Physiological determinants of best performances in human locomotion

Transcription

1 Eur J Appl Physiol (1999) 80:298±307 Ó Springer-Verlag 1999 ORIGINAL ARTICLE Carlo Capelli Physiological determinants of best performances in human locomotion Accepted: 22 March 1999 Abstract In human locomotion, the metabolic power required ( _E) to cover a given distance d, in the time t is set by the product of the energy cost of the locomotion (C), i.e. the amount of metabolic energy spent to move over one unit of distance, and the speed (v =d t )1 ): _E ˆ C m ˆ Cdt 1. Since, for any given d, v is a decreasing function of t and C is either constant or increases with v, it necessarily follows that _E is larger the smaller the value of t. Thus, for any given distance and subject, the shortest time will be achieved when _E is equal to the individual maximal metabolic power ( _E max ). In turn, _E max is a decreasing function of t: it depends upon the subject's maximal aerobic power (MAP) and on the maximal amount of energy derived from the full utilisation of anaerobic energy stores (AnS). So, if the relationship between C and v in the locomotion at stake and the subject's MAP and AnS are known, his best performance time (BPT) over any given distance can be obtained by solving the equality _E max t ˆ _E t. This approach has been applied to estimate individual BPTs in running and cycling. In this paper, the above approach will be used to quantify the role of C, MAP, and AnS in determining BPTs for running, track cycling and swimming. This has been achieved by calculating the changes in BPT obtained when each variable, or a combination thereof, is changed by a given percentage. The results show that in all the three types of locomotion, regardless of the speed, the changes in BPT brought about by changes of C alone account for 45± 55% of the changes obtained when all three variables (C, MAP and AnS) are changed by the same amount. Key words Maximal metabolic power á Energy cost of human locomotion á Best performance time Introduction In human locomotion, the metabolic power required ( _E) to cover a given distance (d) in the time t is set by the product of the energy cost of the locomotion at stake (C) and the speed (v =dt )1 ): _E ˆ C m ˆ Cdt 1 ; 1 where C is the amount of metabolic energy spent to cover one unit of distance (di Prampero 1986). Since v is a decreasing function of t, and C is either constant, or increases with v, it necessarily follows that _E is larger the smaller the value of t. Thus, for any given distance and subject, the shortest time will be achieved when _E is equal to the individual's maximal metabolic power ( _E max ). In turn, _E max is a decreasing function of t: it depends upon the subject's maximal aerobic power (MAP) and on the maximal amount of energy derived from the full utilisation of anaerobic energy stores (AnS, Wilkie 1980). So, if the relationship between C and v, together with the subject's MAP and AnS, are known, his or her best performance time (BPT) over any given distance can be obtained by nding the value of t that solves the equality _E max t ˆ _E t. This approach has been applied satisfactorily to estimate individual BPTs for running and cycling (Capelli et al. 1998b; di Prampero et al. 1993). In the present paper, it will be shown that this approach can be used to quantify the role of C, MAP, and AnS in determining BPTs for three forms of human locomotion, running, track cycling and swimming. Methods Theory C. Capelli (&) Dipartimento di Scienze e Tecnologie Biomediche, School of Medicine, Piazzale Kolbe 4, I-33100, Udine, Italy Metabolic power requirement Cycling The overall energy cost per unit of distance in track cycling (C c ) on at terrain in absence of wind is given by the sum of three terms (Capelli et al. 1993, 1998b):

2 299 C c ˆ C rr M g k 0 m 2 0:5 M m 2 d 1 tot g 1 c 2 where g is the acceleration due to gravity (m á s )2 ), v (m á s )1 ) is the air speed, and the other terms are de ned here below. The rst term (C rr M g) is the metabolic energy spent against rolling resistance; it is proportional to the product of the overall mass M (cyclist, 75 kg; plus bike, 10 kg) and to the coe cient C rr, which is the amount of energy spent over a unit of distance and per unit of overall mass against frictional forces. The value of this term depends upon the characteristics of the tyres and of the terrain and, for M = 85 kg and for C rr = J á m )1 á kg )1 (Capelli et al. 1998b), amounts to 25.8 J á m )1. The second term (k 0 v 2 ) is the metabolic energy spent per unit of distance against air drag (C c,a, J á m )1 ); (k 0 m 2 ) proportional to the square of the air speed (v) and to a constant k 0 which, in turn, is described by (Capelli et al. 1993): k 0 ˆ 0:5 C x q A g c 1 3 C x in Eq. 3 is the dimensionless drag coe cient, which can be considered as a constant at the range of speeds investigated (Pugh 1974); it is equal to 0.75 for a cyclist riding a traditional racing bicycle in a fully dropped position (di Prampero 1986). The air density q, for an air temperature of 20 C, a barometric pressure of 760 mmhg and 50% relative humidity, amounts to kg á m )3 (Weast 1987); A is the frontal area of the subject riding the bicycle, and amounts to 0.42 m 2 for a subject of 175 cm in stature and 75 kg body mass (Capelli et al. 1998b). The overall e ciency of cycling, g c, is equal to 0.22 (Seabury et al. 1977). Thus, k 0 turns out to be 0.86 J s 2 á m )1 á m )2. The speed becomes constant only after the acceleration phase has been completed: therefore, C c,a in the constant speed phase can be calculated as indicated by Olds et al. (1993) as: C c;a ˆ 0:86 m 2 d tot d acc d 1 tot ˆ 0:86 m2 1 d acc =d tot ; 4 where d tot and d acc represent: (1) the total distance, and (2) the distance covered to accelerate from zero to the constant speed, and are assumed to be equal to 100 m, as suggested by Olds et al. (1993). Since the metabolic energy spent against drag depends on the instantaneous speed, C c,a during the acceleration phase is lower than that applied to the second part of the competition covered at constant speed. The overall value of C c,a may be corrected for the initial acceleration phase (100 m long) as follows. First, the total amount of metabolic energy spent against drag during the acceleration phase (E acc,a, J) may be calculated as: E acc;a ˆ 0:86 m 2 d acc 0:5; 5 where E acc,a is the integral mean of the product of C c,a (= 0.86 v 2 ) times d acc, since the acceleration is assumed to be constant over d acc (Olds et al. 1993). Then, the contribution of E acc,a to the overall value of C c,a is calculated by dividing E acc,a by the total distance d. After some simpli cations, the following equation is derived: C c;a ˆ 0:86 m 2 1 d acc =d tot 0:86 m 2 d acc =d tot 0:5 6 ˆ 0:86 m 2 1 d acc =d tot 0:5 d acc =d tot The third term in Eq. 2 (0.5 M m 2 d tot g 1 c ) represents the metabolic energy spent by the cyclist to accelerate the overall mass M from a stationary start to the nal speed v. Thus, if M = 85 kg and assuming g = 0.22 (Capelli et al. 1993; Seabury et al. 1977), the total metabolic requirement per unit distance in track cycling (C c, J á m )1 ) is described by: C c ˆ 25:8 0:86 m 2 1 d acc =d tot 0:5 d acc =d tot 193:2 m 2 d 1 tot 7 The overall metabolic power requirement necessary to progress at the speed v ( _E c ˆ C c m,) (see Eq. 1) is given by: _E c ˆ 25:8 m 0:86 m 3 1 d acc =d tot 0:5 d acc =d tot 8 193:2 m 3 d 1 tot Since in any track competition the distance d tot is xed, d tot t )1 can be substituted for v. As a consequence, Eq. 9 can be nally rearranged to obtain _E c as a function of the time necessary to cover the distance d tot : _E ˆ 25:8 d tot t 1 0:86 d 3 tot t 3 1 d acc =d tot 0:5 d acc =d tot 9 193:2 d 2 tot t 3 Running The overall energy cost per unit of distance of track running (C r ) on at terrain is given by (di Prampero et al. 1993): C r ˆ C r;na k 00 m 2 0:5 m 2 d tot 1g 1 10 where: (1) C r,na (J á m )1 á kg )1 ) is the metabolic energy spent per unit of distance and per kilogram of body mass against nonaerodynamic forces: in elite middle-distance runners the value of this term amounts, on average, to 3.8 J á m )1 á kg )1 (Lacour et al. 1990), (2) k 00 is the proportionality constant between the cost against air drag and the square of the air speed: this amounts to 0.40 J á s 2 á m )3 per m 2 of body surface for a barometric pressure of 760 mmhg and an air temperature of 20 C (di Prampero 1986), (3) the third term is the metabolic energy spent, over a unit of distance, to accelerate 1 kg body mass from zero to the nal speed v: it is given by the kinetic energy per unit of distance and unit of body mass (0.5 v 2 d )1 ) divided by the overall e ciency of running (g r ). As a rst approximation, g can be assumed to be equal to 0.25 since, in the initial acceleration phase, no (or only very small) recovery of elastic energy can take place. Hence, the mechanical ef- ciency approaches that of the concentric muscular contraction (Cavagna et al. 1971). Substituting these values into Eq. 10, the overall metabolic energy spent per unit distance (J á m )1 ) of an elite runner of standard anthropometric characteristics (75 kg, 175 cm) can be described as: C r ˆ 285 m 0:76 m m 2 d tot 1 11 The overall metabolic power output ( _E r ˆ C r m) necessary to run at speed v is obtained by multiplying C r,tot by v. Once again, since in track running the distance of each competition is xed and known, d tot t )1 can be substituted for v in Eq 12: _E r ˆ 266 d tot t 1 0:74 d 3 tot t d 2 tot t 3 12 Swimming In elite swimmers, and for speeds ranging from aerobic to supra-maximal speeds (0.96±1.97 m á s )1 ), the energy cost of front crawl swimming per unit of distance (J á m )1 ) above the metabolic rate at rest may be described as a power function of the speed (Capelli et al. 1998a): C s ˆ 560 m 1:6 13 Thus, the rate of metabolic expenditure per unit of time ( _E s ˆ C s m, W) is given by: _E s ˆ 560 m 2:6 14 Since in swimming competitions the total distance d tot is also known and is constant, the speed v can be replaced by d tot t )1. Equation 14 becomes, therefore: _E s ˆ 560 d 2:6 tot t 2:6 15 During competitions in a swimming pool, however, the swimmers, after the starting dive, glide under water for a short distance. In addition, before and after each turn, the swimming technique is necessarily altered; it has been found that, at least during breaststroke competitions, up to 39% of the total race time is spent on turning (Thayer and Hay 1984). The relationship between C s and speed in these phases is not described by the equations above, so to calculate correctly _E s as a function of t during competitions, the following precautions must be taken. The distance (d s ) covered by the subject swimming canonically is the total distance (d tot ) minus the sum of: (1) the distance covered by the subject gliding after the starting dive (d st ), and (2) the distance covered during each turn (d t ) multiplied by the number of turns (n t ):

3 300 d s ˆ d tot d st R n t d t 16 In turn, the time (t s ) spent swimming canonically is the total time of the race (t) minus: (1) the time of the gliding phase after the start (t st ), and (2) the time of each turning (t t ) multiplied the number of turns: t s ˆ t t st R n t t t 17 If this is so, d s and t s (Eqs. 16 and 17) are the distance and time values that should be introduced into Eq. 15 to yield E s at speed m s ˆ d s ts 1. Moreover, to obtain a better estimate of the overall metabolic power, the metabolic power output for the dive should also be considered. Finally, the average speed maintained by the subject during the race could be calculated by dividing the total distance (d tot ) by the total time (t), [i.e. the sum of t s plus (t st + S n t t t )]. The distances covered by the subject after the start and during the turning motion (d st, d t ), and the corresponding times (t st, t t ), were obtained as follows. We assumed that a subject, after leaving the block at the start covers, without swimming, a distance (gliding included) equal to (Toussaint and Hollander 1994): d st ˆ 7 3 ln m; 18 and that this phase takes an overall time, including a reaction time interval 0.9-s long (Toussaint and Hollander 1994), of: t st ˆ 0:6 3 m 1 19 where v in Eqs. 18 and 19 is the average speed maintained during the competition. We determined d t and t t in a group of elite crawl swimmers: on average, they amounted to 2.0 (0.55) m (mean SD) and 1.0 (0.24) s, respectively and 3.16 (0.58) m and 1.8 (0.54) s, respectively, for the phases before and after the turning motion, respectively (see Appendix below). Moreover, it was also assumed that the subjects spend an amount of metabolic energy equal to 0.04 kj á kg )1 on diving (Toussaint and Hollander 1994). This last, once multiplied by the body mass of the subject (75 kg) and divided by the total performance time, was added to the value resulting from Eq. 15 to obtain a better estimate of the overall E s. Maximal metabolic power The _E max (W) a given subject can sustain at a steady level throughout the e ort is a decreasing function of the exhaustion time (t e, Capelli et al. 1998b; di Prampero et al. 1993; Wilkie 1980). This term be appropriately described by: _E max ˆ AnS t 1 e F MAP F MAP s 1 e tes 1 t 1 e Š 20 where: (1) AnS is the metabolic energy obtained in the time t e from the complete utilisation of anaerobic sources (i.e. from anaerobic glycolysis and from net high energy phosphate splitting). (2) MAP is the equivalent in W of the individual maximal oxygen consumption ( _V O 2 max ) measured above the oxygen consumption measured at rest. (3) s is the time constant with which MAP is attained at the onset of the e ort. (4) F is the MAP, expressed as fraction of MAP, that can be sustained as a function of t e. The third term is due to the fact that at the onset of exercise _V O 2 max is not attained instantaneously, but with a time constant s. Hence, the average aerobic power up to the time t e is given by the quantity in square brackets, i.e. it is reduced below F MAP by an amount equal to the oxygen de cit incurred up to t e [F MAP s (1 e tes 1 )] divided by the time t e itself. The rst and third terms of Eq. 20 become progressively smaller with increasing values of t e. So, Eq. 20 shows that for long-term exercise, the maximal sustainable metabolic power is essentially set by the subject's _V O 2 max and by the fraction of _V O 2 max that can be maintained throughout the e ort. As the time of the exercise becomes shorter, the contribution of the anaerobic energy stores to the overall metabolic power becomes progressively greater because, with decreasing t e : (1) the rst term of Eq. 20 increases hyperbolically, and (2) the amount by which the oxygen de cit a ects (negatively) the actual average aerobic power becomes larger. AnS is the sum of the amount of energy derived from the full exploitation of the anaerobic glycolytic pathway (anaerobic lactic capacity, Lac, J) plus that provided by the maximal splitting of phosphocreatine and adenosine triphosphate (anaerobic alactic capacity, Alac, J) in the working muscles. The former can be set equal to 1000 J á kg )1 of body mass (di Prampero 1981), and the latter increases with the duration of exhausting exercise with a time constant (s al ) of 23.4 s to attain a value of 420 J á kg )1 for exercises lasting longer than 120 s (Medbù and Tabata 1993). Hence, the anaerobic metabolic energy that can be utilised by time t e in a 75-kg body-mass subject (AnS, J) is described by: AnS ˆ e te=sal Š Assuming a gross _V O 2 max value of 5.0 l á min )1, a value typical for elite athletes (di Prampero 1981), and a resting oxygen consumption of 3.5 ml O 2 á min )1 á kg )1, the maximal metabolic power above resting metabolism turns out to be 1650 W for a subject of 75 kg body mass. The time constant of the simple exponential function describing the increase of oxygen consumption at the muscular level at the onset of the maximal exercise (s), is set to 24 s, as measured by means of 31 P nuclear magnetic resonance spectroscopy (Binzoni et al. 1992). F depends, among other factors, upon the duration of the exercise. For a t e longer than 7 min, it may be assumed that F decreases linearly with the logarithm of t e (Pe ronnet and Thibault 1989). Accordingly, F in Eq. 21 takes the following values: F ˆ ˆ 1:00; for 45 s < t e < 420 s ˆ 1 0:0568 ln t e =420 ; for > 420 s Best performance times 22 As shown above: (1) the metabolic power required ( _E) to cover a given distance d, is a decreasing function of the total performance time t, and (2) the maximal metabolic power ( _E max ) that a given subject can sustain at a constant level is described by a decreasing function of the time of exhaustion (t e ). In Fig. 1, the dashed thick Fig. 1 Metabolic power requirement _E c (dashed thick line) to cover 1.0 km in track cycling in the time indicated on the abscissa. The maximal available metabolic power ( _E max ) is also indicated (continuous thick curve) on the same time axis. Data refer to an elite athlete whose characteristics are assumed to be as follows: body mass = 75 kg; maximal aerobic power above rest (MAP) = 1.66 kw; overall anaerobic capacity (AnS) = kj. Best performance times become shorter as MAP is changed over the range from 90% to 110% of 1.65 kw, in 2.5% steps

4 301 curve represents the metabolic power required ( _E c, kw) by a standard elite cyclist (75 kg body mass, 175 cm height) to cover 1.0 km from a stationary start, over a time interval ranging from 65 s to 80 s. In the same diagram, the thick continuous curve represents the _E max that is sustainable at a constant level by the same athlete, as function of t e over the same time interval. For a certain range of t values, _E max is below the function describing the metabolic power requirement ( _E c ). These times will therefore be unattainable by this subject. For longer t values, _E max is above _E c. Therefore, this hypothetical athlete could have covered the distance at stake in a shorter time. The theoretical BPT (t theor ) is obviously given by the abscissa, at the point where the two functions cross. In practice, the time value solving the equalities _E max t e ˆ _E c t (i.e. t theor ), can be obtained either graphically, as in the example given in Fig. 1, or by means of a computerised iterative procedure. It goes without saying that t theor can be also calculated for the other two forms of locomotion on the basis of assumed values of Alac, Lac, _V O 2 max and C. The procedure illustrated above can also be utilised to calculate to what extent the physiological inputs, namely MAP, Lac, Alac and C, may a ect performance for running, cycling and swimming. To this aim, BPTs for the three forms of locomotion were calculated for various distances, applying initial, pre-de ned values for MAP, C, Lac and Alac that are typical of an elite athlete. The procedure was then repeated while modifying one variable at a time by discrete and pre-de ned intervals above and below the initial control value (i.e. the value utilised in the rst run of the simulation). This procedure, if applied for each variable, distance and type of locomotion, allows the calculation of percent increases (or decreases) of BPTs. The four input variables (MAP, C, Lac and Alac) were varied by 2.5% steps over an interval ranging from 90 to 110% of their initial control values. For the overall cost of running, only the non-aerodynamic fraction of the overall energy cost (C r,na ) was altered, whereas for cycling only the aerodynamic component of the energy cost (C c,a ) was modi ed. Finally, in swimming the coe cient A of Eqs. 14±15 were changed over the interval ranging from 90 to 110% of its initial control value. Results The results obtained by applying the procedure illustrated above to a standard elite cyclist riding a traditional racing bike over the distance of 1.0 km from a stationary start are represented in Fig. 1. The thick continuous curve describes _E max as a function of t e when Fig. 2 Percent improvement of the initial best performance time in track cycling as a function of the distance for di erent percent changes of anaerobic lactic capacity (Lac, Squares), anaerobic alactic capacity (Alac, Circles), maximal aerobic pair (MAP, triangles), and overall energy cost of cycling (C c, Crosses)

5 302 MAP was equal to the initial control value given in Eq. 20 (see above). The eight thin curves describe _E max when MAP values ranging from 110 to 90% of the control value were utilised in the same equation. The abscissa values at which these nine _E max functions cross the dashed thick E c function form the set of BPTs obtained when MAP was changed from 90 to 110% of the initial values, while the other three variables (Lac, Alac, C c ) remained constant. The percent improvement of the initial BPTs for cycling is plotted in Fig. 2 as a function of the distance covered (d, km) and for 2.5, 5, 7.5 and 10% changes of the four variables. Each curve describes the results obtained when only the indicated variable was modi ed. These diagrams give an immediate representation of the weight of each of the four variables in determining the performance in track cycling. For instance, the improvement achieved by modifying either the maximal aerobic power or the overall anaerobic capacity (AnS = Lac + Alac) depends upon the time spent to cover the distance. On the contrary, the improvement achieved by modifying only C c remains essentially constant, regardless of the distance covered. Since these increasing distances are covered in longer times, this implies that the improvements brought about by changes of C c, are hardly a ected by the metabolic power output. The foregoing analysis gives rise to similar ndings when applied to running and swimming (Figs. 3 and 4), con rming that the percent improvement brought about by changes in C is almost independent of the distance covered. These results are further summarised in Fig. 5, where the improvement in performance due to a 5% change (decrease) of the overall C is plotted as a function of the performance time, and is expressed as a fraction of the overall improvement obtained when all four variables (C, decrease; MAP, Lac, Alac, increase) were changed by the same amount. The diagram shows that the extent of improvement achieved by decreasing C alone is comparable to that attained by increasing Fig. 3 Percent improvement of the initial best performance time in track running as a function of the distance for di erent percent changes of the four variables (Lac squares, Alac circles, MAP triangles, C r crosses)

6 303 Fig. 4 Percent improvement of the initial best performance time in front crawl swimming as a function of the distance for di erent percent changes of the four variables (Lac squares, Alac circles, MAP triangles, C r crosses) simultaneously MAP, Lac and Alac by the same percentage. Moreover, this e ect is a ected only slightly by the performance time. Figure 5 shows unambiguously that the single variable the changes of which most e ectively in uence performance, is the energy cost C. However, for practical purposes, it is often convenient to know the percentage change in performance brought about by any given predetermined change(s) of all four variables at stake. These calculations can be easily solved algebraically. For convenience, an example of such an approach is reported in Fig. 6, for cycling 1, 5 or 10 km where the linear functions represent the percent increase, or decrease, of the BPT over the appropriate distance when the variable at stake was changed by the amount indicated on the abscissa. The point upon which all lines pivot corresponds to the t theor calculated when all variables are set at their initial value. In Fig. 6, the relationships between the percent increase or decrease of the performance time and the percent change of the independent variable were assumed to be linear. However, since neither _E nor _E max are linear functions of the performance time, the above assumption is an obvious oversimpli cation. Nevertheless, within the range of the independent variable reported in Fig. 6, these relationships could be appropriately described by linear regressions, the slopes of which are reported in Table 1, with a mean r 2 equal to ( ) (n = 68; range: 1.000±0.983). The advantage of this approach, although not strictly correct, is that linear e ects can be added easily. For instance, the e ects deriving from a 5% increase of MAP and from a 3% increase of Lac can be added to yield the overall percent decrease of the t theor in a given distance and/or type of locomotion. The data given in Table 1 make it possible to apply this type of theoretical analysis to the three forms of locomotion studied and to all of the distances considered.

7 304 Discussion Fig. 5 The ratio (%) between the percent improvement of best performance time brought about by decreasing the energy cost of locomotion (C) by 5% and that obtained by changing all four variables by the same amount is plotted as an function of performance time for track cycling (squares), running (circles) and for front crawl swimming (crosses) The results of the above analysis depend upon several assumptions, particularly as far as the form of the equations is concerned. Hence, the few paragraphs that follow will be devoted to a discussion of the dependence of the results upon the assumptions made. Equations similar to that yielding _E max as function of t e (Eq. 20, see Theory) have been applied in the past to calculate t theor in runners and in elite cyclists over several distances (Capelli et al. 1998b; di Prampero et al. 1993). Theoretical and actual times agreed well, and all assumptions upon which Eq. 20 is based have been discussed extensively in previous work (Capelli et al. 1998b; di Prampero et al. 1993). Equation 20 has also been used recently to estimate the energy cost of swimming (Capelli et al. 1998a) and running (Hautier et al. 1994) at supra-maximal speeds. Hence, we are fairly con dent that the physiological premises upon which Eq. 20 is based are su ciently solid to allow us to obtain an accurate description of the individual _E max as a function of t e. MAP has been reported to be signi cantly larger after high-intensity training both in moderately active subjects and in elite athletes. For instance, _V O 2 max increased by Fig. 6 The percent increase and decrease of the initial best performance time is plotted as a function of the percent change of MAP, Alac, Lac and C for three distances in track cycling. The point at which all lines pivot corresponds to the theoretical best time of performance when all variables are set at their initial value (see main text for details)

8 305 Table 1 Percent decrease or increase of theoretical best times of performance obtained by changing by 1% maximal aerobic power (MAP), anaerobic alactic capacity (Alac), anaerobic lactic capacity (Lac), and the energy cost of locomotion (C) in cycling, running and swimming. Time values in seconds indicate the best performance times obtained utilising the initial values of MAP (1.66 kw), Alac (31.53 kj), Lac (75 kj) and C representing the physiological characteristics of a hypothetical elite athlete Exercise Distance (m) Time (s) Dt/DMAP (%) Dt/DAlac (%) Dt/DLac (%) Dt/DC (%) Cycling )0.199 )0.068 ) )0.297 )0.040 ) )0.356 )0.020 ) )0.379 )0.019 ) )0.415 )0.008 ) Running )0.288 )0.202 ) )0.622 )0.128 ) )0.784 )0.065 ) )0.897 )0.034 ) )0.897 )0.020 ) )0.997 )0.010 ) Swimming )0.034 )0.044 ) )0.084 )0.042 ) )0.141 )0.030 ) )0.187 )0.017 ) )0.215 )0.008 ) )0.229 )0.005 ) % after 6 weeks of high-intensity intermittent exercise in a group of seven physical education students (Tabata et al. 1996), and rose signi cantly from winter to the competitive mid-year season (+4.5%) in a group of welltrained, high-level track runners (Svedenhag and SjoÈ din 1985). In addition, AnS can be substantially increased by high-intensity training. This was shown to occur, for instance, in moderately active young male subjects in whom AnS rose by 28% at the end of 6 weeks of intense, intermittent cycloergometric exercise, and in sprinttrained subjects whose AnS was 10% larger at the end of 6 weeks of high-intensity running training (Medbù and Burgers 1990). Finally, peak lactate concentration was found to be in uenced by the training intensity applied at di erent periods of the competitive season in swimmers (Gullstrand and Holme r 1983). Therefore, the range of percent changes imposed upon MAP and to anaerobic capacities reproduce fairly well those brought about by training or detraining in elite athletes. There is far from a consensus as to the e ects of training on running economy (Morgan et al. 1989; Bourdin et al. 1993; Lake and Cavanagh 1996), even though cross-sectional studies have revealed that C is lower in long-distance runners and in adult subjects than in sprint runners and in children (Morgan et al. 1995). Moreover, C seems to be identical between similarly trained male and female athletes (Bourdin et al. 1993). More recently, a retrospective analysis of seven publications spanning a period of 20 years reexamined the di erences in the cost of running among three groups of runners who were strati ed according to their performance level (Morgan et al. 1995). The study revealed that elite, high-level runners were more economical than the less-talented, and that on average, trained subjects had a better running economy than the untrained controls. However, within-group variability was high in all of the groups and there was also a substantial overlap of minimum, mean and maximal values of running economy over the categories. In the four categories of runners, the coe cients of variation of C r and of _V O 2 max turned out to be, on average, of mean S.D., the same magnitude [ _V O 2 max : 6.1 (1.47)%; C r : 6.6 (1.42)%], although they were the lowest (5.6% and 4.2% for C r and _V O 2 max, respectively) in the highly trained subjects. Moreover, C r variation, expressed as a percent ratio between minimum and maximum values, was fairly high, being equal on average, to 24.5 (2.64)% in the four groups (calculated from the values reported in Fig. 1 of Morgan et al. 1995). These data suggest that, in spite of the uncertainty concerning the e ects of training on running economy, C r should always be assessed in runners, especially so because of its high intra-individual variability. Indeed, although the variability of C r and _V O 2 max seems to be comparable, the weight of running economy in a ecting the level of performance of runners ranges from three times as much, over the shortest distances, to the same as that of _V O 2 max in the longest events. Thus, C r di erences existing between athletes endowed with similar values of _V O 2 max may justify the corresponding discrepancy of their levels of performance. The aim of this paper was to quantify, by means of a theoretical model, the role of the energy cost of locomotion (C), of the MAP and of the AnS in determining BPTs for running, track cycling and swimming. This approach should be challenged by comparing the theoretical percents of improvement calculated by utilising the slopes listed in Table 1 with the actual ones induced by training. To this aim, at the beginning and at the end of the training period, we measured _V O 2 max and Lac in crawl swimmers who were training from early September to the end of December. In addition, the relationship between swimming speed and C s was determined on the two occasions, whereas Alac was assumed to be equal to 20 ml O 2 á kg )1 of body mass in a male adult subject, and it was corrected for the subject's age in younger athletes (all details concerning the protocol, methods and calculations utilised in this study have been illustrated and discussed in a recent paper, Capelli et al. 1998a). This allowed us to deter-

9 306 mine in each subject the percentage improvement of MAP, AnS and C s that occurred over this period. The latter was calculated by dividing the values of C s corresponding to the average speeds maintained during competitions in December, and calculated by means of the equation describing C s versus speed obtained in September, by those calculated, at the very same speeds, by means of the equation obtained in December. These data made it possible, by utilising the proper slopes reported in Table 1: (1) to calculate the theoretical percent improvements and (2) to compare them with the actual percent changes of performance that occurred from September to December over various distances (50±400 m). MAP increased, on average, by 4.0 (4.2)%, whereas anaerobic capacity remained almost stable [)0.8 (3.2)%]. C s showed a clear improvement, decreasing on average by 10 (2.0)% with respect to that prevailing, at the very same speed, in early September. The theoretical percent improvement turned out to be, on average, equal to 2.7 (0.74)% in comparison with the actual value of 3.3 (1.33)%. The di erence between the two values, 0.51 (1.32)%, turned out not to be signi cant, as con rmed using a Wilcoxon rank test for non-parametric data (Daniel 1991). In summary, the analysis illustrated in the present paper indicates that C was the major determinant of the maximal speeds attained in the three forms of locomotion studied. Indeed, the results show that in all three types of locomotion, regardless of the speed, the changes in BPT brought about by changes of C alone accounted for 45±55% of the changes obtained when all three variables (C, MAP, AnS = Lac + Alac) were changed by the same amount. Appendix In the present paper, a theoretical model was used to quantify the role of C, MAP, and AnS in determining BPTs for running, track cycling and swimming. The constants and the coe cients that appear in the equations were obtained mostly from the papers quoted in the text. However, to calculate appropriately t theor for swimming, the times spent and the distances covered by the swimmer during one turn had to be assessed experimentally. These were assessed on a group of crawl swimmers whose physiological characteristics have been reported in a previous paper (Capelli et al. 1998a). Brie y, a belt was fastened around the waist of the subject and connected by means of a nylon wire (diameter: 0.5 mm) to a reel through a system of two pulleys that were arranged to maintain the direction of the traction force parallel to the surface of the water. The wire was released from the reel while the subject was leaving the border of the pool either swimming or gliding, and the number of revolutions per unit of time of one of the two pulleys was measured with the aid of a DC motor (ESCAP 28L , Portescap, Hauppauge, N.Y., USA) whose pin was the axis of rotation of the pulley. The output of the DC motor, which was previously calibrated by unwinding the wire at di erent speeds, was recorded on a chart recorder (7DA, Grass, Quincy, Mass., USA; paper speed: 10 cm á s )1 ). During an experimental run, the subject was rst asked to push from the border of the pool, swimming at his maximal speed. The onset of the push and the end of the gliding phase were monitored visually by an operator who marked on the chart recorder the instants corresponding to these two events. 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