Xavier Bonnet

Coralie Villa

Pascale Fode

Francois Lavaste

Helene Pillet
email: helene.pillet@ensam.eu

He ´le `ne Pillet

Mechanical work performed by individual limbs of transfemoral amputees during step-to-step transitions: Effect of walking velocity

Keywords: Locomotion, prosthetics, biomechanics, hip moment, mechanical power

come

Mechanical work performed by individual limbs of transfemoral amputees during step-to-step transitions:

Effect of walking velocity

Introduction

People with transfemoral amputation expend more metabolic energy while walking than non-amputees. It is also known that the more proximal the amputation, the greater the energy consumption. [START_REF] Waters | The energy expenditure of normal and pathologic gait[END_REF] In addition, Schmalz et al. have shown that the metabolic rate increases at faster walking speed and is greater in people with a leg amputation. [START_REF] Waters | The energy expenditure of normal and pathologic gait[END_REF][START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] The quantification of metabolic energy cost, as an evaluation of the overall efficiency of gait, has been widely used to compare prosthetic elements. [START_REF] Gailey | The CAT-CAM socket and quadrilateral socket: a comparison of energy cost during ambulation[END_REF][START_REF] Graham | A comparative study of oxygen consumption for conventional and energy-storing prosthetic feet in transfemoral amputees[END_REF][START_REF] Boonstra | Energy cost during ambulation in transfemoral amputees: a knee joint with a mechanical swing phase control vs a knee joint with a pneumatic swing phase control[END_REF][START_REF] Traballesi | Energy cost of walking in transfemoral amputees: comparison between Marlo Anatomical Socket and Ischial Containment Socket[END_REF] However, the reference method of quantification, oxygen consumption, implies the measurement of exhaled gas and cannot always be easily integrated into conventional gait analysis.

Among possible explanations for the greater energy consumption of amputees, different coping mechanisms, impacting both kinematic and kinetic parameters of gait, have been described in the literature. The stance flexion of the prosthetic knee dramatically decreases and is often totally absent. Prosthetic feet do not allow a powered plantar flexion at the end of the stance phase, [START_REF] Au | Powered ankle-foot prosthesis to assist level-ground and stair-descent gaits[END_REF] producing about 20% less push off work at the prosthetic ankle compared to nonamputees. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF] Kinetic parameters of the contralateral joints are also modified, as a consequence of the lack of prosthetic push off and to control the prosthetic knee. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF][START_REF] Jaegers | Prosthetic gait of unilateral transfemoral amputees: a kinematic study[END_REF][START_REF] Radcliffe | Four-bar linkage prosthetic knee mechanisms: kinematics, alignment and prescription criteria[END_REF] Modifications are especially observed during the double support phase of gait. Nolan and Lees [START_REF] Nolan | The functional demands on the intact limb during walking for active trans-femoral and trans-tibial amputees[END_REF] showed an overuse of the patient's intact limb compared to normal ambulation. In particular, hip moments were greater for people with transfemoral amputation than for asymptomatic subjects. These moments involve increased hip extensor activity in early stance and increased hip flexor 'pull-off' activity by the affected limb during push off. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF] Schmalz et al. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] also showed the influence of prosthetic alignment variations on the hip flexion moment necessary to initiate knee bending in late stance phase and hypothesized that the muscular effort needed was a cause for the increase in oxygen consumption. The calculation of mechanical work from gait analysis data may help us to understand the causes of the variations of metabolic energy expenditure.

It was hypothesized that the kinematic compensations during transfemoral amputee gait modify the centre of gravity path and can explain the metabolic inefficiency of above knee amputee gait. However, although the metabolic cost was 27% higher in amputees compared with normal subjects in the study by Gitter et al., [START_REF] Gitter | A reassessment of center-of-mass dynamics as a determinate of the metabolic inefficiency of above-knee amputee ambulation[END_REF] no significant differences were noted in the mechanical work done on the trunk centre of mass. Moreover, Detrembleur et al. [START_REF] Detrembleur | Relationship between energy cost, gait speed, vertical displacement of centre of body mass and efficiency of pendulum-like mechanism in unilateral amputee gait[END_REF] showed that even if the vertical displacement of the body centre of mass was much higher for transfemoral amputees due to the lack of knee damping, the energy cost was not directly affected thanks to the better efficiency of the pendulum-like mechanism during the single stance phase of gait. It may be that the major part of the energy loss occurs during the step-to-step transition, and thus, the calculation of external total energy of the centre of mass may not be sufficient to explain the variations in metabolic energy.

On the other hand, several inverted pendulum models of walking have been developed to investigate the metabolic cost of human walking from the mechanical point of view. [START_REF] Mcgeer | Passive dynamic walking[END_REF][START_REF] Kuo | The six determinants of gait and the inverted pendulum analogy: a dynamic walking perspective[END_REF][START_REF] Kuo | Energetic consequences of walking like an inverted pendulum: step-tostep transitions[END_REF] Although the lower limb nearly behaves as a pendulum during single support phase, the need to redirect the centre of mass during double support requires muscular work. Thus, Kuo et al. [START_REF] Kuo | Energetic consequences of walking like an inverted pendulum: step-tostep transitions[END_REF] showed that work was needed between steps rather than within each step.

In particular, Donelan et al. [START_REF] Donelan | Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking[END_REF] has shown that the mechanical work needed for the step-to-step transitions is a major determinant of the metabolic cost of walking of asymptomatic subjects. Regarding amputee gait, the step-to-step transition was evaluated by Houdijk et al. [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] for transtibial amputees. They concluded that the increase of mechanical work needed for the transition from the affected limb to the intact limb contributed to the increased metabolic cost.

Until now, no study has investigated this transition for transfemoral amputees. However, it can be hypothesized that this transition is all the more difficult for transfemoral amputees due to the behaviour of their prosthetic knee and foot. This is particularly true during the transition between the affected limb and the intact limb. People with a transfemoral amputation must transfer their centre of mass on the intact limb and to initiate prosthetic knee flexion, in order to release the prosthetic knee joint. This hypothesis may be consistent with the observation of Schmalz et al. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] and Radcliffe et al. [START_REF] Radcliffe | Four-bar linkage prosthetic knee mechanisms: kinematics, alignment and prescription criteria[END_REF] concerning the increase of hip flexion moment during the double stance of gait, considered as an active strategy to induce an external flexion moment at the prosthetic knee. In this framework, the aim of this study was to evaluate mechanical work performed during step-to-step transition for transfemoral amputees compared to able-bodied people at different walking velocities.

Material and method

Subjects

Six active male transfemoral amputees (age: 43 6 13 years; height: 1.80 6 0.07 m; mass: 79 6 6 kg) and six control subjects (three males and three females: age 29 6 5 years; height: 1.77 6 0.13 m; mass 72 6 13 kg) participated in the study. All amputees used their own prostheses. All the prosthetic knees had a regulation of the swing phase, which was controlled by microprocessor for five of them. All patients were amputated for traumatic reasons and were active walkers. The protocol was approved by the local ethics committee and all subjects gave their consent.

Protocol

All experiments were performed on level ground. Each subject first adopted their self-selected walking speed (SSWS). This speed was identified by asking the subject to walk along the 9 m walkway at a comfortable velocity. Then, the subject was asked to walk more slowly on the same walkway, which allows determining his slow walking speed (SWS), and finally, he was asked to walk faster and his fast walking speed (FWS) was obtained. So, each subject performed the experiment at three different speeds. For each speed, at least five successful trials were done by the subjects. The walkway was equipped with two separate force plates (AMTI) in the middle of the path. A trial was considered successful when each lower limb hits the floor in one of the two plates. So, for each trial, three components of the ground reaction force and three components of the ground reaction moment at the centre of the plate were collected under each lower limb at a sampling rate of 120 Hz. During these trials, segmental and articular kinematics were measured with an optoelectronic system (Vicon V8i 8 cameras) at a sampling rate of 120 Hz. Markers were positioned on specific anatomical landmarks in accordance with the protocol described by Pillet et al. [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. Comparison of two anthropometric models[END_REF] In addition, a static acquisition was performed prior to the experiment in order to create a personalized geometric model of the body. The method used to obtain the geometric model was presented in details by Pillet et al. [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. Comparison of two anthropometric models[END_REF]

Data analysis

The average walking velocity was quantified by calculating the average velocity of a marker placed on the first thoracic vertebra. The position of the centre of mass of the body was obtained from the subject-specific geometric model. The volume of each body segment was computed and used to find the mass and the centre of mass of this segment. For sound segments, Dempster's 20 densities were used. Prosthetic components were weighed. The density of each prosthetic segment was adjusted to match the segment mass and was used for the calculation of its centre of mass.

The three components of the velocity of the centre of mass of the body were calculated as the first derivative of its three-dimensional (3D) coordinates using a fourth-order finite difference. Prior to the derivation, the data were filtered with a zero-phase forward and reverse butterworth fourth-order filter at each step. The cut-off frequency was 5 Hz.

Mechanical power was computed using the method put forward by Donelan et al. [START_REF] Donelan | Simultaneous positive and negative external mechanical work in human walking[END_REF] The mechanical powers produced by each lower limb were computed as the dot product of the ground reaction force applied on each limb and the body's centre of mass velocity (see Appendix 1 for more details). The mechanical work was then assessed using the time integral of the mechanical power profiles. In particular, the double support phases of gait were analysed. For each cycle of gait, two double support phases were defined: the mechanical work calculated over the first and the second double supports were called WDS lead and WDS trail , respectively, according to the notations proposed by Houdijk et al. [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] They were divided by the body mass of each subject and expressed in Joule per kilogram. So, for people with amputations due to the asymmetry of gait, it was possible to calculate two values of each parameter depending on the leading lower limb of the cycle (prosthetic or contralateral).

To understand the link with lower limb kinetic adaptations, the peak hip moment during double support of the late stance phase was selected. Using inverse dynamic methods, hip moment was also computed from force plate and segmental kinematic data. From this curve, the maximum during the double support phase (corresponding to the peak of hip flexion moment) was calculated.

Statistics

Statistical analyses were performed using MATLAB. Descriptive statistics for each parameter were first performed. The influences of the speed and the limb (prosthetic, contralateral and control) were tested using a non-parametric test of Friedman for independent observations. Correlation between mechanical work and walking velocity was analysed using Pearson's correlation coefficient for each group (prosthetic leg, intact leg and control group). Statistical significance was set at p \ 0.05.

Results

Walking velocity

For the three walking conditions, Figure 1 shows that walking velocities achieved by transfemoral amputees were slower on average but in the range of speeds collected for the control group.

Mechanical work

Average mechanical power patterns generated by the affected limb, the contralateral limb and the limbs of asymptomatic subjects are depicted in Figure 2 for the Mechanical work calculated over the first and the second double supports for each limb for each speed condition is provided in Table 1. WDS lead and WDS trail were available for all speeds for the intact limb but values at SWS could not be calculated for the affected limbs. Indeed, because of the slow walking velocity, the step lengths of the patients were too small to allow them to strike both force plates during these trials.

Friedman tests showed a significant influence of the speed at p = 0.02 and a significant influence of the limb at p = 0.0001 on the mechanical work of the trailing leg during double support time. The mechanical work provided by the affected limb during double support phases was always less than the one produced by the control group. On the contrary, the mechanical work produced by the intact limb at the end of the stance phase was much greater than that of the control group.

WDS trail produced by the control group during the second double support phase increased with walking speed. This increase was lower for the affected limb and higher for the contralateral limb (Figure 3). The correlation between WDS trail and walking speed was shown to be significant for the affected limb (r = 0.72, p = 0.008) and the contralateral limb (r = 0.75, p = 0.007) of the amputee and for the control group (r = 0.72, p = 0.008).

Hip moment

Mean and standard deviation of maximum hip flexion moments computed for able-bodied subjects and transfemoral amputees are reported in Table 2. For all speeds, the peak hip moment at the prosthetic side in the population of amputees was larger by between 0.10 and 0.20 N m/kg than the one computed at the sound side. It was also larger than the one computed for the control group. Thus, the maximum hip moment was greater in transfemoral amputees on their affected limb.

Discussion

In this study, the aim was to investigate the mechanical work required by the individual limbs during step-tostep transition in a population of transfemoral amputees according to the walking velocity. A control group was used as a reference. The results of mechanical work during the step-to-step transition of the asymptomatic subjects (WDS lead = 20.28 (0.05) J/kg and WDS trail = 0.18 (0.05) J/kg) were consistent with the results of Houdijk et al. [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] for transtibial amputees and for similar walking velocities. In accordance with the literature, the walking velocities chosen by people with amputation were slower on average than those of asymptomatic subjects. However, the subjects were active enough to be able to walk with speeds in the range observed in the control group.

Our results for transfemoral amputees were fundamentally similar in nature to those for transtibial amputees studied by Houdijk et al. [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] but the effects were quantitatively much greater in the transfemoral amputees. The affected limb was able to produce limited mechanical work in late stance (WDS trail = 0.09 (0.03) J/kg vs 0.16 (0.04) J/kg for transtibial amputee [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] ). The mechanical work produced by the intact limb in late stance was much greater for transfemoral amputees (WDS trail = 0.34 (0.10) J/kg) than for transtibial amputees (WDS trail = 0.27 (0.04) W/kg [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] ). It confirmed the finding of Tesio et al. [START_REF] Tesio | The 3-D motion of the centre of gravity of the human body during level walking. II. Lower limb amputees[END_REF] who observed a greater asymmetry of mechanical energy changes of the centre of mass for above knee amputees than for below knee amputees. The results showed that mechanical work produced by the prosthetic leg during the push off phase, when the intact limb was leading, was dramatically less compared to the control group. In parallel, the mechanical work of the intact leading limb increased. These results are consistent with previous study, which showed a decrease of the prosthetic ankle power generated during the propulsion and a greater concentric hip extensor work done in early stance for the contralateral limb. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF] At the same time, the mechanical work produced during the push off of the intact limb significantly increased compared to the control group for comparable ranges of speed. This result is consistent with the findings of Nolan and Lees 11 concerning the functional demand on the intact limb for transfemoral amputees.

The increase of mechanical work production by the intact limb partially compensates the decrease of mechanical work resulting from the affected limb. Donelan et al. [START_REF] Donelan | Simultaneous positive and negative external mechanical work in human walking[END_REF] already showed that the combined limbs method underestimates external mechanical work because during double support positive and negative work is performed simultaneously by the leading and the trailing legs.

One of the principal limitations of the present study is that we did not measure oxygen consumption during gait; it was therefore impossible to calculate the correlation between the increase of mechanical work and the increase of metabolic cost. However, as stated in the introduction, the measurement of gas exchange during gait necessitates a specific instrumentation and is often performed in treadmill. [START_REF] Detrembleur | Relationship between energy cost, gait speed, vertical displacement of centre of body mass and efficiency of pendulum-like mechanism in unilateral amputee gait[END_REF][START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF] In contrast, the quantification of mechanical work can be made using the conventional gait analysis equipment and can therefore be integrated in the follow-up process of people with amputation.

Not surprisingly, average mechanical power during step-to-step transitions increased with walking velocity for both asymptomatic and amputee subjects. This confirms the results of previous studies on asymptomatic subjects [START_REF] Kuo | Energetic consequences of walking like an inverted pendulum: step-tostep transitions[END_REF][START_REF] Donelan | Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking[END_REF] and amputees. [START_REF] Houdijk | The energy cost for the step-to-step transition in amputee walking[END_REF][START_REF] Tesio | The 3-D motion of the centre of gravity of the human body during level walking. II. Lower limb amputees[END_REF] But, as the present research is the first to apply the individual limb method to calculate mechanical work during step-to-step transition for people with transfemoral amputation, it provides an original insight into the asymmetric work produced by each limb for these patients. As already showed by Kuo et al., [START_REF] Kuo | Energetic consequences of walking like an inverted pendulum: step-tostep transitions[END_REF] the total mechanical power is a good estimation of the resulting power from the three major joints of the lower limbs (ankle, knee and hip). Therefore, mechanical power analysis provides an overview of the combined actions of these joints in producing mechanical work useful to the locomotion. [START_REF] Kuo | Energetic consequences of walking like an inverted pendulum: step-tostep transitions[END_REF] As regards the influence of walking velocity, mechanical work produced by the trailing limb during the double stance phase is clearly correlated with walking velocity. For asymptomatic people and for people with a unilateral transfemoral amputation, this work increased with walking velocity. However, the amount of mechanical work that can be produced during the propulsion by the affected limb is further limited at faster walking velocities. On the contrary, the mechanical work that must be produced by the intact limb in late stance rapidly increased with walking velocity showing the additional demand on the intact limb to ensure the progression of the body. This finding is consistent with the increase of metabolic cost observed for transfemoral amputees depending on walking velocity. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] As already stated in the literature, the double stance phase is a critical phase of gait for transfemoral amputees as it corresponds to the period when knee flexion must be initiated. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF][START_REF] Radcliffe | Four-bar linkage prosthetic knee mechanisms: kinematics, alignment and prescription criteria[END_REF] Seroussi et al. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF] observed an increase of concentric hip pull-off for the affected limb referring to the concentric work done by hip flexors during push off. Schmalz et al. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] also noted an increase of hip moment to initiate knee bending when shifting the prosthetic knee joint posteriorly. Considering the mechanical behaviour of a prosthetic knee, knee flexion initiation may require a flexor moment at the joint that can be obtained by generating a flexion moment at the hip. [START_REF] Radcliffe | Four-bar linkage prosthetic knee mechanisms: kinematics, alignment and prescription criteria[END_REF] The results of the present study support the increase of hip moment at the prosthetic hip, which is also correlated with walking velocity. Hip moment values are in accordance with those of Sjo¨dahl et al. [START_REF] Sjo¨dahl | Kinematic and kinetic gait analysis in the sagittal plane of trans-femoral amputees before and after special gait re-education[END_REF] even if they are difficult to compare due to different walking velocities. The hip moment generated at the residual limb reorients the ground reaction force in order to make it go behind the knee joint allowing knee bending. However, reorientation of the force has the direct consequence of decreasing mechanical work of the prosthetic leg, observed for the patients with transfemoral amputation, preventing them from producing as much work as asymptomatic subjects during push off. Thus, we can speculate that the reorientation of the ground reaction force can be achieved with strong muscle activity at the hip that can explain a part of the increased metabolic cost for transfemoral amputees as already hypothesized by Schmalz et al. [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: the influence of prosthetic alignment and different prosthetic components[END_REF] and Seroussi at al. [START_REF] Seroussi | Mechanical work adaptations of above-knee amputee ambulation[END_REF] To conclude, the results confirm that the step-to-step transition is a critical phase of gait, and that measuring the mechanical work of the individual limbs can be a useful method that may correlate with the metabolic demand of walking in transfemoral amputees. In particular, the study shows the limitations undergone by transfemoral amputee, which results from a significant decrease of the mechanical work produced by the affected limb in opposition with an increase of this study at the intact limb during step-to-step transition compared to the control group. We also found that affected limbs were unable to increase the produced mechanical work with walking velocity in the same amount as the contralateral limb or the asymptomatic limbs. These limitations were also related to the increase of the hip flexion moment during the late stance phase, necessary to initiate prosthetic knee flexion. The process of prosthetic fitting and rehabilitation are dedicated to enhance gait to improve patients' autonomy in the society. To validate mechanical work of individual limbs as an indicator of the energetic efficiency of gait, an experiment combining metabolic demand quantification and mechanical work should be designed.

Appendix 1 Details on the calculation of the mechanical power

When walking, three forces are applied to the human body during the double support phase: W the gravity action modelled by a force applied at the centre of mass of the body; Fr ! and Fl ! the right and left ground reactions, respectively. Noting g G=Ro ! the acceleration of the centre of mass and m the total mass of the body, it is possible to express the dynamic equilibrium

mg G=Ro ! = W +Fr ! + Fl ! ð1Þ
The energetic analysis of the system also allows writing

P kin = P ext + P int ð2Þ
where P kin = dEkin=dt is the kinetic power defined as the derivative of the kinetic energy, P ext is the power of external actions and P int is the power of internal actions.

Considering that the feet do not slide on the ground, the external power is reduced to the power of the gravity action, which is equal to P( W) = W 3 V G=Ro ! where V G=Ro ! is the velocity of the centre of mass of the body.

So, equation (2) becomes

dEkin dt = W 3 V G=Ro ! + P int
The total kinetic energy is composed of kinetic energy of the translation of the centre of mass and kinetic energy due to the rotations of the body segments. Making the hypothesis that the kinetic energy due to the rotations of the body segments can be neglected in front of the kinetic energy of the translation of the centre of mass, equation ( 2) can be written as follows

d P segments 1 2 3 m i (V Gi=Ro ! ) 2 " # dt = W 3 V G=Ro ! + P int Giving mg G=Ro ! 3 V G=Ro ! = W 3 V G=Ro ! + P int ð3Þ
Combining equations ( 1) and (3), we obtain

P int =Fr ! 3 V G=Ro ! + Fl ! 3 V G=Ro !
So, the sum of the mechanical powers produced by each lower limb is an estimation of the mechanical power of the body.

The mechanical work of each individual limb was then assessed using the time integral of mechanical power profiles provided by each term of this sum.

Figure 1 .

 1 Figure 1. Average walking speeds of each transfemoral subject compared to the range of speeds observed for the control group for the three walking conditions. SWS: slow walking speed; SSWS: self-selected walking speed; FWS: fast walking speed.

Figure 2 .

 2 Figure 2. Mechanical power produced by affected limb (dashed line) and intact limb (dash-dotted line) of transfemoral amputees compared to asymptomatic subjects (solid line) (W/kg) according to the percentage of gait cycle in the self-selected walking speed condition.

Figure 3 .

 3 Figure 3. Mechanical work of trailing leg during the second double support produced by the affected limb (black diamond) and the intact limb (grey square) of transfemoral amputees compared to the one produced by the right limb of asymptomatic subjects (dark grey triangle) according to walking velocity.

Table 1 .

 1 Mechanical works (J/kg) at three different walking velocities.

	Walking velocity	Parameters (J/kg)	Amputee patients:	Amputee patients:	Control subjects
	conditions		intact limb cycle	prosthetic limb cycle
	SWS	WDS lead	20.12 (0.02)	NA	20.13 (0.03)
		WDS trail	0.22 (0.03)	NA	0.25 (0.05)
	SSWS	WDS lead	20.19 (0.09)	20.12 (0.05)	20.18 (0.04)
		WDS trail	0.34 (0.10)	0.09 (0.03)	0.28 (0.06)
	FWS	WDS lead	20.30 (0.14)	20.23 (0.14)	20.37 (0.11)
		WDS trail	0.48 (0.23)	0.13 (0.05)	0.39 (0.11)

SWS: slow walking speed; SSWS: self-selected walking speed; FWS: fast walking speed; NA: not available data; WDS lead and WDS trail : given for each limb cycle for amputee patients.

Table 2 .

 2 Mean and standard deviation of the maximal hip extension moments (N m/kg) for the three walking velocity conditions for both limbs of transfemoral amputees and control subjects.

	Maximum of external hip	Amputee: intact limb	Amputee: prosthetic limb	Control subject
	extension moment (N m/kg)
	SWS	0.51 (0.10)	0.72 (0.17)	0.57 (0.17)
	SSWS	0.74 (0.31)	0.86 (0.22)	0.80 (0.13)
	FWS	0.86 (0.29)	1.06 (0.34)	1.02 (0.14)
	SWS: slow walking speed; SSWS: self-selected walking speed; FWS: fast walking speed.

Acknowledgements

The authors thank Didier Azoulay and Christian Cazorla, prosthetists at CERAH/INI, and the patients who participated in the study.

Funding

This study was financially supported by Proteor.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.


Dimitris Chatzidimitriou
email: hatzisdimitris@gmail.com

Jean-Florent Raymond
email: jean-florent.raymond@mimuw.edu.pl

Ignasi Sau
email: ignasi.sau@lirmm.fr

Dimitrios M Thilikos

Covering and packing pumpkin models

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Erdős-Pósa theorem, proved in 1965 [START_REF] Erdős | On independent circuits contained in a graph[END_REF], revealed the following min-max relation between coverings and packings of cycles in graphs: every graph that does not contain k disjoint cycles, contains a set of O(k log k) vertices meeting all its cycles. They also proved that this result is tight by giving graph contractions where the O(k log k) bound is realized. Various extensions of this result, referring to different notions of packing and covering, attracted the attention of many researchers in modern Graph Theory (see [START_REF] Birmelé | The Erdős-Pósa property for long circuits[END_REF][START_REF] Geelen | The Erdős-Pósa property for matroid circuits[END_REF]).

A model of a graph H is any graph that can be contracted to H. Given two graphs H and G, we denote by pack H (G) the maximum number of vertexdisjoint models of H in G. We also denote by cover H (G) the minimum number of vertices that intersect all minor models of H in G. We are interested in graphs H for which the following relation holds:

for every G, if pack H (G) ≤ k then cover H (G) = O(k • log k) (1)
Clearly if θ 2 is the graph with two vertices and two edges between them, then Relation (1) holds because of the Erdős-Pósa theorem. In the most general case, Robertson and Seymour proved [START_REF] Robertson | Graph Minors. V. Excluding a planar graph[END_REF] 1) is the best we may expect for non-acyclic planar graphs and the question remains whether the same bound can be achieved for every planar graph H. The most general result in this direction concerns the (parameterized) case where H is the multi-graph containing two vertices and r parallel edges between them. This graph is also known as the r-pumpkin and is denoted by θ r . In [START_REF] Fiorini | Optimal Erdős-Pósa property for pumpkins[END_REF], Fiorini, Joret, and Sau proved that Relation (1) holds for every

H = θ r , r ≥ 2.
Another approach towards improving the bound in Relation ( 1) was to restrict the class of graphs where it applies. In this direction, it was proven by Fomin, Saurabh, and Thilikos in [START_REF] Fedor | Strengthening Erdős-Pósa property for minor-closed graph classes[END_REF] that the log k factor can be dropped in Relation [START_REF] Birmelé | The Erdős-Pósa property for long circuits[END_REF], for all planar H's, in the case when we restrict G to to a graph class that excludes some fixed graph as a minor.

In this paper we provide a common extension of both the results of [START_REF] Fiorini | Optimal Erdős-Pósa property for pumpkins[END_REF] and [START_REF] Fedor | Strengthening Erdős-Pósa property for minor-closed graph classes[END_REF] that indicates how the effect of excluding a graph as a minor is reflected in the transition from O(k • log k) to O(k). Our result is the following: Theorem 1. There exists a function f : N → N such that for every two positive integers r, q, and every graph G excluding K q as a minor, it holds that

cover θr (G) ≤ f (r) • pack θr (G) • log q.
Notice that if pack θr (G) ≤ k then G does not contain a clique of (k+1)(r+1) vertices as a minor. Therefore, if we set q = (k + 1)(r + 1), we conclude that Relation (1) holds when H = θ r . This implies the main result of [START_REF] Fiorini | Optimal Erdős-Pósa property for pumpkins[END_REF].

A consequence of the proof of Theorem 1 is the existence of a polynomial O(log OP T )-approximation algorithm for both cover θr and pack θr (G). This improves the O(log n)-approximation algorithm that was given by Joret, Paul, Sau, Saurabh, and Thomassé in [START_REF] Joret | Hitting and harvesting pumpkins[END_REF] for the same parameters.

The proof

The proof of Theorem 1 is based on three results that we describe in this section.

Given a graph G, a separation of G is a pair (X 1 , X 2 ) such that no edge of G has an endpoint in X 1 \ X 2 and the other endpoint in X 2 \ X 1 . The order of (X 1 , X 2 ) is the cardinality of the set X 1 ∩ X 2 .

We start with an observation in the case where there is no separation

(X 1 , X 2 ) of G for which both G[X 1 \ X 2 ] and G[X 2 \ X 1 ] contain some model of H as a minor.
Observation 1. Let H and G be graphs and let

(X 1 , X 2 ) be a separation of G such that both G 1 = G[X 1 \ X 2 ] and G 2 = G[X 2 \ X 1 ] contain H as a minor. Then • cover H (G) ≤ cover H (G 1 ) + cover H (G 2 ) + |X 1 ∩X 2 | • pack H (G 1 ) + pack H (G 2 ) ≤ pack H (G) • cover H (G i ) < cover H (G), i = 1, 2. • pack H (G i ) < pack H (G), i = 1, 2.
The following lemma reduces the problem to the case where for each separation (X 1 , X 2 ) of G of order at most 2r -2, if G[X 1 ] does not contain a model of H, then |X 1 | is bounded by some function depending only on H. Lemma 1. For every q ∈ N and graph H there exists a function f H : N → N such that for every K q -minor free graph G there is a K q -minor free graph G such that

• pack H (G) = pack H (G ) • cover H (G) = cover H (G ) • for every separation (X 1 , X 2 ) of G , if G [X 1 \ X 2 ] is a H-minor-free graph in G , then |X 1 | ≤ f H (|X 1 ∩X 2 |).
The proof of Lemma 1 is too long to fit in this extended abstract. It uses protrusion replacement techniques that have been developed in [START_REF] Bodlaender | Meta) kernelization[END_REF] (see also [START_REF] Hans | Reduction algorithms for graphs of small treewidth[END_REF] and [START_REF] Borie | Automatic generation of lineartime algorithms from predicate calculus descriptions of problems on recursively constructed graph families[END_REF]) that permit the replacement of the part of graph induced by X 1 by another graph so that none of the parameters pack H and cover H change in the new graph.

We say that a graph G is (α, β)-loosely connected if for every separation

(X 1 , X 2 ) of G, |X 1 ∩X 2 | ≤ α ⇒ min{|X 1 |, |X 2 |} ≤ β.
Observation 1 and Lemma 1, applied for separators of order at most 2r -2 reduce the proof of Theorem 1 to the case where G is (2r -2, f θr (2r -2))-loosely connected. The second lemma derives from this assumption and is able to detect in such a graph a model of θ r that contains O(log q) vertices.

Lemma 2. There is a function g : N → N such that every (2r -2, f θr (2r -2))loosely connected K q -minor free graph G contains a model of θ r with at most g(r) • log q vertices. The proof of Lemma 2 is the core of our results and is technical. We enumerate below its main steps.

1. Take into account that every K q -minor free graph has some vertex of degree < c • q • √ log q (see e.g., [START_REF] Kostochka | Lower bound of the hadwiger number of graphs by their average degree[END_REF][START_REF] Thomason | The extremal function for complete minors[END_REF]). Let d = f θr (2r -2) • log((r -1) • c • q • √ log q) and our target is to find a model of θ r in G with at most 4 • r • d + 2 vertices. 2. Let S = {s 1 , . . . , s l } be a maximal 2d-scattered set (all its elements are in distance ≥ 2d from each other) and consider a partition

V 1 , . . . , V l of V (G) such that • ∀ i,j ∀ u∈Vi dist G (u, s i ) ≤ dist G (u, s j ) • ∀ i,j V i ∩ V j = ∅. 3. Let G in i = G[B d G (s i )] and G i = G[V i ]. (B i G (x)
are the vertices that are in distance at most i from x.) 4. Let D i = ({X t } t∈V (Ui) , U i ) be an s i -rooted tree-distance decomposition of G i (for the definition of a rooted tree-distance decomposition, see [START_REF] Yamazaki | Isomorphism for graphs of bounded distance width[END_REF]). 5. Observe that ∀ i ∀ u,v∈Vi dist G (u, v) ≤ 2d and that all edges between G i and G j have endpoints from their d-th layer and then. 6. Prove that, unless we are done, all bags of D i have ≤ r -1 vertices. 7. Prove that, unless we are done, in each G i,j = G[V (G i ) ∪ V (G j )] there are at most r -1 paths from the d-th level of G i to the d-th level of G j . 8. Deduce from the previous step and the (2r -2, f θr (2r -2))-loose connectivity of G that there are at least 2 d/f θr (2r-2) pairwise vertex-disjoint paths between vertices in the d-th level of G i and the union of the vertices of the d-th levels of all other G j 's. 9. Deduce from the previous two steps that there is a collection of pairwise vertex-disjoint paths P in G, such that • if for some i = j there is a path P i,j ∈ P joining a vertex from G in i with a vertex of G in j then this is unique path in P with this property and • G i contains the endpoints of at least 2 d/f θr (2r-2) r-1 paths from P. 10. Contract all edges of every G i , contract all but one edge of the paths in P and remove all other edges and isolated vertices. Let H be the resulting minor of G. Prove that, from Step 8, δ(H) ≥ 2 d/f θr (2r-2) r-1 = c•q • √ log q, a contradiction to what we took into account in the 1st step.

We now require the following observation. It is now easy to verify that, taking into account Observation 2, Theorem 1 follows by successively applying Observation 1, and Lemmata 1 and 2.

Observation 2 .

 2 If k = pack θr (G) and W is the set of vertices of a model as in Lemma 2, then• pack θr (G \ W ) ≤ pack θr (G) -1 ≤ k -1 • cover θr (G) ≤ cover θr (G \ W ) + |V (W )| ≤ cover θr (G \ W ) + g(r) • log q.

The first and the last authors were co-financed by the E.U. (European Social Fund -ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) -Research Funding Program: "Thales. Investing in knowledge society through the European Social Fund".


MSc Karine Langlois

MSc Coralie Villa

PhD Xavier Bonnet

PhD François Lavaste

MD, MSc Pascale Fodé

MD Noel Martinet

PhD Hélène Pillet
email: helene.pillet@ensam.eu

Influence of physical capacities of males with transtibial amputation on gait adjustments on sloped surfaces

Keywords: biomechanics, gait analysis, joint power, kinematics, kinetics, muscle strength, rehabilitation, slope walking, stump length, transtibial amputation 6MWT = 6-minute walk test, PPA-LCI = Prosthetic Profile Amputee-Locomotor Capacities Index, SF-

The aim of the study was to investigate how kinematic and kinetic adjustments between level and slope locomotion of persons with transtibial amputation are related to their individual muscular and functional capacities. A quantified gait analysis was conducted on flat and slope surfaces for seven patients with transtibial amputation and a control group of eight subjects to obtain biomechanical parameters. In addition, maximal isometric muscular strength (knee and hip extensors) and functional scores were measured. The results of this study showed that most of the persons with transtibial amputation could adapt to ramp ascent either by increasing ankle, knee, and hip flexion angles of the residual limb and/or by recruiting their hip extensors to guarantee enough hip extension power during early stance. Besides, 6-minute walk test score was shown to be a good predictor of adaptation capacities to slope ascent. In ramp descent, the increase of knee flexion moment was correlated with knee extensor strength and residual-limb length. However, no correlation was observed with functional parameters. Results show that the walking strategy adopted by persons with transtibial amputation to negotiate ramp locomotion mainly depends on their muscular capacities. Therefore, muscular strengthening should be a priority during rehabilitation.

INTRODUCTION

The activity level of persons with transtibial amputation depends upon their physical condition, their prosthetic components, and how they use their prosthesis. Different kinds of evaluations are currently used to assess locomotion outcomes of the patients. For example, muscular capacities of the patients are evaluated by quantifying maximal isometric muscular strengths [START_REF] Maffiuletti | Assessment of hip and knee muscle function in orthopaedic practice and research[END_REF]. Functional evaluations are also used to give global scores based on self-reported questionnaires such as the Prosthetic Profile Amputee-Locomotor Capacities Index (PPA-LCI) and the 36-Item Short Form Health Survey (SF-36) or on performance measures such as the 6-minute walk test (6MWT) [START_REF] Raya | Impairment variables predicting activity limitation in individuals with lower limb amputation[END_REF]. Finally, quantitative analysis of gait provides detailed biomechanical parameters useful to the understanding of gait limitations [START_REF] Sagawa | Biomechanics and physiological parameters during gait in lower-limb amputees: A systematic review[END_REF]. Most often, these parameters (muscular, functional, and biomechanical) are studied independently. Only a few studies have attempted to correlate these parameters during both level surface and stair locomotion [START_REF] Vanicek | Lower limb kinematic and kinetic differences between transtibial amputee fallers and non-fallers[END_REF][START_REF] Lloyd | Strength asymmetry and osteoarthritis risk factors in unilateral transtibial, amputee gait[END_REF].

In the literature, the modifications of biomechanical parameters that occur when walking on slopes compared with level walking have already been well described for asymptomatic subjects. In brief, during ramp ascent, the major adjustments are the increase of ranges of motion of lower-limb joints and of hip external flexion moment [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF][START_REF] Hansen | Roll-over characteristics of human walking on inclined surfaces[END_REF]. These alterations were associated with a higher generated ankle power at the end of the stance phase and a higher generated hip power at the beginning of the stance phase [START_REF] Mcintosh | Gait dynamics on an inclined walkway[END_REF]. During ramp descent, an increase of knee flexion angle, knee external flexion moment, and absorbed knee power have been reported [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF][START_REF] Hansen | Roll-over characteristics of human walking on inclined surfaces[END_REF][START_REF] Mcintosh | Gait dynamics on an inclined walkway[END_REF].

Literature about persons with transtibial amputation during level walking shows that the peaks of power at the lower-limb joints of persons with transtibial amputation are different from those of asymptomatic subjects because of the loss of ankle muscles. Sadeghi et al. showed a decrease in absorbed knee power together with an increase in generated hip power [START_REF] Sadeghi | Muscle power compensatory mechanisms in below-knee amputee gait[END_REF] in the residual limb in order to compensate for the absence of active ankle plantar flexion of prosthetic feet. These impairments are expected to worsen when walking uphill with the need to lift the body [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF][START_REF] Devita | Muscle work is biased toward energy generation over dissipation in non-level running[END_REF]. To date, three studies have analyzed the locomotion of persons with transtibial amputation in slopes. Vrieling et al. analyzed kinematical alterations between level walking and slope ascent and descent [START_REF] Vrieling | Uphill and downhill walking in unilateral lower limb amputees[END_REF]. During uphill walking, they observed an increase in hip and knee flexions at heel strike on the prosthetic side. During downhill walking, the main alteration was the increase in residual knee flexion in late stance. However, in their study, the inclination of the slope was limited to 5 percent and dynamic parameters were not estimated. On the contrary, Vickers et al. [START_REF] Vickers | Elderly unilateral transtibial amputee gait on an inclined walkway: A biomechanical analysis[END_REF] and Fradet et al. [START_REF] Fradet | Biomechanical analysis of ramp ambulation of transtibial amputees with an adaptive ankle foot system[END_REF] calculated both kinematic and kinetic parameters for persons with transtibial amputation compared with asymptomatic subjects. Vickers et al. showed that persons with transtibial amputation reduced their speed and hip and knee flexion ranges of motion and generated less moment in the knee, particularly during descent [START_REF] Vickers | Elderly unilateral transtibial amputee gait on an inclined walkway: A biomechanical analysis[END_REF]. In these two studies, the inclinations of the slopes were 9 percent and 13 percent, but these authors did not perform any comparison with level walking as a reference.

Previous studies aimed at establishing relationships between muscular parameters and gait characteristics have already shown some significant correlations. However, all studies have been performed on flat surfaces to date [START_REF] Raya | Impairment variables predicting activity limitation in individuals with lower limb amputation[END_REF][START_REF] Lloyd | Strength asymmetry and osteoarthritis risk factors in unilateral transtibial, amputee gait[END_REF][START_REF] Powers | The influence of lower-extremity muscle force on gait characteristics in individuals with below-knee amputations secondary to vascular disease[END_REF].

To our knowledge, no studies have investigated the link between muscular forces, functional scores, and bio-mechanical parameters during transtibial amputation gait on slopes. Considering the adjustments already observed for asymptomatic subjects during ramp ambulation, it can be hypothesized that the muscular capacities of persons with transtibial amputation, in particular at the hip in ascent and at the knee in descent, will have a high effect on their ability to biomechanically adapt their gait on slopes. It is also to be expected that this ability will be related to functional scores traditionally used to evaluate the activity level of persons with lower-limb amputation.

In this framework, the aim of this study was to investigate how kinematic and kinetic adjustments between level and slope locomotion of persons with transtibial amputation are related to their individual muscular and functional capacities.

MATERIALS AND METHODS

Subjects

Seven active male patients with transtibial amputation (mean age 48 ± 13 yr, height 1.74 ± 0.05 m, mass 83 ± 10 kg) participated in the study. All patients were amputated for traumatic reasons, except one for tumoral cause, and were wearing their own prosthetic foot. Eight asymptomatic subjects (mean age 46 ± 19 yr, height 1.77 ± 0.10 m, mass 68 ± 13 kg) were recruited to obtain reference data of the biomechanical parameters.

Protocol

Data collection was conducted in two different motion analysis laboratories with the same instrumentation, protocol, and operators.

Regarding the biomechanical analysis, patients and control subjects were equipped with 54 markers whose positions were recorded with an optoelectronic motion analysis system (Vicon V8i; Oxford, United Kingdom) in accordance with the protocol described by Pillet et al. [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. Comparison of two anthropometric models[END_REF]. The inertial parameters of the body used in inverse dynamics to compute kinetic parameters were obtained according to the method proposed by Pillet et al. [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. Comparison of two anthropometric models[END_REF]. Subjects were asked to walk at a comfortable self-selected speed on a flat pathway and on a 12 percent inclined slope. Two force platforms (100 Hz, AMTI; Watertown, Massachusetts) were used to measure ground reaction forces. Five successful trials (when each lower limb hit one force platform) were recorded for each condition.

Regarding the muscular and functional evaluations, only patients with amputation were tested during a clinical examination conducted prior to gait analysis. The isometric strength of hip extensors, hip abductors, and knee extensors was assessed using a handheld dynamometer according to the protocol described by Maffiuletti [START_REF] Maffiuletti | Assessment of hip and knee muscle function in orthopaedic practice and research[END_REF]. The average force over three measurements of isometric maximal voluntary contraction was scaled by the lever arm (taken from the dynamometer to the joint axis) and the patient's weight for each group of muscles.

The residual-limb length was measured from the lateral epicondyle of the knee to its bone limb and normalized according to the sound-limb length measured from the lateral epicondyle of the knee to the lateral malleolus.

In addition, all subjects with amputation completed PPA-LCI and SF-36 self-reported questionnaires [START_REF] Loiret | Evaluation of amputees[END_REF][START_REF] Miller | Lower extremity prosthetic mobility: A comparison of 3 self-report scales[END_REF]. A functional score out of 70 was calculated from the PPA-LCI, and a score out of 30 was calculated from the mobility item of the SF-36. The functional capacities of subjects with amputation were tested with the 6MWT, measuring the distance covered in 6 min by the patients on level surface [START_REF] Raya | Impairment variables predicting activity limitation in individuals with lower limb amputation[END_REF].

Data Analysis

Spatiotemporal parameters, anatomical frames, and segmental and articular kinematics and kinetics of the lower limbs were computed according to Pillet et al. [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. Comparison of two anthropometric models[END_REF]. All parameters were estimated for level walking and for the ascent and the descent of the slope.

Several biomechanical parameters identified as being significantly modified for asymptomatic subjects when walking on slopes were selected based on previous literature [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF][START_REF] Mcintosh | Gait dynamics on an inclined walkway[END_REF]. Thus, for each situation (level, uphill, and downhill walking) the following spatiotemporal parameters were computed: gait speed; step length; percentage of stance phase; and the amplitude of flexion angles of the ankle, knee, and hip joints. Then, the following parameters were quantified during level and uphill walking: the ankle flexion angle at the end of the weight-acceptance phase (20% of the gait cycle), the knee flexion and hip flexion angles at heel strike, the peak generated hip power in early stance, and the peak generated ankle power in late stance. Likewise, the following parameters were quantified during level and downhill walking: the knee flexion angle at the end of the single-stance phase (40% of gait cycle), the hip flexion angle at heel strike, the peak knee moment in early stance, and the peak knee power in early stance. All these parameters were evaluated on the residual-limb side for subjects with amputation and as the mean of right and left sides for control subjects. Variations of biomechanical parameters between level and slope walking were expressed by computing the difference of the value of each biomechanical parameter during ramp ascent/descent minus its value during level walking (Figure). These variations were considered as quantitative parameters characterizing the adjustment strategy.

Statistics

A Lilliefors test was used to test normality on all parameters. A correlation analysis was performed between muscular strengths, functional scores, and all quantitative biomechanical parameters defined to characterize the adjustment strategy. Pearson correlation coefficients were calculated in case of normality and Spearman correlation coefficients in case of nonnormality. The level of significance was set at α = 0.05.

RESULTS

Muscular Parameters and Functional Scores

The PPA-LCI score (considered as the reference test for the locomotion of persons with amputation) results were between 62 and 70 out of a maximal score of 70 (with 5 patients who scored 70). This finding suggests that for our population, the PPA-LCI was not discriminatory enough; this limit has been previously observed by Miller et al. [START_REF] Miller | Lower extremity prosthetic mobility: A comparison of 3 self-report scales[END_REF]. The scores for SF-36 and 6MWT are reported in Table 1.

Concerning muscular parameters, the strengths of gluteus maximus and quadriceps of residual and intact limbs are summarized in Table 1. For all patients, there was no deficiency of gluteus maximus, whereas a decrease of the muscular strength of the quadriceps was observed. Pearson correlation showed that quadriceps strength was correlated with stump length (r = 0.95, p = 0.001).

Correlations Among Biomechanical Parameters Variations and Muscular and Functional Outcomes

Ramp Ascent

Table 2 presents the variation in the biomechanical parameters of the residual limb between slope ascent and level walking.

Subjects with transtibial amputation walked at an average speed of 1.32 ± 0.08 m/s on the flat surface. For the control group, the walking velocity was 1.36 ± 0.12 m/s. During ramp ascent, the average speed was 1.14 ± 0.25 m/s for the transtibial amputation group against 1.24 m/s for the control group. The decrease in gait speed was correlated with the muscular strength of hip extensors (gluteus maximus) (r = 0.78, p = 0.04).

An increase in the peak hip power during early stance was observed in the population with transtibial amputa-tion. The increase in the peak hip power was correlated with the muscular strength of hip extensors (gluteus maximus) (r = 0.82, p = 0.03), with the SF-36 score (r = 0.88, p = 0.01), and with the 6MWT score (r = 0.83, p = 0.02).

Concerning kinematic parameters, hip and knee flexion angles increased at initial contact and ankle dorsiflexion angle increased at 20 percent of gait cycle. Moreover, these increases were correlated for knee and hip (r = 0.94, p = 0.002) and for ankle and hip (r = 0.92, p = 0.003). Kinematic variations were correlated with the 6MWT score (r = 0.81, p = 0.03 for the ankle; r = 0.81, p = 0.03 for the knee; and r = 0.84, p = 0.02 for the hip).

Ramp Descent

Table 3 presents the variation in the biomechanical parameters between slope descent and level walking.

As concerns the residual knee, we found an increase in the flexion angle at mid-stance, in the peak flexion moment, and in the absorbed power in early stance.

The increase in the first peak knee flexion moment in early stance was strongly correlated with muscular strength of the quadriceps (r = 0.80, p = 0.03) and to the residuallimb length (r = 0.89, p = 0.01). It was also correlated with the variation of the first peak knee power in early stance (r = 0.90, p = 0.01).

No correlation was found between the variation of biomechanical parameters and the functional scores or the variation of spatiotemporal parameters.

DISCUSSION

The present study aimed at identifying the relationships between muscular and functional abilities of persons with transtibial amputation with the biomechanical adaptations when walking on inclined surfaces compared to level surfaces.

Ramp Ascent

The situation of ramp ascent requires a high hip power in early stance to lift the body [START_REF] Mcintosh | Gait dynamics on an inclined walkway[END_REF]. In the present study, an increase in the peak hip power in early stance was observed among subjects with amputation. The value of the peak of hip extensors power was highly correlated with the maximal isometric strength of the residual hip extensor (gluteus maximus). Thus, when available, the strength of the hip extensor was used to increase the power supplied to the body. In addition, the decrease of walking velocity observed during ramp ascent compared with level walking [START_REF] Vickers | Elderly unilateral transtibial amputee gait on an inclined walkway: A biomechanical analysis[END_REF] was reduced for the patients exhibiting higher hip extensor strength in their residual limb. This velocity variation could be proposed as an individual parameter characterizing the ability of the patient to climb the slope. Finally, the hip extensor strengthening already recommended to compensate for the loss of ankle muscles during level walking [START_REF] Sadeghi | Muscle power compensatory mechanisms in below-knee amputee gait[END_REF][START_REF] Neptune | Contributions of the individual ankle plantar flexors to support, forward progression and swing initiation during walking[END_REF][START_REF] Neptune | Muscle force redistributes segmental power for body progression during walking[END_REF] would be all the more beneficial during slope ascent.

Regarding kinematic parameters variations, the prosthetic ankle was more dorsiflexed at the end of the bearing period, and the flexions of the residual knee and hip were greater at heel strike. However, these increases were of different magnitudes among patients with amputation. As already described in the literature, the increases of these kinematic parameters were higher for the control group than for the transtibial patients of this study [START_REF] Vickers | Elderly unilateral transtibial amputee gait on an inclined walkway: A biomechanical analysis[END_REF]. Considering the adaptations of asymptomatic subjects as a reference, it is assumed that the capacity to increase lower-limb mobility would be directly related to the ability to adapt to slope ascent.

The results also showed that a lack of mobility at the ankle would affect kinematics of the overlying joints as knee and hip parameters were correlated with ankle dorsiflexion parameter. Using the Proprio Foot prosthesis set in ramp mode, Fradet et al. have previously observed that having a larger angle of ankle flexion during ramp ascent improved knee kinematics [START_REF] Fradet | Biomechanical analysis of ramp ambulation of transtibial amputees with an adaptive ankle foot system[END_REF].

The results of this study showed that most of the subjects with transtibial amputation could adapt to ramp ascent either by altering ankle, knee, and hip flexion angles of the residual limb and/or by recruiting their hip extensors to guarantee enough hip extension power during early stance to walk uphill.

The increase of kinematic parameters and the increase of hip power in early stance were correlated with the 6MWT scores. Although the 6MWT is not specific to this situation, a good performance in this endurance test was a predictor of an adaptation close to the one of asymptomatic people from a kinematic point of view.

Ramp Descent

The adaptations to ramp descent were observed to be mainly realized by the knee for asymptomatic subjects [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF][START_REF] Mcintosh | Gait dynamics on an inclined walkway[END_REF]. In fact, the knee flexion during stance phase enables subjects to lower their center of gravity. However, this increase of flexion induces an increase in the peak flexion moment that must be controlled by the quadriceps [START_REF] Lay | The effects of sloped surfaces on locomotion: A kinematic and kinetic analysis[END_REF]. It must be noted that the maximum isometric quadriceps strength of the residual side was less than half of the intact side in seven transtibial patients, thus jeopardizing the control of the knee. The increase in the peak knee flexion moment in early stance found for our group of subjects varied between patients and was correlated with the quadriceps strength of the residual knee. It could be assumed that patients with low quadriceps force limit their knee flexion moment in descent.

Our results also showed that maximal quadriceps strength was highly correlated with stump length. Indeed, Fraisse et al. previously demonstrated that muscular strength loss was more important in the case of short residual limbs [START_REF] Fraisse | Muscles of the below-knee amputees[END_REF]. Contrary to the findings for ramp ascent, no correlation between the previously mentioned parameter, functional score, and variation of spatiotemporal parameters was found during ramp descent.

Finally, qualitative observation of strategies for walking downhill showed important variations among subjects: some were able to control their descent increasing knee peak moment like the control group, some reduced gait speed and step length, and others swept along the ramp. This is consistent with the findings of Hunter et al., who suggested that walking downhill resulted from a compromise between the necessity of ensuring stability and the concern for minimizing energy, which can be more difficult to achieve for people with gait disorders [START_REF] Hunter | The cost of walking downhill: Is the preferred gait energetically optimal?[END_REF].

Study Limitations

Few subjects were recruited in the study and they wore different prosthetic components. This heterogeneity, although inherent to this type of study, could affect the patterns of gait of the patients, although only to a certain extent because all prosthetic feet were energy storing types [START_REF] Schmalz | Energy expenditure and biomechanical characteristics of lower limb amputee gait: The influence of prosthetic alignment and different prosthetic components[END_REF]. This study is the first to investigate the relationship between functional and muscular capacities and adaptation to ramp ambulation in persons with transtibial amputation. The heavy experimental procedure proposed in this explorative study should now be simplified to complete the results through an extensive study on the selected parameters.

CONCLUSIONS

The present study linked muscular capacities of the residual limb with biomechanical walking adjustments between level and inclined surfaces for patients with transtibial amputation. Results suggested that muscular strengthening of hip and knee extensors during rehabilitation could enhance ability to negotiate ramps. This is all the more true for patients with short residual limbs who usually present reduction of the quadriceps strength in the residual limb.

Concerning functional evaluations of patients, the results of the study showed that 6MWT and the decrease of gait velocity were good predictors of ability to walk uphill. On the contrary, no such predictor was found for ramp descent.

Figure .

 . Figure. Patterns of kinematic and kinetic parameters (average of control group) during ramp (a)-(c),(f) ascent and (b),(d)-(e) descent. Biomechanical parameters of interest were extracted and represented on each graph.

Table 1 .

 1 Functional scores and muscular strengths of gluteus maximus and quadriceps for both limbs and residual-limb length. 6MWT = 6-minute walk test, SF-36 = 36-Item Short Form Health Survey.

Table 2 .

 2 Variation of biomechanical parameters between slope ascent and level walking. ROM = range of motion, SD = standard deviation.

	Subject	Walking Velocity (m/s)	Step Length (m)	ROM of Prosthetic Ankle (°)	ROM of Residual Knee (°)	ROM of Residual Hip (°)	Knee Flexion at Midstance (°)	Peak of Knee Moment in Early Stance (Nm/kg)	Peak of Knee Power in Early Stance (W/kg)
	Patients
	1	0.20	0.08	0	7	28	18	0.21	0.11
	2	0.12	0.07	0	4	16	25	0.27	0.10
	3	0.11	0.08	0	7	10	16	0.64	1.05
	4	0.13	0.09	1	4	15	19	0.48	1.46
	5	0.05	0.08	2	7	2	7	0.14	0.06
	6	0.56	0.22	4	8	16	10	0.08	0.13
	7	0.46	0.15	2	3	12	6	0.12	0.03
	Mean ± SD	-0.22 ± 0.21	-0.11 ± 0.05	-1 ± 2	2 ± 6	-14 ± 8	14 ± 7	0.28 ± 0.21	-0.38 ± 0.61
	Controls	-0.11 ± 0.18	-0.04 ± 0.05	-1 ± 5	7 ± 4	-9 ± 3	13 ± 4	0.64 ± 0.22	-2.24 ± 0.96
	(Mean ± SD)

Table 3 .

 3 Variation of biomechanical parameters between slope descent and level walking.

ROM = range of motion, SD = standard deviation.

ACKNOWLEDGMENTS

The authors have declared that no competing interests exist. Funding/Support: This material was based on work supported by the French National Research Agency (grant ANR-2010-TECS-020). Institutional Review: The protocol was approved by the local ethics committee and all participants gave their consent. Participant Follow-Up: The authors plan to inform participants of the publication of this study.


J Martin
email: jose.martin@cirad.fr

M Chabalier
email: chabalier@ercane.re

P Letourmy

J.-L Chopart
email: jean-louis.chopart@cirad.fr

E Arhiman
email: arhiman@ercane.re

D Marion
email: marion@ercane.re

COMPETITION AND FACILITATION EFFECTS OF DIFFERENTIAL INTRA-AND INTER-ROW WEED MANAGEMENT IN SUGARCANE

Keywords: sugarcane, differential weed management, competition, facilitation, root profile canne à sucre, désherbage différencié, concurrence, facilitation, profil racinaire

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Differential intra-and inter-row weed management can be a mean to reduce herbicide use in sugarcane. For example, herbicides could be applied as a stripe over the rows and mechanical cultivation implemented between the rows. Inter-cropping of non-creeping legumes as cover-crops or managing the spontaneous vegetation between the rows can be further options. Most studies show that the critical period of weed competition for sugarcane is the period of crop establishment, from spiking to the "out-of-hand stage" (McMahon et al., 2000), i.e. mainly between one and four months after planting for plant-cane crops [START_REF] Marion | Nuisibilité de l'enherbement sur une culture de canne à sucre[END_REF][START_REF] Azania | Plantas daninhas[END_REF]. During this period before canopy closure, yield losses commonly ranged between 10 and 12% per month of weed competition. These results were obtained from conventional trials with mixed weed populations and without separation of intra-and inter-row competition. A field experiment was designed in La Reunion Island to assess inter-row weed competition on sugarcane. The weediness and weed flora of sugarcane fields in Reunion Island are quite well known (Le [START_REF] Bourgeois | Caractérisation des enherbements en culture de canne à sucre à la Réunion. 19ème Conférence Internationale du Columa[END_REF][START_REF] Lebreton | Effect of harvesting season of sugar cane on weed growth dynamic in La Reunion Island. Effet de l'époque de coupe sur la dynamique de développement de l'enherbement de la canne à sucre à la Réunion[END_REF]. Weed management relies mainly on chemical control [START_REF] Marnotte | A network of sugarcane herbicide weeding trials in La Reunion Island. Un réseau d'essais sur le désherbage de la canne à sucre à La Réunion[END_REF][START_REF] Martin | L'IFT herbicides canne à sucre à La Reunion : premières estimations[END_REF].

MATERIALS AND METHODS

The experiment was conducted in 2011 in sub-humid lowlands to assess sugarcane yield losses from inter-row weed competition by keeping the intra-rows free of weeds. The experiment was established in a field left as a weedy fallow for 10 years, submitted to moderate grazing and periodic crushing-mowing, where broadleaves weeds became dominant. Conventional cultivation was used for soil preparation. NPK fertilizers were pre-plant applied in the furrows according to soil testing recommendations. The sugarcane cultivar used was 'R579'. The field was sprinkler-irrigated as needed throughout the growing season. Four treatments T1, T2, T3, and T4 were compared, with inter-row weed competition occurring for (and stopped at) one, two, three and four months after planting, respectively in a randomized complete block design with 5 replicates. Each of these 20 plots was paired at random with an adjacent control plot, whose inter-rows were not weeded for the entire season. Just after planting, a broadcast application of a non residual, contact herbicide killed the first emergence of weed seedlings before planting, in order to measure the effect of weed competition from planting to harvest. Row spacing was 1.5 m and space allocated to intra-and inter-rows was equally divided. All intra-rows (including those of the control plots) where kept weed-free all season long, using labeled PRE and POST herbicides in stripe applications over the rows (three applications at 14, 43 and 65 days after planting). By mid-season, intra-row colonization by weeds coming from inter-rows in T4 and control plots -mainly vines -was prevented by pushing them back to the interrows (up to three times on control plots). For T1 to T4 treatments, inter-rows weed competition was stopped at the respective treatment time and kept weed-free by hand hoeing and hand pulling after weed removal. Weeds species present in the field are listed in Table I. The percentage of ground covered by weeds in the inter-rows and the percentage of each weed species were assessed visually in each plot, every month until the seventh month (with the exception of the sixth month). The weed biomass in the inter-rows was assessed by harvesting and measuring the dry weight of 3.75 m² areas for T2, T3 and T4 treatments at the end of their respective time of weed competition for cane. Regarding the control plots, dry weight of weeds was also assessed in a lateral inter-row at five, six and seven months after planting. The most abundant species were weighted separately as far as possible. Tillering and stalk elongation were measured monthly; sugarcane was harvested 9.5 months after planting and cane yields were assessed by weighing the two central rows of each plot. Tillering, stalk elongation, cane yield (and difference with control plots) were statistically analyzed with linear models (analysis of variance or covariance and regression analysis), using the GLM procedure of SAS software (SAS Institute Inc. 2002[START_REF] Azevedo | Sugarcane root length density and distribution from root intersection counting on a trench-profile[END_REF].

Table I:

Botanical names and family of weeds present at the study location. Functional groups are relative to their contribution to global coverage (Figure 1) and global biomass (Table II). Les adventices de l'étude. Les groupes fonctionnels renvoient à leur contribution au recouvrement global (figure 1 Root mapping and distribution were assessed 5.4 months after planting in two adjacent plots including a T1 treatment and its adjacent control using the trench-profile method [START_REF] Chopart | Estimating sugarcane root length density through root mapping and orientation modelling[END_REF][START_REF] Azevedo | Sugarcane root length density and distribution from root intersection counting on a trench-profile[END_REF]. Roots interceptions (RI) were counted using a grid with a 5 x 5 cm mesh, down to a depth of 1.2 m over 1.5 m wide soil profiles. Sugarcane roots and weed roots were not distinguished at counting. RI values were processed using RACINE2 software application in order to assess root length densities (RLD) [START_REF] Chopart | RACINE2: Software application for processing spatial distribution of root length density from root intersections on trench profiles[END_REF].

RESULTS

Weeds covered 96% of the inter-rows two months after planting, reaching 100% cover at around 3 months and then declining to 96% at 7 months in non-weeded treatments (Figure 1). Weed composition was quite diverse (Table I), but at 4 months N 2 -fixing weeds (9 among 10 Fabaceae) and broomweeds (4 among 5 Malvaceae) became dominant, accounting for approximately 30 and 32% of total coverage at 7 months, respectively (Figure 1). Weed biomass (dry matter) reached a maximum of 1.8 kg/m² in the inter-rows of the control plots five months after planting; when adjusted to a curvilinear model and extrapolated, this maximum value becomes 7.1 t/ha (Table II). At 6 months, broom weeds and Crotalaria retusa accounted for 62 and 19% of weed biomass, respectively (Table II). N 2 -fixing Fabaceae other than C. retusa were not separated from other weeds because of their voluble, spiny or sticky traits.

Figure 1: Inter-row weed coverage (%) and number of species (red bars) of treatment plots before their weeding at 1, 2, 3 and 4 months plus control plots. All intra-rows, including those of control plots, were kept weed-free using chemical plus manual control.

Recouvrement moyen (en %) et nombre d'espèces (barres rouges) dans les inter-rangs avant leur désherbage à 1, 2, 3, 4 mois et des parcelles témoins. Tous les rangs de canne, y compris ceux des témoins, ont été maintenus propres par voie chimique ou manuelle.

Number of species

Table II:

Total biomass of weeds in the inter-rows and contribution of some specific weeds or group of weeds to the total biomass (%). Missing values correspond to an apparent declining contribution of the referred species or groups. Biomasse d'adventices dans les interlignes et contribution de certaines espèces ou groups d'espèces à la biomasse totale (%). Les valeurs manquantes correspondent à une contribution apparemment en baisse de l'espèces ou groupe considéré.

Sugarcane height was far above weed canopy by mid-season, completely closing the canopy between the rows for all treatments. Overall, sugarcane tillering (Table III) and production were significantly reduced by weed competition. Stalk elongation and sugar content were not affected by weed competition (results not shown).

The effect of the duration of weed competition was highly significant on sugarcane tillering (p < 0.0001 at 3 and 6 months after planting) despite one significant effect of the initial shoot emergence rate on tillering (p = 0.006 and p = 0.015). Effects other than initial shoot emergence rate and duration of weed competition were not significant (not shown).

Table III: Sugarcane tillering at 3 and 6 months after planting. Tallage de la canne à sucre 3 et 6 mois après plantation. Sugarcane yield was 131 t/ha for T1 and significantly decreased by 13.5 t/ha/month for consecutive months of early competition through T4 (p = 0.0015, Figure 2-A). The differences observed between the treatments and their respective control plots ranged from +33 t/ha for T1 to -23 t/ha for T4; a significant decrease (p = 0.0029) of 17.2 t/ha/month was recorded for consecutive months of early competition from T1 to T4 (Figure 2-B). The checking of treatments minus control yield differences showed (Table IV) that T1 was significantly superior to control (T4-control > 0) and T4 was significantly inferior to control (T4-control < 0) whereas T3 minus control and T2 minus control were not significantly different from zero. Figure 2:

A -Average sugarcane yields (blue bars) and linear regression of treatment yields over the 20 plots (y, t/ha) against time of weed control (x, months); B -Average yield differences between treatments and their control plots (red bars) and linear regression of the 20 yield differences (y, t/ha) against time of weed control (x, months). MAP: months after planting. A -Rendements moyens en canne à sucre (barres bleues) et régression linéaire des rendements individuels (20 parcelles) (y, t/ha) sur la durée de concurrence des adventices (x, mois) ; B -moyennes des différences de rendements entre les traitements et leurs témoins adjacents (barres rouges) et régression linéaire des 20 différences de rendements (y, t/ha) sur la durée de concurrence des adventices (x, mois). MAP : mois après la plantation.

Table IV RLD profiles observed in a single T1 plot (as weeded sugarcane) and its adjacent control plot (unweeded inter-rows) at 5.4 months after planting are shown in Figure 3. The mean RLD values for the T1 plot were 1.20, 0.42 and 0.06 cm/cm 3 in the 0-20, 20-40 and 40-120 cm layers, respectively. In contrast, the control plot (sugarcane plus weeds roots) exhibited higher RLD values over the profile, with +156%, +150% and +236% in the 0-20, 20-40 and 40-120 cm layers, respectively. However, these differences are not supported by statistical analysis (no replicated lectures). These differences can obviously be due to the weeds, particularly those with strong taproots dominant in this trial (N-fixing weeds and broomweeds). However, this assumption is not supported by differential counting of sugarcane roots versus weeds roots. The RLD of the control plot reached +200% when compared with the T1 plot in the 0-10 cm layer (Figure 3), with a maximum value of 2.92 cm/cm 3 in the upper 5 cm layer. This contrasted distribution pattern in the upper soil supports the existence of a strong early weed competition for sugarcane in this surface layer. This assumption is also consistent with the depth of the furrows opened for sugarcane plantation (around 15 cm): sugarcane cannot develop significant rooting upon this depth. In contrast, sugarcane roots could have a stronger development in depth, as a response to the strong early weed competition in the surface layer. RLD values decreased rapidly in depth, with values lower than 0.5 cm/cm 3 at 25 cm depth in the T1 plot, and at 50 cm depth in the control plot. Thus, competition relations between roots, either belonging to sugarcane or weeds in the case of the control plot, probably decreased too. Finally, the main trait resulting from this comparison of RLD profiles in a couple of trenches by midseason is that the association of strong sugarcane in the intra-rows with limited weeds in the interrows probably originated a higher density of mixed roots over the profile.

Figure 3: Profiles of root lenght density (RLD) expressed as cm of roots by cm3 of soil, in a couple of adjacent plots belonging to T1 treatment (weeded sugarcane in the intra-rows plus weeded inter-rows) and the control treatment (weeded sugarcane in the intra-rows plus unweeded inter-rows) at 5.4 months after planting. For the control plot, RLD included both sugarcane and weed roots without distinguishing them. Profils de longueur de racines (RLD, en cm de racines par cm3 de sol) observés sur un couple de parcelles adjacentes, correspondant à la modalité T1 (propre) et à son témoin adjacent (rangs de canne propres et inter-rangs non désherbés) 5,4 mois après la plantation. Sans distinction des racines de canne et d'adventices pour le témoin.

DISCUSSION AND CONCLUSION

Results from this study were similar to those of [START_REF] Marion | Nuisibilité de l'enherbement sur une culture de canne à sucre[END_REF] in terms of weed and maximum cane biomass accumulation, despite the fact that their study was conducted in quite different conditions and harvested at 12.7 months versus 9.5 for this study. The cane yield reduction documented with total intra-row weed control in this study is almost as high as [START_REF] Marion | Nuisibilité de l'enherbement sur une culture de canne à sucre[END_REF] recorded without intra-row weed control (-13.5 versus -14.5 t/ha/month of exposure to weed competition). Thus, mitigation of weed competition by intra-row weed control above ground is probably less than proportional to the space allocated to the intra-row: we observed in some soil profiles evidence that weed roots coming from inter-rows colonize intra-row space and meet sugarcane roots. However, the average yield of our control plots (whose inter-rows were kept unweeded over the cycle), logically expected to be lower than T4 (whose inter-rows were weeded at four months after planting), were significantly higher than T4 (p = 0.05), reaching the yield level of T3. This finding suggests a late facilitation effect of the inter-row vegetation in control plots which partially compensated for an early competition effect. This may be due to the particular flora of our study, with high populations of N 2 -fixing weeds and broomweeds, probably associated with a higher global root length density over the soil profile, as observed in this study by the observation of a single couple of trench-profiles by mid-season. Could this higher density of mixed roots over the profile be involved in and responsible for the partial yield compensation significantly recorded in the control plots? Additional research is needed (i) to corroborate the results in terms of yields over several cropping seasons, (ii) to assess the root distribution and the potential root extraction ratio for water and nutrients for sugarcane and weeds separately, and (iii) to investigate the diversity of facilitation processes involved.

  : Confidence intervals at p=0.05 for the yield differences predicted by the equation of regression from figure 2-B [y (yield difference, t/ha) = -17.16x (months) + 47.24]. Intervalles de confiance à p=0.05 des différences de rendement calculées avec l'équation de régression de la figure 2-B [y (différence de rendement, en t/ha) = -17.16x (en mois) + 47.24].

  ) et à leur biomasse(tableau II)

	Botanical name	Family	Functional groups
	Cyperus rotundus	Cyperaceae	Geophyte plant (as sedge)
	Brachiaria nana	Poaceae	Grasses
	Cenchrus biflorus	Poaceae	Grasses
	Digitaria ciliaris	Poaceae	Grasses
	Eleusine indica	Poaceae	Grasses
	Melinis repens	Poaceae	Grasses
	Panicum maximum	Poaceae	Grasses
	Paspalum dilatatum	Poaceae	Grasses
	Rottboellia cochinchinensis	Poaceae	Grasses
	Sorghum verticilliflorum	Poaceae	Grasses
	Commelina benghalensis	Commelinaceae	Commelina benghalensis
	Achyranthes aspera	Amaranthaceae	Amaranthaceae + Solanaceae
	Amaranthus dubius	Amaranthaceae	Amaranthaceae + Solanaceae
	Amaranthus viridis	Amaranthaceae	Amaranthaceae + Solanaceae
	Bidens pilosa	Asteraceae	Asteraceae + others broadleaf weeds
	Parthenium hysterophorus	Asteraceae	Asteraceae + others broadleaf weeds
	Sigesbeckia orientalis	Asteraceae	Asteraceae + others broadleaf weeds
	Cleome viscosa	Brassicaceae	Cleome viscosa
	Ipomoea eriocarpa	Convolvulaceae	Vines
	Ipomoea hederifolia	Convolvulaceae	Vines
	Ipomoea nil	Convolvulaceae	Vines
	Ipomoea obscura	Convolvulaceae	Vines
	Acalypha indica	Euphorbiaceae	Euphorbiaceae
	Croton bonplandianus	Euphorbiaceae	Euphorbiaceae
	Euphorbia heterophylla	Euphorbiaceae	Euphorbiaceae
	Euphorbia hypericifolia	Euphorbiaceae	Euphorbiaceae
	Phyllanthus amarus	Euphorbiaceae	Euphorbiaceae
	Cajanus scarabaeoides	Fabaceae	N-fixing Fabaceae
	Centrosema pubescens	Fabaceae	N-fixing Fabaceae
	Crotalaria retusa	Fabaceae	N-fixing Fabaceae
	Desmanthus virgatus	Fabaceae	N-fixing Fabaceae
	Desmodium intortum	Fabaceae	N-fixing Fabaceae
	Desmodium tortuosum	Fabaceae	N-fixing Fabaceae
	Indigofera hirsuta	Fabaceae	N-fixing Fabaceae
	Mimosa invisa	Fabaceae	N-fixing Fabaceae
	Mimosa pudica	Fabaceae	N-fixing Fabaceae
	Senna occidentalis	Fabaceae	within others broadleaf weeds
	Hibiscus surattensis	Malvaceae	Vines (as creeping plant)
	Malvastrum coromandelianum	Malvaceae	Broomweeds Malvaceae
	Melochia pyramidata	Malvaceae	Broomweeds Malvaceae
	Sida acuta	Malvaceae	Broomweeds Malvaceae
	Sida glutinosa	Malvaceae	Broomweeds Malvaceae
	Oxalis corniculata	Oxalidaceae	Geophyte plant
	Argemone mexicana	Papaveraceae	Asteraceae + other broadleaf weeds
	Passiflora foetida	Passifloraceae	Vines
	Portulaca oleracea	Portulacaceae	Asteraceae + other broadleaf weeds
	Cardiospermum halicacabum	Sapindaceae	Vines
	Solanum americanum	Solanaceae	Amaranthaceae + Solanaceae
	Solanum lycopersicum	Solanaceae	Amaranthaceae + Solanaceae
	Solanum nigrum	Solanaceae	Amaranthaceae + Solanaceae
	Striga asiatica	Scrofulariaceae	none (parasitic plant, scarse)
	51 species	17 families	10 functional groups

ACKNOWLEDGMENTS

This research was conducted by ERCANE and CIRAD in La Reunion. It was supported by funding from Europe (FEADER) and France (ONEMA) for MAGECAR Project, and from CIRAD and ERCANE. Authors acknowledge Régis Goebel, Guillaume Lestrelin, Florent Tivet and Krishna Naudin for the revision of the manuscript.

REMERCIEMENTS

Cette expérimentation conçue et conduite par le Cirad et eRcane à La Réunion a été majoritairement financée par l'Union européenne (FEADER) et la France (ONEMA) à travers le projet MAGECAR. Les auteurs tiennent à remercier Régis Goebel, Guillaume Lestrelin, Florent Tivet et Krishna Naudin pour la relecture du manuscrit.