# Simulated front crawl swimming performance related to critical speed and critical power

Purpose: Competitive pool swimming events range in distance from 50 to 1500 m. Given the difference in performance times (±23-1000 s), the contribution of the aerobic and anaerobic energy systems changes considerably with race distance. In training practice the regression line between swimming distance and time (Distance = critical velocity × time + anaerobic swimming capacity) is used to determine the individual capacity of the aerobic and anaerobic pathways. Although there is confidence that critical velocity and anaerobic swimming capacity are fitness measures that separate aerobic and anaerobic components, a firm theoretical basis for the interpretation of these results does not exist. The purpose of this study was to evaluate the critical power concept and anaerobic swimming capacity as measures of the aerobic and anaerobic capacity using a modeling approach.

Methods: A systems model was developed that relates the mechanics and energetics involved in front crawl swimming performance. From actual swimming flume measurements, the time dependent aerobic and anaerobic energy release was modeled. Data derived from the literature were used to relate the energy cost of front crawl swimming to swimming velocity. A balance should exist between the energy cost to swim a distance in a certain time and the concomitant aerobic and anaerobic energy release. The ensuing model was used to predict performance times over a range of distances (50-1500 m) and to calculate the regression line between swimming distance and time.

Results and Conclusions: Using a sensitivity analysis, it was demonstrated that the critical velocity is indicative for the capacity of the aerobic energy system. Estimates of the anaerobic swimming capacity, however, were influenced by variations in both anaerobic and aerobic energy release. Therefore, it was concluded that the anaerobic swimming capacity does not provide a reliable estimate of the anaerobic capacity.

*Department of Kinesiology, Faculty of Human Movement Sciences, Vrije Universiteit, Amsterdam, THE NETHERLANDS; Laboratory of Health and Sport Science, Nara University of Education, JAPAN; Swimming Performance Laboratory, National Institute of Fitness and Sports, Kanoya, Kagoshima-Prefecture, JAPAN*

Submitted for publication December 1996.

Accepted for publication June 1997.

Competitive pool swimming events involving the front crawl are held over a range of distances (50-1500 m). The 50-m sprint takes about 23 s for a world-class sprinter and requires, among other things, considerable strength and power. The best 1500-m swimmers need about 14 min and 40 s to cover this distance and do so on the basis of a high endurance capacity and excellent technique. Depending on the race distance, factors like the aerobic and anaerobic energy systems, power, and technique seem to play a more or less dominant role in determining performance. Especially the contribution of the aerobic and anaerobic systems to the power output changes with the race distance. The anaerobic contribution is 80% or more in the 50- and 100-m events, whereas the aerobic contribution predominates at the 400-, 800-, and 1500-m events ^{(8)}. These differences make designing a swimming training program eliciting the ideal metabolic adaptations for optimal swimming performance a challenging task ^{(34)}. In the planning of training programs the various demands for a specific competitive distance should be taken into account. Furthermore, the coach should consider whatever deficiencies there may be in the swimmers' resources or capabilities to meet these demands ^{(2)}. This requires the identification of those performance factors that are “weak links” in the performance chain and therefore assume a position of greater concern.

A test was developed to determine the capacity of the aerobic and anaerobic metabolic pathways by Wakayoshi et al. ^{(38,40)}. The test estimates the swimming velocity that theoretically can be maintained forever without exhaustion. This so-called “critical swimming velocity” is the slope of the regression line between swimming distance and time. It relies on the concept of critical power, first proposed by Monod and Scherrer ^{(16)} for synergistic muscle groups and extended to whole body cycle ergometer exercise by Moritani et al.^{(17)}. The concept is based on a hyperbolic relationship between power output and time to exhaustion. It incorporates both the anaerobic and aerobic contribution to the total power output. The anaerobic part is estimated as the total work that can be performed anaerobically(anaerobic work capacity, AWC). The aerobic part is considered inexhaustible, and is called the critical power (CP). At a given power output (Po) the time to exhaustion will then equal: eq. 1 This relationship can be transformed to a power-time relationship where: eq. 2 This power-time relationship in cycle ergometry was extended to a velocity-time relationship in swimming^{(6,38,39)}: eq. 3 The critical velocity (V_{crit}) is the velocity at which one performs indefinitely, and thus V will approach b. Integration yieldseq. 4 Accordingly, V_{crit} equals b, i.e., the slope of the regression line between swimming distance and time, while a is indicative of the anaerobic parameter, i.e., the anaerobic swimming capacity(ASC) ^{(6)}.

The critical velocity parameter has been shown to be well correlated with the ˙VO_{2} swimming at the anaerobic threshold, the swimming velocity at which blood lactate starts to accumulate, and velocities of 200- and 400-m front crawl ^{(38,40)}. However, in some studies V_{crit} overpredicted the velocity that could be maintained during endurance swims ^{(6,18)}. The anaerobic swimming capacity was correlated with peak lactate ^{(6)}. Although there is confidence that critical velocity and anaerobic swimming capacity are fitness measures that separate aerobic and anaerobic components^{(6)}, a firm theoretical basis for the interpretation of these results does not exist. An approach to gain a better understanding of the role the aerobic and anaerobic components play in the performance on various distances is to develop a systems model. A systems model is established to explain theoretically the process that produces the data rather than to describe the data themselves ^{(19)}. In this model the mechanical power needed to swim at a specific speed is related to the finite and time dependent metabolic power input ^{(28)}. This requires an analysis of both the mechanics and energetics of front crawl swimming. In this approach, the mechanical power required to swim a distance at a specific speed was linked to the time dependent liberation of metabolic power.

## MECHANICS OF SWIMMING

A considerable part of the energy cost in swimming is used to overcome water resistance or drag ^{(21)}. It was established that the resistance is related to the square of the swimming velocity (v in m·s^{-1}): eq. 5 in which F_{d} (N) denotes drag force, A is a constant incorporating the density of water, the drag coefficient, and the greatest cross-sectional area. A was estimated to be±30 for males and ±23 for females in front crawl swimming^{(27)}. The mechanical power to overcome drag (P_{d} in J·s^{-1} or W) will equal the drag-force times the velocity, and thus eq. 6 The power to overcome drag appears to be dependent on the drag factor A and the cube of the velocity.

To swim at a constant speed, the swimmer has to generate a propulsive force that equals drag: eq. 7 Unlike on land activities the propulsion in swimming cannot be generated by pushing off from a fixed point. Thrust is provided by pushing water (more or less) backward. This pushed back amount of water (m_{i}) is given a velocity change (Δv_{i}) and thus an impulse equal to: eq. 8 Thus to create propulsion in swimming water is given a velocity change, which implies that it acquires at the same time kinetic energy (E_{kin} = 1/2Σm_{i}(Δv_{i})^{2}). This kinetic energy is given by the swimmer to the water while generating the propulsive force. The generation of the propulsive force in swimming actually costs additional energy apart from the energy to overcome the drag force^{(24,26,30)}. Consequently, the energy flow to the accelerated water should be included in the power bookkeeping. Hence, the total mechanical power (P_{o}) produced by the swimmer equals not only power to overcome drag (P_{d}) but also power expended in giving pushed away masses of water a kinetic energy change (P_{k}):eq. 9 The ratio of the useful power (to overcome drag) to the total power output has been defined as the propelling efficiency(e_{p}) ^{(1,24,26)}:eq. 10 The propelling efficiency was estimated for well trained swimmers to be 46-77% ^{(26)}. Given these various mechanical factors involved in front crawl swimming the total power(P_{o}) to swim at a specific speed will equal: eq. 11

**Modeling of the aerobic and anaerobic power production in swimming**. In other sports like ice skating^{(11,13)} and running ^{(20)}, the power production of the aerobic and anaerobic systems were modeled and related to the power production necessary to overcome friction with the environment. It seems justified to assume that the power production of the anaerobic and aerobic systems in swimming will yield a power-time curve similar to those modeled for speed skating or running, hence:eq. 12 where P_{aer} equals the metabolic power liberated aerobically and P_{an} the metabolic power liberated anaerobically. P_{aer,max} and P_{an,max} equal the maximal aerobic and anaerobic power, while λ denotes a time constant defining the rate of increase and decrease of aerobic and anaerobic power, respectively. The total amount of aerobic and anaerobic energy over a certain amount time τ can be calculated by integration of the P_{aer} and P_{an} functions, respectively: eq. 13 eq. 14 Not all the metabolic power is converted to mechanical power. In the transformation process part of the chemical power present in foodstuff is converted to heat. The gross efficiency (e_{g}) of this process is quantified by the power output divided by the rate of energy expenditure:eq. 15 The gross efficiency appeared not to be different in groups of elite male and female competitive swimmers, nor was it different from the values found in a group of triathletes^{(24,31)}. The average regression equation describing the process of converting (Ė) into P_{o} while swimming appears to be eq. 16

**An energy balance applied to front crawl swimming**. The energy production of the aerobic and anaerobic systems is used to swim a specific event. The power input (Ė) swimming at a specific velocity can be quantified when taking into account the drag constant, the velocity cubed, the gross efficiency, and the propelling efficiency: eq. 17 Integration of this equation gives the energy expenditure necessary to swim a certain distance (d) at a specific speed (v) or in a certain time (t):eq. 18 The energy generated should balance the energy necessary to swim a distance in a certain time. In other words (E_{an} + E_{aer}) at time t must equal the total energy expended to swim a distance at a specific velocity. This equality can be used to predict the performance times over different distances, hence eq. 19

This modeling approach was illustrated using data collected on a group of experienced male college swimmers. This model was successfully used to predict swimming performance at the 50- and 100-yard front crawl^{(29)}. In the present study, this modeling approach was used to predict the critical power and the anaerobic work capacity, i.e., the critical speed and anaerobic swimming capacity, respectively. To achieve a better understanding of the relation between the aerobic and anaerobic metabolic pathways and the critical velocity and anaerobic swimming capacity, a sensitivity analysis of the model was performed.

## METHODS

**Subjects**. Eight male college swimmers (weight, 65.74 ± 8.23 kg; height, 1.79 ± 0.058 m; and ˙VO_{2max(swimming)}, 3.54± 0.67 L·min^{-1}) participated after informed consent was obtained. All subjects were experienced competitive swimmers.

**Equipment**. All testing was performed in a swimming flume(Unidyne, Minneapolis, MN) at the International Center for Aquatic Research, Colorado Springs, CO. The water flow had been calibrated previously, and the water temperature was kept at 26.5°C for all trials. Before testing, subjects were familiarized with the swimming flume, the procedures, and equipment. Oxygen uptake (˙VO_{2}) and minute ventilation(˙V_{E}) were measured with an automated open circuit system that incorporated electronic O_{2} and CO_{2} gas analyzers (Model S-3A/II and CD-3A, Ametek, Pittsburgh, PA) and a pneumotachometer (Model 3813, Hans Rudolf, Kansas City, MO) interfaced with a PC. The gas analyzers were calibrated with gases of known concentration. A specially designed respiratory valve was used to collect expired gases while swimming^{(33)}.

**Measurement of the maximal oxygen uptake while swimming(˙VO2max)**. Each subject completed a continuous incremental swimming test to volitional exhaustion for the determination of the maximal oxygen uptake (˙VO_{2max}) according to Trappe et al.^{(35)}.

**Estimating λ**. The time constant λ describes the rate of increase and rate of decrease of the aerobic and anaerobic power production. Following Margaria ^{(14)} and Ward-Smith^{(41)}, it was assumed that the power production of the anaerobic and aerobic pathways can be described with one time constant, suggesting a causal relation between them. The magnitude of λ can then be estimated from repetitious oxygen uptake measurements of maximal exercise bouts. Subjects performed 60-s all-out swims in the swimming flume. Every 10 s the oxygen uptake was determined. The ˙VO_{2} values were converted to watts according to Garby and Astrup ^{(5)}. The data were least square fitted to the function *Paer* =*P*_{aer,max}(1 -*e*^{-λt}), where*P*_{aer,max} was set equal to the individual power equivalent of the ˙VO_{2max}.

**Estimating Pan,max**. The estimation of P_{an,max} relies on a method to measure the total anaerobic energy production, which was described by Medbø ^{(15)} for running. This method was applied in the context of swimming by Troup et al^{(36)}. In a series of submaximal swims, the relationship between oxygen consumption and the cube of the swimming velocity was determined. From this regression equation the oxygen consumption was estimated(˙VO_{2(100m)}) to swim at the speed equal to the individual 100-m best performance. The subject was then invited to swim at this specific speed for 60 s in the swimming flume. During this (maximal) exercise all oxygen consumed is measured by integrating the measured ˙VO_{2}. The accumulated oxygen deficit can then be calculated from the integrated amount of oxygen necessary to swim at this speed(duration·˙VO_{2(100m)}) minus the total oxygen consumed. This deficit reflects the total anaerobic energy production^{(15,36)}.

**Estimating the drag coefficient, gross efficiency, and propelling efficiency**. For this group of American college swimmers it was impossible to measure active drag, gross efficiency, and propelling efficiency directly. Such measurements require (in part) a system to measure active drag (see for extensive discussion ^{(25,26,28)}). Therefore, the values for e_{g} and e_{p} were assumed to be equal to the average values found for trained swimmers, hence e_{g} = 9.2%^{(31)} and e_{p} = 60% ^{(24)}. The drag coefficient A (see eq. 5) was estimated given the mass of the subject (kg) according to A = 0.35·mass + 2^{(27,32)}.

**Further considerations for calculating the energy cost**. To improve the validity of the energy balance, some biomechanical considerations were added. A front crawl swimming race is started with a dive, turns are made, and the race is finished when the hand touches the wall. These factors influence the actual swimming distance. In the present calculations it was assumed that every turn and the final touch reduce the distance of a lap of 25 m by 1.1 m. For the start it was assumed (see Behncke^{(3)}) that 40 J·kg^{-1} was expended for the dive and that a distance without swimming was covered (including the glide) of 7.5 - 3 ln(v) ^{(23,42)}, which took a total time of 0.6 + 3/v (including the reaction time), where v equals the average swimming velocity of the event.

**Predicting critical velocity and anaerobic swimming capacity**. The average parameter values for the energy production (P_{aer,max}, P_{an,max}, and λ) were used to estimate the time dependent energy production (according to eq. 19). The energy cost for the 50, 100, 200, 400, 800, and 1500 m was calculated using the values for e_{g} = 9.2% and e_{p} = 60%. Given that the average body mass equaled 65.7 kg, A was set to 25 kg·m^{-1}. The best times for the distances can be estimated by equating the time dependent energy cost of these distances with the energy production. The ensuing results were least square fitted to produce the regression line between swimming distance (D) and time, yielding equations of the form D = V_{crit}·time + b. A sensitivity analysis of the model was performed to assess what input parameters influence V_{crit} and the anaerobic swimming capacity. The effect of a 20% increase and decrease for each factor (P_{aer,max}, P_{an,max}, and λ) on the performance for the 50, 100, 200, 400, 800, and 1500 m and on the regression equation of distance on time was determined.

**Predicting critical power and anaerobic work capacity**. The critical power and anaerobic work capacity was estimated in two ways: (i) by relating the energy cost for the 50, 100, 200, 400, 800, and 1500 m to the performance time including the time for the start and turns as if swimming in a 25-m pool, and (ii) the energy cost excluding the cost for dive and turns related to the net time to exhaustion (as if swimming in a flume). For both approaches the regression equation E = CP·time + AWC was determined.

**Simulations**. The simulations were performed with time intervals at Δt = 0.01 s using Matlab (The Math Works, Inc., Natick, MA) on a Power Macintosh 7600.

## RESULTS

**Estimates of λ and Paer,max**. The ˙VO_{2} values obtained during the 60-s all-out swims were converted to watts^{(5)} and are presented for the group inTable 1. The data clearly suggest that the oxygen consumption increases with time. It should, however, be noted that the standard deviation is considerable. This has to do with the short sampling time (10 s), where one breath more or less will have considerable influence on the outcome. The maximal oxygen consumption is attained after a longer time(>200 s ^{(2)}). At that time (1 - e^{-λt}) will reach 1, and thus P_{aer,max} will equal the ˙VO_{2max}. The group average was estimated to be 1240 W. The data fromTable 1 were least square fitted to 1240·(1 - e^{-λt}) and λ was calculated to be 0.0323 s^{-1}. This is slightly lower than values reported for bicycle ergometry (λ = 0.0398 s^{-1}) ^{(10,12)} and indicates a less rapid development of the aerobic capacity. This could be because in contrast to bicycle ergometry which involves mainly leg work, in swimming the arms provide more than 85% of the total thrust in the crawl stroke^{(4,7)}.

**Estimate of Pan,max**. From the 60-s swim at the estimated 100-m swimming speed, the accumulated oxygen deficit was calculated^{(36)}, which can be assumed to reflect the anaerobic energy production ^{(15)}. This yielded an average of 1862.6 J. Thus, for this group the aerobic and anaerobic power production in swimming was estimated to be: eq. 20 and the energy delivered at time τ will equal: eq. 21

**Individual estimations of the aerobic and anaerobic power production**. The individual results for P_{aer,max}, λ, and P_{an,max} are presented in Table 2. The data forλ and P_{an,max} show considerable variation. It remains to be determined whether this variation is the result of a not yet fully developed test protocol or whether considerable variation exists among individuals forλ and P_{an,max}. If the latter case is true, it would open new possibilities for diagnosing the performance capacities of swimmers.

**Prediction of critical velocity and anaerobic swim capacity**. The times on the 50, 100, 200, 400, 800, and 1500 m were estimated for races in a 25-m pool using the group averaged data. This was accomplished by determining the intersection of the graphs representing the time dependent energy cost of these distances with the energy production graph (Fig. 1). At the intersection the energy necessary to swim that distance is in balance with the maximal energy produced during that time. The estimated times for the distances were 50 m: 31.4 s; 100 m: 65.2 s; 200 m: 2 min 14.4 s; 400 m: 4 min 34.0 s; 800 m: 9 min 13.6 s; 1500 m: 17 min 22.9 s.Figure 2 presents these times for the distances together with the regression line between swimming distance (D) and time. The regression equation was D = 1.43·time + 6.61, yielding a critical velocity of 1.43 m·s^{-1} and an anaerobic swim capacity (ASC) of 6.6 m.

**Sensitivity of the model to variation of input parameters**. The effect of a 20% increase and decrease of P_{aer,max}, P_{an,max}, andλ on the performance for the 50, 100, 200, 400, 800, and 1500 m and the regression equation of distance on time was determined(Table 3). The 20% decrease and increase in P_{aer,max} resulted in a 5.7% decrease and 6.2% increase, respectively, of v_{crit}. This suggests that changes in P_{aer,max} are reflected in v_{crit}, albeit to a considerable lesser extent than the 20% variation in P_{aer,max}. The changes in P_{an,max} had no influence on v_{crit}, but strongly affected ASC and the y-intercept of the regression equation (-54.7% and 48.5%, respectively). This is in line with Hill et al.^{(6)} who suggested that the y intercept is a measure of the anaerobic capacity. However, from Table 3 it becomes apparent that besides P_{an,max}, variations in P_{aer,max}, andλ have an influence on the ASC. Thus the model calculations suggest that variations in ASC cannot be attributed to variations in P_{an,max} only. This agrees with previous observations in which the test-retest reliability of the anaerobic work capacity was qualified as ambiguous^{(18)}. The variations in λ hardly affected the predicted swimming times and, consequently, v_{crit}. Moderate changes in ASC were observed. Apparently, λ is not a predominant factor in the model.

**Prediction of critical power and anaerobic work capacity**. The critical power and anaerobic work capacity were estimated twice (i) on the basis of the energy cost for the distances (50-1500 m) in 25-m pool, including the cost for the dive and making turns that was related to the overall time; and (ii) on the basis of the energy cost to swim a specific distance at the highest speed possible, excluding start and turns related to the time to exhaustion (excluding reaction time, dive, etc.) as if swimming in a flume. The first approach resulted in the regression equation Energy = 114.4·time + 1184. The second approach yielded the equation Energy = 114.5·time + 1462 (see Fig. 3). This suggests a critical power of 114.5 W and an anaerobic work capacity of 1184 J in the“pool” condition and of 1462 J for flume swimming.

## DISCUSSION

The simulated performance times (Table 3) are well in range with times that can be expected from a group of experienced college swimmers with an average weight of 65.7 kg. In a previous study, the model was used to predict performance for the 50- and 100-yard front crawl. The results of the model were compared with the actual swimming times. The coefficients of correlation were for the 50-yard, 0.94, (adjusted coefficient of determination: 0.86) and for the 100-yard, 0.91 (adjusted R^{2}: 0.80)^{(29)}. The coefficients of correlation indicate that reasonable predictions of individual swim performances can be made with a power equation model based on the individual kinetics of the anaerobic and aerobic pathways. Thus, although for all subjects the same mean values were used for the gross efficiency (9.2%) and propelling efficiency (60%), and drag was estimated on the basis of subject mass, more than 80% of the variance in“sprint” performance could be explained with the model. Even better predictions can be expected when individual determined values for these parameters are incorporated in the model. Apart from the mass dependent drag coefficient, the subject specific input of the model related only to the metabolic factors. This suggests that these factors determine sprint performance for swimmers with reasonable skill (i.e., a propelling efficiency equal to ±60%) given their drag profile. This underlines the importance for the training practice of determining the capacity of the aerobic and anaerobic energy systems.

Although no data are available for performance on the longer distances for the subjects studied, the simulated performance can be compared with data of Swaine ^{(22)}. In his study the 1500-m swim performance in a group of 12 male competitive swimmers (age 18.4 ± 2.8 yr) was related to critical power measured using an isokinetic swim bench. The mean swimming time for the 1500 m was 1036 s and mean critical power was 120.7 W. The 1500-m swimming times significantly correlated with the critical power (r = -0.89). In the present study the 1500-m time was 1043 s, whereas the critical power was estimated to be 114.5 W. The similarity of the data in the two studies is striking. It provides additional, albeit inconclusive, evidence for the validity of the model. However, as stated before, individual data concerning A, e_{g}, e_{p}, and performance times were not available for the group; thus the final verification of this approach remains to be determined.

**Critical velocity and anaerobic swim capacity**. Values reported in the literature for the critical velocity range from 0.97 m·s^{-1} for an age group 8-10 yr ^{(6)}, 1.22 m·s^{-1} for an age group 14-18 yr ^{(6)}, 1.33 for male trained swimmers (18-21 yr) ^{(9)}, 1.57 m·s^{-1} for 26 swimmers (mean age 15.1 yr) ^{(6)} to 1.60 m·s^{-1} for a Japanese top swimmer ^{(37)}. The observed value of 1.43 m·s^{-1}, therefore, seems a little bit high, whereas the observed value for the anaerobic swim capacity (6.6 m) is low in comparison to the cited studies. Three factors can be put forward to explain this rather high value. (i) The present calculations incorporated biomechanical considerations related to races in a 25-m pool reducing the actual swimming distance. The distance related to the dive and the turns is energetically cheaper given the fixed push-off point (e_{p} = 100%). Furthermore, it is covered faster than the remaining distance traveled swimming. In previous studies subjects swam in a swimming flume or a 50-m pool, where the mentioned effect is absent or reduced. (ii) In the model it is assumed that the aerobic metabolism after reaching the plateau phase(˙VO_{2}), can continue at this level incessantly. However, with real swimmers competing in real events, ˙VO_{2max} may not truly represent a continuously attainable maximal rate of aerobic work performance. Highly trained adult athletes can maintain a level representing 100% of the oxygen uptake capacity for about 15 min. Events of longer duration involving leg work can only be completed if the oxygen uptake is reduced to 90% of the maximum^{(2)}. The question is what is realistic for this group of moderately trained college swimmers. If the simulation is repeated while reducing the P_{aer,max} to 90% for the 800 m and to 80% for the 1500 m, the critical speed is reduced to 1.35 m·s^{-1}. This value is still rather high. It could indicate that in real swimmers, as opposed to this computer model, factors such as local muscle fatigue have a performance limiting effect in events of longer duration. (iii) In the model it is assumed that at exhaustion all anaerobic capacity is fully used irrespective of the intensity of the exercise. Obviously, this is not the case. This consideration led to the formulation of a three-parameter critical power model^{(18)} in which the maximal power that could be developed at any instant is proportional to the amount of AWC remaining at that instant. In formula ^{(18)}: in which P_{max} represents maximal instantaneous power. This power time relationship can be extended to a velocity time relationship for swimming:eq. 23 in which V_{max} represents the maximal instantaneous swimming velocity. If eq. 23 is used to fit the data, V_{crit} is again estimated to be 1.43 m·s^{-1} when the 25-m pool condition was simulated, while V_{crit} reduces to 1.40 m·s^{-1} when exercise was simulated under swimming flume conditions (see Fig. 4). The maximal instantaneous velocity is estimated at 1.91 and 1.67 m·s^{-1}, respectively. The anaerobic swimming capacity yielded values of 8.6 (pool) and 6.6 m (flume).(Note that the simulation still assumed a continuous 100% ˙VO_{2max} contribution.) In Morton's study ^{(18)} it was demonstrated that the three-parameter model (eq. 22) resulted in a significant improvement in the goodness of fit to data over the standard two-parameter critical power model equation. Whether this also applies for the swimming situation (eq. 23) remains to be determined.

**Critical power and anaerobic work capacity**. Similar critical power estimates were obtained for the “pool” and“flume” condition (114.4 and 114.5 W, respectively). The rather different values observed for the anaerobic work capacity (1184 and 1462 J, respectively) are in accordance with the large variation observed in the sensitivity analysis (Table 3). The values should be an estimate of the anaerobic work capacity, i.e., the total work that can be performed using the limited energy resources. In the present model this is equal to e_{g}·E_{an} for τ approaching infinity(eq. 14). This amount equaled 5305 J. Apparently as a consequence of the nonlinearity of the power production, the estimate of the anaerobic work capacity is negatively biased. The estimation of the anaerobic work capacity using eq. 22 yielded a value of 2316 J, still less than 50% of the actual value. This suggests that the estimation of the true anaerobic work capacity remains far from reliable usingeq. 2 or eq. 22, if in reality the anaerobic energy release is nonlinear (eq. 14).

The value for the critical power (114.5 W) can be compared with the critical swimming speed in the flume condition (1.3987 m·s^{-1}). In this condition the mechanical power to swim at this velocity will equal(see eq. 11) 25·1.3987^{3}/0.6 = 114 W. So it could be concluded that critical speed and critical power must be related according to eq. 24 In this simulation the rate of aerobic energy liberation was modeled as P_{aer}(t) = 1240·(1 - e^{-0.0323t}). For t → ∞ this will develop into P_{aer}(∞)= 1240 W. Therefore, in this model ˙VO_{2crit} equals˙VO_{2max}. As discussed, this is not attainable in human exercise. Nevertheless, if in general the critical velocity and consequently CP reflects endurance, CP should equal e_{g}·˙VO_{2crit}, (thus, in this case ˙VO_{2max}). Indeed, the estimated critical power of 114.5 W compared very well with the mechanical power equivalent of the˙VO_{2max}·e_{g}, i.e., 0.092·1240 = 114.1 W. Thus the critical swimming velocity is a measure for endurance and relates to the oxygen uptake according to: eq. 25

**Conclusion**. The model calculations illustrate that the critical swimming velocity is a measure of swimming endurance. As a consequence of the nonlinearity of the power production, the estimates of the anaerobic work capacity were negatively biased and are therefore unreliable.

Address for correspondence: Huub M. Toussaint, Department of Kinesiology, Faculty of Human Movement Sciences, v d Boechorststraat 9, Amsterdam, The Netherlands. E-mail: H_M_Toussaint@fbw.vu.nL.

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**Keywords:**

HUMAN; COMPUTER SIMULATION; AEROBIC CAPACITY; ANAEROBIC CAPACITY; POWER BALANCE; DRAG; ENERGY COST; BIOMECHANICS; 3-PARAMETER CRITICAL POWER MODEL