Abstract
This study examined the scaling relationships of net O_{2} uptake [V˙o
_{2(net)}= V˙o
_{2} − restingV˙o
_{2}] to body mass (M
_{B}) and combined mass (M
_{C}= M
_{B} + bicycle) during uphill treadmill bicycling. It was hypothesized thatV˙o
_{2(net)}(l/min) would scale proportionally withM
_{C} [i.e.,V˙o
_{2(net)}∝
 allometry
 regression
the net external power demand (W˙_{D}, W) of endurance sport performance can be modeled as the product of the net resistance (R_{net}, N) to forward motion and the average maximal rate, or ground speed (s˙_{max}, m/s), at which R_{net} is resisted (6)
During outdoor bicycling the external forces impeding forward motion include aerodynamic drag (R_{D}, N), gravitational resistance (R_{G}, N), and the rolling friction (R_{R}, N) between the tires and road surface (7). During bicycling at a level grade, R_{D} is the dominant resistive force (7). In contrast, during bicycling up steep hills or on an inclined treadmill, R_{G} is the dominant resistive force and R_{D}can be considered negligible (7). Thus, for steep uphill or inclined treadmill bicycling, it follows that R_{net} = R_{G} + R_{R} andW˙_{S} for a givens˙ is provided by (from Eq.3
)
Interestingly, research involving the energetic demands of uphill bicycling have mostly been limited to issues of pedal cadence and body position (26, 27). Thus the relationship between submaximalV˙o _{2} andM _{C} during uphill bicycling has never been addressed experimentally.
The related issue of V˙o
_{2}demand during uphill bicycling as a function of body mass (M
_{B}) was evaluated by Swain (25) using allometric scaling procedures. Swain concluded that the V˙o
_{2} cost of uphill bicycling was proportional toM
_{B} raised to the 0.79 power (i.e., V˙o
_{2} ∝
The above review outlines a theoretical framework for predicting the scaling relationship between submaximalV˙o
_{2} andM
_{C} andM
_{B} during uphill bicycling. The theoretical dependence ofV˙o
_{2} onM
_{C}(V˙o
_{2} ∝
METHODS
Subjects.
Volunteer competitive cyclists from the local area read and signed an informed consent document, as well as a cycling history questionnaire, before any testing in the Human Performance Laboratory at the University of Massachusetts (Amherst, MA). Subjects refrained from strenuous activity on the day before each visit and abstained from caffeine ingestion for ≥3 h before arriving at the laboratory.
Testing of peak V˙o_{2}.
On the first laboratory visit, each subject completed a continuous, incremental cycle ergometry test to exhaustion (model 829E cycle ergometer, Monark Bodyguard Fitness, Varberg, Sweden). Before each test, the ergometer was calibrated according to procedures outlined by the manufacturer. In addition, seat height and handlebar position were set according to each subject’s preference. RestingV˙o _{2} was measured first with subjects sitting quietly on the ergometer (no pedaling) over a 5min period. This was followed by a standardized warmup of 3 min at 80 W while pedaling 80 rpm, 3 min at 150 W and 80 rpm, and finally 3 min at 180 W and 90 rpm. The peakV˙o _{2}(V˙o _{2 peak}) test began immediately thereafter by increasing power output by 30 W at 1min intervals during pedaling 90 rpm until volitional exhaustion. Each subject’s V˙o _{2 peak}was defined as an average of the highest two or three values within 2.0 ml ⋅ kg^{−1} ⋅ min^{−1}of each other. TheV˙o _{2 peak} values were considered valid if at least two of the three following criteria were satisfied: 1) a leveling ofV˙o _{2}, despite an increase in power output, 2) a maximal heart rate >10 beats below each subject’s agepredicted maximal heart rate (220 − age in years), and 3) a respiratory exchange ratio ≥1.1.
Graded treadmill bicycling.
On the second laboratory visit, body height (m), as well as separate mass measures for the body, the bike, and the cyclists’ extra gear for riding (i.e., helmet and cycling cleats), was obtained. Mass was determined using a standard beam scale to the nearest 0.1 kg. Bicycles were stripped of extraneous equipment such as tire pumps, spare tubes, and water bottles before mass measurements.
On the basis of observations during pilot testing and reports by other researchers (26), a separate laboratory visit for practice riding on the treadmill was not necessary. Thus subjects practiced and warmed up before testing by riding their own bicycles on the laboratory treadmill (Trackmaster TM500E, JAS Fitness Systems). The treadmill’s surface measured 2.3 m long × 1.8 m wide, with speed and incline ranges of 1–11 m/s and 0–12.7°, respectively. The practice session also served to acquaint each subject with the specific treadmill speed and grades to be tested. Practice and testing on the treadmill were limited to the left side of the treadmill, where a handrail was installed down the entire length of the treadmill. The amount of practice time on the treadmill, which varied between 10 and 25 min, depended on how quickly each subject became comfortable with the task of treadmill bicycling. As an added safety measure, two mattresses were placed directly behind the treadmill to cushion the subject in the event of a fall.
Before treadmill practice and testing, all bicycle tires were inflated to the manufacturers’ suggested pressure (i.e., 69–83 N/cm^{2}). The four treadmill bicycling conditions corresponded to treadmill grades of 1.7% (1°), 3.5% (2°), 5.2% (3°), and 7.0% (4°), all at a treadmill speed of 3.46 m/s. Pilot testing indicated that these combinations of speed and grade would elicit a wide range of steadystate energetic demands in moderately trained cyclists. Subjects began their test session with a 2 to 3min warmup on the treadmill at a speed of 3.46 m/s and grade of 1.7%, which was followed immediately with an adjustment of the grade to match the first condition being tested. The four grades were tested successively, with 6 min of riding at each grade, the order of which was counterbalanced across subjects. Subjects received verbal feedback during all treadmill bicycling and were encouraged to maintain a steady position on the treadmill that was centered lengthwise but within reach of the handrail. Subjects were also required to maintain the same gripping position (i.e., hands on the brake hoods of handlebars) on their handlebars during all four conditions to minimize changes in body position relative to the bicycle.
Because the subjects’ bicycles were equipped with various gear combinations, it was not feasible for all subjects to use the same gearing without major equipment modifications to many of the bicycles. Alternatively, the subjects used the gearing available on their own bicycles to achieve similar gear ratios and thus similar pedal cadences. The gear ratios actually used were 1.75 (42/24 = T_{F}/T_{R}, where T_{F} is the number of teeth on the front chain ring and T_{R} is the number of teeth on the rear cog), 1.62 (42/26), 1.70 (39/23), and 1.63 (39/24).
Pedal cadence and treadmill speed were measured twice near the end of each condition; grade was measured at the beginning of each condition. Cadence was determined by timing 10 pedal revolutions; a digital hand tachometer (Biddle Instruments, Blue Bell, PA) was used to measure treadmill speed. Treadmill grade was measured within ±0.5° using an inclinometer on a flat surface adjacent to the treadmill belt.
Estimating μ_{D}.
The μ_{D} was determined for each subject at each grade for use as a covariate in the regression analyses. The R_{net} to treadmill bicycling was computed as the sum of R_{G} and R_{R} (7, 16)
Values for R_{net} were measured directly as the towing force required to maintain a stationary position on the treadmill. After the metabolic testing described above, the head tube of each subject’s bicycle was attached via a lightweight cable to a handheld digital dynamometer (model DFIS 100, range 0.5–500 N, Chatillon, Greensboro, NC) that was zeroed before each measurement. Subjects maintained a balanced position on the moving belt of the treadmill for 5–10 s while the researcher held and visually read the digital display on the dynamometer. The most stable dynamometer reading was recorded within 0.5 N.
Anthropometry.
Percent body fat and lower limb mass (M _{LL}) were also determined for use as potential covariates in the statistical analyses. Percent body fat was estimated from hydrostatic measures of body density (8) and the formula derived by Brozek et al. (4). Lower limb volume for each subject was also estimated using a geometric modeling technique validated by Sady et al. (23) and Freedson et al. (10). All lower limb anthropometric measures were taken on the right side of the body by the same investigator using standard anthropometers (lengths and breadths) and cloth tape measures (circumferences) according to the procedures outlined by Lohman et al. (18). TotalM _{LL} (kg) was computed as follows:M _{LL} = 2(ρ_{T}V_{T}+ ρ_{L}V_{L}+ ρ_{F}V_{F}), where the subscripts T, L, and F refer to estimated segment densities (ρ, g/cm^{3}) and segment volumes (V, liters) for the thigh, leg, and foot, respectively. Segment densities were estimated as 1.06, 1.08, and 1.10 g/cm^{3} for the thigh, leg, and foot, respectively (30).
V˙o_{2} instrumentation.
Standard indirect calorimetry procedures were used to determine submaximal V˙o _{2} andV˙o _{2 peak}. Expired gases were continuously sampled (250 Hz) from a 3liter mixing chamber and analyzed for O_{2} and CO_{2} concentrations via a computerbased system (286 Leading Edge computer using VO2Plus Software from Exeter Research, Brentwood, NH) interfaced with Ametek O_{2} (model S3AI) and CO_{2} (model CD3A) analyzers. The gas analyzers and Rayfield Equipment dry gas meter (for measuring inspired gas volumes) were interfaced to the computer via an analogtodigital board. The computer system compiled O_{2} information at 60 and 30s intervals for the submaximalV˙o _{2} and V˙o _{2 peak} protocols, respectively. The metabolic system analyzers were calibrated using standardized gases of verified O_{2}and CO_{2} concentrations before each test. Heart rate was monitored continuously during theV˙o _{2 peak} test with a Vantage heart rate monitor (Polar CIC).
Statistical analyses.
All submaximal V˙o _{2} values were converted toV˙o _{2(net)}values by subtracting subjects’ sitting restingV˙o _{2} from their respective submaximal V˙o _{2} values from the four conditions. ComputedV˙o _{2(net)}> 0 l/min were then assumed to represent the energetic needs of the bicycling task above those required for sitting at rest. The internal consistency of reliability of replicateV˙o _{2(net)}measures across minutes 3–5 for restingV˙o _{2(net)}and across minutes 4–6 for each test grade was assessed using a twofactor repeatedmeasures intraclass correlation (R_{xx} ) model, as described by Baumgartner (3). MeanV˙o _{2(net)}values were determined by averaging across the last 3 min of measurement. Measured values for treadmill speed, pedal cadence, and meanV˙o _{2(net)}were analyzed for differences across treadmill grades using singlefactor repeatedmeasures ANOVA procedures. The above significance tests were performed at the 0.05 alpha level.
Standard loglinear regression analysis techniques (19, 20) were used to determine the dependence ofV˙o
_{2(net)}on M
_{C} andM
_{B}. The loglinear model forV˙o
_{2(net)}takes the following form
RESULTS
The 25 subjects (23 men and 2 women) averaged 24.7 ± 5.7 (range 19–40) yr old, 1.80 ± 0.09 (range 1.57–1.96) m body height, 11.7 ± 4.5% (range 6–22%) body fat, 4.61 ± 0.79 (range 2.5–5.98) l/minV˙o _{2 peak}, and 6.2 ± 3.4 (range 0.5–13) yr of endurance activity experience and were riding 262 ± 126 (range 100–523) km/wk at the time of testing. Mass measurements averaged 73.9 ± 8.8 (range 56.48–97.39) kg for M _{B}, 10.1 ± 0.66 (range 8.86–11.00) kg for bike mass, 1.13 ± 0.19 (range 0.80–1.48) kg for all additional mass (helmet and cleats), and 85.0 ± 9.0 (range 66.93–108.86) kg for M _{C}. Measures of treadmill speed (P = 0.95) and pedal cadence (P = 0.88) did not differ across the four test grades. Pedal cadence averaged 59.9 ± 1.6 rpm, while individual pedal cadences ranged from 57 to 63 rpm (this was a result of the slightly different gear ratios available on each subject’s bicycle).
Data for three subjects on the steepest grade (7.0%) were dropped from all analyses, because the subjects could not maintain a steadystateV˙o _{2(net)}. With use of the remaining data (n = 97), all intraclass correlations forV˙o _{2(net)}during uphill bicycling were high (R_{xx} = 0.96–0.99) with no significant differences between mean minute values over the 3 min of measurement (P > 0.255). Therefore, meanV˙o _{2(net)}values were computed over the last 3 min of measurement for use in all ensuing analyses.
MeanV˙o _{2(net)}values for treadmill grades of 1.7% [1.10 ± 0.17 (SD) l/min], 3.5% (1.67 ± 0.22 l/min), 5.2% (2.26 ± 0.25 l/min), and 7.0% (2.88 ± 0.32 l/min) differed significantly from each other (P < 0.001). Slopes for the regression of log[V˙o _{2(net)}] on log(M _{B}) (Fig. 1) and log(M _{C}) (Fig.2) did not differ significantly across the four test grades (P > 0.344) (13). This indicated that the log[V˙o _{2(net)}] data for all four test grades could be pooled for the final regression analyses.
The resulting coefficients from the pooled regression ofV˙o _{2(net)}on M _{B} are provided in Table 1. The only consistently significant covariate across all regression analyses was μ_{D}, an increase of which was associated with a positive increase inV˙o _{2(net)}. The results in Table 1 suggest that, after controlling for differences in treadmill grade and μ_{D},V˙o _{2(net)}increased positively with an increase inM _{B} raised to the 0.89 power (95% confidence interval = 0.72–1.07;R ^{2} = 0.95,P < 0.001). The nominal scaled subject variables were not significant and thus were dropped from the final regression model (P > 0.08). The same analysis was performed for the regression ofV˙o _{2(net)}on M _{C} (Table2), which found thatV˙o _{2(net)}increased in proportion toM _{C} raised to the 0.99 power (95% confidence interval = 0.80–1.18;R ^{2} = 0.95,P < 0.001). Neither the exponent forM _{B} (0.89) nor that for M _{C}(0.99) differed statistically from 1.0. Finally, neither regression model’s residuals demonstrated a lack of normality (P > 0.20) (24).
DISCUSSION
The energetic demands of uphill bicycling have been modeled by a number of researchers (7, 16, 21), each utilizing some form ofEq. 1 . The exact scaling relationship betweenV˙o _{2(net)}demand and M _{B} orM _{C} for uphill bicycling, however, has never been verified or explained experimentally. Thus the purpose of this study was to evaluate these issues using logarithmically based multiple regression analysis and allometric scaling procedures as analytic tools.
It was hypothesized that, for a given grade and speed, the energetic cost of overcoming gravitational resistance would be directly proportional toM
_{C} (i.e.,V˙o
_{2} ∝
The present study also found thatV˙o _{2(net)}scaled with M _{B} to the 0.89 power. This value is higher, although not significantly, than the 0.79 M _{B}exponent reported by Swain (25) forV˙o _{2} during uphill treadmill bicycling at a 10% grade. These differences may be the result of different approaches to the statistical analysis. In the present study, for example, it was necessary to use computed values of μ_{D} as a covariate in the analysis, whereas Swain did not report the use of any covariates for deriving the 0.79 exponent. Values for μ_{D} in the present study averaged 2.06E03 ± 9.13E04 (SD), which is much higher than 3.4E05 reported for highpressure sewup racing tires on a smooth surface (16). These high μ_{D} values are attributed to the treadmill surface, which was specifically designed with a high rolling friction so that inline skating at steep grades was possible. When the regression model in Table 1 forV˙o _{2(net)}was recomputed without μ_{D} as a covariate, theM _{B} exponent decreased from 0.89 to 0.75, which is similar to Swain’s reported value of 0.79. Therefore, Swain’s 0.79M _{B} exponent may be due, in part, to a lack of statistical control over high μ_{D} values as a covariate.
Initially, there was some doubt concerning the physiological significance of the 0.89M
_{B} exponent, since it did not actually differ statistically from 1.0. This issue was addressed by using various energetic equations of locomotion from the literature (1, 7, 12) to verify the experimental derivation of the 0.89 exponent. For example, an equation for the metabolic cost of walking with varioussize loads carried on the back is given by (12)
Although the simulations described above support the present study findings, the simulations appear to contradict reports in the literature (22, 28). Rogers et al. (22), for example, determined that an M
_{B} exponent of 0.75 was more appropriate than 1.0 for comparing the submaximal energetic cost of treadmill running between prepubertal children, circumpubertal children, and adults. The authors noted that the 0.75 exponent was probably a function (in part) of the children having a greater stride frequency than the adults. Similar observations were reported by Taylor et al. (28) for an interspecies comparison of submaximal energetic data on 62 avian and mammalian species. Taylor et al., however, followed up their observations with a computation of the energy required per stride per unit mass at the relative speed where a quadruped changes gaits from a trot to a gallop. This analysis revealed that the quadrupeds, with a fourfold range inM
_{B} (0.01–100 kg), consumed a nearly constant 5 J ⋅ stride^{−1} ⋅ kg^{−1}when compared at a physiologically similar running speed (i.e., speed corresponding to gait transition). Thus, when compared by relative rates of limb movement, the submaximal energetic cost of running at any given speed was directly proportional toM
_{B} (e.g.,V˙o
_{2(net)}∝ M
_{B}, whereM
_{B} =
Swain (25) suggested that the relatively lighter bicycle mass, as a percentage of M
_{B}, for heavier cyclists should decrease theM
_{B} exponent below 1.0 for combined mass. To investigate this issue in the present study, the extra mass (M
_{EM}) of the bicycle, cleats, and helmet worn by each cyclist was calculated as follows: M
_{EM} =M
_{C} −M
_{B}. With use of the same statistical procedures described earlier andM
_{EM} as the dependent variable (no covariates),M
_{EM} in the present group of cyclists scaled to the 0.11 power ofM
_{B}(M_{EM} ∝
Interestingly, the estimatedM
_{LL} did not enter either regression model (Tables 1 and 2) as a significant covariate. Initially, this was unexpected, because segmental energy analyses (29) and physiological evaluations of pedaling efficiency (9, 11) have demonstrated how influential movement of the lower limb segments during pedaling can be on the total energy demand of a cycling task. A closer evaluation of theM
_{LL} data suggests two reasons for its exclusion from the regression models. First, theM
_{B} andM
_{C} regression models already had 95% of the total variance explained with the inclusion of M
_{B}, μ_{D}, and treadmill grade as dependent variables (Tables 1 and 2). Second, even ifM
_{LL} could have entered the models as a significant covariate, it would not have changed the M
_{B} orM
_{C} mass coefficients. By use of the same loglinear regression statistical procedures described earlier, it can be shown thatM
_{LL} for the group of cyclists studied scaled withM
_{B} raised to the 1.01 power (i.e.,M
_{LL} ∝
In summary, the results of this study support the premise by others (7,21) that the submaximal energetic demand of uphill bicycling increases proportionally withM
_{C} [i.e.,V˙o
_{2(net)}∝ W˙_{net}∝
Acknowledgments
The author acknowledges the assistance of Edward Debold and Bill Stekle, as well as the enthusiastic participation of the University of Massachusetts Cycling Team members, in the successful completion of this study.
Footnotes

Address for reprint requests: D. P. Heil, Dept. of Health and Human Development, 103 Romney, Montana State University, Bozeman, MT 597173540.
 Copyright © 1998 the American Physiological Society
Appendix
The results of this study can be used to predict the influence of mass on the submaximal energetics and performance of uphill timetrial cycling.
Submaximal energetics.
From Table 2 it is given thatV˙o
_{2(net)}∝ M
_{C} for a constant grade and velocity on steep uphill climbs (influence of R_{R} assumed constant, R_{D} assumed negligible). It follows, therefore, that a decrease inM
_{C} should cause a proportional decrease inV˙o
_{2(net)}for any given grade and velocity. By use of the coefficients from Table2 and insertion of the mean value for μ_{D} (0.00206), a generalized description of the contribution ofM
_{C} toV˙o
_{2(net)}at a 7% grade is given by
Interestingly, ifM _{C} is decreased by an absolute amount through a decrease in equipment mass or percent body fat (the source of mass is not important so long as the cyclist is in an energetic steady state), the smaller cyclist will actually gain an advantage (Table 3). By use ofEq. EA1 , the percent decrease inV˙o _{2(net)}expected with absolute decreases inM _{C} between 0.5 and 3.0 kg are provided forM _{C} values between 50 and 100 kg (Table 3). For example, a 50kg cyclist can decreaseV˙o _{2(net)}by 3.0% by decreasingM _{C} by 1.5 kg, but a 100kg cyclist must decreaseM _{C} by 3.0 kg to realize the same decrease inV˙o _{2(net)}. The percentages provided in Table 3 should apply so long as the cyclists are at the same speed and grade and at an energetic steady state (Eq. 7 ) and should not be dependent on gender.
Uphill timetrial cycling performance.
The influence of mass on uphill timetrial cycling performance can also be evaluated theoretically by determining the mass exponent for the ratio of metabolic power [W˙_{S(max)}] to R_{G}. RearrangingEq. 2
to solve fors˙_{max} and substituting R_{G} for R_{net} gives
Predicted and theoretical performance exponents for steep uphill cycling are consistent with anecdotal observations that lighter cyclists tend to win uphill time trials and stage races that end with a long steep climb. The −0.223 depends completely, however, on the present experimental finding that R_{G} ∝