Could a female athlete run a 4-minute mile with improved aerodynamic drafting?

1. Introduction

In 1954, two milestones in track and field history were reached. On 6 May, Roger Bannister became the first human to run one mile (1609.344 m) in under 4 min (3:59.4) [1]. Shortly thereafter, Diane Leather ran a mile in 4:59.6, becoming the first female athlete to run under 5 min [2]. Over the ensuing seven decades, the female mile record continued to progress and in 2023, Faith Kipyegon ran 4:07.64 [3]. In this article, we quantitatively explore if improved aerodynamic drafting could allow a top female athlete, like Kipyegon, to run just 3.19% faster and thus break the 4-minute mile barrier.

Aerodynamic drafting occurs when other athletes (pacers) run in formation around a designated athlete to reduce the aerodynamic drag force acting on the designated runner [4]. Drafting allows the designated athlete to run at a given speed with a slower rate of metabolic energy consumption; or, more relevant here, drafting allows the designated athlete to run faster at the same rate of metabolic energy consumption [5]. When Bannister first broke the 4-minute mile, he drafted closely behind two different pacers for more than 80% of the race. By contrast, during her world record mile race, Kipyegon ran behind pacers for just the first ~900 m (~56% of the race) and thereafter ran solo, with no drafting. Further, she did not consistently run closely behind her pacers, and thus the drafting was suboptimal.

Scientists and engineers have quantified the air resistance force experienced by a solo runner in several ways: scaled manikins in wind tunnels [6,7], actual human runners on treadmills in a wind tunnel [8–10] and using computational fluid dynamics simulations (CFD) [11–14]. The classic equation [15] for calculating aerodynamic drag force (F_aero) is:

where F_aero is in units of newtons, A_f is the frontal area of the runner in m², C_d is the dimensionless coefficient of drag, ρ is the air density in kg m⁻³ and v is the running velocity in m s⁻¹. Using equation (1.1), Bannister’s height (1.88 m) and body mass (70 kg) along with anthropometric equations¹, reasonable values for C_d (0.9) and air density 1.2 kg m⁻³, we calculate that at 4-minute mile pace (6.706 m s⁻¹), the aerodynamic drag force acting on him when running solo was approximately 12.58 N or 1.83% of his body weight (BW). For the smaller Kipyegon (1.57 m and 42 kg), the corresponding values are 8.89 N and 2.16% of BW, notably a greater percentage of her body weight owing to her greater surface area : volume ratio.

Classic [8] and recent [18,19] empirical physiological measurements concur that the rate of metabolic energy consumption (i.e. metabolic power) required to overcome horizontal resistive forces comprises approximately 6% of the total metabolic power required to run per BW of resistive force. Using da Silva et al.’s [19] average value of 6.13% per BW, and the values in the previous paragraph, we calculate that overcoming air resistance when running solo at 4-minute mile pace comprised 11.4% of Bannister’s metabolic power and the corresponding value for Kipyegon is 13.5%. Wind tunnel and CFD methods can estimate how much various drafting formations of pacers reduce the drag force. Estimates of drag force reduction (drafting effectiveness) in the literature vary widely, ranging from 9.7 to 85% depending on the pacer formation, spacing and measurement/simulation methods, and running speeds [11–14,17,18,20–22]. Because middle-distance running performance is largely determined by an athlete’s ability to supply metabolic energy to their muscles [23], reducing aerodynamic drag and hence the rate of metabolic energy required, can enhance performance considerably [5].

We hypothesized that given sufficient aerodynamic drafting, an elite female athlete could run a 4-minute mile. Our specific goal in this article was to model and calculate what drafting effectiveness would be needed.

2. Methods

Our overall approach was to first estimate Kipyegon’s total rate of metabolic energy consumption (i.e. metabolic power) during each lap of her mile world-record performance. Then, we used our empirically established relationship between horizontal resistive force and metabolic power [19] to estimate how much faster she could run at the same metabolic power if the aerodynamic force was reduced via drafting. Middle-distance running speeds require the supply of metabolic energy to the muscles via both oxidative (aerobic) and non-oxidative (glycolytic) metabolic pathways [24,25]. Using indirect calorimetry, it is possible to accurately measure the oxidative metabolic rate at sustainable running speeds. However, measuring the rate of non-oxidative metabolism during high-speed running would require radioactive tracers [26] and, to our knowledge, has not yet been done. One can, however, extrapolate from the rates of oxidative metabolism measured at slower speeds to estimate the total rate of metabolic energy consumption (metabolic power) at faster speeds.

Batliner et al. [27] measured the oxidative metabolic power (P_met) in watts per kilogram body mass required for treadmill (tm) running in sub-elite athletes over a range of running speeds (1.78–5.14 m s⁻¹). Their curvilinear regression equation is:

where v is the running velocity in units of m s⁻¹. Note that, during treadmill running, there is no significant aerodynamic force to overcome.

We then used da Silva et al.’s [19] finding that for each 1% of BW of horizontal impeding force, metabolic power during running is, on average, 6.13% greater than during normal treadmill running. Conveniently, the da Silva et al. value applies at different running speeds. Thus, for solo overground (og) running, total P_met (in W kg⁻¹) is equal to:

where F_aero is expressed as decimal BW, i.e. 1% of BW is 0.01. Equation (2.2) can be modified to account for the drafting effectiveness (eff_draft) where 0.50 would represent a 50% reduction in drag force

Substituting equations (2.1) and (2.2) into equation (2.3) yields:

By calculating F_aero for a solo runner and assuming a certain eff_draft, equation (2.4) reduces to a quadratic equation in terms of v:

where a = 0.5929 × (1 + (6.13 × (1 − eff_draft) × F_aero)),

b = 0.1186 × (1 + (6.13 × (1 − eff_draft) × F_aero)) and

c = 5.6986 × (1 + (6.13 × (1 − eff_draft) × F_aero)) − P_{met og}.

The discriminant, d, is equal to (b² − 4ac). The quadratic equation can then be solved for v as:

The next step is to solve equation (2.4) for P_{met og} based on an actual race performance and the inferred drafting effectiveness. Then, it is possible to explore how much velocity would increase with different drafting effectiveness values.

Using that approach, we focused on the 1 mile world record performance of Faith Kipyegon in Monaco on 21 July 2023 [28]. The ambient temperature was 29°C and the relative humidity was 57%. The barometric pressure at the start time (20:35) was 751.8 mm Hg [29] equating to an air density of 1.146 kg m⁻³ [30]. A trackside windsock visible in the broadcast video of the race indicated the wind was nearly dead calm. Kipyegon started on the inside of lane 1. Because one mile is 1609.344 m, on the Monaco 400 m per lap track, the starting line was 9.344 m before the finish line so that ‘lap 1’ was 409.344 m.

At the start, two designated pacemakers sprinted ahead in single file but there was a large spacing of ~4 m between Kipyegon and the second pacemaker through ~300 m. The spacing narrowed to about 3 m from ~300 to ~409 m. We used the 1.0 m spacing between the pacing lights visible in the broadcast video to estimate the spacing between the pacer and Kipyegon. At ~409 m, the spacing was reduced to ~2.5 m and maintained at ~809 m, with the first pacemaker exiting at ~700 m. From ~809 to about 909 m, the spacing was ~2 m and at 909 m, the remaining pacemaker moved out to lane 2, no longer providing any substantial drafting and then left the race at ~1009 m. Kipyegon ran the remainder of the mile solo. According to the World Athletics website [31], Kipyegon reached the 409.344 m mark in 1:02.60 and the 809.344 m mark in 2:04.60. Thus, her average velocity for lap 1 (0–409.344 m) was 6.539 and 6.452 m s⁻¹ for lap 2 (409.344–809.344 m). Kipyegon reached 1209.344 m in 3:06.80 and thus lap 3 time was 62.20 s, which equates to a velocity of 6.431 m s⁻¹. She then ran lap 4 (1209.344–1609.344 m) in 60.84 s or an average velocity of 6.575 m s⁻¹. Using those pacer-designated runner spacings and two previous studies of drafting effectiveness [7,14], we estimated the impeding aerodynamic forces and hence Kipyegon’s total metabolic power during her record-breaking run using equation (2.4). Although other previous studies have estimated drafting effectiveness for one specific pacer-designated runner spacing, only Marro et al. [7] and Schickhofer & Hanson [14] estimated drafting effectiveness across a range of spacings. We provide these calculations as a spreadsheet (electronic supplementary material, appendix S1) so that the reader can explore different assumptions. In these calculations, we calculate the mile times if Kipyegon was provided with optimal drafting for one, two, three and four laps. In §4, we suggest options for how that drafting might be achieved.

Marro et al. [7] used scaled manikins in a small, instrumented wind tunnel to estimate drag forces. Both Marro et al. and Schickhofer & Hanson [14] denoted the configuration of one pacer in front of the designated runner as formation 1 (figure 1). Marro et al.’s data suggest that in formation 1, with a pacer–designated runner spacing of ~3–4 m, the drafting effectiveness was about 22% for Kipyegon’s lap 1, and for lap 2, the spacing of ~2.5 m provided about a 30% drafting effectiveness (table 1). From 809 to 909 m, the spacing was ~2 m and Marro et al.’s findings indicate a drafting effectiveness of about 32% but then as soon as the pacer moved to lane 2 (initiated at 909 m), the drafting effectiveness probably went to zero. If we average 100 m at 32% and 300 m at 0%, we get 8% average drafting effectiveness for lap 3. Because Kipyegon ran lap 4 solo, the drafting effectiveness was 0. Marro et al. estimated that when the pacer-designated runner spacing is 1.3 m, drafting effectiveness would be 39.5%. We used that 39.5% value in the spreadsheet to calculate how fast Kipyegon would be able to run with such optimal drafting for the full one mile race distance (electronic supplementary material, appendix S1).

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Using CFD, Schickhofer & Hanson [14] found considerably greater drafting effectiveness values (table 1). Their simulations suggest that in formation 1, with a spacing of ~3–4 m, the drafting effectiveness would be ~43% for lap 1 and ~48% for lap 2. For 809 to 909 m, Schickhofer & Hanson predict ~50% drafting effectiveness, which when averaged with zero drafting for ~909–1209 m equates to 12.5% drafting effectiveness for lap 3. Again, lap 4 had zero drafting. For formation 1, with 1.2 m spacing, Schickhofer & Hanson found that drafting effectiveness is 70.1%. Again, we used that value in the spreadsheet to calculate how fast Kipyegon would be able to run with better drafting for the full one mile race distance. Furthermore, Schickhofer & Hanson also simulated drafting effectiveness for one pacer 1.2 m in front combined with one pacer 1.2 m directly behind the designated runner (formation 2) (figure 1). That formation increases drafting effectiveness to 75.6% and we also made our calculations using that value.

We have retained a precision of 0.01 s (approx. 0.004%) in our calculations. We recognize that as being potentially misleading since our assumptions certainly have more than 1 s of uncertainty, which is ~0.4%. Our justification follows here. We began our analysis with World Athletics times, which are recorded to the 0.01 s. Rather than sequentially rounding the numbers in our calculations, we retained that precision. Running one mile (1609.344 m) in exactly 4 min (240.00 s) is an average velocity of 6.7056 m s⁻¹. Running a mile in 240.01 s equates to a velocity of 6.7053 m s⁻¹, and a 4:00.01 mile would not be as historically notable. That is, a difference of 0.01 s (or 0.001 m s⁻¹) is relevant in this situation. Thus, we have used four significant figures (e.g. 6.705 m s⁻¹) in all velocity calculations.

3. Results

We encourage the reader to use table 2 and the spreadsheet provided in the electronic supplementary material, appendix S1 through the following examples. For lap 1 (409.344 m), Kipyegon’s actual time was 62.60 s or an average velocity of 6.539 m s⁻¹. Substituting that velocity into equation (2.4) and using a drafting effectiveness value of ~22% for a pacer–designated runner spacing of ~3–4 m from Marro et al. [7] gives a metabolic power of 35.01 W kg⁻¹. If we keep that metabolic power value but improve the drafting effectiveness to 39.5% as Marro et al. [7] suggest for a spacing distance of 1.3 m in formation 1, then we can solve equation (2.4) for the faster v. In this case, lap 1 velocity increases to 6.618 m s⁻¹ or a time for lap 1 of 61.86 s, an improvement of 0.74 s. For lap 2 (400 m), the Monaco time was 62.00 s (6.452 m s⁻¹) and drafting effectiveness was ~30% which equates to a metabolic power of 33.86 W kg⁻¹. Increasing drafting effectiveness to 39.5% would allow velocity to increase to 6.493 m s⁻¹ and a lap 2 time of 61.60 s, an improvement of 0.40 s. Thus, providing improved drafting for just two laps yields a mile time of 4:06.82. For lap 3 (400 m), the Monaco time was 62.20 s (6.431 m s⁻¹) and drafting effectiveness averaged ~8% yielding a metabolic power of 34.52 W kg⁻¹. Increasing drafting effectiveness to 39.5% would allow velocity to increase to 6.566 m s⁻¹ and a lap 3 time of 60.92 s, an improvement of 1.28 s. Thus, if improved drafting was provided for three laps, Kipyegon’s mile time would be 4:05.22. For lap 4 (400 m) in Monaco, the time was 60.84 s (6.575 m s⁻¹) and there was no drafting. Thus, according to equation (2.2), the metabolic power was 36.27 W kg⁻¹. Increasing drafting effectiveness to 39.5% would allow velocity to increase to 6.753 m s⁻¹ and a lap 4 time of 59.23 s, an improvement of 1.61 s. Summing those four lap times equals 4:03.61 for the full mile.

If instead, we use the drafting effectiveness values from Schickhofer & Hanson [14], we calculate that Kipyegon could run substantially faster (table 3). For lap 1, the Monaco velocity was 6.539 m s⁻¹ and using Schickhofer & Hanson’s formation 1 drafting effectiveness value of 43% gives a metabolic power of 34.15 W kg⁻¹. Improving the drafting effectiveness to 70.1% as Schickhofer & Hanson suggest for formation 1 with a pacer–designated runner spacing of 1.2 m, lap 1 velocity increases to 6.665 m s⁻¹ or a time of 61.42 s, an improvement of 1.18 s. For lap 2 (400 m), Kipyegon’s Monaco velocity was 6.452 m s⁻¹ and with a drafting effectiveness value of 48%, we calculate a metabolic power of 33.16 W kg⁻¹. Improving drafting effectiveness to 70.1% increases lap 2 velocity to 6.551 m s⁻¹ or a time of 61.06 s, an improvement of 0.94 s. Thus, providing drafting for two laps yields a time of 4:06.03. For lap 3 (400 m), Kipyegon’s Monaco velocity was 6.431 m s⁻¹, which combined with a drafting effectiveness value of 12.5%, gives a metabolic power of 34.34 W kg⁻¹. Improving drafting effectiveness to 70.1% increases lap 3 velocity to 6.684 m s⁻¹ or a time of 59.84, an improvement of 2.36 s. Thus, providing drafting for three laps yields a mile time of 4:03.16. For lap 4 (400 m), Kipyegon’s Monaco velocity was 6.575 m s⁻¹ and with zero drafting that required a metabolic power of 36.27 W kg⁻¹. Improving drafting effectiveness to 70.1% increases lap 4 velocity to 6.899 m s⁻¹ or a time of 57.98 s, an improvement of 2.86 s. Thus, for the full mile distance, her time would be 4:00.30. Keeping all the same assumptions but changing drafting effectiveness to 75.6% for four laps of drafting in formation 2 yields a time of 3:59.37, breaking the 4-minute mile barrier!

Next, we used the spreadsheet to calculate how fast Kipyegon could have run in two hypothetical situations: solo with no drafting and with zero air resistance, such as on a treadmill. If we retain the Marro et al. [7] formation 1 drafting effectiveness values for Monaco, but zero out the hypothetical drafting effectiveness values, the calculations predict that running solo, Kipyegon could run 4:10.14 versus her actual 4:07.64, a difference of 2.50 s. Retaining the Schickhofer & Hanson [14] formation 1 drafting effectiveness values for Monaco and again zeroing out the hypothetical drafting effectiveness, we calculate that she would have run 4:12.04 solo, a difference of 4.40 s. On the other hand, if we retain the Marro et al. drafting effectiveness values for Kipyegon in Monaco but input 100% hypothetical drafting effectiveness in formation 1, we calculate a mile time of 3:53.51. Using Schickhofer & Hanson formation 1 values, we obtain 3:55.24. In both examples, the discrepancy between the two predictions is owing to Marro et al.’s finding of generally lower drafting effectiveness values than Schickhofer & Hanson.

4. Discussion

Our calculations support the hypothesis that with optimal aerodynamic drafting, an elite female athlete could break the 4-minute mile barrier. Combining the Schickhofer & Hanson [14] aerodynamic data with our physiological model, we calculate that a female athlete would need to be provided with at least 71.9% drafting effectiveness for the full mile distance to run 3:59.99 (table 3). Schickhofer & Hanson found that a drafting effectiveness of 75.6% could be achieved in formation 2, i.e. one pacer 1.2 m in front of a designated athlete combined with a second pacer 1.2 m behind the designated athlete. With 75.6% drafting effectiveness, our calculations predict that Kipyegon could run 3:59.37. Coincidentally, that is essentially the same time that Bannister ran in his first 4-minute mile. Using our same approach but with the scaled wind tunnel estimates of Marro et al. [7] and four laps of drafting, we calculated a best mile time of 4:03.61.

The prospect of a female athlete breaking the 4-minute mile is exciting but the drafting effectiveness values vary considerably between studies. It is not clear whether Schickhofer & Hanson [14] or Marro et al.’s [7] aerodynamic estimates are more accurate. A CFD analysis specific to the body dimensions of Kipyegon and potential pacers at 6.706 m s⁻¹ and across a range of spacing distances would help to clarify the likelihood of a 4-minute mile.

Because our calculations depend on several assumptions, we performed sensitivity tests using the Schickhofer & Hanson [14] drafting effectiveness values. We assumed a drag coefficient of 0.90 as per [6,7,9] but C_d values in the literature range from 0.80 to 1.1. We found that in formation 1, decreasing C_d to 0.80 would add less than a second to the final time and increasing C_d to 1.00 would reduce the final time by less than a second (table 4; figure 2). Such changes in C_d are large, yet the model predictions were robust.

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It should be noted that the aerodynamic analyses by Marro et al. [7] and Schickhofer & Hanson [14] assumed ‘static’ body positions, that is, the runners in their models did not incorporate reciprocating limb movements relative to the body. However, Crouch et al. [32] compared cycling with static and dynamic limb positions/movements and found only small differences in aerodynamic drag force.

As per da Silva et al. [19], we assumed a value of 6.13% change in metabolic power per BW of horizontal resistive force. That value was the mean empirical finding but individual values ranged from 4.17 to 8.14%. We do not have any empirical measurements for Kipyegon and she may be an average, high or low responder to horizontal resistive forces. Thus, we tested the sensitivity of our calculations by assuming a variety of metabolic cost factors from 4 to 8% (table 5 and figure 3). We found that decreasing the metabolic cost factor from 6.13%/BW to 4.0%/BW increased the mile time by ~2.5 s. Conversely, we calculated that increasing the metabolic cost factor to 8%/BW would decrease the mile time by ~2 s. We recognize that the fastest running speed we studied in da Silva et al., 4.4 m s⁻¹, is slower than 4-minute mile pace (~6.5 m s⁻¹). However, given that the metabolic cost factor did not substantially differ from 3.3 to 4.4 m s⁻¹, we have some confidence in applying the average da Silva et al. metabolic factor (6.13%/BW) to 6.5 m s⁻¹ running, even though it is an extrapolation.

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Our frontal area estimate could also have affected our calculations. We used surface and frontal area equations that were developed for males [16,17] but of course, females and males are not all the same shape. Fortunately, Schickhofer & Hanson [14] provided empirical data that allow us to test our frontal area estimate for Kipyegon. They measured the height and body mass of 59 female athletes and measured their frontal area from photos. They used those data to calculate the frontal area of a 55 kg, 1.65 m tall female athlete as 0.44 m². If we input 55 kg and 1.65 m into the Du Bois & Du Bois [16] and Pugh [17] equations, we calculate a remarkably similar projected frontal area (0.43 m²). Such close correspondence gives us confidence that our frontal area estimate for Kipyegon is reasonable and accurate.

When Kipyegon set the world record for the mile in 2023 in Monaco, the air had a low density owing to the high temperature and humidity. Hotter or more humid conditions would provide a lower air density that might be advantageous, but the thermoregulation challenges would probably make it more difficult to run a 4-minute mile [33]. Similarly, running at altitude would reduce air resistance but the lower partial pressure of oxygen would have a much greater negative effect on aerobic capacity [34].

In practice, adequate aerodynamic drafting for a female 4-minute mile attempt could be achieved in several ways. First, the designated elite female could run a time trial with two elite males (i.e. established 4-minute milers) as designated pacers for the full distance in formation 2. That would not conform to World Athletics eligibility regulations for a female-only world record. Second, the designated elite female could have elite female middle-distance runners as pacers in formation 2 for the first two laps. Then, towards the end of the second lap, the first pair of pacers could move off the track and a second pair could pace the designated runner for the third and fourth laps. That would mimic the method pioneered in the Breaking2 project during which Eliud Kipchoge nearly broke the 2 h marathon barrier; but again, World Athletics rules would not recognize such a performance as eligible for a world record because pacers are not allowed to enter a race en route.

In all these scenarios, pacing lights on the track curbing would be helpful and such light systems (i.e. Wavelight Technologies, Nijmegen, The Netherlands) are now commonplace in elite competitions. Further, in any of these scenarios, the athletes would need to devote significant practise time to coordinate the choreography of their movements and minimize the designated athlete-pacer fore-aft spacing. It would also be important for athletes to practise running directly behind the runner in front of them because any lateral positioning decreases drafting effectiveness [7]. Any attempt to break the 4-minute mile barrier should target a location/day/time with minimal wind. Both Marro et al. [7] and Schickhofer & Hanson [14] assume that the designated athlete and the pacers have the same body dimensions. However, Kipyegon is small in stature (1.57 m) compared with many elite female 800 m specialists [35]. For example, the Tokyo Olympic champion, Athing Mu has a height of 1.78 m and the 2024 Paris Olympic champion, Keely Hodgkinson is 1.70 m tall. If tall pacers were selected, Kipyegon might experience better drafting effectiveness than we have assumed, though a taller pacer with longer legs might make it more difficult to maintain close spacing. Finally, more effective pacer formations are also possible. The arrowhead or reverse arrowhead elite female midd formations used to break the 2 h marathon could be implemented on the unusual track configuration of Franklin Field at the University of Pennsylvania [36]. On that track, the designated runner could run in lane 4, which is 400 m in circumference, with pacers running in an arrowhead formation in lanes 1−3 and 4−7.

A possible limitation to our method/model is that several studies we rely on did not include any female participants. Specifically, Batliner et al. [27] only studied males. However, running economy (expressed per kilogram body mass) does not differ substantially between males and females [37]. Further, even if there was a difference, our model only uses relative changes in velocity at a given metabolic power and it is not dependent on absolute values for metabolic power. We also rely on da Silva et al. [19] who only studied male participants. However, the key factor from da Silva et al. is scaled to body weight and it is difficult to imagine why such a scaled factor would differ between, e.g. a 60 kg male and a 60 kg female. Unfortunately, we are not alone; the field of exercise physiology needs to improve in terms of including female study participants [38]. An organized scientific effort to facilitate the first female 4-minute mile would be an excellent opportunity to study elite female runners and create excitement for more female-based studies in the field of exercise physiology.

5. Conclusions

Our calculations, based on Schickhofer & Hanson’s [14] drafting effectiveness value of 75.6%, suggest that with greatly improved (but reasonable) aerodynamic drafting, the current record holder, Faith Kipyegon, could break the 4-minute mile barrier. We find that she could feasibly run ~3:59.37 with two teams of female pacers (one 1.2 m in front and one 1.2 m in the back) who change out at 800 m. However, it should be noted that there is considerable variability among the drafting effectiveness values reported in the literature. Hopefully, Ms Kipyegon can test our prediction on the track.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

Our analysis begins with the lap times of Faith Kipyegon during her 2023 mile world record race. Those data are in the public domain and we cite the source [31]. We use equations and values from other sources which we cite: e.g. Batliner et al. [27] and da Silva et al. [19]. We also extracted values for drafting effectiveness from Marro et al. [7] and Schickhofer & Hanson [14]. Finally, we submitted our Excel spreadsheet as the electronic supplementary material. That spreadsheet comprises our 'code' for making our calculations. Thus, all data are publicly accessible or submitted as the electronic supplementary material with this article [39].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

E.S.d.S.: conceptualization, data curation, formal analysis, methodology, validation, visualization, writing—original draft, writing—review and editing; W.H.: conceptualization, formal analysis, investigation, methodology, supervision, validation, writing—original draft, writing—review and editing; S.K.: formal analysis, methodology, writing—original draft, writing—review and editing; R.K.: conceptualization, formal analysis, methodology, project administration, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

No funding has been received for this article.