


ABSTRACT 
For highvelocity running or swimming, the relationship between velocity (v) and its sustainable duration (t) can be described by a hyperbolic relationship: (v  V_{crit})·t = D’, where V_{crit} is termed critical velocity, and D’ is defined as a curvature constant of the hyperbolic curve. The purposes of this study were to examine whether the V_{crit} could be applied to evaluate shortdistance breaststroke swimming performance and to evaluate the relative contribution of D’ in shortdistance swimming performance. Eleven male swimmers performed a series of time trials corresponding to 75, 100, and 150m in an indoor 50m swimming pool. The observed records were calculated into average velocities of each event to determine V_{crit} and D’. After the determination of V_{crit} and D’, all subjects performed 50m time trial on another day. A maximal anaerobic power test using cycle ergometer was also performed in the laboratory. The average velocity of the 50m time trial significantly correlated with the obtained V_{crit}, but not with D’. D’ was significantly correlated with the residual error, calculated from the regression analysis for the relationship between V_{crit} and the average velocities of 50m time trial. A cluster analysis showed that most of the subjects were classified as V_{crit} dependency when performing 50m time trial. Those results indicated that V_{crit} could be applied to evaluate shortdistance swimming performance, and it determined around 80% of the shortdistance breaststroke swimming performance. 
Key words:
Critical swimming velocity, D’, hyperbolic curve, cluster analysis

Key
Points
 For highvelocity running or swimming, the relationship between velocity (v) and its sustainable duration (t) can be described by a hyperbolic relationship: (v  V)·t = D’, where V is termed critical velocity, and D’ is defined as a curvature constant of the hyperbolic curve. The D’ contributed only around 20% of the breaststroke swimming performance even in a shortdistance event.
 Critical velocity determined around 80% of 50m breaststroke swimming performance, and it could be a useful tool for evaluating shortdistance swimming performance.
 Most of the swimmers showed characteristics for critical velocity dependent physical fitness even in shortdistance swimming event.

Monod and Scherrer, 1965 have found a linear relationship between total work done (kJ) at several work intensities and time to exhaustion during highintensity cycle ergometer exercise. It means that a hyperbolic relationship exists between power output (watt) and time to exhaustion (sec) during highintensity cycle ergometer exercise. Such a hyperbolic relationship possesses a curvature constant with an asymptote, which is termed “critical power”. Critical power has been utilized as a “fatigue threshold” in consideration of its practical application for highintensity endurance sports, such as running (Florence and Weir, 1997; Hughson et al., 1984; Kolbe et al., 1995), swimming (Wakayoshi et al., 1992a; 1992b; 1992c; 1993), and cycling (Housh et al., 1989; Smith et al., 1997). However, the hyperbolic relationship between power output and time to exhaustion has been applied to endurance sports event, completed from, at least, several minutes to 1 hour (e.g., Hughson et al. , 1984). Compared to treadmill running or cycling in the laboratory, it is inconvenient to measure blood lactate concentrations or gas exchange parameters in the swimming pool, so that the application of critical power or “critical velocity (V_{crit})” will give valuable information for swimmers to estimate swimming potential as a noninvasive and inexpensive method. To date, previous studies regarding critical velocity and swimming performance were quite limited (Dekerle et al., 2002; Martin and Whyte, 2000; Wakayoshi et al., 1992a; 1992b; 1992c; 1993). Almost all of the previous studies focused on middle and/or longdistance freestyle swimming. In other words, it has not been revealed whether the critical velocity can be applied to shorter swimming events with other swimming styles. Fukuba et al., 1996 suggested that previously defined indices for evaluating physical fitness (e.g. maximal oxygen uptake) always have a physiological background, on the other hand, critical velocity is obtained from the actual performance done by the subject. The first purpose of this study was to examine whether the critical velocity could be applied to estimate shortdistance swimming performance in a breaststroke style. It was hypothesized that critical velocity could be a useful predictor even in shortdistance swimming events. The curvature constant of the hyperbolic curve between power output and time to exhaustion during highintensity cycle ergometer exercise has attracted a great deal of research attention (Fukuba et al., 2003; Miura et al., 1999, 2000, 2002). Indeed, the curvature constant of the hyperbolic curve can be expressed by the product of velocity and time above V_{crit}, thus, it was suggested that the curvature constant related to a possible distance being performed by anaerobic working capacity. However, as far as we know, nothing has been discussed regarding curvature constant in swimming yet. If the curvature constant is solely reflecting anaerobic working capacity, then it will significantly contribute to the short distance swimming performance. The second purpose of this study was to evaluate the relative contribution of the curvature constant of the hyperbolic curve in swimming. It was also hypothesized that the curvature constant of the hyperbolic curve could be explained by the anaerobic working capacity, and it would significantly relate to the shortdistance swimming performance.
SubjectsEleven male swimmers belonging to the varsity swimming team participated in this study. The physical characteristics and other observed variables of each subject were summarized in Table 1. The best record of 50 m breaststroke swimming of each subject was expressed in percentage of the current world record. Most of the subjects were ranked at a regional competition level. After being informed of the purpose and possible risks of this study, the subjects gave their written consent. An approval of the local Ethical Review Committee was obtained for all procedures.
Protocols for V and D’ determinationA series of time trials for the determination of V_{crit} and curvature constant (D’) of the hyperbolic relationship between swimming velocity and time to exhaustion took place in a 50 m indoor swimming pool. On all measurement days, water temperature was set at 28~29 degrees celsius. All data samplings were performed from late April to early May, just before the first competition of each individual’s annual race schedule. Based on a previous study done by Wakayoshi et al., 1992a, each subject performed 75, 100, and 150m time trials in breaststroke style to determine V_{crit} and D’. The subjects sufficiently stretched and warmed up before the trials. The subjects were instructed to perform each predetermined distance as quickly as they could. Each time trial was performed with a sufficient resting period (~ 2hr). The order of each trial was randomized. At each time trial the same three timekeepers checked the performance time with a stopwatch. Each time measured by three timekeepers was averaged, and the averaged time was regarded as the time of the event. The obtained performance time of each event was further converted into average velocities of each event to calculate V_{crit} and D’. Three days later, each subject performed 50m time trial in breaststroke style in the same swimming pool, and then the observed time was converted into the average velocity of its event (V_{50}). The relationship between swimming velocity (v) and its time to exhaustion (t) is well characterized by the following equation:
Number of trialsHill, 1993 proposed that four trials be recommended when obtaining critical power and curvature constant during highintensity cycle ergometer exercise. Wakayoshi et al., 1993 obtained two data sets to determine V_{crit} in highly trained competitive swimmers. The present study employed subjects who were well accustomed to swimming regardless of performance level, and the possible risks of repeatable maximal efforts were also considered for the subjects. For a confirmation of number of trials, an additional subject performed four repeatable measurements at maximal effort (Figure 1), even though this additional subject was not included in the group of subjects because the level of performance was much greater than that of other subjects. The result clearly showed that three time trials seemed to be sufficient to obtain V_{crit} and D’, so the other remaining subjects performed three time trials.
Laboratory testOn the hypothesis that D’ represents anaerobic working capacity and it will mainly explain the shortdistance swimming performance, each subject performed a maximal anaerobic power (MAnP) test using a cycle ergometer (PowerMaxVII, Combi, Tokyo) based on a standardized procedure (Bulburian et al., 1996; Nakamura et al., 1984). It was recognized that this MAnP test was not performed in the water, thus, the direct comparison between the results obtained from the laboratory test and those in the water might require some caution. However, it is known that this MAnP test has a reliable reproducibility (McCartney et al., 1983), so that it was assumed that the following procedure would have a validity to estimate MAnP in swimmers. After warmingup at 50 W for 10 minutes, subjects were instructed to perform a premaximal trial at 1kp. When subjects were accustomed to cycle ergometer exercise, the first maximal trial started. The initial workload for determining maximal anaerobic power was determined based on the subjects’ body mass (less than 60 kg, 3 kp, n = 2; 61 to 80 kg, 4 kp, n = 7; above 81 kg, 5 kp, n = 2). The second work load was determined based on the rotations at the initial stage (less than 149 rpm, + 1 kp, n = 1; 150 to 179 rpm, + 2 kp, n = 8; above 180 rpm, + 3 kp, n = 2). The final workload was determined with the same criteria as the second stage. These maximal efforts were performed within 10 seconds at each stage with 120 seconds resting. Maximal anaerobic power (MAnP) was calculated based on the observed maximal rotations at each stage with Nakamura et al., 1984’s calculation.
Statistical analysesThe observed values were presented as mean and standard deviation (SD). The relationships between V_{50} and V_{crit} and/or D’ as well as between V_{crit} and MAnP were evaluated using a simple linear regression analysis. The residual error of V_{50} was obtained from the relationship between V_{50} and V_{crit}. The relationship between D’ and its residual error was also evaluated by a simple linear regression analysis. Statistical significance was established at the 0.05 probability level. A cluster analysis, particularly in Ward method with a squared Euclidean distance, was further applied for the observed values of V_{crit} and D’ to classify the characteristics of subjects’ physical fitness either V_{crit} or D’ dependency. The result of the cluster analysis was described as a dendrogram.
A high correlation was found in the relationship between 1/t and v in each subject when obtaining D’ and V_{crit} using eq. 2. The correlation coefficient values between 1/t and v of each subject ranged from 0.97 to 1.00. Table 1 shows the physical characteristics of the subjects and data obtained from the tests in the present study. The average V_{crit} was 0.855 ± 0.106 m/sec, ranging from 0.670 to 0.989 m/sec. The average velocity of 50m breaststroke swimming was 1.064 ± 0.124 m/sec, ranging from 0.878 to 1.232 m/sec. Linear regression analysis indicated that there was a strong relationship between V_{crit} and V_{50} (r = 0.85, p < 0.05; Figure 2). V_{crit} was not significantly correlated with MAnP. D’ widely ranged from 4.96 to 18.88 m, with an average value of 10.19 ± 3.73 m. D’ was significantly correlated with neither V_{50} nor MAnP. D’ was significantly correlated with the residual error, calculated from the regression analysis for the relationship between V_{crit} and V_{50} (r = 0.84, p < 0.05; Figure 3). Figure 4 showed that only one subject was classified as D’ dependency, while the remaining ten subjects were classified as V_{crit} dependency if the subjects were classified into two groups. Moreover, if the subjects were classified into three groups, two subjects were classified as “intermediate between V_{crit} and D’”.
In support of our first hypothesis, the strictly new finding of the present study was that V_{crit} was significantly correlated with the average V_{50} even though the average V_{50} was 24.4% faster than the observed V_{crit} (Figure 2). This result simply proposed that V_{crit} be of potential to estimate shortdistance swimming event done in breaststroke style. Many previous studies have employed various physical fitness relating indices to evaluate endurance sports performance. However, it has been pointed out that those previously defined indices cannot always explain the variation of endurance sports performance (Abe et al., 1998; 1999). It is assumed that those previously employed indices for evaluating physical fitness are solely defined under consideration of its physiological mechanism (Fukuba et al., 1996). In contrast, observed hyperbolic relationship between swimming velocity and its time to exhaustion in this study directly measured performance itself done by the athletes. In relation to V_{crit}, it was possible to estimate a predictable V_{crit} from the time required to swim 50m at the time trial (t50) and observed D’ with an average value of 10.19 ± 3.73 m. The equation for predictable V_{crit} was calculated as follows: yielding a very close value with the observed V_{crit} (0.855 m, see Table 1). The difference between predictable V_{crit} and observed V_{crit} was less than 1%, meaning that the observed V_{crit} from a series of time trials has a high reliability. It has been considered that the anaerobic contribution for 50m or 100m swimming event was 80% or more (Holmer, 1983), thus, it was quite surprising to note that D’ contributed only 20.4% to the 50m breaststroke swimming performance in the present study, being consistent with 29.6% in 50m freestyle swimming performance (Dekerle et al., 2002). The swimming velocity associated with V_{crit} was identical with that associated with onset of blood lactate accumulation (Wakayoshi et al., 1992c; 1993). Moreover, in cycle ergometer exercise, the curvature constant of the hyperbolic relationship between power output and its tolerable duration was significantly decreased under glycogendepleted condition (Miura et al., 2000). Miura et al., 1999 also reported that the oral creatine supplementation increased the curvature constant by 25%. Those previous results and the current result that D’ was not significantly associated with MAnP might give a possible explanation that the curvature constant of such a hyperbolic relationship was consisted of both anaerobic glycolysis and ATPPCr shuttle. It was also interesting to note that the relative contribution of D’ (20.4%) to the swimming performance might explain that the average V_{50} was 24.4% faster than the observed V_{crit}. Figure 3 showed the relationship between D’ and the residual error of V_{50} calculated from the regression analysis for the relationship between V_{crit} and V_{50}. Those results indicated that V_{50} could be considerably explained by V_{crit} (Figure 2), however, unexplained residual error could be explained by D’ (Figure 3). As discussed above, little attention has been paid for the physiological implication of D’ in sports performance. This study found that D’ was not significantly correlated with neither V_{50} nor MAnP, although, as hypothesized above, the curvature constant of the hyperbolic relationship has been recognized as an anaerobic working capacity that can be performed above the critical power (Fukuba et al., 2003; Miura et al., 2002). Indeed, the result of the present study showed that D’ still contributed for the performance by 20%, suggesting that the appropriate training program to improve both critical velocity and D’ simultaneously could bring a better performance for breaststroke swimmers. In addition to investigations concerning energetics of swimming, some recent studies have examined the mechanics of the breaststroke swimming to improve the performance (Leblanc et al., 2005; Seifert and Chollet 2005; Takagi et al., 2004). This could be because a greater potential to improve the performance will be available in breaststroke swimming than in freestyle swimming due to an existence of a large active drag during swimming. Future investigations considering both mechanics and energetics of swimming will bring a further understanding of swimming, resulting in a development of an efficient training program for swimmers. The obtained MAnP consisted of three maximal pedalling at different workloads for, at most, 10 seconds each, meaning that the test must be too brief to utilize glycolytic ATP production system completely. This interpretation was in consistent with Bulburian et al. (1996). It means that MAnP observed in this study would reflect only breakdown of PCr (ATPPCr shuttle), not anaerobic glycolysis (Figure 5). It was also hypothesized that the observed V_{crit} in swimming was not necessary to reflect pure aerobic working capacity. Indeed, Fukuba et al., 1996 showed that respiratory responses did not reach the steadystate during cycle ergometer exercise at around the critical power, indicating that the exercise intensity associated with critical power exceeds the ‘lactate threshold’, which, in theory, represents the highest metabolic rate where a steady state response can be achieved during prolonged exercise. The results of the present study and those previous studies suggest that exercise intensity corresponding to V_{crit} be sustained in part by the pure aerobic work capacity and anaerobic glycolysis (Figure 5), supporting an interpretation obtained from the model analysis (Toussaint et al., 1998). The present study further performed cluster analysis to classify the subjects’ physical fitness for either D’ or V_{crit} dependency. It was worth noting that ten of eleven subjects showed V_{crit} dependency even in shortdistance swimming event (Figure 4). If the cluster was divided into three groups, two subjects were classified as an intermediate situation between V_{crit} and D’ dependency, meaning that most of the subjects were still classified as V_{crit} dependency. The result of the present study clearly suggested that V_{crit}, not D’, mainly determined athletes’ success even in shortdistance breaststroke swimming. In other words, the higher the V_{crit}, the better the performance in shortdistance breaststroke swimming. Although Dekerle et al., 2002 suggested that D’ could not be employed by coaches to control an anaerobic swimming training program, we hereby pointed out that it might be relevant to the maximal swimming velocity, which would be performed at the end and/or start of the race.
In conclusion, V_{crit} could be a potential index to evaluate shortdistance swimming performance in breaststroke style. V_{crit}, but not D’, determined most of 50m breaststroke swimming performance.
ACKNOWLEDGEMENTS 
This study was supported in part by GrantinAid from The Japan Ministry of Education, Culture, Sports, Science and Technology (17770217 to D.A. and 16500407 to S.U.) and Athletic Performance Promotion in Kumamoto, Japan (to Y.F.). 

AUTHOR BIOGRAPHY 

Daijiro Abe 
Employment: Assistant Professor, Faculty of Integrated Cultures and Humanities, University of East Asia, Japan. 
Degree: MSc, MEd 
Research interests: Biomechanics, ergonomics and exercise physiology 
Email: daijiro@touau.ac.jp 


Hiroaki Tokumaru 
Employment: Undergraduate Student, Faculty of Integrated Cultures and Humanities, University of East Asia, Yamaguchi Japan. 
Degree: BSc 
Research interests: Sports science in swimming. 
Email: 


Shigemitsu Niihata 
Employment: Professor, Faculty of Welfare and Health, Fukuyama Heisei University, Fukuyama, Hiroshima, Japan. 
Degree: PhD 
Research interests: Coaching science and sports medicine 
Email: 


Satoshi Muraki 
Employment: Associate Professor, Department of Human Living System Design, Faculty of Design, Kyushu University, Fukuoka, Japan. 
Degree: PhD 
Research interests: Ergonomics 
Email: 


Yoshiyuki Fukuoka 
Employment: Professor, Faculty of Environmental and Symbiotic Sciences, Prefectural Univ. of Kumamoto, Kumamoto, Japan. 
Degree: PhD 
Research interests: Respiration and circulation physiology 
Email: 


Sachio Usui 
Employment: Associate Professor, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashihiroshima, Hiroshima, Japan. 
Degree: MEd 
Research interests: Mathematical analysis for human movement 
Email: 


Takayoshi Yoshida 
Employment: Professor, Department of Health and Sport Sciences, Graduate School of Medicine, Osaka University, Japan. 
Degree: PhD 
Research interests: Evaluation for sports performance 
Email: 



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