Modelling the effect of curves on distance running performance

Gross metabolic energy expenditure as a function of body weight

Using a spring and harness system, Teunissen, Grabowski & Kram (2007) quantified how simulated reduced gravity decreased the gross rate of metabolic energy expenditure during treadmill running. We utilized their data to calculate the fractional change in the rate of gross metabolic energy expenditure f as a function of the average axial leg force (${\overline{F}}_a$) (1)${\displaystyle f=0.6234{\overline{F}}_a+0.3766}$ where ${\overline{F}}_a$ is expressed as multiples of body weight (BW) and is calculated over an entire stride cycle (from touch down of one leg to the next touch down of the same leg). While Teunissen, Grabowski & Kram (2007) only measured metabolic energy expenditure in normal and simulated reduced gravity (${\overline{F}}_a\le 1BW$), we assume that the slope in Eq. (1) extrapolates to ${\overline{F}}_a>1BW$. According to Eq. (1), when a person is running in a straight line (F_a=1 BW), f = 1 (i.e., no change). If ${\overline{F}}_a=1.25$ (a 25% increase in average axial force), f = 1.16 (a 16% increase in gross metabolic energy expenditure). Note that we calculated Eq. (1) based on the data of Teunissen, Grabowski & Kram (2007) but to calculate the gross rate of metabolic energy expenditure, we added Teunissen et al.’s standing metabolic rate value of 1.87 W/kg to the net values reported in their tables. Further, we forced the regression to have an exact value of f =1 when ${\overline{F}}_a$=1.

Axial leg force as a function of velocity and curve radius

A person with a body mass m (kg), running with a tangential velocity v (m/s) on a curve of radius r (m) is subject to two forces in the frontal plane (Greene, 1985): ${\overline{F}}_v$ the average force in the vertical direction due to gravity, (2)${\displaystyle {\overline{F}}_v=mg}$ where g is gravitational acceleration, and ${\overline{F}}_c$ the average centripetal force, (3)${\displaystyle {\overline{F}}_c=\frac{m{v}^2}{r}}$ The vector sum of F_v and F_c is the average axial leg force: (4)${\displaystyle {\overline{F}}_a=\sqrt{{\overline{F}}_v^2+{\overline{F}}_c^2}}$ where ${\overline{F}}_a$ is measured in newtons (N). Dividing Eq. (4) by the body weight of the runner (1 BW = mg) and combining it with Eqs. (2) and (3), the average axial force ${\overline{F}}_a$ can be calculated in multiples of body weight: (5)${\displaystyle {\overline{F}}_a=\sqrt{1+\frac{v^4}{{\left(gr\right)}^2}}}$

Gross rate of metabolic energy expenditure during curve running

By inserting Eq. (5) into Eq. (1), it is possible to calculate the fractional increase, f, in the gross rate of metabolic energy expenditure for a runner with a tangential velocity v, on a curve of radius r compared to running straight-ahead at the same velocity: (6)${\displaystyle f=0.6234\sqrt{1+\frac{v^4}{{\left(gr\right)}^2}}+0.3766}$ As r → ∞ (straight running), ${\overline{F}}_a\to 1$ in Eq. (5) (Fig. 1A) and therefore Eq. (6) reduces to f → 1 (Fig. 1C) irrespective of running velocity v. At slower velocities, as v → 0, ${\overline{F}}_a\to 1$ in Eq. (5) (Fig. 1B) and therefore Eq. (6) reduces to f → 1 (Fig. 1D) irrespective of curve radius r.

Running velocity on straight and curved paths

The following equation, derived by Kipp, Kram & Hoogkamer (2019), expresses the relationship between gross rate of oxygen uptake ($\dot{V}{O}_{2s}$) and overground running velocity (v_s) on a straight path: (7)${\displaystyle \dot{V}{O}_{2s}=0.02724v_s^3+1.7321v_s^2-0.4538v_s+18.91}$ where $\dot{V}{O}_{2s}$ is measured in mlO₂/min/kg. The cubic term in Eq. (7) takes into account air resistance (Pugh, 1970). The Kipp, Kram & Hoogkamer (2019) equation is based on submaximal (below lactate threshold, LT), steady state measurements of oxidative metabolic rates. Beyond the lactate threshold, indirect calorimetry calculations (derived from oxygen uptake and carbon dioxide production rates) do not represent all of the metabolic energy required. However, we believe that extrapolating to faster speeds (beyond LT) provides a reasonable measure of the total metabolic energy required even though in reality that energy is supplied by both oxidative and non-oxidative metabolism.

To calculate the gross rate of oxygen uptake of a person running on a curve ($\dot{V}{O}_{2c}$) with a tangential velocity along the curve v_c, we can combine Eqs. (6) and (7): (8)${\displaystyle \dot{V}{O}_{2c}=\left(0.6234\sqrt{1+\frac{v_c^4}{{\left(gr\right)}^2}}+0.3766\right)\left(0.02724v_c^3+1.7321v_c^2-0.4538v_c+18.911\right)}$ where $\dot{V}{O}_{2c}$ is measured in mlO₂/min/kg. Note: we and others prefer to express running economy in units of energy or power (e.g., W/kg or kcals/min/kg) (Beck et al., 2018; Fletcher, Esau & Macintosh, 2009; Kipp, Byrnes & Kram, 2018; Shaw, Ingham & Folland, 2014) to account for differences in substrate utilization and therefore, in the amount of energy liberated per liter of oxygen uptake. However, Pugh (1970) used oxygen uptake rates. For our purpose here, assuming equivalence between rates of metabolic energy utilization and oxygen uptake incurs an insignificant error because we are only considering small changes in metabolic rate between curve and straight-line running.

A runner maintaining a constant velocity on both straight and curved portions (v_s = v_c), would therefore alternate their gross rates of metabolic energy expenditure according to Eqs. (7) and (8) respectively, where $\dot{V}{O}_{2c}$>$\dot{V}{O}_{2s}$. A runner performing at the maximal sustainable percentage of their aerobic capacity on the straight portion of a race cannot sustain an equal tangential velocity on the curve, since this would increase their rate of metabolic energy expenditure. Rather, in order to maintain the same metabolic energy expenditure throughout the race, running velocity on the curve must be reduced (v_c < v_s) so that $\dot{V}{O}_{2c}$= $\dot{V}{O}_{2s}$.

To calculate the running velocity on the curve (v_c) for a given velocity on the straight (v_s), we used numerical approximation methods (see Appendix for algorithm 1). To calculate the increased time during a single gradual 180° turn in an out-and-back race, we first used Eq. (7) to calculate the required $\dot{V}{O}_{2s}$ for a straight racecourse. We then calculated the running velocity v_c on the curved portion according to Eq. (8) given the same metabolic energy expenditure ($\dot{V}{O}_{2c}$ = $\dot{V}{O}_{2s}$) for a range of radii from 0.3 m (minimum radius according to IAAF rules (IAAF)) up to 36.8 m (outdoor track (IAAF, 2008b)). The increased time during the curved portion is calculated as: (9)${\displaystyle \Delta t_{180°}=\frac{d_c}{v_c}-\frac{d_c}{v_s}}$ where d_c is the distance run in the 180° turn, corresponding to πr.

In order to calculate the time difference between a straight race course (t_straight) and the same racing distance on indoor or outdoor tracks, we used the same approach described above, i.e.: we assumed that an athlete maintains the same metabolic energy expenditure on the straight and on the curved portion of the track ($\dot{V}{O}_{2c}$ = $\dot{V}{O}_{2s}$), with the curve radii set at 17.5 m for indoor track (IAAF, 2008a) and 36.8 m for the outdoor track (IAAF, 2008b). The total time (t_track) on the track is then calculated as: (10)${\displaystyle t_{track}=\frac{d_s}{v_s}+\frac{d_c}{v_c}}$ where d_s and d_c are the total distances run on the straight and curved portions of the track, respectively, and the total racing distance is d_tot = d_s + d_c.

Vice-versa, when a certain time t_trackon the track is known, assuming that $\dot{V}{O}_{2c}$ = $\dot{V}{O}_{2s}$, it is possible to calculate the respective velocities on the straight and curved paths, v_s and v_c, that satisfy Eq. (10) (see Appendix for algorithm 2). The respective time on a straight racecourse would then be: (11)${\displaystyle t_{straight}=\frac{d_{tot}}{v_s}}$ The same procedure can be used to convert times between tracks with different curve radii and/or sizes: for example between indoor (r_indoor = 17.5 m, distance of one lap d_lap,indoor=200 m) vs. outdoor tracks (r_outdoor = 36.8 m, d_lap,outdoor=400 m).

For a given racing distance, it is possible to calculate the time difference Δt as follows: (12)${\displaystyle \Delta t=t_{indoor}-t_{outdoor}}$ to compare indoor vs. outdoor tracks,

and: (13)${\displaystyle \Delta t=t_{straight}-t_{outdoor}}$ to compare straight racecourses vs. outdoor tracks.

Given that r_indoor < r_outdoor, Δt > 0 in Eq. (12) represents the increased amount of time for running on an indoor track while keeping the same rate of oxygen uptake maintained on the outdoor track. On the other hand, given that r_straight → ∞, Δt < 0 in Eq. (13) represents the increased amount of time for running on a straight racecourse while keeping the same rate of oxygen uptake maintained on the outdoor track. We used the outdoor 400 m track as a reference because the majority of racing distances (1,500 m, 3,000 m, 5,000 m and 10,000 m) are commonly run on outdoor tracks, compared to indoor tracks (1,500 m, 3,000 m and 5,000 m) and very few races are contested on straight racecourses.

We also determined the ideal geometry of an outdoor track, where we kept the track lap distance constant (d_lap,outdoor=400 m) and changed curve radii from r=6 m, corresponding to an oval track with a total straight portion of 362.3 m and a total curved portion of 37.7 m per lap, to r = 63.66 m, corresponding to a perfectly circular track with all 400 m run on the curved portion.

We selected the world record times t_WR on a standard outdoor track for 1,500 m, 3,000 m, 5,000 m and 10,000 m as a reference and then calculated the total racing time t(r) as a function of the different curve radii r according to Eq. (10). The time difference Δt: (14)${\displaystyle \Delta t=t\left(r\right)-t_{WR}}$ is the increased time (Δt> 0) or decreased time (Δt< 0) as a function of radius r compared to the respective world record. The ideal track geometry corresponds to the curve radius that allows the biggest time savings.

More generally, these algorithms can be used to convert times between straight racecourses and the same distance run on a path with a series of curves with different radii: (15)${\displaystyle t_{path}=\frac{d_s}{v_s}+\sum \frac{d_{c,i}}{v_{c,i}}}$ where d_c,i is the distance ran on the i-th curve, with a given radius r_i, and v_c,i is the velocity on the i-th curve.

Breaking 2 on a straight path

Using Eq. (15), we analyzed the racetrack in Monza, Italy used for the “Breaking 2” marathon exhibition (https://en.wikipedia.org/wiki/Breaking2). We divided the total lap distance, d_lap,Monza = 2424.4 m (17.4 laps to run a full marathon: 42,195 m) into a straight portion d_s=1907 m and 6 different curves (see Fig. 2, racetrack blueprints: personal communication, Brett Kirby, Ph.D.). We divided the “Curva parabolica” into three different portions in order to account for the non-constant radius of this specific section. All other curves were assumed to have a fixed radius throughout each section. We then applied the same algorithm described in the previous paragraph: we calculated the running velocities on the straight and on each of the curved portions of the track assuming that Eliud Kipchoge maintained a constant $\dot{V}{O}_{2s}$ = $\dot{V}{O}_{2c}$. We then converted the total time t_Monza =7225 s (2 h and 25 s) to the time t_straight that Kipchoge might have run on a straight path with a length d_tot=42195 m, while maintaining all the other factors (drafting, shoes, hydration etc.) adopted during the Breaking 2 attempt.

Increased time for a single 180° turn

We report the increased time Δt_180° as a function of radius r according to Eq. (9) for three different representative velocities (v₁ = 7.3 m/s, corresponding to Hicham el Guerrouj’s 1,500 m world record; v₂ =6 m/s, corresponding to the men’s half marathon world record; and a recreational running velocity, v₃ =4 m/s) in Fig. 3. The radius r influences both the distance run on the curve d_c and the velocity on the curve v_c in Eq. (9). As r → ∞, d_c → ∞, but given that v_c → v_s (Eq. (8)), the increased time Δt_180° → 0. As r decreases, v_c < v_s and Δt_180° starts to increase up to a specific radius $\hat{r}$, different for each velocity. In particular ${\hat{r}}_1=2.7$ m and Δt_180° = 0.261 s for v₁; ${\hat{r}}_2=1.9$ m and Δt_180° = 0.232 s for v₂; ${\hat{r}}_3=0.8$ m and Δt_180° = 0.193 s for $v_3.\hat{r}$, therefore, represents the worst radius in terms of velocity reduction (v_c < v_s) and non-trivial distance run on the curve (d_c > 0). As r further decreases ($r<\hat{r}$), d_c → 0,  leading to an overall decrease in Δt_180°.

Outdoor tracks vs. indoor tracks vs. straight races

We report the time difference Δt a function of running velocity v in Fig. 4A (1,500 m, 5,000 m and 10,000 m) and Fig. 4B (half marathon and marathon) respectively, the maximum velocity v for each distance corresponds to the respective current men’s world record. In both figures, Δt > 0 represents the increased amount of time for running on an indoor track (r = 17.5 m) compared to the outdoor track (r = 36.8 m), while Δt < 0 represents the decreased amount of time for running on a straight racecourse compared to an outdoor track. The increased or decreased amount of time compared to an outdoor track increases non-linearly with velocity v and is inversely proportional to the curve radius r (see Appendix for step-by-step algorithms). In addition, we selected four racing distances commonly contested on outdoor oval tracks (1,500 m, 3,000 m, 5,000 m and 10,000 m). Based on the actual outdoor 400 m track world records for both males and females, we calculated the respective time the same athlete would have run on an indoor track and on a straight racecourse, while keeping the same rate of oxygen uptake maintained on the outdoor track (Table 1). For comparison, we report the actual world record on the indoor track for 1,500 m, 3,000 m and 5,000 m distances. 10,000 m is not officially run on an indoor track (IAAF, 2018).

According to our model, an athlete running a marathon in 2:01:39 (corresponding to the actual world record, v = 5.78 m/s) on an outdoor track, would run 2:01:32 on a straight path and 2:02:00 on an indoor track. According to our model, an athlete running a half marathon in 58:01 (corresponding to the actual world record, v = 6.06 m/s) on an outdoor track, could run 57:57 on a straight path and 58:13 on an indoor track (Fig. 4B).

Ideal geometry of 400 m track

For a track constrained to comprise a 400 m lap, we report the time difference Δt a function of curve radius r in Fig. 5 for 1,500 m, 3,000 m, 5,000 m, and 10,000 m respectively. The plots intersect at (r = 36.8 m, Δt=0 s), where $t\left(r\right)=t_{WR}$. For r<36.8 m, Δt>0 s, indicating that a reduction in curve radius, compared to standard outdoor tracks, is detrimental for performance. For example, when r=6 m, Δt values range between +1.18 s for 1,500 m and +7.14 s for 10,000 m. On the other hand, for r > 36.8 m, Δt < 0 s for all distances, indicating that an increase in curve radius, compared to standard outdoor tracks, favors performance; in particular, at the maximum radius (r = 63.66 m) Δt equals −0.15 s for 1,500 m, −0.31 s for 3,000 m, −0.48 s for 5,000 m and −0.86 s for 10,000 m

Breaking 2 on a straight path

We report the velocities v_c on each of the curve portions and the velocity v_s on the straight portions, calculated assuming that Kipchoge maintained a constant oxygen uptake ($\dot{V}{O}_{2c}$ = $\dot{V}{O}_{2s}$) in Table 2. Note that combining each velocity with the respective distance, the time for a full lap (2424.4 m) is t_lap,Monza = 415.1s, and the total time for a full marathon (17 full laps plus the remaining 0.4 laps, i.e., 980.2 m on the last straight portion) coincides with t_Monza =7225 s.

To calculate the time t_straight that Kipchoge could have run on a straight marathon course, it is sufficient to divide the total distance by the velocity on the straight: (16)${\displaystyle t_{straight}=\frac{d}{v_s}=\frac{42195}{5.8414}=7223.48s}$ leading to an overall time difference of only Δt = 1.52 s.

Limitations and future studies

Running economy is affected by a multitude of biomechanical factors. In combination with the axial leg force that drives our model, contact time of the foot with the ground and the rate of force production (Roberts et al., 1998), antero-posterior ground reaction forces (Chang & Kram, 1999), stride length (Cavanagh & Kram, 1989) and stride frequency (Snyder & Farley, 2011) all affect the energetic cost of running. When running at maximum speed on curves with small radii (r ≤ 6 m), runners increase their contact time, decrease antero-posterior ground reaction forces and stride length compared to straight running (Chang & Kram, 2007). It is unclear if these biomechanical differences are maintained at sub-maximal speeds and at the larger radii. We have no knowledge of studies that measured biomechanics and/or, more crucially, energetics of curve running that could validate our model. In the future, we intend to empirically test a key assumption of our model - that athletes run slower on curves compared to straight portions of a track during races. Indeed, it is not clear if athletes can accurately sense their speed and metabolic rate with the precision and time resolution required. It may be that athletes run at the same speed on straight and curved sections and thus do not maintain a constant metabolic rate.

The data collected by Teunissen, Grabowski & Kram (2007) that allowed us to derive Eq. (1) were collected at one fairly slow velocity (3 m/s) on a treadmill, but to our knowledge there are no equivalent data for faster running velocities and none for overground running under different gravity conditions. In addition, we extrapolated Eq. (1) beyond normal gravity, assuming the same slope is maintained when the average axial force acting on the runner is increased (${\overline{F}}_a>1BW$). Additional experiments are needed to quantify the effects of different velocities and increased gravity on Eq. (1) and verify our assumption. In addition, our model does not distinguish between male and female athletes. While Eq. (1) can be applied to both male and female athletes, given that Teunissen, Grabowski & Kram (2007) included both sexes in their study, Eq. (7) was derived for male runners only (Kipp, Kram & Hoogkamer, 2019). Generally, studies find that males are slightly more economical than females at matched absolute running velocities (Daniels & Daniels, 1992). Equation (7) should therefore be adapted for female athletes with a different set of parameters that take in account these differences.

Our extrapolation of the Kipp, Kram & Hoogkamer (2019) equation seems reasonable but it would be preferable to obtain empirical measurements of oxygen uptake or metabolic rate for elite athletes at faster running speeds that are closer to the world record performances. In order to run 42,195 m on the Monza racetrack in a total time t_Monza =7225 s or 5.84 m/s, Eq. (7) predicts that Eliud Kipchoge sustained a rate of oxygen uptake ${\dot{VO}}_{2s}=80.87$ mlO₂/min/kg (see algorithm 2 in Appendix for details). This incredible value suggests that either that the Pugh (1970) factor for air resistance is too large or that Kipchoge is much more economical runner that the subjects tested by Kipp, Kram & Hoogkamer (2019). Fortunately, the absolute value does not affect our calculations of the effects of curve running.

When we model an athlete transitioning from straight to curved running, such as when running on a track or on a non-straight road race, we assume that the change in velocity (from v_s to v_c and vice-versa) is instantaneous, i.e., there is no deceleration or acceleration phase between straight and curved portions. This assumption may be reasonable for larger radii, such as outdoor or indoor racing tracks. If r = 17.5 m, when the velocity on the straight is v_s = 7.00 m/s, the velocity on the curve is reduced to v_c = 6.91 m/s, allowing an athlete to decelerate and re-accelerate in one single step. But, for much smaller radii (e.g., r = 1 m) when the velocity on the straight is v_s = 7.00 m/s, the velocity on the curve is v_c = 4.69 m/s, an athlete would likely need more than one step to decelerate and then re-accelerate). Non-trivial decelerations and accelerations increase the metabolic cost of running (Di Prampero et al., 2005) and should therefore be factored into our model, especially for smaller (<6 m) curve radiuses. This approach, while theoretically possible, can lead to accurate calculations only if the exact values of deceleration and accelerations are known. Future studies (from video and/or from lab-based measurements) could provide such information and fill this gap to create a more realistic model.

When an athlete is running on a curve with large radius, even for faster (>7 m/s) velocities, the increase in axial force ${\overline{F}}_a$ is relatively small (see Fig. 1B) and it is reasonable to assume that, as modeled in this paper, the limiting factor on v_c is mainly the metabolic cost of running. However, at smaller radii the increase in ${\overline{F}}_a$ is much more marked (for r = 1 m, when v_s = 7.00 m/s and v_c = 4.69 m/s, ${\overline{F}}_a=2.45$ BW). ${\overline{F}}_a$ is the axial force calculated over a full step, assuming a duty factor of 45% (Chang & Kram, 2007) the average force during contact reaches an even higher value of 2.72 BW. Chang & Kram (2007) measured velocities and ground reaction forces of recreational athletes sprinting on the straight and on curves of small (6 m or less) radii. While subjects were able to reach v_s = 7.70 m/s on the straight, the maximum velocity on a curve when r = 1 m was only 2.99 m/s, well below the velocity predicted by our model. In addition, the peak axial forces reached only 1.87 BW for the inside leg and 2.25 BW for the outside leg. In the Chang & Kram (2007) study, subjects were instructed to run as fast as possible but only for a very limited amount of time. Therefore, the maximum velocity they were able to attain on curves was not limited by metabolic cost, but by other constraints. Chang & Kram (2007) concluded that during small radius curve sprinting, the ability to generate force, in particular from the inside leg, limits maximum curve velocity. When athletes run on curves at sub-maximal speeds (i.e., for a prolonged period of time), it is likely that both mechanisms play roles. When transitioning from straight to curved running, at larger radii, the main driving factor in the velocity reduction is maintaining a constant metabolic rate. But, at progressively smaller radii, the increase of centripetal, and therefore axial forces, is amplified and velocity is further reduced as the athlete is limited by his/her ability to generate forces.

A third effect not included in our model is the difference between flat and banked curves. Equations (1) and (7), in particular, apply only to straight running. When we calculate the average axial leg force ${\overline{F}}_a$ of an athlete running on a curve, we assume that the body is aligned with ${\overline{F}}_a$ (see Fig. 1, inset) and the ankle remains in the sagittal plane, similar to straightline running. In other words, we assume that the curve is banked and ${\overline{F}}_a$ is always perpendicular to the running surface. Greene (1987) provided evidence that this is the ideal condition for sprint running and showed that any deviation of the banking angle, resulting in a misalignment of ${\overline{F}}_a$ and the running surface, results in a further reduction of velocity on the curve v_c. For example, if v_c=10 m/s and r = 17.5 m, the angle of ${\overline{F}}_a$ with the vertical is 30°. A sprinter running on a flat (unbanked) curve would slow down by 1.1 m/s compared to a curve with a 30° bank (Greene, 1987). Since this effect is much less important at distance running velocities (<7 m/s) and for large radii (e.g., Monza racetrack), we did not account for this further reduction in v_c on flat, or non ideally-banked, curves.