}and S

_{2sub}= C

_{2}S

_{2}

_{lim1 sub}= C

_{1}S

_{1}t

_{lim1}= C

_{1}D

_{lim1}and D

_{lim2 sub}= C

_{2}S

_{2}t

_{lim2}= C

_{2}D

_{lim2}

_{1sub}= C

_{1}P

_{1}and P

_{2sub}= P

_{2}S

_{2}

_{lim1 sub}= C

_{1}P

_{1}t

_{lim1}= C

_{1}W

_{lim1}and W

_{lim2 sub}= C

_{2}P

_{2}t

_{lim2}= C

_{2}W

_{lim2}

_{Crit}= (D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})

_{Crit sub}= (D

_{lim2 sub}− D

_{lim1sub})/(t

_{lim2}− t

_{lim1}) = (C

_{2}D

_{lim2}− C

_{1}D

_{lim1})/(t

_{lim2}− t

_{lim1})

_{Crit sub}/S

_{Crit}= [(C

_{2}D

_{lim2}− C

_{1}D

_{lim1})/(t

_{lim2}− t

_{lim1})]/[(D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})]

_{Crit sub}/S

_{Crit}= (C

_{2}D

_{lim2}− C

_{1}D

_{lim1}) /(D

_{lim2}− D

_{lim1})

_{Crit sub}/P

_{Crit}= (C

_{2}W

_{lim2}− C

_{1}W

_{lim1})/(W

_{lim2}− W

_{lim1})

#### 2.3.3. Computation in the constant-distance protocol

_{1sub}= C

_{1}S

_{1}and S

_{2sub}= C

_{2}S

_{2}

_{1sub}= C

_{1}P

_{1}and P

_{2sub}= C

_{2}P

_{2}

_{lim1sub}= D

_{lim1}/S

_{1sub}= D

_{lim1}/C

_{1}S

_{1}= t

_{lim1}/C

_{1 }

_{lim2sub}= D

_{lim2}/S

_{2sub}= D

_{lim2}/C

_{2}S

_{2}= t

_{lim2}/C

_{2}

_{Crit}= (D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})

_{Crit sub}= (D

_{lim2}− D

_{lim1})/(t

_{lim2sub}− t

_{lim1sub}) = (D

_{lim2}− D

_{lim1})/(t

_{lim2}/C

_{2}− t

_{lim1}/C

_{1})

_{Crit sub}/S

_{Crit}= [(D

_{lim2}− D

_{lim1})/(t

_{lim2}/C

_{2}− t

_{lim1}/C

_{1})]/[(D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})]

_{Crit sub}/S

_{Crit}= (t

_{lim2}− t

_{lim1})/(t

_{lim2}/C

_{2}− t

_{lim1}/C

_{1})

_{Crit sub}/P

_{Crit}= (t

_{lim2}− t

_{lim1})/(t

_{lim2}/C

_{2}− t

_{lim1}/C

_{1})

## 3. Results

_{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.

#### 3.1. Results for Protocol 1 (Constant Speed or Power Output Protocol)

_{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.

_{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).

_{Crit sub}/S

_{Crit}are lower when the range of t

_{lim}is longer (1–7 instead of 1–4).

_{Crit sub}/S

_{Crit}are equal to 0.9567 and 1.0295, respectively.

_{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):

_{Crit sub}/S

_{Crit}= [(C

_{2}D

_{lim2}− C

_{2}D

_{lim1})/(C

_{2}t

_{lim2}− C

_{2}t

_{lim1})]/[(D

_{lim2}− D

_{lim1})/(t

_{lim2}− t

_{lim1})] = 1

#### 3.2. Results for Protocol 2 (Constant-Time Protocol)

_{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.

_{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).

_{Crit sub}/S

_{Crit}are lower when the range of t

_{lim}is longer (1–7 instead of 1–4).

_{Crit sub}/S

_{Crit}are equal to 0.9254 and 1.0246, respectively.

_{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):

_{Crit sub}/S

_{Crit}= (C

_{2}D

_{lim2}− C

_{2}D

_{lim1})/(D

_{lim2}− D

_{lim1}) = C

_{2}(or C

_{1})

#### 3.3. Results for Protocol 3 (Constant-Distance Protocol)

_{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.

_{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.

_{Crit sub}/S

_{Crit}are equal to 0.9321 and 1.0206, respectively.

_{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):

_{Crit sub}/S

_{Crit}= (t

_{lim2}− t

_{lim1})/(t

_{lim2}/C

_{2}− t

_{lim1}/C

_{2}) = C

_{2}(t

_{lim2}− t

_{lim1})/(t

_{lim2}− t

_{lim1}) = C

_{2}(or C

_{1})

## 4. Discussion

_{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.

_{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).

_{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:

- -
- -
- 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.

_{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}.

_{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).

_{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].

_{Crit sub}/S

_{Crit}are lower in the high-endurance athlete for constant-speed and constant-time protocols (Figure 1 and Figure 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 Figure 1, Figure 2 and Figure 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.

_{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].

_{lim}= k t

_{lim}

^{g}and S = k t

_{lim}

^{g−1}power-law model

_{lim}/t

_{MAS}) logarithmic model

## 5. Conclusions

_{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.

## Funding

## Conflicts of Interest

## References

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**Figure 1.**effects of submaximal performances in protocol 1 (constant speed or constant power) on ratios S

_{Critsub}/S

_{Crit}or P

_{Critsub}/P

_{Crit}for the different values of C

_{1}and C

_{2}. Figures

**A**and

**B**correspond to the range t

_{lim1}-t

_{lim2}(1–4) whereas Figures

**C**and

**D**correspond to the range t

_{lim1}-t

_{lim3}(1–7). The specifications of the lines are presented in Figure 1A. Empty circles correspond to C

_{1}equal to C

_{2}.

**Figure 2.**effects of submaximal performances in protocol 2 (constant time-protocol) on ratios S

_{Crit sub}/S

_{Crit}for the different values of C

_{1}and C

_{2}. Figures

**A**and

**B**correspond to the range t

_{lim1}-t

_{lim2}(1–4) 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}.

**Figure 3.**Effects of submaximal performances in protocol 3 (constant distance-protocol) for the different values of C

_{1}(specification of the lines in 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}.

**Figure 4.**Comparison of the effects of submaximal performances in the 3 protocols. Black dot: maximal performance. Red dot: submaximal performance in protocol 1. Blue dot: submaximal performance in protocol 2. Green dot: submaximal performance in protocol 3. The dotted line corresponds to the relationship between distance (D) and time (t) at a given running speed (S).

**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 | ||||||||
---|---|---|---|---|---|---|---|---|---|

S_{1} | S_{2} | t_{lim 1} | t_{lim 2} | S_{Crit} | S_{Critsub} | S_{Critsub}/S_{Crit} | S_{Critsub} | S_{Critsub}/S_{Crit} | |

m.s^{−1} | m.s^{−1} | s | S | m.s^{−1} | m.s^{−1} | m.s^{-1} | |||

A | 4 | 3.0316 | 180 | 720 | 2.709 | 2.750 | 1.015 | 2.659 | 0.982 |

B | 5 | 3.7895 | 180 | 720 | 3.386 | 3.438 | 1.015 | 3.324 | 0.982 |

C | 6 | 4.5474 | 180 | 720 | 4.063 | 4.126 | 1.015 | 3.989 | 0.982 |

D | 4 | 3.0316 | 240 | 960 | 2.709 | 2.750 | 1.015 | 2.659 | 0.982 |

E | 5 | 3.7895 | 240 | 960 | 3.386 | 3.438 | 1.015 | 3.324 | 0.982 |

F | 6 | 4.5474 | 240 | 960 | 4.063 | 4.126 | 1.015 | 3.989 | 0.982 |

**Table 2.**Athletes A, B and C have the same values of t

_{lim1}and t

_{lim2}but different power-outputs (P

_{1}and P

_{2}). Athletes D, E and F have the same values of power-outputs 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 | ||||||||
---|---|---|---|---|---|---|---|---|---|

P_{1} | P_{2} | t_{lim 1} | t_{lim 2} | P_{Crit} | P_{Critsub} | P_{Critsub}/P_{Crit} | P_{Critsub} | P_{Critsub}/P_{Crit} | |

W | W | S | S | W | W | W | |||

A | 240 | 182 | 180 | 720 | 163 | 165 | 1.015 | 160 | 0.982 |

B | 320 | 243 | 180 | 720 | 217 | 220 | 1.015 | 213 | 0.982 |

C | 400 | 303 | 180 | 720 | 271 | 275 | 1.015 | 266 | 0.982 |

D | 240 | 182 | 240 | 960 | 163 | 165 | 1.015 | 160 | 0.982 |

E | 320 | 243 | 240 | 960 | 217 | 220 | 1.015 | 213 | 0.982 |

F | 400 | 303 | 240 | 960 | 271 | 275 | 1.015 | 266 | 0.982 |

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**MDPI and ACS Style**

Vandewalle, H. Effects of Submaximal Performances on Critical Speed and Power: Uses of an Arbitrary-Unit Method with Different Protocols. *Sports* **2019**, *7*, 136.
https://doi.org/10.3390/sports7060136

**AMA Style**

Vandewalle H. Effects of Submaximal Performances on Critical Speed and Power: Uses of an Arbitrary-Unit Method with Different Protocols. *Sports*. 2019; 7(6):136.
https://doi.org/10.3390/sports7060136

**Chicago/Turabian Style**

Vandewalle, Henry. 2019. "Effects of Submaximal Performances on Critical Speed and Power: Uses of an Arbitrary-Unit Method with Different Protocols" *Sports* 7, no. 6: 136.
https://doi.org/10.3390/sports7060136