A Pilot Study Using Entropy for Optimizing Self-Pacing during a Marathon

roup of marathon participants with minimal prior experience encounters the phenomenon known as “hitting the wall,” characterized by a notable decline in velocity accompanied by the heightened perception of fatigue (rate of perceived exertion, RPE). Previous research has suggested that successfully completing a marathon requires self-pacing according to RPE rather than attempting to maintain a constant speed or heart rate. However, it remains unclear how runners can self-pace their races based on the signals received from their physiological and mechanical running parameters. This study aims to investigate the relationship between the amount of information conveyed in a message or signal, RPE, and performance. It is hypothesized that a reduction in physiological or mechanical information (quanti ed by Shannon Entropy) affects performance. The entropy of heart rate, speed, and stride length was calculated for each kilometer of the race. The results showed that stride length had the highest entropy among the variables, and a reduction in its entropy to less than 50% of its maximum value (H = 3.3) was strongly associated with the distance (between 22 and 40) at which participants reported “hard exertion” (as indicated by an RPE of 15) and their performance

kilometer of the race. The results showed that stride length had the highest entropy among the variables, and a reduction in its entropy to less than 50% of its maximum value (H = 3.3) was strongly associated with the distance (between 22 and 40) at which participants reported “hard exertion” (as indicated by an RPE of 15) and their performance (p< 0.001). These ndings suggest that integrating stride length's Entropy feedback into new cardioGPS watches could improve marathon runners' performance. Keywords:marathon running; hitting the wall; Shannon entropy; stride length; performance 1. Introduction Paris 2024 is set to create history by offering the public a unique opportunity to participate in the Olympic Marathon, following the exact same route as the elite athletes. The Mass Participation Marathon, also known as “Marathon Pour Tous”, has no entry fee and provides 2024 slots for participants. This democratization of the marathon is leading to the emergence of new marathon runners, many of whom experience physical duress commonly referred to as “hitting the wall” (HTW) [1,2]. HTW is primarily in uenced by running strategy, speci cally by maintaining an excessively fast pace during the initial half marathon [3–5]. The most prevalent physiological indicators associated with HTW are overall fatigue and fatigue localized in the legs [1]. Research has demonstrated that muscular power output is regulated in an anticipatory manner to prevent uncontrolled disruptions in physiological homeostasis [6]. Pacing has a signi cant impact on energy production from both aerobic and anaerobic energy systems. The goal of the pacing strategy is to optimize these energy systems accordingly. Although the effects of various physiological regulators overlap, the conscious brain integrates their net input using the rating of perceived exertion (RPE) [7–10]. Changes in homeostatic status, re ected by momentary RPE, allow for alteration of pacing strategy (power output) in both Entropy2023,25, 1119.

Entropy2023,25, 1119 2 of 13 an anticipatory and responsive manner based on pre-exercise expectations and peripheral feedback from different physiological sensors [11,12]. Recent studies have examined the continuous physiological response and RPE during marathons, revealing a similar decrease in the ratio between RPE and speed, heart rate, and . VO2for all recreational runners [4]. Our current understanding of how runners adapt their marathon pace to account for various cardiorespiratory and biomechanical factors remains incomplete. One hypothe- sis proposes a strong connection between the rating of perceived exertion (RPE) and a physiological and mechanical message that is crucial for maintaining accurate adjustments in running speed. More speci cally, it is essential to achieve a balance between stride amplitude and stride frequency, in the same way that a cyclist needs to dose his equip- ment. Furthermore, the signal must possess a suf cient level of uncertainty to effectively convey information. The concept of information, as originally de ned by Shannon [13], represents a physi- cal quantity associated with systems capable of existing in multiple states. All life-related functions can be described in physical terms as information processing. The sensory organs of higher organisms perceive information from the environment and transmit it to the nervous system for further processing. The neural network then facilitates the exchange of information between the different parts of the organism and the central nervous system. Living organisms rely on a constant supply of information, which they continually process to maintain their organized structure. Without this ow of information, which is necessary for their various functions, the structured organization of living organisms would gradually disintegrate as their constituent materials undergo irreversible changes towards a state of randomness and disorganization. Consequently, living organisms can be viewed as reservoirs of physical information that is constantly maintained and transformed. Entropy plays a pivotal role in this context. It serves as a measure of the level of uncertainty or randomness within a given dataset. In information theory, entropy is measured in bits and calculated based on the probabilities of the various possible outcomes. If a message contains a set of possible outcomes with

information that is constantly maintained and transformed. Entropy plays a pivotal role in this context. It serves as a measure of the level of uncertainty or randomness within a given dataset. In information theory, entropy is measured in bits and calculated based on the probabilities of the various possible outcomes. If a message contains a set of possible outcomes with equal probabilities, the entropy of the message will be high due to a signi cant degree of uncertainty or randomness. Conversely, if the message contains only one possible outcome, the entropy will be low because there is no uncertainty or randomness. We propose a hypothesis that suggests marathon runners experience the phenomenon known as “hitting the wall” when their physiological and mechanical parameters fall below a threshold associated with their maximum entropy, as determined through a maximal test. This threshold could represent an entropy reserve that enables runners to remain aware and adjust their speed accordingly. The objective of this study is twofold: (1) to examine whether entropy values derived from heart rate (HR), stride length (SL), or speed (which are currently accessible through cardio-GPS watches) can serve as indicators of an increase in the rating of perceived exertion (RPE), and (2) to determine the level of entropy decrease in HR, SL, or speed (referred to as the “entropy threshold decrease”) at which marathon runners typically experience hitting the wall. 2. Materials and Methods 2.1. Subject Six non-elite male marathoners with the following characteristics (mean standard deviation (SD)): age: 37 8.8 years, weight: 73.4 6.1 kg, and height: 179.3 3.6 cm participated in this study (Table). For organizational reasons, we chose to have only one gender in the present study in order to avoid introducing an additional factor that could in uence the statistical analysis. All the subjects volunteered to participate in the study and were asked not to modify their usual training.

Entropy2023,25, 1119 3 of 13 Table 1. The age of the subjects, their personal record, and the year of this performance. * Subjects who beat their personal best during the Lake Annecy Marathon. Runners id Age (Years) Fastest Marathon Time (Years) Lake Annecy Marathon (2019) 1 34 02 h 55 0 03 00 (2018) 02 h 50 0 00 00 * 2 33 02 h 53 0 43 00 (2017) 02 h 51 0 36 00 * 3 23 03 h 10 0 12 00 (2019) 03 h 21 0 40 00 4 42 03 h 32 0 23 00 (2018) 03 h 31 0 27 00 * 5 44 03 h 35 0 59 00 (2017) 03 h 31 0 34 00 * 6 47 03 h 12 0 46 00 (2018) 03 h 32 0 58 00 They were selected for their homogeneous physiological and endurance characteris- tics [14–16], and all runners had already run at least two marathons. All subjects reported training three to four times per week (50–80 km/week) for more than 5 years. Once a week, they performed high-intensity interval training of 6 1000 m at 90–100% of their maximal heart rate and tempo training (15–25 km) at 100–90% of their average marathon speed. The study's objectives and procedures were approved by an institutional review board (CPP Sud-Est V, Grenoble, France; reference: 2018-A01496-49). All participants received information about the study and gave their written consent to participate. 2.2. Experimental Design: RABIT ® Test and the Marathon Race All participants performed the RABIT ® (Running Advisor Billat Training) test, to determine individual . V O2max and HRmax values. Three days later, all participants ran a marathon in an of cial race (Annecy Lake Marathon, France). The race started at 9 a.m. The temperature was between 6 and 8 C (between 9 a.m. and 1 p.m.), with no precipitation and an average humidity of 55%. Blood lactate was measured on the nger (Lactate PRO2 LT-1730; ArKray, Japan) just after the warm-up (15 min at an easy pace) and three minutes after crossing the nish line. The RABIT

race started at 9 a.m. The temperature was between 6 and 8 C (between 9 a.m. and 1 p.m.), with no precipitation and an average humidity of 55%. Blood lactate was measured on the nger (Lactate PRO2 LT-1730; ArKray, Japan) just after the warm-up (15 min at an easy pace) and three minutes after crossing the nish line. The RABIT ® test was performed outdoors using a portable gas exchange system to determine . V O2max and HRmax. The RABIT ® test has been validated as a valid eld test of maximal and functional aerobic capacity [17,18]. This RABIT ® test consisted of 3 incremental exercise stages, adjusted to a prescribed RPE (rating of perceived exertion) [7] equivalent to “light” (RPE 11) for 10 min, “somewhat hard” (RPE 14) for 5 min, and “very hard” (RPE 17) for 3 min. Exercise intensity is assessed subjectively by the person exercising. The corresponding written descriptions range from “very light” to “very, very hard”. The scale correlates well with cardiorespiratory and metabolic variables such as minute ventilation, heart rate, and blood lactate levels [7]. Each step was followed by a 1 min rest period. Participants were instructed to modify their running speed on a moment- to-moment basis in line with the prescribed RPE (rather than the endpoint of the task), so that their RPE (not their speed) remained constant for each stage. The test was conducted outdoors on a hard dirt path. The RPE scale could be consulted by the participant at regular intervals (i.e., every 100 m) because he was followed by the investigator on a bike. 2.3. Experimental Measurements RABIT ® Test Respiratory gases (oxygen uptake, . V O2) were continuously measured using a tele- metric, portable, breath-by-breath sampling system (K5, Cosmed, Rome, Italy). A GPS running watch (Garmin [19], Olathe, KS, USA) paired with the K5 [20] was used to measure heart rate (HR), stride length (SL), and the speed (V) response (using 5 s data averages) throughout each trial. We used the same cardiac belt for the Garmin and K5 because it was compatible for both.

Cosmed, Rome, Italy). A GPS running watch (Garmin [19], Olathe, KS, USA) paired with the K5 [20] was used to measure heart rate (HR), stride length (SL), and the speed (V) response (using 5 s data averages) throughout each trial. We used the same cardiac belt for the Garmin and K5 because it was compatible for both.

Entropy2023,25, 1119 4 of 13 Marathon The same watch used for the RABIT ® test was used for the marathon to measure V, SL, and HR. Given that performance in a marathon has recently been shown to be dependent on pacing oscillations [21], we encouraged runners to self-pace their run without focusing on the cardio-GPS whose dial was hidden. During the marathon, refreshment points (offering water, dry and fresh fruit, and sugar) were located every 5 km and at the nish line. Sponge stations were located every 5 km from km 7.5 Runners were allowed to remove their masks at the refreshment points to eat and drink. All runners were given a drink at each refreshment point (with at water and fruits). The RPE was recorded by runners using a small microphone attached to the jersey. Runners recorded an RPE at least every km or more frequently if they felt the need. We used the Borg 6–20 scale [7] to assess fatigue during the marathon in correlation with physiological stress indicators. Runners were familiarized with the scale in the two weeks prior to the race. 2.4. Data Analysis We analyzed the variables V, HR, RPE, and SL for every kilometer of the marathon. We consider that there is a steady state change when the entropy value differs from the mean by 2 standard deviation. We call this phenomenon the entropy falling. 2.5. Statistical Analysis Shannon's Entropy The Shannon entropy is an indicator of the average information embedded in a message. In fact, in this research, a message is considered a time series. A higher Shannon entropy indicates that the information contained in a message is more important [22]. The entropy H(x) of a discrete random variable X is a measure of its average uncertainty. Entropy is calculated by the equation [23]: H(X)= å x i p(x i)log 2 (x i) (1) where in this research,p(x i)indicates the probability of a state (V, HR, SL) with the value ofx i, whereican be change along with the signal. In our case, we have 9 states de ned in our RABIT ® test.

of its average uncertainty. Entropy is calculated by the equation [23]: H(X)= å x i p(x i)log 2 (x i) (1) where in this research,p(x i)indicates the probability of a state (V, HR, SL) with the value ofx i, whereican be change along with the signal. In our case, we have 9 states de ned in our RABIT ® test. We calculated the quartiles (Q) for the 3 running sensations, namely easy, medium, and hard. Each sensation is divided into 4 equal parts that each represent 25% of the area. In this study, we only used Q1, Q2, and Q3 which delimit 2 zones for each sensation. States 1 and 2 represent the two zones of easy pace (RPE 11), states 4 and 5 for medium pace (RPE 14), and states 7 and 8 for hard pace (RPE 17). States 0, 3, 6 and 9 represent the values between our three incremental exercise steps, as shown in the graph below (Figure). Figure 1. Distribution of a subject's speed according to the 9 possible states (N = 9). The green box represents the easy sensation, the orange box the medium sensation, and the red box the hard sensation.

Entropy2023,25, 1119 5 of 13 Entropy was calculated for each kilometer of the marathon. The maximum entropy (Hmax) is calculated according to the following equation: Hmax=log 2 (N) (2) whereNindicates the number of possible states in the signal (N= 9). Therefore, to illustrate how entropy is calculated for speed, the values are quanti ed in nine states according to the RPE values obtained in the RABIT ® test (Figure, Table). Table 2. The 9 states of each variable are calculated using the RABIT ® test, with a summary of the RPE's and associated HR: heart rate (bpm), speed (km.h 1 ), and stride length (m) (example for the runner n 3). States Speed HR Stride Length 1 x < 12.4 x < 145 x < 1.10 2 12.4 x < 12.9 145 x < 150 1.10 x < 1.15 3 12.9 x < 13.6 150 x < 154 1.15 x < 1.21 4 13.6 x < 14.2 154 x < 161 1.21 x < 1.36 5 14.2 x < 14.8 161 x < 164 1.36 x < 1.41 6 14.8 x < 16.5 164 x < 167 1.41 x < 1.52 7 16.5 x < 16.8 167 x < 170 1.52 x < 1.62 8 16.8 x < 17.3 170 x < 176 1.62 x < 1.71 9 x 17.3 x 176 x 1.71 Global Tendency of Pace and Its Asymmetry The coef cients of variation are calculated using all V, HR, and SL values, i.e., the values calculated every 5 s. The trend in speed time series (i.e., Kendall'stnon-parametric rank correlation coef cient) [24] and the pacing design (i.e., asymmetry characteristics of the race) [3] were compared. The equation of Kendall'st: K(vi,vj)=K(vj,vi)= 8 < : 1 ifi<j and vi<vj 0 ifvi=vj 1 ifi<j and vi>vj (3) t= 2 n(n 1) å i<j K(vi,vj) (4) vi= ith value of a speed;vj=jth value of a speed;i<j=iindicates a period of time prior toj; sum being performed over then(n 1)/2 distinct unordered couples of indices {i,j}, so that takes values in between 1 and 1. Furthermore, we used the Pearson median skewness to

0 ifvi=vj 1 ifi<j and vi>vj (3) t= 2 n(n 1) å i<j K(vi,vj) (4) vi= ith value of a speed;vj=jth value of a speed;i<j=iindicates a period of time prior toj; sum being performed over then(n 1)/2 distinct unordered couples of indices {i,j}, so that takes values in between 1 and 1. Furthermore, we used the Pearson median skewness to calculate the skewness value of the speed distribution as 3 (mean median)/SD. Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. A negative coef cient indicates a distribution shifted to the right of the median and thus a distribution tail spread to the left. Its value can be positive, negative, or unde ned. A positive skewness means that the mean is greater than the median, while a negative skewness means the mean is less than the median. In this case, it means that the marathon runner covered more kilometers above the nal average speed due to the decrease in his speed in the last part of the race [3]. The variability of the signal was determined by calculating the coef cient of variation using this formula [25]: Coef cient of variation= std mean 100 (5)

Entropy2023,25, 1119 6 of 13 Where std = the standard deviation of the signal over the marathon; mean = the average of the signal over the marathon. 3. Results All subjects completed the race, and four of them ran their personal best times despite the weight of the devices (Table). The nal blood lactate value was equal to 2.6 0.5 mM vs. 1.4 0.5 mM after the warm-up (p< 0.05). 3.1. Statistical Characteristics of the Variables throughout the Marathon (Speed, Heart Rate, and Stride Length) The average time to complete the race was 196 20.3 min. Table scriptive data of speed, heart rate, and stride length for the marathon. The average speed was 13.0 1.4 km.h 1 , the heart rate average was 155 8 bpm, and the stride length average was 1.3 0.17 m. The coef cient of variation speed in this study was averaged at 6.0 1.4%. Table 3. Performance during the marathon race. Speed: speed for the marathon (km.h 1 ), HR: heart rate (bpm), SL: stride length (m). Runners id Marathon Time Speed HR SL 1 2 h 50 min 00 s Mean 14.9 146 1.5 SD 0.6 9 0.2 Coef cient of variation 4.3% 6.4% 13% 2 2 h 51 min 36 s Mean 14.8 150 1.5 SD 1.8 6 0.1 Coef cient of variation 5.1% 3.7% 12% 3 3 h 21 min 40 s Mean 12.5 159 1.1 SD 0.8 5 0.1 Coef cient of variation 6.9% 3.5% 7.7% 4 3 h 31 min 27 s Mean 12.1 149 1.4 SD 1.3 7 0.2 Coef cient of variation 6.7% 4.6% 16% 5 3 h 31 min 34 s Mean 12.0 167 1.2 SD 0.7 6 0.1 Coef cient of variation 5.8% 2.8% 11.3% 6 3 h 32 min 58 s Mean 11.9 159 1.2 SD 0.9 5 0.1 Coef cient of variation 7.5% 3.3% 7.7% All subjects ran a large positive gap race, as indicated by a negative Kendall's for speed and stride length (SL) in contrast to heart rate (HR), which had a positive Kendall's (Table). Speed and SL decreased signi cantly throughout

The study explores how entropy affects marathon runners' pacing and performance.