Vol.:(0123456789)
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European Journal of Applied Physiology (2020) 120:2507–2515 
https://doi.org/10.1007/s00421-020-04474-7
ORIGINAL ARTICLE
Biomechanical and metabolic aspects of backward (and forward) 
running on uphill gradients: another clue towards an almost inelastic 
rebound
L. Rasica1  · S. Porcelli1,2  · A. E. Minetti3  · G. Pavei3 
Received: 23 June 2020 / Accepted: 10 August 2020 / Published online: 25 August 2020 
© The Author(s) 2020
Abstract
Purpose On level, the metabolic cost (C) of backward running is higher than forward running probably due to a lower elastic 
energy recoil. On positive gradient, the ability to store and release elastic energy is impaired in forward running. We studied 
running on level and on gradient to test the hypothesis that the higher metabolic cost and lower efficiency in backward than 
forward running was due to the impairment in the elastic energy utilisation.
Methods Eight subjects ran forward and backward on a treadmill on level and on gradient (from 0 to + 25%, with 5% step). 
The mechanical work, computed from kinematic data, C and efficiency (the ratio between total mechanical work and C) 
were calculated in each condition.
Results Backward running C was higher than forward running at each condition (on average + 35%) and increased linearly 
with gradient. Total mechanical work was higher in forward running only at the steepest gradients, thus efficiency was lower 
in backward running at each gradient.
Conclusion Efficiency decreased by increasing gradient in both running modalities highlighting the impairment in the elastic 
contribution on positive gradient. The lower efficiency values calculated in backward running in all conditions pointed out 
that backward running was performed with an almost inelastic rebound; thus, muscles performed most of the mechanical 
work with a high metabolic cost. These new backward running C data permit, by applying the recently introduced ‘equiva-
lent slope’ concept for running acceleration, to obtain the predictive equation of metabolic power during level backward 
running acceleration.
Keywords Backward acceleration · Efficiency · Mechanical work · Metabolic cost · Metabolic power
Abbreviations
C 
 Metabolic cost
CBA 
 Metabolic cost of backward running acceleration
BCoM  Body centre of mass
PE 
 Potential energy of BCoM
KE 
 Kinetic energy of BCoM
TE 
 Total energy of BCoM
WEXT 
 Positive external work
WINT 
 Positive internal work
WTOT 
 Total work
WEXT
− 
 Negative external work
Introduction
Backward running is commonly used in rehabilitation and 
as an injury prevention strategy (e.g. Soligard et al. 2008; 
Gilchrist et al. 2008; Heiderscheit et al. 2010; Rössler et al. 
2016), thanks to the reduced knee joint forces and lower 
vertical peak of the ground reaction force compared with 
forward running (Flynn and Soutas-Little 1995; Sussman 
et al. 2000; Roos et al. 2012). Moreover, the reverse direc-
tion of the movement gives the possibility to involve and 
Communicated by Jean-René Lacour.
 * G. Pavei 
 
gaspare.pavei@unimi.it
1 
Institute of Biomedical Technologies, National Research 
Council, Segrate, Italy
2 
Department of Molecular Medicine, University of Pavia, 
Pavia, Italy
3 
Laboratory of Physiomechanics of Locomotion, Department 
of Pathophysiology and Transplantation, Physiology 
Division, University of Milan, Via Mangiagalli 32, 
20133 Milan, Italy
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train different muscles groups (DeVita and Stribling 1991; 
Flynn and Soutas-Little 1995; Sterzing et al. 2016); for a 
comprehensive review on backward running see Uthoff et al. 
(2018). An increasing number of backward running competi-
tions have also been organised all over the world (also the 
RetroRunning world championship), with athletes training 
specifically backward for improving their performance.
On level, the metabolic demand of backward running 
is higher than forward running (Reilly and Bowen 1984; 
Flynn et al. 1994; Wright and Weyand 2001) probably due 
to a higher muscle activation (Flynn and Soutas-Little 1993, 
1995; Wright and Weyand 2001; Sterzing et al. 2016) and/
or a reduced elastic energy utilisation (Cavagna et al. 2011, 
2012). This lower elastic contribution could be caused by 
the inverse approach of the foot on the ground that does not 
allow to store and recoil the energy from Achilles tendon or 
foot arch. Up to now, on level, no studies have analysed the 
mechanical work and metabolic cost of backward running 
concurrently so that conclusions about efficiency and elastic 
energy were inferred only indirectly.
When moving on positive gradient, the energy saving 
mechanism of forward running is impaired (Minetti et al. 
1994). When running uphill the downward trajectory of the 
body centre of mass is reduced and less energy can be stored 
in the elastic elements of the lower limbs, which decreases 
the overall running efficiency (Minetti et al. 1994). There 
are no studies on the metabolic aspects (or efficiency) of 
backward running on gradient yet. However, it has been 
shown that the difference in metabolic cost between forward 
and backward walking was 100% on level, and decreased to 
5–8% at gradients steeper than + 15% (Minetti and Ardigò 
2001) and this decrement was addressed to the impairment 
in the pendulum like motion while walking uphill.
Based on this general knowledge, the analysis of mechan-
ical and metabolic aspects of backward running on gradi-
ent would test the hypothesis of the higher metabolic cost 
and the possible decreased efficiency in backward than for-
ward running due to the impairment in the elastic energy 
utilisation.
Materials and methods
Subjects
Eight male endurance runners (age: 25.6 ± 3.2 year, height: 
1.76 ± 0.07 m, mass: 68.4 ± 6.6 kg, ̇V  O2max: 65.7 ± 6.2  mlO2 
 kg−1 min−1; mean ± SD) took part in the study. Each sub-
ject was fully informed about the aims, methods, and risks 
associated with participation and gave his written informed 
consent before the start of the study. All procedures were 
in accordance with the Declaration of Helsinki and the 
study was approved by the local ethics committee. Subjects 
undertook three familiarisation sessions with backward run-
ning at all speeds and gradients to get used with balance and 
proprioception while moving backward. After familiarisa-
tion, subjects came to the laboratory six times to complete 
the entire protocol.
Experimental protocol
Subjects visited the laboratory on six different not-con-
secutive days. This protocol was designed to avoid any 
fatigue effect due to the high metabolic and neuromuscu-
lar demand of each acquisition; the comparison between 
forward and backward running on the same subject was 
performed to avoid any mechanical or metabolic con-
founding factors; a number of speeds were tested to check 
the metabolic cost behaviour. On day 1, subjects ran for-
ward on level at 2.78 m s−1, on gradient + 5% at 2.5 m s−1 
and + 10% at 2.22 m s−1, with 15 min of recovery among 
trials. On day 2, subjects ran forward on gradient + 15% at 
1.94 m s−1 and + 20% at 1.67 m s−1, with 15 min of recovery 
between trials. On day 3, subjects ran backward on level at 
1.67 m s−1, on gradient + 5% at 1.53 m s−1 and + 20% at 
1.11 m s−1, with 15 min of recovery among trials. On day 
4, subjects ran backward on gradient + 10% at 1.11 m s−1, 
1.39 m s−1 and 1.67 m s−1, with 15 min of recovery among 
trials. On day 5, subjects ran backward on gradient + 15% 
at 1.11 m s−1, 1.25 m s−1, 1.39 m s−1 and 1.67 m s−1, with 
15 min of recovery among trials. All acquisitions lasted 
5 min. On day 6, kinematics data for all conditions were 
recorded (see below). The mechanical parameters (and 
efficiency) were compared between backward and forward 
running at each slope pairwise at these speeds: 1.67, 1.53, 
1.39, 1.25, 1.11, 0.97 m s−1 for backward running and 2.78, 
2.50, 2.22, 1.94, 1.67, 1.39 m s−1 for forward running at 
0, + 5, + 10, + 15, + 20 and + 25% gradient, respectively.
Metabolic measurements
Each experimental session was preceded by an 8-min 
stand resting oxygen consumption ( ̇VO2,  mlO2  kg−1 min−1) 
assessment after which subjects started running on the 
treadmill. Data acquisition lasted 5 min in order to reach 
a steady state ̇VO2. Pulmonary ventilation, oxygen con-
sumption and carbon dioxide production were analysed 
breath by breath by a metabolic cart (Vmax229, Sensor-
Medics, The Netherlands). The metabolic cost of running 
(C, J  kg−1 m−1, Margaria et al. 1963) was calculated from 
the data collected during the last minute of exercise by 
dividing the measured net ̇VO2 (total – resting ̇VO2) by the 
running speed. The unit conversion from  mlO2 to meta-
bolic J was achieved by considering the mean respiratory 
exchange ratio ( ̇VCO2 ̇VO2
−1) for each acquisition. At 
rest and during recovery (3rd and 5th minute) 20 μL of 
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capillary blood was obtained from a preheated earlobe for 
the determination of blood lactate concentration  ([La−]b) 
by an enzymatic method (Biosen 5030, EKF, Germany).
Kinematics
Three-dimensional (3D) body motion was collected by 
an 8-camera system (6 Vicon MX 1.3, 2 T20-S, Oxford 
Metrics, UK), by sampling at 100 Hz the spatial coordi-
nates of 18 reflective markers located on the main joint 
centres (Minetti et al. 1993; Pavei et al. 2017), while the 
subject was running on a treadmill (Ergo LG Woodway, 
Germany). Marker positions were filtered through a ‘zero-
lag’ second-order Butterworth low pass filter with a cutoff 
frequency detected by a residual analysis on each marker 
coordinate (Winter 1979). Each acquisition lasted 1 min 
and the time course of the 3D body centre of mass (BCoM) 
position was computed from an 11-segment model 
(Minetti et al. 1993; Pavei et al. 2017) based on Dempster 
inertial parameters of body segments (Winter 1979). From 
the BCoM 3D trajectory, the time course of potential (PE) 
and kinetic (KE) energies was computed to obtain the total 
mechanical energy (TE = PE + KE). The summation of all 
increases in TE time course constitutes the positive exter-
nal work (WEXT, J  kg−1 m−1), the work done to accelerate 
and lift the BCoM (Cavagna et al. 1963; Cavagna et al. 
1976). The work necessary to rotate and accelerate limbs 
with respect to BCoM (WINT, J  kg−1 m−1) (Cavagna and 
Kaneko 1977; Willems et al. 1995) was also calculated 
(according to Minetti et al. 1993) and summed to WEXT to 
obtain the total mechanical work (WTOT, J  kg−1 m−1). The 
frictional component of WINT (Minetti et al 2020) was not 
included in the present calculation. The negative exter-
nal work (WEXT
−, J  kg−1 m−1), the decreases in TE time 
course, was analysed as percentage of ‘comprehensive’ 
external mechanical work (= (WEXT) + (WEXT
−)) in gradi-
ent locomotion, as suggested by Minetti et al. (1994). The 
ratio between WTOT and C was used to estimate locomotion 
efficiency. Elastic energy contribution was estimated at 
each step as the difference between the mechanical equiva-
lent of C and WTOT. C was converted into WTOT by multi-
plying by an efficiency value of the positive work of 0.28 
(Woledge et al. (1985) reported a range of 0.25–0.30 for 
positive work muscle efficiency), then the measured WTOT 
was subtracted from it. The result, multiplied by the pro-
gression speed and divided by step frequency, provides an 
estimate of the elastic energy stored in a step. The elastic 
energy value of forward running on level was set to 1, and 
all the other conditions are reported as (sub)multiples. All 
data were analysed with custom-written Labview programs 
(release 10, National Instruments, USA).
Statistics
Data were presented as mean ± SD and compared between 
running conditions using paired t test; difference among 
speeds were compared using one-way ANOVA for repeated 
measures and Bonferroni post hoc test; significance level 
was set at p < 0.05. Statistical analyses were performed with 
SPSS version 20 (IBM).
Results
Metabolic cost
Forward running C increased with slope and present data are 
comparable with Minetti et al. (2002) values (Fig. 1). Back-
ward running C was significantly higher than forward run-
ning at each slope (P < 0.01, Fig. 1) and speed independent 
at the analysed gradients. Backward running C (J  kg−1 m−1) 
can be computed as a function of gradient (with same units 
as in Fig. 1) with the equation: C = 0.31*gradient + 4.9 
(R2 = 0.99). The difference between forward and backward 
running was almost constant among gradients 35 ± 7%.
Biomechanical parameters
The mechanical WEXT, WINT, and WTOT of backward run-
ning in all gradient conditions are plotted as a function of 
speed in Fig. 2. WEXT was the major determinant of WTOT 
and decreased with speed, but increased with gradient. 
WINT was almost gradient independent due to the decrease 
Fig. 1  Metabolic cost (J  kg−1  m−1) as a function of gradient (%). 
Black circles represent backward running, and white circles repre-
sent forward running. The superimposed dotted line represents the 
Minetti et al. 2002 equation of metabolic cost on gradient and well fit 
the experimental data. Backward running cost is always higher than 
forward running (*p < 0.01) on average of 35%. Data are mean ± SD
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of speed. In Fig. 3, the mechanical parameters of back-
ward and forward running are shown at each slope. Data 
were collected and presented at these identical gradients 
(0, + 5, + 10, + 15, + 20 and + 25%), however, at different 
speeds: 1.67, 1.53, 1.39, 1.25, 1.11, 0.97 m s−1 for back-
ward running and 2.78, 2.50, 2.22, 1.94, 1.67, 1.39 m s−1 
for forward running. WEXT was greater in backward run-
ning from 0 to 10%, whereas  WINT was significantly lower 
in backward running at all gradients (p < 0.01) and WTOT 
turned to be greater in forward running only at maximal 
gradients (20–25%, p < 0.05) (Fig. 3). Stride frequency 
(SF, Hz, Fig. 4) was statistically higher in backward than 
forward running at all slopes (p < 0.01).  
Locomotion efficiency (Fig. 5) was greater in forward 
than backward running (p < 0.001) and decreased with gradi-
ent. Backward running reached values close to the muscular 
efficiency (0.25–0.30) at the steepest gradient where both 
metabolic and mechanical variable were measured.
Estimated elastic energy contribution (Fig. 6) was higher 
in forward than backward running in all gradient conditions 
(p < 0.001) and decreased with gradient. Backward running 
approached no elastic energy contribution at the steepest 
gradient.
Discussion
The metabolic cost of backward running was higher than 
forward running in all the investigated gradients, whereas 
the total mechanical work was similar in the two gaits at all 
gradients. Thus, the lower locomotion efficiency of back-
ward than forward running (also on gradient) seems to be 
explained by the lower elastic energy contribution that does 
Fig. 2  The mechanical external (WEXT), internal (WINT) and total (WTOT) work (J  kg−1 m−1) as a function of speed (m  s−1) in backward running is 
represented at the different investigated gradients. Data are mean ± SD
Fig. 3  The mechanical external (WEXT), internal (WINT) and total 
(WTOT) work (J  kg−1 m−1) as a function of gradient (%) is represented 
in backward (black circles) and forward (white circles) running. Sta-
tistical difference between backward and forward running: #p < 0.05; 
*p < 0.01. Data are mean ± SD
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not assist muscles in performing mechanical work, which is 
carried out with a higher metabolic cost.
The metabolic cost of backward running was already 
shown to be higher than forward running on level over a 
range of speeds (Flynn et al. 1994; Wright and Weyand 
2001) and the percentage difference is close to that reported 
in the present study. The novelty of this work consists in 
extending the previous knowledge also to gradients, where 
we found that the difference in metabolic cost was almost 
constant between the two running modalities at the different 
slopes, with a similar increase among gradients (Fig. 1). This 
behaviour differs from walking, since Minetti and Ardigò 
(2001) reported a decrease in delta cost between forward 
and backward walking on gradient, down to a + 5–8% dif-
ference at gradients steeper than 15%. They ascribed this 
decrease in delta cost to the impairment of the pendulum-
like energy-saving mechanism of forward walking (energy 
recovery decreased in parallel with the metabolic cost) on 
gradient. Running does not rely on this mechanism, there-
fore a direct comparison cannot be performed; we will 
discuss later the energy-saving mechanism of running and 
its implication on the metabolic cost. The high metabolic 
power required for running backward forced us to test dif-
ferent speeds in the two running modalities, and to decrease 
speed (in both modalities) by increasing the gradient. The 
metabolic cost of forward running is speed independent on 
level and on gradient (Margaria 1938; Margaria et al. 1963; 
Minetti et al. 2002). Backward running C showed the speed 
independency on level (Wright and Weyand 2001), and here 
we extended this speed independency also on gradient [in 
Fig. 4  Stride frequency (Hz) as a function of gradient (%). Black cir-
cles represent backward running, and white circles represent forward 
running. Statistical difference between backward and forward run-
ning: #p < 0.05; *p < 0.01; §p < 0.001. Data are mean ± SD
Fig. 5  Running efficiency, calculated as the ratio between total 
mechanical work (WTOT, J  kg−1  m−1) and metabolic cost (C, J 
 kg−1 m−1), as a function of gradient (%) is represented in backward 
(black circles) and forward (white circles) running. Statistical differ-
ence between backward and forward running: *p < 0.01; §p < 0.001. 
Data are mean ± SD
Fig. 6  Estimated elastic energy contribution is represented as a func-
tion of gradient (%) in backward (black circles) and forward (white 
circles) running. The mean elastic energy of forward running on level 
is considered as 1 (see Material and methods for details), and all the 
other conditions are represented as submultiple. Statistical differ-
ence between backward and forward running: §p < 0.001. Data are 
mean ± SD
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the tested range of speeds (1.11–1.67 m s−1) and gradients 
(+ 10%, + 15%)]; thus, this speed difference between run-
ning modalities should not affect our metabolic conclusion.
The mechanical work values of backward running on 
level of present investigation showed the same pattern as in 
Cavagna et al. (2011) values (Fig. 2), whereas no data have 
been previously reported for backward running on gradi-
ent. WEXT decreased with running speed, WINT increased by 
increasing speed, but its contribution was small, and then 
WTOT decreased in the investigated range of speeds at all 
gradients. The mechanical work data for forward running 
(Fig. 3) revealed similar trend compared with Minetti et al. 
(1994) values up to + 15%, which was the steepest gradient 
analysed in that study, whereas data on steeper slopes are 
not reported in the literature. At the two steepest gradients 
(+ 20 and + 25%), forward running WEXT increased with the 
same trend as the previous gradients (Fig. 3). However, WINT 
that was gradient independent until + 15% (present data and 
Minetti et al. 1994) showed a tendency to increase probably 
due to an increased duty factor and more extended limbs that 
increased the inertia during the swing (thus the compound 
factor q of the predictive equation for WINT (Minetti 1998) 
is increased). This WINT tendency to increase at the steep-
est gradients is similar to data reported by Nardello et al. 
(2011). A similar behaviour in the increase of WINT and q 
factor has been reported at the beginning of the acceleration 
phase in sprint running (Pavei et al. 2019) and reinforces the 
idea that the mechanics of constant speed uphill running can 
be assimilated to running acceleration (di Prampero et al. 
2005; Minetti and Pavei 2018). When comparing forward 
and backward running, albeit not at the same speed, on the 
different slopes the same trend in WEXT and WTOT was found, 
with WEXT that increased linearly with gradient (on level 
WEXT is higher in backward running, as reported by Cavagna 
et al. (2011)) and was the main determinant of WTOT. WINT 
was slope independent in backward running, but showed a 
tendency to increase in forward running, which caused a 
higher WTOT in forward than backward running at the steep-
est gradients. Stride frequency was higher in backward than 
forward running at all slopes (Fig. 4). On level, a higher 
stride frequency in backward compared with forward run-
ning at paired speed was already reported (Threlkeld et al. 
1989; Flynn et al. 1994; Wright and Weyand 2001; Cav-
agna et al. 2011, 2012). Our results on level showed that 
speed (1.39–2.22 m s−1 range) was increased with a constant 
stride frequency and an increased stride length, similar to the 
results of Cavagna et al. (2012). The higher stride frequency 
would increase WINT, but we found higher values in forward 
than backward running. Other kinematics parameters concur 
in the computation of WINT: duty factor, defined as the frac-
tion of foot contact within the stride duration, mean velocity 
and a compound q factor that accounts for the limb mass and 
spatial configuration during the stride (Minetti 1998). When 
analysing the differences of each WINT component between 
backward and forward running on gradients, we found the 
already mentioned increase in stride frequency (+ 9%), an 
increase in duty factor (+ 27%), together with a decrease in 
velocity (− 36%) and q (− 32%), which led to a decreased 
WINT (− 35%) in backward running.
Running has been classically represented as a bouncing 
ball (Cavagna et al. 1964) or a spring mass model (Blick-
han 1989), where the lowering trajectory of the BCoM dur-
ing the first half of the contact time compresses the spring 
(or deforms the ball) that can store elastic energy, which is 
then released to assist muscles while lifting and accelerat-
ing BCoM for the next step. Thanks to this elastic recoil of 
the muscle–tendon structures, running efficiency values are 
higher than the muscle efficiency (25–30%) and it is also 
termed ‘apparent efficiency’. In the present study, forward 
running apparent efficiency on level was ~ 60%, in line with 
the literature (Cavagna and Kaneko 1977), and decreased 
with increasing gradient, ~ 40% at + 20%, losing most of 
the ‘apparent’ part (Fig. 5). This is in accordance with and 
expand the results of Minetti et al. (1994). Apparent effi-
ciency of backward running decreased similarly to forward 
running, but with about − 20% value in the slope range from 
level to + 20% (Fig. 5). These results showed that the energy-
saving mechanism of running (the storage and release of 
elastic energy) is impaired on gradient. One explanation 
can be found by looking at the trajectory of the BCoM and 
the fraction of positive (WEXT) and negative external work 
(WEXT
−) (Fig. 7). On level, positive (WEXT) and negative 
(WEXT
−) external work equally contributes to the ‘compre-
hensive’ external mechanical work (= (WEXT) + (WEXT
−)). 
By moving uphill, WEXT
− reduced its contribution as the 
BCoM trajectory became more ascending (as an effect of the 
slope) than descending (Minetti et al. 1994). Since the spring 
is compressed, and elastic energy is stored, during the lower-
ing part of the trajectory, and this part is smaller by increas-
ing gradient, less elastic energy can be stored. The muscles 
had then to perform the positive work to lift the BCoM, 
which increases with slopes, with less assistance from ten-
dons; this required more metabolic energy that increased C 
(which is the denominator of the efficiency equation) and the 
efficiency decreased (Fig. 5). Since muscles are required to 
perform more work, a higher sEMG activity can be expected 
in backward than forward running; we did not assess sEMG, 
but higher activity was found when running backward on 
level (Flynn and Soutas-Little 1993, 1995; Sterzing et al. 
2016). The partitioning between positive and negative exter-
nal work was similar between the two running modalities 
(Fig. 7), highlighting the same behaviour of the BCoM tra-
jectory on gradient. The estimated elastic energy contribu-
tion showed the same decreasing tendency with gradient 
of efficiency (Fig. 6), reinforcing the aforementioned idea 
that the energy-saving mechanism is impaired. Backward 
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running values were always lower than forward running, 
and while at the steepest gradient forward running main-
tained some kind of elastic contributions, backward running 
relied only on muscle capability to perform work and power 
(Fig. 7). The mechanical inefficacy of backward running was 
already described by Cavagna et al. (2011, 2012) with the 
reversed landing take-off asymmetry, which resulted in a 
greater muscle activation during positive work and a lower 
ability to store and release elastic energy. These mechanical 
premises for inefficiency were tested here (since Cavagna 
et al. did not measure metabolic cost), confirmed in their 
original theory (elastic energy) and extended to the gradient, 
where we already knew that forward running energy saving 
was impaired (Minetti et al. 1994). Backward running with 
a reversed use of the lever system of the limbs that already 
impaired the efficiency on level showed the same impair-
ment of forward running on gradient. However, starting from 
a lower level of ‘apparent efficiency’, at the steepest gradi-
ent backward running reached values of the ‘pure’ muscular 
efficiency, very likely with no elastic component.
Backward running is performed also in various sport 
activities, e.g. in soccer it has been reported to be as frequent 
as high-speed running (Mohr et al. 2003). However, up to 
now, backward running bouts are only counted (frequency 
of occurrence) and/or considered for their duration. The 
‘Equivalent Slope’ concept has been an ingenious idea to 
infer the metabolic cost of running acceleration (di Pramp-
ero et al. 2005) from the metabolic cost of the steady-state 
uphill running (Minetti et al. 2002). With the present meta-
bolic cost data of backward running on gradient (Fig. 1), we 
can calculate the metabolic cost of backward running over a 
range of 0–2 m s−2 acceleration (CBA, J  kg−1 m−1). However, 
since the metabolic cost increased linearly with gradient in 
backward running (as occurred in forward running), we can 
expect that the proposed equation can be used over a wider 
range of accelerations. Rearranging the Minetti and Pavei 
(2018) equation for the metabolic cost in forward running 
acceleration with present data of backward running C on 
gradient, the cost of backward running acceleration can be 
computed as:
where ab is the absolute backward acceleration (a positive 
value, e.g. + 1.5 m s−2, even if it is performed backward, 
because the negative value is usually given to deceleration).
With this new equation, the metabolic power (= instanta-
neous CBA × instantaneous speed) of backward acceleration 
can be computed, with the acceleration and speed values 
obtained from any GPS system, and added to the metabolic 
power for forward running acceleration and deceleration 
(Minetti and Pavei 2018) to obtain a more precise estimate 
of the metabolic power during different types of sports and 
activities.
Conclusions
The metabolic cost of backward running on level and uphill 
gradient is higher than for forward running, with a similar 
difference between the two running modalities. This higher 
cost was not determined by an increased mechanical work; 
thus, the locomotion efficiency was lower in backward than 
forward running. When analysing the trajectory of the body 
centre of mass, the two running modalities showed a similar 
impairment in the spring mass model behaviour; however, 
backward running relied less on the elastic energy. With less 
elastic contribution, the muscles have to perform ‘alone’ 
the work to lift and accelerate BCoM with a higher meta-
bolic demand. With the metabolic cost of backward running 
on gradient, and the concept of equivalent slope, the new 
equation for the metabolic cost of backward running accel-
eration was computed. The metabolic power of backward 
acceleration can be now calculated and integrated with the 
well-known equations for forward running acceleration and 
deceleration to obtain a more precise estimate of the meta-
bolic demand of the sport activities.
CBA = (a2
b + 96.2)0.5 × (3.14ab + 4.9)
,
Fig. 7  Negative external work (WEXT
−) as a percentage of ‘compre-
hensive’ external mechanical work (= (WEXT) + (WEXT
−)) is repre-
sented as a function of gradient (%). Black circles represent back-
ward running, White circles represent forward running. Data are 
mean ± SD
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Author contributions GP, SP, and AEM conceived and designed the 
study. GP and LR conducted the experiments. GP and LR analysed 
the data. GP and AEM interpreted the results of the experiments. GP 
wrote the manuscript. All authors read and approved the manuscript.
Funding Open access funding provided by Universitá degli Studi di 
Milano within the CRUI-CARE Agreement.
Compliance with ethical standards 
Conflict of interest The authors report no conflict of interest.
Ethical approval All procedures were performed in accordance with 
the ethical standards of the institutional research committee and with 
the 1964 Helsinki Declaration and its later amendments or comparable 
ethical standards.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long 
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Biomechanical and metabolic aspects of backward (and forward) running on uphill gradients: another clue towards an almost inelastic rebound.
Rasica, L,Porcelli, S,Minetti, A E,Pavei, G