The present paper is concerned with high-intensity exercise, that is, exercise that either leads to exhaustion or if continued would lead to exhaustion. During such vigorous exercise, measurements of pulmonary oxygen uptake and the carbon dioxide content of expired gas are relatively straightforward and so both the build-up and subsequent quasi-steady energy contributions associated with aerobic metabolism are readily quantifiable. In contrast, the magnitude of the energy contribution from anaerobic metabolism is far more difficult to determine (^{2,13} ). Experimental techniques involve one of two basic kinds of measurement. In the direct approach the depletion of ATP and phosphocreatine and the increase in lactate production in muscle are measured. In the indirect approach, information on anaerobic metabolism is inferred from measurements of oxygen consumption during exercise. This latter approach has been widely used (^{3,7,9,12} ) to determine anaerobic capacity, which is inferred from the measurement of maximal accumulated oxygen deficit (MAOD). Both the direct and indirect techniques are subject to difficulties of interpretation. In particular, the results of the direct technique cannot readily be related to whole-body energy conversion (^{3} ).

One of many forms of vigorous exercise is competitive running, and the body of track statistics that exist, particularly those relating to world record and Olympic performances, provides a further method of gaining insights into anaerobic metabolism . This is an important source of data which does not appear to have been exploited by physiologists. In recent years several successful mathematical analyses of running have been published. The mathematical analyses can be subdivided into two distinct categories: i) those based on Newton’s laws of motion, and ii) those based on satisfying the first law of thermodynamics. The second group is of interest in the present context. These mathematical methods model running performance by setting down the contributions to the whole-body energy balance during running. In conformity with the first law of thermodynamics, the energy released by chemical conversion in the muscles, which passes through a number of intermediate phases, must ultimately be accounted for either in the form of external mechanical work or as mechanical energy degraded into thermal energy. The latter quantity is also referred to simply as heat loss or heat production and is often expressed by sports physiologists in terms of the energy cost of running. Several successful analyses based on the bioenergetics of running and covering distances from 60 m up to the marathon distance have been put forward to study the performance of male and female athletes (^{5,11,15,16} ). More focused studies have also been published; van Ingen Schenau et al. (^{14} ) have analyzed sprinting, and di Prampero et al. (^{4} ). have investigated middle-distance running. In direct methods of analysis, running performance can be predicted by substituting appropriate physiological data into theoretical relationships describing mathematical models of running; in the inverse process, the mathematical model of running is used to deduce physiological data from running performance. These types of analysis are remarkably successful. For example, Peronnet and Thibault (^{11} ) correlated 1987 male world records covering 16 different distances from 60 m to 42195 m (the marathon), with a maximum percentage error between actual and estimated running times of 2.3% (at 3000 m), and an average absolute error of only 0.73%. Inherent in all these methods of analysis is the separate estimation of the aerobic and anaerobic contributions to the energy balance for each race distance, embracing short distances, dominated by anaerobic metabolism , to long-distance endurance running, for which aerobic metabolism is the principal energy source. Because there is excellent agreement between prediction and track performance, there are good grounds for confidence that the aerobic and anaerobic energy contributions are modeled with considerable precision.

The purpose of this paper is to extract information on anaerobic and aerobic metabolism from published mathematical analyses of world record and Olympic running performances, including short-duration through to endurance events (^{4,5,11,14–16} ). This information will be compared with data from a variety of other sources, including laboratory measurements and estimates from biochemical data. The data assembled here are applicable to those forms of exercise dominated by the musculature of the legs, i.e., running, cycling, skating, cycle ergometry, and treadmill running. It is worth noting that other forms of exercise, such as weight-lifting, swimming, and rowing, can involve the utilization of musculature in the upper body and/or arms to perform external work. Consequently, during the first few seconds of exercise, greater levels of anaerobic energy conversion take place within the body than in the forms of exercise considered here.

Mathematical relationships for aerobic and anaerobic energy release.
We define C _{aer} as the energy contribution from aerobic metabolism . Based on the experimental work reported by Margaria (^{6} ) and others, the rate of aerobic energy release under conditions of maximum steady mechanical power output can be written in the form of an exponential relationship as MATH 1 For competitive running races, provides an accurate description of the kinetics of aerobic metabolism and here R is interpreted as the steady-state maximum sustainable power available from aerobic metabolism , measured relative to the resting state, λ_{1} is a parameter governing the rate of aerobic energy release, and t is the time from the start of high energy expenditure. Defining T as the time required to run a distance X , it has been found that the maximum sustainable aerobic power is equal to the maximum aerobic power for T in the range 0 <T <T _{map} . Peronnet and Thibault (^{11} ) adopted a value for T _{map} of 420 s. For running times in excess of T _{map} the maximum sustainable aerobic power progressively declines.

Different energy utilization strategies are used by athletes, depending on running distance. For short sprints over 100 m and 200 m, an athlete exerts maximum effort throughout the period of running to capitalize as quickly as possible on the high initial power levels available from anaerobic metabolism . For most other distances, certainly those greater than about 800 m, the optimal utilization of the available energy, both aerobic and anaerobic, is brought about by minimizing the energy expenditure terms in the overall energy balance. This minimum corresponds to running at constant speed.

During the first few seconds of vigorous exercise, there are anaerobic energy contributions from the utilization of phosphocreatine and oxygen-independent glycolysis. A small amount of energy is also provided by the endogenous ATP store in muscle (^{13} ). So far, the kinetics of these energy contributions during the first one or two seconds of exertion have defied both accurate measurement and precise mathematical representation. However, the build-up of oxygen-dependent glycolysis takes place very rapidly and is fully active within the first few seconds from the start of high-intensity exercise. Beyond this initial period, mathematical relations have been proposed to describe the overall energy conversion associated with anaerobic metabolism . We define C _{an} as the energy contribution from anaerobic metabolism . Based on theoretical considerations, the rate of energy release by anaerobic metabolism under conditions of maximum exertion can be represented (^{15} ) by a general relation of the form MATH 2 In this equation, P _{max} represents the maximum power available from anaerobic metabolism , and λ_{2} is a parameter governing the rate of anaerobic energy release. Because the contributions from endogenous ATP and phosphocreatine are both transitory and a small proportion of the total anaerobic energy yield, best describes the period when oxygen-independent glycolysis prevails. Nevertheless, the relation can conveniently be used as an approximation for all t > 0. It must be recognized, however, that the relationship breaks down at t = 0, since the relationship implies that maximum anaerobic power is established instantaneously, whereas in practice it takes a finite time, albeit no more than a second or two, for the power to build up to a maximum. Also, because of the particular musculature activated, the relationship used here is applicable to running, etc., but not to activities such as weightlifting or rowing. Hence the quantity P _{max} defined above is an approximate indication of the maximum anaerobic power generated by a sprinter, but it is unrepresentative of the power generated by, say, a weightlifter. Incorporating ideas advanced by Margaria (^{6} ), namely, that the buildup of aerobic power and the decline of anaerobic power are inter-related, Ward-Smith (^{15} ) also argued that equations 1 and 2 could be simplified with the substitution λ_{1} = λ_{2} = λ. However for the purposes of generality this paper initially considers the condition λ_{1} ≠ λ_{2} .

Although originally justified by theoretical arguments (^{15} ), the mathematical expressions 1 and 2 for the aerobic and anaerobic powers in terms of the exponential function have received confirmation from two separate sources. First, there is the experimental work of van Ingen Schenau et al. (^{14} ). Three elite sprinters performed supramaximal tests using a cycle ergometer, and these test results were correlated (^{14} ) using equations 1 and 2 with values for λ_{1} of 0.0384 s^{−1} for the aerobic power and λ_{2} = 0.0403 s^{−1} for the anaerobic power, respectively. The difference between these values is less than 5%. For the five elite skaters tested in the same program, values for λ_{1} of 0.0384 s^{−1} for the aerobic power and λ_{2} = 0.0379 s^{−1} for the anaerobic power were obtained. The closeness of these values can be interpreted as providing support for the theoretical arguments for the use of a single value of λ applicable to both the aerobic and anaerobic contributions. Second, the successful incorporation of equations 1 and 2 into mathematical models of running provides further confidence in their reliability as models of the kinetics of both anaerobic and aerobic metabolism . Excellent correlations of predicted and actual running times have been obtained (^{5,11,15,16} ), indicating that the overall energy balances are accurately described. Furthermore, there is a particularly sensitive test of the mathematical integrity of the term describing anaerobic metabolism . The velocity-time history of an athlete running over the course of a 100-m event has been modeled (^{15} ); not only is the acceleration phase of the athlete’s progress reproduced but so also is the decline in running speed resulting from fatigue during the latter stages of the sprint. The ability to reproduce these detailed characteristics of sprinting is further evidence that the mathematical models describe the overall kinetics of anaerobic and aerobic metabolism very adequately. It is recognized, however, that there is scope for further detailed improvements in the mathematical modeling, in particular replacing by an expression that more accurately reflects conditions for small values of t .

Special case for R, P _{max} , λ_{1} and λ_{2} independent of t.
If R and λ_{1} are independent of t , integration of between the limits t = 0 and t =T yields the energy provided by aerobic metabolism MATH 3 where R is, in general, a function of T but does not depend on t . For the large T applicable to middle- and long-distance running, exp( −λ_{1} T) → 0 and in the limit simplifies to MATH 4 If P _{max} and λ_{2} are independent of t , integration of between the limits t = 0 and t =T yields the energy provided by anaerobic metabolism MATH 5 which, for large T , can be represented by MATH 6 where E _{0} is the anaerobic capacity. Combination of equations 3 and 5 yields MATH 7 where C _{tot} = C _{aer} + C _{an} and MATH 8

simplifies in certain circumstances. If λ_{1} = λ_{2} = λ then is replaced by MATH 9 Furthermore for large T , which in practice means for T greater than about 150 s, exp (- λT) → 0, and then reduces to MATH 10 Most of the models of running considered previously in this paper employ the above mathematical formulations of aerobic and anaerobic metabolism , equations 1, 2, 3, and 5. Ward-Smith (^{15} ), di Prampero et al. (^{4} ), and Ward-Smith and Mobey (^{16} ) use these formulas and also take λ_{1} = λ_{2} = λ. van Ingen Schenau et al. (^{14} ) use equations 1 and 2 with λ_{1} ≠ λ_{2} . Finally the method of Peronnet and Thibault (^{11} ) is broadly based on this approach but incorporates some modifications in the detail.

Variation of anaerobic and aerobic energy contributions with time.
The energy contributions from aerobic and anaerobic metabolism defined in existing mathematical models of running applied to the performance of male athletes (^{4,11,14,15} ) are considered here. Excluded from consideration are the work of Ward-Smith and Mobey (^{16} ), which relates to female athletes, and the mathematical method of Lloyd (^{5} ), which does not readily lend itself to precise compartmentalisation of the two terms. The results, which relate to elite male athletes, are presented in Table 1 in the form of running time T and the ratio (C _{aer} /C _{tot} ) × 100, as a function of running distance. Here C _{tot} =C _{aer} + C _{an} where C _{aer} and C _{an} are the energy contributions from aerobic and anaerobic metabolism , respectively. The precise sources of the data are as follows. Column 1: Ward-Smith (^{15} ), extracted from Table 5; Column 2: Peronnet and Thibault (^{11} ), extracted from Table 1; Column 3: van Ingen Schenau et al. (^{14} ), computed by integration of power terms in ; Column 4: di Prampero et al. (^{4} ), compartmentalization of . For comparison data from Newsholme et al. (^{10} ), based on biochemical information, are given in Column 5.

Table 1: The ratio of energy from aerobic metabolism to total energy (%) as a function of time from the initiation of vigorous exercise.

Measurements of the ratio of energy from aerobic metabolism to total energy as a function of time have been reported (^{3,7–9,12} ). The subjects of these five studies had backgrounds different from the elite athletes to whom the mathematical studies applied. Medbo and Tabata (^{10} ) studied “17 healthy men.” Bangsbo et al. (^{3} ) studied eight healthy males, all habitually physically active but none trained for competition. The subjects of the investigation of Scott et al. (^{12} ) included three sprinters, five middle-distance athletes, four distance runners, all of NCAA Division I standard, plus four healthy college-aged males with no active participation in track and field athletics. McArdle et al. (8) quote the results of cycle ergometer tests, but the background of the subjects investigated is not described. Finally, the subjects of Maxwell and Nimmo (^{7} ) were “18 male students from a variety of sporting backgrounds.” Medbo and Tabata (^{9} ), Bangsbo et al. (^{3} ), and McArdle et al. (^{8} ) all measured the energy ratio (C _{aer} /C _{tot} ) at specific intervals of time after the start of vigorous exercise; their results are set out in Table 2 .

Table 2: The ratio of energy from aerobic metabolism to total energy (%), as a function of time from the initiation of vigorous exercise. Experimentally-determined data.

On the other hand, the test methods of Scott et al. (^{12} ) and Maxwell and Nimmo (7) involved the measurement of the ratio (C _{aer} /C _{tot} ) over a range of times extending between 120 s and 180 s, so their results are not sufficiently precise for inclusion in the present analysis. Nevertheless, the data from these two sources display broad general consistency with the trends of other data. Scott et al. (^{12} ) measured values of (C _{aer} /C _{tot} ) in the range between 61% and 70% for T in the range between 120 s and 180 s, while Maxwell and Nimmo (7) derived a value of 64% for (C _{aer} /C _{tot} ), corresponding to a range of T between 120 s and 180 s. Also included in Table 2 are the estimates of the variation of the ratio (C _{aer} /C _{tot} ) with time T , made by Astrand and Rodahl (^{1} ), based on the scientific evidence available at that time.

DISCUSSION
Three of the four mathematical models considered in Table 1 yield very consistent results. The odd one out is the method of van Ingen Schenau et al. (^{14} ) which, although it is restricted to short distances, clearly overestimates the role of anaerobic metabolism . The authors were aware that their method could not be applied to longer running distances, and Table 1 indicates that this is a result of the shortcomings in the modeling of the relative contributions of the anaerobic and aerobic mechanisms. The remaining data from mathematical modeling exhibit the same general trends as the biochemical data although, overall, the latter results exhibit a greater dependency on anaerobic metabolism , which is further emphasized if the data from Table 1 of their paper is substituted for the value from Table 2 used here.

Figure 1 brings together for comparison: (a) data derived from the mathematical models of running, Table 1 ; (b) the experimental data from Table 2 , and (c) the estimates produced by Astrand and Rodahl (^{1} ), also set out in Table 2 . Also shown in Figure 1 are curves derived from . The mathematical data indicate the spread of results from three analyses (^{4,11,15} ) of the running performance of male elite athletes, but with the data of van Ingen Schenau et al. (^{14} ), which exhibit different trends (see Table 1 ), excluded. The mathematical data indicate a good internal consistency, whereas detailed examination of the experimental data shows that they encompass a wide spread of values, with the disparity between the various results particularly evident at T = 30 s, where the measurements span values of (C _{aer} /C _{tot} ) from 20% to 40%. Also evident from Figure 1 for values of T < 100 s, is the fact that the experimental data display values of (C _{aer} /C _{tot} ) significantly higher than the figures derived from the mathematical analysis of running performance. This may indicate that the experimental techniques are underestimating the anaerobic contribution inferred from measurements of MAOD. A comparison of the data from running performance analysis with the estimates of Astrand and Rodahl (^{1} ) indicates that the data of Astrand and Rodahl overestimate the aerobic contribution for T < 10 s, and overestimate the anaerobic contribution for T > 100 s.

Figure 1: The ratio of energy from aerobic metabolism to total energy (%) as a function of time from the start of high-intensity exercise. Experimental data: ♦, Medbo and Tabata (9); ▴, Bangsbo et al. (3); ▾, McArdle et al. (8). Spread of data from mathematical analyses of the running performance of male elite athletes, (▪), based on Ward-Smith (15), Peronnet and Thibault (11) and di Prampero et al. (4). •, estimates of Astrand and Rodahl (1). The curves were obtained by solving . References:^{9} . Medbo, J. I. and I. Tabata. Relative importance of aerobic and anaerobic energy release during short-lasting exhausting bicycle exercise. J. Appl. Physiol. 67:1881–1886, 1989; 3. Bangsbo, J., P. D. Gollnick, T. E. Graham, et al. Anaerobic energy production and O_{2} deficit-debt relationship during exhaustive exercise in humans. J. Physiol. 422:539–559, 1990; 8. McArdle, W. D., F. I. Katch, and V. I. Katch. Essentials of Exercise Physiology . Philadelphia: Lea and Febiger, 1994, p. 122; 15. Ward-Smith, A. J. A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes. J. Biomech. 18:337–349, 1985; 11. Peronnet, F. and G. Thibault. Mathematical analysis of running performance and world running records. J. Appl. Physiol. 67:453–465, 1989; 4. Di Prampero, P. E., C. Capelli, P. Pagliaro, et al. Energetics of best performances of middle distance running. J. Appl. Physiol. 74:2318–2324, 1993; 1. Astrand P.-O. and K. Rodahl. Textbook of Work Physiology, 3rd Ed. New York: McGraw-Hill, 1986, p. 325.

For running times below 50 s, corresponding to distances below 400 m, anaerobic metabolism is the dominant contribution to the overall energy account, whereas above 100 s, corresponding to a distance of about 800 m, aerobic metabolism makes the greater contribution to energy requirements. Over the very limited range of running times between 50 s and 100 s, or distances between 400 m and 800 m, there is a dramatic change in dependency from anaerobic metabolism to aerobic metabolism .

The ratio of maximum anaerobic power to maximum sustainable aerobic power, denoted by Q and defined in , varies to some extent from one individual to another; furthermore it depends on an athlete’s specialization. Consequently, it is in general a function of T . This is easily demonstrated from the mathematical analysis of world records. Sprinting performance is particularly dependent on anaerobic metabolism , so sprinters have higher values of Q than middle and long-distance runners. With increase in running distance, athletes depend increasingly on aerobic metabolism , so Q decreases with increase in running distance and T . However, for running distances beyond 10,000 m, an athlete cannot rely solely on the aerobic conversion of glycogen, and free fatty acids become increasingly important. Under such conditions the value of maximum sustainable aerobic power declines, and so once again the value of Q tends to increase with increasing T . These trends are qualitatively consistent with the measurements of Scott et al. (^{12} ), who tested athletes on a treadmill for 2 to 3 min, and obtained average values for the ratio (C _{aer} /C _{tot} ) ranging from 61% for sprinters, through 63% for middle distance runners, to 70% for long-distance runners. A control group of individuals who did not participate in sport had an average value of 66%.

The solution to , for three different combinations of Q and λ, is plotted on Figure 1 for comparison with the various data points. A more detailed examination of the solution of for different values of Q and λ reveals that, besides the obvious and fundamental effect of T , the most important variables defining curves which are broadly consistent with the data points are λ and Q/ λ, but not Q itself.

Taking λ = λ_{2} , and combining equations 6 and 8 , we obtain MATH 11 showing that Q/ λ, which has the dimensions of time, is the ratio of anaerobic capacity to maximum sustainable aerobic power. Broadly, curves for Q/ λ ≈ 40 s provide an upper bound to the majority of the mathematical and experimental data, while the value Q/ λ ≈ 100 s defines the lower bound. Provided Q/ λ is in the identified range, values for λ between 0.03 s^{−1} and 0.05 s^{−1} give results within the defining envelope. The data derived from the main group of mathematical analyses fall within a tighter envelope defined by the approximate range 55 s <Q/ λ < 72 s. It can now be seen why the results of van Ingen Schenau et al. (^{14} ), which correspond to Q/ λ of about 130 s, do not conform to the remaining data from mathematical models. A good average fit for the ratio (C _{aer} /C _{tot} ) is given by evaluating , taking Q = 2.5 and λ = 0.04 s^{−1} .

Although di Prampero et al. (^{4} ) incorporate a value for λ of 0.1 s^{−1} , other results (^{11,14,15} ) for λ appear to be converging toward the more restricted range between 0.03 s^{−1} and 0.042 s^{−1} corresponding to an average value of about 0.036 s^{−1} . These data suggest that Q , the ratio of maximal anaerobic power to maximum sustainable aerobic power, is in the range 2.0 < Q < 2.6, a result which is consistent with the view (^{13} ) that the magnitude of Q lies between 2 and 4.

CONCLUSIONS
A study has been made of the anaerobic and aerobic contributions to energy conversion during intense exercise. It is shown that data derived from mathematical analyses of world and Olympic running records are a valuable resource that can be used to supplement data obtained experimentally. If the results of van Ingen Schenau et al. (^{14} ) are excluded, data from other mathematical models of running (^{4,11,15} ) are more consistent than the body of experimental data. There are three factors contributing to this difference. First, the mathematical models of running are internally highly consistent; second, the subjects of the mathematical analyses, world and Olympic record holders, are a more homogeneous group than the rather disparate range of experimental subjects; and third, there are inherent difficulties in the experimental techniques. It is shown that the ratio (C _{aer} /C _{tot} ) depends principally on T , λ, and the parameter Q/ λ, which is the ratio of anaerobic capacity to maximum sustainable aerobic power. Values in the range 55 s <Q/ λ < 72 s and 0.030 s^{−1} < λ < 0.0.042 s^{−1} provide the best correlation of data. The analysis indicates that the ratio of maximal anaerobic power to maximum sustainable aerobic power lies in the range of 2.0 to 2.6.

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