Marathon Performance Depends on Pacing Oscillations between Non-Symmetric Extreme Values

on was recently run in less than 2 h by a man who ran the three fastest marathons ever recorded in a span of three years—Eliud Kipchoge—in the Tokyo Olympic games. Here, we demonstrate that the best marathons were run according to a pace distribution that is statistically not constant and with negative asymmetry. The concept of mirror race enables us to show that the sign of asymmetry is not due to sampling uctuations. We show that marathon performance depends on pacing oscillations between extreme values, and that even the best marathons ever run differ and can be improved upon. The utilization of extreme values and oscillations allows for recovery and optimization of the complementary aerobic and anaerobic metabolisms. Our ndings suggest new ways to approach the pacing for optimizing endurance performance. Keywords:extreme values; symmetry breaking; pacing strategy; optimization 1. Introduction For centuries, the limits of physiology and athletic records have fascinated scientists. In 2019, the sub-two-hour marathon barrier was broken with a time of 1:59:40.2. In 1925, Nobel laureate Archibald Vivian (AV) Hill published “The Physiological Basis of Athletic Records” [1]. Nowadays, one century later, advances in wearable sensor technology have enabled real-time measurement of

pacing strategy; optimization 1. Introduction For centuries, the limits of physiology and athletic records have fascinated scientists. In 2019, the sub-two-hour marathon barrier was broken with a time of 1:59:40.2. In 1925, Nobel laureate Archibald Vivian (AV) Hill published “The Physiological Basis of Athletic Records” [1]. Nowadays, one century later, advances in wearable sensor technology have enabled real-time measurement of physiological data during exercise [2]. Future directions in train- ing are going to be about encouraging marathon runners. To achieve this, we must provide ways to improve satisfaction regarding training progress and with the feeling that it has been optimized. The results show that according to our hypothesis, we are testing the hy- pothesis that the ideal race, with the world's best marathon runners, can have several degrees of optimization. This can be observed in Kipchoge's last ve marathons, since Kipchoge ran the three fastest marathons at the time across a span of three years, and even winning Olympic gold medals in the process (2016, 2021). Hill's and Kennelly's approach to running the fastest marathon was based on the con- stant pace paradigm. In Hill's reference to Kennelly's 1906 paper, he stated that the ideal way to run a race was not necessarily to win, but to achieve a new athletic record for the distance, and to do it by running it at a constant speed. Fifty years later, the physicist Joseph Keller modelled the predominating view that a runner should maintain a constant pace to achieve the shortest time in his paper: “A Theory of Competitive Running“ [3]. To determine the optimal race strategy, he used simple physics and mathematics to correlate the phys- iological attributes of runners with world track records. He derived the optimal speed variationt7!v(t) by formulating and solving a problem in the optimal control theory. For distances greater than 291 m, his theory predicted a maximum acceleration for 1 or 2 s, then constant speed throughout the race, until the nal 1 or 2 s, and nally, a slight slowing Int. J. Environ. Res. Public Health2022,19, 2463.

formulating and solving a problem in the optimal control theory. For distances greater than 291 m, his theory predicted a maximum acceleration for 1 or 2 s, then constant speed throughout the race, until the nal 1 or 2 s, and nally, a slight slowing Int. J. Environ. Res. Public Health2022,19, 2463.

Int. J. Environ. Res. Public Health2022,19, 2463 2 of 19 down. At the time, his results con rmed the recognized view that a runner should maintain a constant speed to achieve the shortest time (Keller, 1973). Importantly, Keller admits his theory omitted several important variables, such as the up-and-down motion of the limbs, internal and external resistance, the depletion of fuel, and the accumulation and removal of waste products. These ideas were perhaps valid in 1925, when the fastest marathon time was only 2 h 29 min (by Albert Michelson, an average speed of 16.99 km/h, Port Chester, NY, USA). In the same way, Arthur Kennelly desired to measure all points of a race. Delv- ing deeper into Kennelly's paper: “It is to be noted that all these speeds are average speeds over the courses. There is no evidence among the records to show what the speed was at different points in the course. So far, as concerns anything appearing in the data, the speed of a runner, for example, which averages 7.17 ms per s over a 1 km course, might be 10 m per s in the rst part and 5 in the last part, or vice versa. Evidence is lacking to show what the facts are, and they are of great importance to the science of athletics. The speed of a world's record type of trained runner might be determined at any or all points of a course, either by securing a light recording chronograph on the back of his belt, with a thread paid out as he ran, or by pacing the runner with a light motor-car carrying an automatic speed“ [4], p. 328. Nowadays, thanks to microchip technology, it is possible to measure every point during a marathon. Physiologic processes are inherently never constant. French physiologist Claude Bernard wrote, in his classic book [5], that: “The use of averages in physiology and medicine most often gives only a false precision to the results by destroying the biological character of the phenomena“. In the same way, the observed paces in a marathon are far from constant

during a marathon. Physiologic processes are inherently never constant. French physiologist Claude Bernard wrote, in his classic book [5], that: “The use of averages in physiology and medicine most often gives only a false precision to the results by destroying the biological character of the phenomena“. In the same way, the observed paces in a marathon are far from constant at an average speed. The best marathons are run with an asymmetric distribution of paces [6]. Nowadays, it is possible to test the hypothesis that the best marathon performance is run with a variable pace and to challenge the paradigm of a constant pace. Statistically, a race is run with an oscillating pace between extreme values that are inter-played by optimizing fuel, recovery, and avoiding VO2 and heart rate drift. Using mathematical statistics, we aim to show that an asymmetric distribution is optimal for marathon running, to delve deeper into the pace distribution, and to nd the exact outline of the optimal pacing signature by analysing the of cial best world marathon performances by Dennis Kimoto (Berlin 2014) and Eliud Kipchoge (Monza 2017, Berlin 2018, 2019, Vienna 2019, and the Tokyo Olympics Games, 2021). Furthermore, previous marathon studies reported, also run in actual conditions, that physiological oscillations were the subjacent of real pace variations, even when the marathon pace is planned to be constant [6]. The Berlin marathons were of cial competitions and are of cial world records. While Monza 2017 and Vienna 2019 were exhibition marathons accomplished with ro- tating pacers, an electric pace vehicle, a laser beam projecting the ideal position on the road and ideal conditions. At Vienna, Kipchoge ran at a consistent average pace of 2:50 min per km (4:33.5 min per mile) and was 11 s ahead of schedule halfway through the marathon. He even accelerated in the nal kilometre. Given that the best marathon performances, includ- ing the Olympics, were run by a single man (Eliud Kipchoge), this shows the possibility of adding a new dimension of running performance beyond the chronometer and propos- ing a qualitative way of optimizing the physiological capacity for

s ahead of schedule halfway through the marathon. He even accelerated in the nal kilometre. Given that the best marathon performances, includ- ing the Olympics, were run by a single man (Eliud Kipchoge), this shows the possibility of adding a new dimension of running performance beyond the chronometer and propos- ing a qualitative way of optimizing the physiological capacity for running a marathon at21 km/h. Our approach represents a powerful possibility for the marathon runner to validate the optimal aspects of his/her performance. However, before exploring the pos- sibilities of providing feedback to a marathon runner, we must test the consistency of such a model in Eliud Kipchoge, who is purported to have a robust “performance template“. They have shown this to be primarily related to the increased con dence that the distance in question can be completed without unreasonable levels of exertion or injury [7]. This can apply to real-world conditions with the aid of mathematical modelling and wearable technology, as also already proposed in the journal Nature by the physicist Emig Thorsten, see the Ref. [2].

Int. J. Environ. Res. Public Health2022,19, 2463 3 of 19 2. Materials and Methods Our data consists of six marathon races: Berlin 2014, Monza 2017, Berlin 2018, Berlin 2019, Vienna 2019, and Tokyo 2021. They were performed by Dennis Kimetto, D.K. (Berlin 2014), Eliud Kipchoge, E.K. (Monza 2017, Berlin 2018, Vienna 2019 and Tokyo 2021), and Kenenise Bekele, K.B. (Berlin 2019). For a given marathon, our data were collected from September 2021). The original dataset includes marathons that provide timing data for 1km race seg- ments (plus the nal 41–42,195 km segment); the requirement for 1 km segments is based on the need to track changes in pacing during different stages of the marathon. The ac- curacy of the data is, then, higher than those collected by strava from the individual GPS measurement which depend on GPS models (between 0.47 and 1.65%, see the Ref. [8]). Thus, we will assume our data consist of a sequence, denoted by(p) = (p i) 1 i 42 , where p iis the pace, that is, the duration measured in s, taken to cover theith km, all values being integers. Following standard notations from the eld of probability and statistics, if this se- quence is arranged in order of magnitude and rewritten as p (1) p (42) , (1) thenp (i), for1 i 42, it is called theith order statistic (see §11.4 in the Ref. [9], or §1.1 in the Ref. [10]). We callaverage pacethe real number p computed from the sequence by the two equivalent relations p= 1 42 42 å i=1 p i, 42 å i=1 (p i p) =0. (2) It is important to note that this value may differ slightly from what can be called theof cial average paceobtained by dividing the of cial total time by the distance 42.195 km. In our data, this only occurs for Vienna 2019. The resulting difference may, as usual, come from the phenomenon of rounding error, with the result that a total of rounded numbers is not equal to the rounded version of the original total. Furthermore, in our case, a rather precise

dividing the of cial total time by the distance 42.195 km. In our data, this only occurs for Vienna 2019. The resulting difference may, as usual, come from the phenomenon of rounding error, with the result that a total of rounded numbers is not equal to the rounded version of the original total. Furthermore, in our case, a rather precise value of the sum of the paces (minus the time taken to run the last 195 m) is known as the total time. The difference may also, in the case of a marathon, be accounted for, at least partially, by a very fast or slow pace during the last 195 m, that are not included in our measurements. Since our aim is to provide computational methods for big data that have more avail- able, we will therefore describe the stages of a statistical study starting after the sequence (p i) 1 i 42has been produced. The standard deviations=s(p)of the pace sequence is de ned to be s 2 = 1 42 42 å i=1 (p i p) 2 . (3) A coef cient of variation used extensively is Karl Pearson's dimensionless coef cient of variation given by V=100 s p , (4) and is one of the the standards for quantifying the relative pace variation. In our study, a fundamental role will be played by the skewness coef cient g1= å 42 i=1 (p i p) 3 /42 få 42 i=1 (p i p) 2 /42g 3/2 , (5) see formula(3.89)in the Ref. [9].

Int. J. Environ. Res. Public Health2022,19, 2463 4 of 19 Computations from our data are summarized in Table. Table 1.Meanp, standard deviations, coefficient of variationVand skewnessg 1for the six marathons. Race Total Time p s V g1 Tokyo 2021 (E. K.) 2 h 08 min 38 s 182.4 s 7.2 4% 0.31 Berlin 2014 (D.K) 2 h 2 min 57 s 174.8 4.1 2% 0.47 Berlin 2019 (K.B) 2 h 01 min 41 s 173.0 3.0 2% 0.55 Berlin 2018 (E.K) 2 h 01 min 39 s 173.1 3.0 2% 0.82 Vienna 2019 (E.K) 1 h 59 min 40 s 169.7 1.9 1% 2.32 Monza 2017 (E.K) 2 h 00 min 25 s 171.2 1.8 1% +0.56 The races are not listed chronologically, but according to the performance and the sign ofg1. It might seem, in view of the small values ofVthat the two performances, Vienna 2019 and Monza 2017, approached the uniform pace strategy. We will see, in fact, that these two performances and neither the other four marathon were run uniformly. This is discussed in the next paragraph. 3. Results 3.1. The Pace Is Not Uniform As in everyday language, uniformity of the pace will mean for us that the race is run at an approximately constant pace. Mathematically, the sequence(p)is, in this case, a con- stant, or a sample of measurements drawn from a distribution highly concentrated around the average pace, for example, a normal distribution with mean p and very small variance. Non-uniformity of the pace can be measured by the range of the observed pace distribution, that is, the difference between the greatest and smallest values observed, given by r1=p (42) p (1) . (6) Non-uniformity of the pace may also be measured by the difference between the fastest and the slowest 10 km, r10= 10 å i=1 fp (43 i) p (i)g, or by the difference between the fastest and the slowest half-marathon r21= 21 å i=1 fp (43 i) p (i)g= (p (22)+p (23)+ p (42)) (p (1)+p (2)+ +p (21)). Note that the well-known negative split is given by formula r 0

difference between the fastest and the slowest 10 km, r10= 10 å i=1 fp (43 i) p (i)g, or by the difference between the fastest and the slowest half-marathon r21= 21 å i=1 fp (43 i) p (i)g= (p (22)+p (23)+ p (42)) (p (1)+p (2)+ +p (21)). Note that the well-known negative split is given by formula r 0 21 = 21 å i=1 fp 43 i p ig= (p22+p23+ p42) (p1+p2+ +p21), so that the sum is calculated from paces ordered chronologically, whereas our sums inr21 are performed after reordering the paces. Our ranges are known in the eld of mathematical statistics as linear rank statistics. The ratios V1= r1 p ,V10= r10 10p ,V21= r10 21p (7) are the corresponding variation coef cients for1,10, and21km. The values are given in Table.

Int. J. Environ. Res. Public Health2022,19, 2463 5 of 19 Table 2.Absolute and relative ranges computed from order statistics. Race min =p (1) max=p (42) r1 r10 r21 V1 V10 V21 Tokyo 2021 167 194 27 189 254 15% 10% 7% Berlin 2014 165 181 16 106 137 9% 6% 4% Berlin 2019 165 179 14 76 98 8% 4% 3% Berlin 2018 164 178 14 73 124 8% 4% 3% Vienna 2019 161 172 11 43 43 6% 2% 1% Monza 2017 168 176 8 43 58 5% 3% 2% In view of this table, it is apparent that none of the six races manifests a range at- tributable to random uctuations about an average target speed. In other words, none was run at an approximately constant pace, in spite of the fact that two of them, Monza 2017 and Vienna 2019 were planned to be run at a constant pace. The smallest ranges are achieved(as expected at Monza)at races designed to be run at a constant pace. As a matter of fact,8s, even for a non-elite runner, is a signi cant time-difference for a single kilometre, as well as 43 s for 10 km. The most spectacular variation occurred for Tokyo 2021'sr21, with more than 4 min between the fastest and the slowest half-marathon in the same race. 3.2. The Pace Is Not Symmetric Once the spread of the pace has been shown to be signi cant, it is legitimate to raise the issue of its shape, in particular, its symmetry or lack of symmetry. A rst insight into the general shape of our distribution is provided by a diagrammatic representation of the data, see the bar charts of grouped data in Figures–5. 5–67–89–1011–1213–1415–1617–1819–2021–22>(p 160)sec 2 2 2 6 8 7 6 7 2 Figure 1. Berlin 2014. Frequency of paces. For instance 8 km were run at the pace 173 or 174 s. Slow (resp. fast) km in green (resp. red). The black bullet on thepaxis marks the average pace.

6 8 7 6 7 2 Figure 1. Berlin 2014. Frequency of paces. For instance 8 km were run at the pace 173 or 174 s. Slow (resp. fast) km in green (resp. red). The black bullet on thepaxis marks the average pace.

The research explores how pacing strategies affect marathon performance.