Modeling Optimal Cadence as a Function of Time during Maximal Sprint Exercises Can Improve Performance by Elite Track Cyclists

k cycling sprint events, optimal cadence PRoptis a dynamic aspect of fatigue. It is currently unclear what cadence is optimal for an athlete's performance in sprint races and how it can be calculated. We examined fatigue-induced changes in optimal cadence during a maximal sprint using a mathematical approach. Nine elite track cyclists completed a 6-s high-frequency pedaling test and a 60-s isokinetic all-out sprint on a bicycle ergometer with continuous monitoring of crank force and cadence. Fatigue-free force-velocity (F/v) and power-velocity (P/v) pro les were derived from both tests. The development of fatigue during the 60-s sprint was assessed by xing the slope of the fatigue-free F/v pro le. Fatigue-induced alterations in PRoptwere determined by non-linear regression analysis using a mono-exponential equation at constant slope. The study revealed that PRoptat any instant during a 60-s maximal sprint can be estimated accurately using a mono-exponential equation. In an isokinetic mode, a mean PRoptcan be identi ed that enables the athlete to generate the highest mean power output over the course of the effort. Adding the time domain to the fatigue-free F/v and P/v

at constant slope. The study revealed that PRoptat any instant during a 60-s maximal sprint can be estimated accurately using a mono-exponential equation. In an isokinetic mode, a mean PRoptcan be identi ed that enables the athlete to generate the highest mean power output over the course of the effort. Adding the time domain to the fatigue-free F/v and P/v pro les allows time-dependent cycling power to be modelled independent of cadence. Keywords: force-velocity pro le; fatigue modelling; optimal pedaling rate; maximum power output; track cycling 1. Introduction In cycling, models are used to investigate the factors that determine performance and optimize competition outcomes. The sprint events in track cycling (i.e., team sprint, sprint, keirin, and 1000- or 500-m time trials) are approximately 15–60 s in duration and require maximal production of power over distances of 200–1000 m [1]. In races that are executed with all-out effort, such as sprint time trials or decisive phases of other sprint events in track cycling [1–4], the mechanical power applied by the athlete at any given time equals the maximum power achievable at the current cadence and state of exhaustion. A decisive physiological determinant of performance in track cycling sprints is the ability to produce fatigue-free muscular power, which can be described by maximal force- velocity (F/v) and power-velocity (P/v) pro les (e.g., [1,4–7]). Although Hill [8] modelled the velocity with which muscle elements contract as a rectangular hyperbola, a strong linear relationship between the force generated and velocity of movement was established in the case of movements of large muscle groups engaging more than one joint [9]. The F/v relationship in cycling of mean pedal force and pedaling rate (PR) was also identi ed as linear and can be derived from short maximal sprints in the laboratory and in the eld [5,10–13]. Multiplying the mean pedal force by the cadence results in a parabolic relationship between power output (P) and cadence (ibid.). Appl. Sci.2021,11, 12105.

maximal sprints in the laboratory and in the eld [5,10–13]. Multiplying the mean pedal force by the cadence results in a parabolic relationship between power output (P) and cadence (ibid.). Appl. Sci.2021,11, 12105.

Appl. Sci.2021,11, 12105 2 of 13 In cycling, the fatigue-free F/v and P/v pro les represent the maximal resistance an athlete can overcome at a certain cadence and the maximal power output at that cadence and allow important parameters of neuromuscular performance to be determined, including the theoretical maximal mean crank force (Fmax,as the F/v intersection of the y-axis) and cadence (PRmax, as the F/v intersection of the x-axis), maximum power output (Pmax, as the P/v apex), and optimal cadence (PRopt, as the pedaling rate corresponding to Pmax) [5]. With increasing duration of maximal exercise and with the onset of fatigue-induced impairment of performance, metabolic performance, interpreted as resistance to fatigue, becomes more and more important [14]. In contrast to the generalized representation of fatigue-free performance by F/v and P/v pro les, there is no mathematical model that describes the change in performance with the onset of fatigue in a generalized form to date, so that time-dependent maximal performance cannot be analyzed independent of the cur- rent cadence. To control for the effects of cadence on power decay during maximal sprints, an isokinetic test design is often used to diagnose anaerobic performance [15]. This limits current performance diagnostics to the analysis of an effort under xed conditions and does not allow the general cadence-independent description of an athlete's performance capacity. The development of such a model was recently re-identi ed as a speci c research gap in the eld of track cycling [1]. When performing at maximal level, elite track sprinters leave their fatigue-free F/v pro le after less than 3 s in maximal sprints [16] and power output declines in an approxi- mately exponential manner as a result of increasing fatigue [4]. In this process, accumula- tion of metabolites and the associated reduction in cellular pH may lead successively to an almost synchronous and uniform decline in force development (due to attenuated energy ow) and the velocity of contraction (re ecting reduced membrane excitability). Physio- logically, this systematic reduction re ects the mechanical and metabolic properties of the different ber types in the main propulsive muscles and their time-dependent

and the associated reduction in cellular pH may lead successively to an almost synchronous and uniform decline in force development (due to attenuated energy ow) and the velocity of contraction (re ecting reduced membrane excitability). Physio- logically, this systematic reduction re ects the mechanical and metabolic properties of the different ber types in the main propulsive muscles and their time-dependent contribution to power output (ibid.). Mathematically, these fatigue-induced changes can be described by a parallel shift in the F/v pro le towards the origin [17,18], whose time course should correspond to the exponential fatigue behavior described. By adding a time domain and incorporating the duration of exercise, two-dimensional F/v pro les should form a three-dimensional model. We proposed that it will be possible to describe the effect of fatigue and motion velocity on power output for any individual athlete, using a function of time in a mono- exponential equation. This model would provide a deeper insight into the physiology of fatigue-induced decline in performance and offer a mathematical approach to optimizing an athlete's competitive performance. With the parallel shift of the initial fatigue-free F/v pro le, its characteristic parameters also decrease [7]. The optimal cadence is, therefore, not a static parameter, but a dynamic aspect of fatigue. Since it has not been possible to determine the changes in optimal cadence over the course of a race, it is currently unclear what cadence is optimal for an athlete's performance in sprint races and how it can be calculated. As a special application of our approach with high practical relevance, we aimed to calculate the time-dependent optimal cadence and determine power output reserves in order to reveal unused potential for performance optimization. 2. Methods Nine male elite track cycling sprinters (22.1 4.2 yrs, 184.4 4.6 cm, 89.2 6.0 kg (means standard deviations)) performed a high-cadence, low-resistance pedaling test (motoric test), as well as a 60-s all-out test (sprint test) in a seated position on a bicycle ergometer. As very high neuromuscular and metabolic performance was required for the study, only athletes whose F/v pro les showed suf ciently high linearity (R 2

yrs, 184.4 4.6 cm, 89.2 6.0 kg (means standard deviations)) performed a high-cadence, low-resistance pedaling test (motoric test), as well as a 60-s all-out test (sprint test) in a seated position on a bicycle ergometer. As very high neuromuscular and metabolic performance was required for the study, only athletes whose F/v pro les showed suf ciently high linearity (R 2 > 0.95) in previous tests and who had experience with sprint time trials at national or international championships were tested.

Appl. Sci.2021,11, 12105 3 of 13 All subjects used their own cycling shoes and pedals during these sprints. The settings of the ergometer were chosen to resemble the demands faced during actual competition. The subjects were requested to refrain from consuming alcohol and from intense training during the 24-h period prior to the experimental session and asked to maintain their normal drinking and eating habits. All provided their written, informed consent to participate in this study, which was approved by the institute's ethical committee and performed in accordance with the Declaration of Helsinki. 3. Exercise Protocol The warm-up prior to both tests consisted of 6 min of low-intensity cycling (1–1.5 W kg 1 bodyweight), followed by a 3-s maximal sprint. Each athlete rested passively for 10 min between warm-up and testing. Participants rst performed 6 s of maximal high-cadence, low-resistance cycling (motoric test) on a specially prepared ergometer, from which the internal ywheel had been removed to minimize resistance. Reducing the overall momentum of the ergometer enabled the athlete to reach a cadence 160 rpm within the rst 3 s to generate additional data points in the high-frequency cadence range for a valid determination of the F/v pro les by combining this data with the fatigue-free pedal revolutions from the acceleration phase of the 60-s sprint [19]. In order to determine the fatigue-induced drop in performance, after a further10 min of rest, each athlete completed a 60-s sprint test in an isokinetic mode at a xed cadence of 120 rpm, on another SRM cycle ergometer (Schoberer Radmesstechnik GmbH, Jülich, Germany) with a 9-kg ywheel, in seventh gear. As the test corresponded to the require- ments of sprint time trials and the athletes were familiar with the settings from previous performance tests in the laboratory, no further familiarization was required. The participants accelerated as rapidly as possible from a simulated rolling start (approx. 20 rpm) to 120 rpm at a self-selected instant and continued sprinting at this rate until the end of each effort, while receiving emphatic verbal encouragement to maintain maximal power output throughout the test. In both tests, crank

tests in the laboratory, no further familiarization was required. The participants accelerated as rapidly as possible from a simulated rolling start (approx. 20 rpm) to 120 rpm at a self-selected instant and continued sprinting at this rate until the end of each effort, while receiving emphatic verbal encouragement to maintain maximal power output throughout the test. In both tests, crank force and cadence were monitored continuously with a SRM power meter (Schoberer Radmesstechnik GmbH, Jülich, Germany). 4. Data Processing Raw data from the ergometer were obtained at a sample rate of 10 Hz. The mean tangential force F (N) at both pedals, averaged over one revolution, as well as the corre- sponding mean pedal rate PR (rpm) were derived from these values. From the rst 3 s of the acceleration phase of the 60-s sprint, three or four cycles with linear decay in pedal force and one or two cycles at pedal rates above 160 rpm from the motoric test were evaluated to establish the fatigue-free F/v and P/v pro les. Since PR is directly proportional to the tangential speed of motion v at the pedal, they were based on the mean cadence PR and corresponding mean crank force F. The force–velocity continuum was subjected to linear (F/v pro le) and non-linear (P/v pro le) regression analysis. The function F(v)=a PR+b (1) approximates the relationship between mean pedal force F and the movement velocity PR in the absence of fatigue. P(v) was calculated by multiplying F(v) by PR: P(v)=a PR 2 +b PR (2) For purposes of statistical analysis, the following indices of performance were calcu- lated: theoretical maximal force Fmax=F(0)=b,

Appl. Sci.2021,11, 12105 4 of 13 theoretical maximal velocity of movement PRmax= b a 1 , optimal cadence PRopt= b (2a) 1, and maximum power output Pmax= b 2 (4a) 1 . To analyze the fatigue-induced performance loss during the 60s-sprint data, the slope derived from the fatigue-free force-velocity pro le (1) was kept xed, in accordance with Buttelli and colleagues [17]. When a time domain was added and the exercise duration was included, the two-dimensional F/v pro les yielded a three-dimensional model F:R 2 !R in maximal sprints no longer than ~ 60 s in duration: F(v,t)=F(v) e t tF+c (3) where F(v) is the fatigue-free F/v pro le and F denotes the time constant of its decline due to fatigue. Since the results of Burnley and Jones [20], among others, described a steady state of power output after an exponential drop, a limiting value c was included in the function to represent this steady state. As the F/v pro le shifted, the characteristic parameters of the fatigue-free pro les also decreased, allowing the three-dimensional model to be reduced to a two-dimensional model by eliminating the dependence on cadence. The fatigue-induced changes in the F/v can, therefore, be characterized by the alteration of a single parameter such as the optimal cadence. The calculation of the instantaneous F/v pro le and theoretical optimal cadence for each data point i was based on the associated cadencePRiand corresponding mean crank force F(PRi): b i=F(PR i) a PR i)PR opt,i= b i (2a) 1 (4) The decay in PRoptduring the sprint was approximated by an exponential function of time t: PRopt(t)=A e t+TD t+c (5) Its parameters were determined by non-linear regression withA=PRopt(0)-c ,cas the limiting value, and as the time constant. A time delay (TD) was used to compensate for any delays in the onset of fatigue. Theoretical mean power output Pmean(PRopt) was calculated as the mean power output that could be generated at the calculated mean optimal cadence. To investigate the meaning of the mean optimal cadence mathematically, the cadence that maximizes power output was determined as follows by non-linear optimization:

time delay (TD) was used to compensate for any delays in the onset of fatigue. Theoretical mean power output Pmean(PRopt) was calculated as the mean power output that could be generated at the calculated mean optimal cadence. To investigate the meaning of the mean optimal cadence mathematically, the cadence that maximizes power output was determined as follows by non-linear optimization: max v2R + P(v)=a PR 2 +b PR (6) 5. Statistical Analyses All data were checked for normality using the Shapiro–Wilk test and are presented as means SD. Linear regression analysis was used to study the relationship between measured and modelled data. The relative difference between the measured and mod- elled data was used to determine the bias of the measurements. The mean differences between parameters were compared usingt-tests for dependent samples. The Pearson product–moment correlation test was used to evaluate interrelationships between vari- ables. Multiple regression analyses were conducted to identify the factors which exerted a major in uence on a dependent variable. Pearson correlation coef cient r (small 0.1; medium 0.3; large 0.5) and Cohen's d (small = 0.2; medium = 0.5; large = 0.8) were employed as a measure of effect size, with statistical signi cance being set at an alpha level of <0.05. The quality of the regression analyses was examined by calculating the coef cient of determination R 2 . For all tests, a post hoc power analysis was performed to determine the retrospective power of the observed effect based on the sample size at the indicated level of signi cance. All mathematical analyses and statistical tests were processed using

Appl. Sci.2021,11, 12105 5 of 13 IBM SPSS statistics version 24 Software for Windows (SPSS Inc., Chicago, IL, USA) and Of ce Excel 2016 (Microsoft Corporation, Redmond, WA, USA). 6. Results Anthropometric data and the parameters of the fatigue-free F/v pro le with corre- sponding model quality R 2 are presented in Table Table 1. Anthropometric data and model parameters of the linear fatigue-free F/v pro le with corresponding model quality R 2 of all participants. Part. Age (yrs) Height (cm) Weight (kg) PR opt(rpm) F max(N) a R 2 1 20 192 96.2 142.74 1459.65 5.11 1.00 2 29 178 81.6 144.35 1293.20 4.48 1.00 3 20 190 91.4 171.68 1076.19 3.13 0.99 4 25 177.5 83.1 138.71 1222.92 4.41 1.00 5 21 184 87.8 141.76 1426.53 5.03 0.99 6 18 181 80.1 160.39 1262.81 3.94 1.00 7 19 186.5 92 152.63 1202.52 3.94 1.00 8 18 183.5 97.9 163.07 1144.68 3.51 1.00 9 29 186 92.4 148.99 1313.48 4.41 0.98 The mean maximal force was 1266.89 116.86 N, mean maximal crank velocity was 303.18 21.23 rpm, and optimal pedaling rate was151.59 10.61rpm, as determined from the fatigue-free F/v pro le, which had a slope ofa= 4.22 0.59. The coef cient of determination R 2 for this pro le was >0.98 for all athletes. The results of the procedure for generating the fatigue-free F/v and P/v pro les are depicted in FigureAppl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 13 level of <0.05. The quality of the regression analyses was examined by calculating the co- efficient of determination R 2 . For all tests, a post hoc power analysis was performed to determine the retrospective power of the observed effect based on the sample size at the indicated level of significance. All mathematical analyses and statistical tests were pro- cessed using IBM SPSS statistics version 24 Software for Windows (SPSS Inc., Chicago, IL, USA) and Office Excel 2016 (Microsoft Corporation, Redmond, WA, USA). 6. Results Anthropometric data and the parameters of the fatigue-free F/v profile with corre- sponding model quality R 2 are presented in Table 1

indicated level of significance. All mathematical analyses and statistical tests were pro- cessed using IBM SPSS statistics version 24 Software for Windows (SPSS Inc., Chicago, IL, USA) and Office Excel 2016 (Microsoft Corporation, Redmond, WA, USA). 6. Results Anthropometric data and the parameters of the fatigue-free F/v profile with corre- sponding model quality R 2 are presented in Table 1 for all athletes. Table 1. Anthropometric data and model parameters of the linear fatigue-free F/v profile with corresponding model qual- ity R 2 of all participants. Part. Age (yrs) Height (cm) Weight (kg) PR opt (rpm) F max (N) a R 2 1 20 192 96.2 142.74 1459.65 −5.11 1.00 2 29 178 81.6 144.35 1293.20 −4.48 1.00 3 20 190 91.4 171.68 1076.19 −3.13 0.99 4 25 177.5 83.1 138.71 1222.92 −4.41 1.00 5 21 184 87.8 141.76 1426.53 −5.03 0.99 6 18 181 80.1 160.39 1262.81 −3.94 1.00 7 19 186.5 92 152.63 1202.52 −3.94 1.00 8 18 183.5 97.9 163.07 1144.68 −3.51 1.00 9 29 186 92.4 148.99 1313.48 −4.41 0.98 The mean maximal force was 1266.89 ± 116.86 N, mean maximal crank velocity was 303.18 ± 21.23 rpm, and optimal pedaling rate was 151.59 ± 10.61 rpm, as determined from the fatigue-free F/v profile, which had a slope of a=–4.22 ± 0.59. The coefficient of deter- mination R 2 for this profile was >0.98 for all athletes. The results of the procedure for gen- erating the fatigue-free F/v and P/v profiles are depicted in Figure 1 for one athlete. Figure 1. The fatigue-free force-velocity and power-velocity profiles for a professional track cycling sprinter calculated on the basis of the force and velocity data derived from the first 3 s of acceleration during the motoric and the sprint test. The solid line represents the F/v and the dotted line represents the P/v function. The maximal mean pedal force was 1223 N Figure 1. The fatigue-free force-velocity and power-velocity pro les for a professional track cycling sprinter calculated on the basis of the force and velocity data derived from the rst 3 s of acceleration during the motoric

The study investigates how optimal cadence affects performance in elite track cycling sprints.