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

Abstract

The effects of submaximal performances on critical speed (S_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 C₁ for the shortest exercises and C₂ 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

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:

a = Pt_lim − bt_lim = (P − b) t_lim

t_lim = a/(P − b) = a/(P − P_Crit)

where b is a critical power (P_Crit).

In 1966, Ettema applied the critical-power concept to world records in running, swimming, cycling, and skating exercises [7] and proposed a linear relationship between the distances (D_lim) and world records (t_lim) from 1500 to 10000 m:

D_lim = a + b_tlim

It was assumed that the energy cost of running was almost independent of speed (1 kcal.kg⁻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).

D_lim = ADC + S_Crit t_lim

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].

1.2. 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.

1.3. 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: C₁ for the shortest performances and C₂ 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).

2. Methods

2.1. Arbitrary Units

The values of t_lim1, D_lim1 and S₁ in arbitrary units in protocols 1, 2 and 3 are equal to 1:

t_lim1 = 1   and   S₁ = 1   and   D_lim1 = 1

2.1.1. Arbitrary Units in Protocols 1 and 2

The values of t_lim2 and t_lim3 in arbitrary units in protocols 1 and 2 are equal to 4 and 7, respectively.

t_lim2 = 4   and   t_lim3 = 7

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:

D_lim1 = k t_lim1^g = k (1^g) = 1 then k = 1   and   D_lim = t_lim^g and   S = t_lim^g−1

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, S₂ and S₃ in arbitrary units are equal to:

D_lim2 = t_lim2^g = 4^0.95 = 3.7321   and   D_lim3 = t_lim3^g = 7^0.95 = 6.3510

S₂ = t_lim2^g−1 = 4^−0.05 = 0.9330   and   S₃ = t_lim3^g−1 = 7^−0.05 = 0.9073

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, S₂ and S₃ in arbitrary units are equal to:

D_lim2 = 4^0.80 = 3.0314   and   D_lim3 = 7^0.80 = 4.7433

S₂ = 4^−0.2 = 0.7579   and   S₃ = 7^−0.2 = 0.6776

2.1.2. Arbitrary Units in Protocol 3

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:

D_lim = t_lim^g

(t_lim^g)^1/g = t_lim = D_lim^1/g

t_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

For the low-endurance athlete:

t_lim2 = 3.3819^1/0.8 = 4.5862    and   t_lim3 = 5.5471^1/0.80 = 8.5130

For the high-endurance athlete:

t_lim2 = 3.3819^1/0.95 = 3.6059   and   t_lim3 = 5.5471^1/0.95 = 6. 0705

2.2. Coefficients C₁ and C₂

The ranges of coefficients C₁ and C₂ 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 C₁/C₂ 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 C₁ or C₂ and the variability of performances are higher [27]. Therefore, the ranges of C₁ and C₂ 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)

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:

t_{lim1 sub} = C₁ t_lim1   and   t_{lim2 sub} = C₂ t_lim2

Therefore, the submaximal values of D_lim (D_{lim1 sub} and D_{lim2 sub}) are equal to:

D_{lim1 sub} = S₁ t_{lim1 sub} = S₁C₁ t_lim1 = C₁ D_lim1

D_{lim2 sub} = S₂ t_{lim2 sub} = S₂C₂ t_lim2 = C₂ t_lim2

For running:

S_Crit = (D_lim2 − D_lim1)/(t_lim2 − t_lim1)

S_{Crit sub} = (C₂ D_lim2 − C₁ D_lim1)/(C₂ t_lim2 − C₁ t_lim1)

S_{Crit sub}/S_Crit = [(C₂ D_lim2 − C₁ D_lim1)/(C₂ t_lim2 − C₁ t_lim1)]/[(D_lim2 − D_lim1)/(t_lim2 − t_lim1)]

(1)

For cycling:

P_{Crit sub}/P_Crit = [(C₂ W_lim2 − C₁ W_lim1)/(C₂ t_lim2 − C₁ t_lim1)]/[(W_lim2 − W_lim1)/(t_lim2 − t_lim1)]

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

The submaximal performances are the result of submaximal speeds.

S_1sub = C₁S₁    and    S_2sub = C₂S₂

D_{lim1 sub} = C₁S₁ t_lim1 = C₁D_lim1    and    D_{lim2 sub} = C₂S₂ t_lim2 = C₂D_lim2

For cycling, the submaximal performances correspond to lower powers.

P_1sub = C₁P₁    and    P_2sub = P₂S₂

W_{lim1 sub} = C₁P₁ t_lim1 = C₁ W_lim1    and    W_{lim2 sub} = C₂P₂ t_lim2 = C₂ W_lim2

For running:

S_Crit = (D_lim2 − D_lim1)/(t_lim2 − t_lim1)

S_{Crit sub} = (D_{lim2 sub} − D_lim1sub)/(t_lim2 − t_lim1) = (C₂ D_lim2 − C₁ D_lim1)/(t_lim2 − t_lim1)

S_{Crit sub}/S_Crit = [(C₂ D_lim2 − C₁ D_lim1)/(t_lim2 − t_lim1)]/[(D_lim2 − D_lim1)/(t_lim2 − t_lim1)]

S_{Crit sub}/S_Crit = (C₂ D_lim2 − C₁ D_lim1) /(D_lim2 − D_lim1)

(2)

For cycling:

P_{Crit sub}/P_Crit = (C₂ W_lim2 − C₁ W_lim1)/(W_lim2 − W_lim1)

2.3.3. Computation in the constant-distance protocol

The submaximal performances are the result of submaximal speeds.

S_1sub = C₁S₁    and    S_2sub = C₂S₂

For cycling, the submaximal performances correspond to lower powers.

P_1sub = C₁P₁    and    P_2sub = C₂P₂

Hence:    t_lim1sub = D_lim1/S_1sub = D_lim1/C₁S₁ = t_lim1/C₁

and    t_lim2sub = D_lim2/S_2sub = D_lim2/C₂S₂ = t_lim2/C₂

For running:

S_Crit = (D_lim2 − D_lim1)/(t_lim2 − t_lim1)

S_{Crit sub} = (D_lim2 − D_lim1)/(t_lim2sub − t_lim1sub) = (D_lim2 − D_lim1)/(t_lim2/C₂ − t_lim1/C₁)

S_{Crit sub} /S_Crit = [(D_lim2 − D_lim1)/(t_lim2/C₂ − t_lim1/C₁)]/[(D_lim2 − D_lim1)/(t_lim2 − t_lim1)]

S_{Crit sub} /S_Crit = (t_lim2 − t_lim1)/(t_lim2/C₂ − t_lim1/C₁)

(3)

For cycling:

P_{Crit sub} /P_Crit = (t_lim2 − t_lim1)/(t_lim2/C₂ − t_lim1/C₁)

3. Results

The interest of the use of arbitrary units is demonstrated in

Table 1. Athletes A, B and C have the same values of t_lim1 and t_lim2 but different running speeds (S₁ and S₂). 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.

						C1 = 0.9 and C2 = 1		C1 = 1 and C2 = 0.9
	S1	S2	tlim 1	tlim 2	SCrit	SCritsub	SCritsub/SCrit	SCritsub	SCritsub/SCrit
	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

and

Table 2. Athletes A, B and C have the same values of t_lim1 and t_lim2 but different power-outputs (P₁ and P₂). 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.

						C1 = 0.9 and C2 = 1		C1 = 1 and C2 = 0.9
	P1	P2	tlim 1	tlim 2	PCrit	PCritsub	PCritsub/PCrit	PCritsub	PCritsub/PCrit
	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

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 S₂/S₁ (0.7579) have the same effects (S_Critsub/S_Crit) corresponding to the same coefficients C₁ and C₂ (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 P₂/P₁ are similar as in running exercises.

The effects of submaximal performances on endurance indices are presented in Figure 1 (constant-speed protocol), Figure 2 (constant-time protocol) and Figure 3 (constant-distance protocol).

3.1. 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 C₁ (0.80, 0.85, 0.90, 0.95 and 1.00) were computed from equation 1 with an increment of C₂ 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 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 S_{Crit sub}/S_Crit (or ratio P_{Crit sub}/P_Crit) according to Equation (1):

S_{Crit sub}/S_Crit = [(C₂ D_lim2 − C₂ D_lim1)/(C₂ t_lim2 − C₂ t_lim1)]/[(D_lim2 − D_lim1)/(t_lim2 − t_lim1)] = 1

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

The values of coefficients C₁ and C₂ were limited from 0.95 to 1. Six curves of ratio S_{Crit sub}/S_Crit corresponding to six values of C₁ (0.95, 0.96, 0.97, 0.98, 0.99 and 1.00) were computed from Equation (2) with an increment of C₂ 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 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 S_{Crit sub}/S_Crit (or P_{Crit sub}/P_Crit) are equal to C₂ (or C₁) according to Equation (2):

S_{Crit sub}/S_Crit = (C₂ D_lim2 − C₂ D_lim1)/(D_lim2 − D_lim1) = C₂ (or C₁)

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

Six curves of ratio S_{Crit sub}/S_Crit corresponding to six values of C₁ (0.95, 0.96, 0.97, 0.98, 0.99 and 1.00) were computed from equation 3 with an increment of C₂ 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 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 S_{Crit sub}/S_Crit (or P_{Crit sub}/P_Crit) are equal to C₂ (or C₁) according to Equation (3):

S_{Crit sub} /S_Crit = (t_lim2 − t_lim1)/(t_lim2/C₂ − t_lim1/C₂) = C₂ (t_lim2 − t_lim1)/(t_lim2 − t_lim1) = C₂ (or C₁)

4. 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:

-

C₁ is equal to C₂ for protocol 1(empty circles in Figure 1);

-

D_lim1 (1 − C₁) is equal to D_lim2 (1 − C₂) for protocol 2_;

-

t_lim1 (1 − 1/C₁) is equal to t_lim2 (1 − 1/C₂) 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.

In protocols 2 and 3, when C₁ is equal to C₂ (empty circles in Figure 2A, Figure 3B) ratios S_{Crit sub} /S_Crit are equal to C₁ and are not very low (S_{Crit sub}/S_Crit ≥ 0.95). When C₂ is lower than C₁, 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 C₂ = 0.95 and C₁ = 1). However, although the range of C₁-C₂ is much larger (0.8–1.0), ratios S_{Crit sub}/S_Crit are not very low (≥ 0.9567) in protocol 1, even when C₂ is lower than C₁ (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 (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.

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 C₁ and C₂ are equal and lower than 1 (empty circles in Figure 1) but increases when C₁ is equal to 1 and C₂ 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:

D_lim = k t_lim^g  and   S = k t_lim^g−1    power-law model

100 S/MAS = 100 − EI log(t_lim/t_MAS)          logarithmic model

5. 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.