François Bonnetblanc
email: francois.bonnetblanc@u-bourgogne.fr

Guillaume Herbet

Pom Charras

Mitsuhiro Hayashibe

David Guiraud

Hugues Duffau

Bénédicte Poulin-Charronnat

M Hayashibe •

D Guiraud Inria

G Herbet •

Asymmetric interhemispheric excitability evidenced by event-related potential amplitude patterns after "wide-awake surgery" of brain tumours

Keywords: Wide-awake surgery, slow-growing infiltrative brain tumours, Plasticity, Interhemispheric imbalance, cortical excitability F. Bonnetblanc, P. charras

the ERP amplitude at several sites, with few occurrences in the contralesional hemisphere. For smaller lesions (and under certain experimental conditions), cortical excitability in the injured hemisphere may increase in order to maintain local compensation. In addition, we observed and increased excitability in the contralesional frontal homologue for one patient who underwent an extensive resection. Postoperative monitoring of interhemispheric asymmetries in ERP amplitudes is of value for determining task constraints inducing electrophysiological imbalance and guiding rehabilitation.

Introduction

Real-time functional mapping of the brain with direct electrical stimulation (DEs) is used to guide the resection of slow-growing infiltrative tumours during wide-awake surgery. the DEs technique reduces the probability of resecting essential areas near or within the tumour [START_REF] Duffau H | lessons from brain mapping in surgery for lowgrade glioma: insights into associations between tumour and brain plasticity[END_REF]. During wide-awake surgery, patients perform a number of neuropsychological tests while DEs is applied both cortically and sub-cortically, in order to detect and thus preserve connectivity online (especially for critical white matter pathways). this technique enables functional mapping of the area near the tumour so that as much as non-eloquent infiltrated tissue as possible can be removed while minimizing sequelae.

slow-growing lesions prompt much greater functional brain reorganization than acute lesions do. More precisely, the recruitment of remote brain areas in the ipsi-and Abstract slow-growing, infiltrative brain tumours may modify the electrophysiological balance between the two hemispheres. to determine whether and how asymmetry in interhemispheric excitability might occur following "wide-awake surgery" for this type of tumour, we recorded electroencephalograms during a simple visuo-manual reaction time paradigm performed by five patients between 3 and 12 months after surgery. Interhemispheric excitability asymmetries were computed by comparing the amplitudes of event-related potentials (ERPs) in the injured hemisphere to those in the healthy hemisphere. For the two patients with the smallest lesions (7.1 and 11.5 cm 3 , respectively), increased excitability within the ipsilesional hemisphere was evidenced by characteristics increases in 1 3 contralesional hemispheres may be more efficient. as such, the assessment of recovery in slow-growing versus acute lesions may offer new interesting insights into brain plasticity [START_REF] Finger S | lesion momentum and behaviour[END_REF][START_REF] Finger | Brain damage and recovery: research and clinical perspectives[END_REF]stein et al. 1983;[START_REF] Bonnetblanc | low grade gliomas and cerebral plasticity: fundamental and clinical implications[END_REF][START_REF] Desmurget | contrasting acute and slow-growing lesions: a new door to brain plasticity[END_REF]. the brain's impressive reorganizational capacity enables patients with slow-growing tumours to recover substantially within days, hours [START_REF] Bonnetblanc | low grade gliomas and cerebral plasticity: fundamental and clinical implications[END_REF][START_REF] Desmurget | contrasting acute and slow-growing lesions: a new door to brain plasticity[END_REF][START_REF] Bonnetblanc | Is the right frontal cortex really crucial in the mentalizing network? a longitudinal study in patients with a slow-growing lesion[END_REF] or even minutes of surgery (sallard et al. 2012a). these impressive, rapid compensations may be partially explained by the progressive emergence of compensatory networks in the contralesional hemisphere as those of the ipsilesional hemisphere are disrupted by slow-growing tumour. hence, surgical removal of the tumour may rapidly unmask homologous contralesional networks.

although direct observations [using functional magnetic resonance imaging (fMRI) or transcranial magnetic stimulation (tMs)] have suggested that the contralesional hemisphere is involved in functional recovery, the topic needs further investigation. In particular, [START_REF] Krainik | Role of the healthy hemisphere in recovery after resection of the supplementary motor area[END_REF] used fMRI to show the involvement of a contralesional homologue after resection of the supplementary motor area. however, the significance of these functional translocations is subject to debate. Indeed, recruitment of the intact contralesional homologue may reflect a decrease in transcallosal inhibition [START_REF] Netz | hemispheric asymmetry of transcallosal inhibition in man[END_REF][START_REF] Kessler | Disturbance and recovery of language function: correlates in PEt activation studies[END_REF]). In contrast, a more recent study has rebutted these concerns by showing that tMs induces speech disruption when applied to contralesional activation sites (thiel et al. 2005). taken as a whole, these results do not mean that the ipsilesional hemisphere has a minor or inexistent role in the recovery process. In fact, one hypothesis concerning functional compensation during slow tumour growth suggests that compensatory networks could switch progressively from the ipsilesional hemisphere to the contralesional hemisphere as the lesion increases in size [START_REF] Bonnetblanc | low grade gliomas and cerebral plasticity: fundamental and clinical implications[END_REF][START_REF] Desmurget | contrasting acute and slow-growing lesions: a new door to brain plasticity[END_REF]. For focal acute lesions (induced experimentally in animals or observed in stroke patients), early increases in cortical excitability in the ipsilesional hemisphere have been observed in the vicinity of the lesion (and more specifically in the motor cortex), which is suggestive of acute functional reorganization in this area [START_REF] Jacobs | Reshaping the cortical motor map by unmasking latent intracortical connections[END_REF][START_REF] Nudo | Neural substrates for the effects of rehabilitative training on motor recovery after ischemic infarct[END_REF][START_REF] Sanes | Plasticity and primary motor cortex[END_REF]hummel andcohen 2005, 2006). however, to the best of our knowledge, changes in excitability have not been monitored in patients harbouring slow-growing tumours, and interhemispheric imbalance has not been evaluated after the resection of large volumes from a single hemisphere. On the behavioural level, some preliminary research on the responses to a visual "go" signal three or 4 days after surgery has evidenced longer reaction times for the impaired hand than for the healthy hand (sallard et al. 2012b). these hand function asymmetries are suggestive of some degree of interhemispheric imbalance.

to further document (1) the role of each hemisphere in this compensation and (2) possible electrophysiological imbalances after surgery, we measured the electroencephalographic (EEG) activity of patients performing an analogue visuo-manual reaction time paradigm between 3 and 12 months after surgery. We hypothesized that the postlesional interhemispheric imbalance could be evidenced by characteristic changes in cortical excitability (as measured by EEG) in general and interhemispheric imbalance in event-related potential (ERP) amplitudes in particular. Importantly, we measured cortical excitability by recording ERPs (rather the classical motor or sensory evoked potentials) in an ecological visuo-manual reaction time paradigm. In theory, each hemisphere's involvement in the latter task varies as a function of the location of the stimulus and the hand used to react. For instance, the role of the left hemisphere in task performance will supposedly be greatest for a stimulus displayed in the right hemifield and for a response given with the right hand. the use of other combinations of the visual stimulus location and the responding hand will supposedly enable the experimenter to access each hemisphere to a different extent. consequently, we also sought to determine whether the measured changes in interhemispheric excitability were correlated (at least to some extent) with estimations provided by various stimulus location/hand-used combinations.

the brain reorganization that occurs during slow lesion growth and then after resection of a large part of the brain is critical in many aspects. On the electrophysiological level, dynamic aspects of the brain may be considerably altered, and EEG signals are necessarily modified by the presence of the cavity. the EEG analyses that are performed usually on healthy subjects or patients with a complete brain volume (as well as pre-vs. post-operative comparisons in resected patients) can only be performed if the signal distortion (due to the cavity) is removed by complex modelling of the lesion and signal conduction pathways. to circumvent these problems and concentrate on excitability changes in the injured and healthy hemispheres, we focused on post-operative comparisons. this approach to understanding interhemispheric dynamics is fully valid, since it is also performed in stroke patients (from whom no pre-lesion data are available). to exclude complex changes in the dynamics of ERP signals (due to signal distortion and changes in sources location), we simply computed an asymmetry index by comparing the ERP amplitudes at pairs of homologous electrodes. hence, simple post-operative analyses of ERP amplitudes enabled us to investigate asymmetries in interhemispheric excitability.

Materials and methods

Participants

Five right-handed patients (P1 to P5; 4 males and 1 female; mean ± sD age 39 ± 5.34, range = [33; 45]) and eight agematched, right-handed healthy controls (7 males and 1 female; mean age 39. 75 ± 4.30, range = [33; 45]) participated in the study. None of the controls had a history of neurological disease (including motor, visual hemifield or visuospatial impairments). the study complied with the World Medical association's code of Ethics (the 1964 Declaration of helsinki and its amendments). the patients' socio-demographic and clinical characteristics are summarized in table 1.

P1 (a 38-year-old male) presented with a right temporal tumour (Fig. 1a). P2 and P3 (a 44-year-old female and a 33-year-old male, respectively) presented with a right frontal tumour (Fig. 1b,c). P4 (a 35-year-old male) presented with a tumour at right middle and inferior temporal area. P5 (a 45-year-old male, respectively) presented with a tumour in the right temporal pole (Fig. 1d,e). all patients underwent the surgical resection of low-grade glioma (grade II, according to the World health Organization classification) with intra-operative DEs.

lesion mapping was based on whole-brain three-dimensional t1 anatomical MRI data sets acquired 3 months after surgery (on a 3t siemens skyra machine, siemens Medical systems, Erlangen, Germany). these individual MRI data sets were then normalized against the Montreal Neurological Institute template by applying cost function masking [START_REF] Brett | spatial normalization of brain images with focal lesions using cost function masking[END_REF]. this method has been very well characterized and is known to be the best option for normalization ), and two had much smaller resections (7.1 and 11.5 cm 3 ). P2 was undergoing her second surgical operation on the frontal lobe. the first operation had been performed in 2004 by another neurosurgeon, but the lesion had continued to grow in the intervening time.

the behavioural task Participants comfortably sat in front of a laptop computer on which different visual stimuli (including the "go" signal) were displayed with Presentation ® software (http://www.neurobs.com). the screen was about 60 cm from the body in the sagittal axis and 45 cm below eye level. two switches were positioned on each side of the sagittal axis (15 cm apart and 45 cm from the body). the motor responses consisted of pressing one or both switches with the corresponding index finger(s). a fixation cross appeared in the middle of the screen and served as the warning signal. after a variable time interval, one or two arrows appeared at different locations (represented the "go" signal); the participant sawing one or two arrows then had to press the corresponding button(s) as soon as possible. the arrow(s) could appear to the left of the initial fixation cross (designated as ls), to the right (Rs), to both the left and the right at the same time (R + ls) or at the fixation cross's central position (centre, cs). the time interval between the appearance of the fixation cross and the appearance of the "go" signal (i.e. the stimulus onset asynchrony) varied randomly from one trial to another in the range 1,000 to 2,000 ms. this interval comprised two phases. Firstly, the fixation cross was displayed for a duration between 1,000 and 1,500 ms. after the disappearance of fixation cross, the stimulus appeared after a variable interval of between 0 and 500 ms. the four combinations of visual stimuli (RS, LS, R + LS and CS) were randomized within three blocks that depended on the hand(s) used (i.e. Rh, lh and R + lh), giving a total of twelve conditions. a total of 360 trials (30 trials × 12 conditions) were performed in two separate sessions.

Event-related potential recording and pre-processing the electroencephalogram was recorded at 64 scalp sites using a 10-20 system (Biosemi activetwo). horizontal eye movements were recorded with electrodes placed on the outer left and right canthi. Vertical eye movements were recorded with an electrode placed under the right eye. two additional electrodes were placed on the left and right mastoids (a1, a2). During recording, the Biosemi system's common mode sense electrode served as the reference electrode. Electrophysiological signals were digitized at a 2,048 hz sampling rate and acquired with actiView software. the EEG signals were analysed using the EEGlaB toolbox for Matlab. the data were re-referenced against the mean of electrodes a1 and a2, downsampled to 250 hz and high-pass filtered with a cut-off frequency of 1 hz. In order to correct for eye movement artefacts, EEGlaB's binica routine was used to perform independent component analyses by removing the first component. all the corrections underwent visual inspection. For each condition, each patient's ERPs were averaged over a latency window going from -1,000 ms prestimulus to 400 ms post-stimulus.

Data analyses

Behavioural and ERP data for each individual patient were compared with the 95 % confidence interval (95 % cI) determined for the control group. In order to compare the ERP amplitudes in the right (injured) and left (healthy) hemispheres, we computed an asymmetry index (aI) for each pair of homologous electrodes and in each experimental condition: aI = -[|Right ERP| -|left ERP on the homologous site|]/the peak ERP at any site. to assess possible interhemispheric imbalances in cortical excitability in each experimental condition, we determined whether a given patient's aI was outside the 95 % cI derived from the control group for a cumulative duration of 100 ms during a time window ranging from the stimulus onset (t 0 = 0 ms) to the motor response (t ≈ 400 ms). as shown in Fig. 2, a negative value (above the blue line, since the scales are inverted) meant that the normalized ERP amplitude in the right (injured) hemisphere (at c4, for instance) was higher than that at the homologous site (c3) and that the asymmetry exceeded that observed in healthy subjects. conversely, a positive value (below the red line) meant that 1 3

the normalized ERP amplitude in the right hemisphere (at c4, for instance) was lower to that of the homologous site (c3) and that the asymmetry exceeded that observed in healthy subjects. It is important to note that the aI was then used qualitatively to determine the location and number of homologous pairs of electrodes that showed a right versus left imbalance in the ERP amplitude. the method is illustrated in Fig. 2.

Results

Behavioural data

For each experimental condition, we compared each patient's individual performances with the 95 % cI from the control group. as shown in Fig. 3, the post-operative values indicated a general slowing of reaction times for patients P1 (with extensive temporal lobe resection) and P2 (with extensive frontal lobe resection). Despite having undergone extensive resection of the frontal lobe, P3 exhibited normal reaction times post-operatively (but not pre-operatively). For the two patients with smaller lesions, reaction times were normal for P5 and even abnormally shorter for P4. By comparing data for the right and left hands, we found that reaction times for the impaired (left) hand were abnormally slow for P1 in the unimanual condition (p < .05), but not in the bimanual condition. Functional asymmetry between the right and left hands was much less marked for patients P2 to P5.

Event-related potential data

For each patient and each experimental condition, we first computed the overall number of electrodes with an aI outside the 95 % cI for controls. these data are summarized in Fig. 4. We found that there were more electrode sites with increased right (injured) hemisphere amplitudes than with increased left hemisphere amplitudes. this was particularly true for the two patients with the smallest lesions in the l + Rh -LS and l + Rh -CS conditions (black circles in Fig. 4). In contrast, the two patients with the frontal lesions (P2 and P3) had more sites with increased left hemisphere amplitudes in the Rh -RS and Rh -L + RS conditions (grey circles in Fig. 4); this was especially true for P2.

Next, we investigated the increased excitability of the injured hemisphere observed in P4 and P5. In particular, we sought to determine the topological organization of the aIs that were outside the 95 % cI in the two most characteristic experimental conditions (l + Rh -LS and l + Rh -CS). as shown in Fig. 5, increased excitability in an area centred on the right motor cortex was observed in both patients.

similarly, we sought to determine which areas of P2's healthy (left) hemisphere exhibited increased excitability. as shown in Fig. 6, we found that the main increase in cortical excitability was located in the left hemisphere. several sites in the contralesional frontal areas demonstrated increased excitability after surgery, although similar increases were also present before surgery (Fig. 4). It must be borne in mind that this patient had already undergone brain surgery prior to the present study. One could hypothesize that this contralesional increase in cortical excitability was artifactual and was due to the difference between ERPs measured over non-resected and resected areas. however, this is unlikely because ERP amplitudes measured over the contralesional frontal areas were even greater than those measured over the dorsal areas in the same experimental condition (Fig. 7). Furthermore, the fact that the increase in cortical excitability in the contralesional frontal part was observed (1) before surgery and (2) for a small subset of the (randomized) experimental conditions indicates that it was not an artefact in P2.

Discussion

We studied five patients after wide-awake surgery for slowgrowing tumours, in order to establish whether interhemispheric imbalance and asymmetries in cortical excitability could be measured by comparing left and right hemisphere ERP amplitudes. Furthermore, we sought to determine whether the measured changes in interhemispheric excitability would mirror (at least to some extent) estimations given by combination for the stimulus site and the hand used to react to the stimulus.

the methodology that we developed to measure changes in cortical excitability is easy to use in a clinical setting, since we simply computed interhemispheric differences in ERP magnitudes during an ecological paradigm. to the best of our knowledge, this is the first attempt to use this type of a method. this ERP-based method may be especially useful for patients having undergone wide-awake surgery, in whom interhemispheric dynamics are considerably modified.

Our results demonstrated that increased ERP magnitudes could be observed in the central part of the right (injured) hemisphere for the two patients with the smallest lesions. Few signs of increased excitability were observed in the healthy hemisphere for these two patients. In contrast (and to a lesser extent), increased excitability was observed in the contralesional homologue for the patients with much more extensive lesions. P2 (with a frontal lesion) exhibited clear signs of hyperexcitability in the frontal part of the healthy hemisphere both before and after the present surgery. It is important to note that two of the three patients with large resections were the most impaired in terms of reaction times. conversely, the two patients with the smallest lesions exhibited faster reaction times than most of the control subjects (Fig. 3). For P1 (with an important temporal resection), most of the increased excitability was observed in the right hemisphere. this patient had abnormally slow reaction times, as previously observed for patients with lesions in the parietal lobe or the temporoparietal junction (sallard et al. 2012b). In agreement with the latter study, we observed functional asymmetry (i.e. longer reaction times for the left (impaired) hand than the right (lesioned) hand) in the single-handed conditions only. the opposite functional asymmetry was observed in bimanual conditions (i.e. longer reaction times for the right (healthy) hand (Fig. 3). On the whole, behavioural performances were independent of lesion size but hand responses appeared to be more affected when the parietal lobe or the temporoparietal junction had been extensively resected. this observation suggests that the deeper the surgery in this part of the brain, the more the white matter pathways required for hand rapid responses to visual stimuli are disconnected. In other words, when the resection mainly involves "U" fibres (i.e. local connectivity) at the surface of the parietal lobe or in the temporoparietal junction, behavioural performances are not impaired-as demonstrated for the two patients with the smallest lesions. Our results show that increases in ERP magnitudes and cortical excitability can be observed in the injured hemisphere after wide-awake surgery when the volume of tumour resection is low (7.1 or the right and left hemisphere. a negative value (above the blue line, since the scales are inverted) meant that the normalized ERP amplitude in the right (injured) hemisphere was higher than that at the homologous site and that the asymmetry exceeded that observed in healthy subjects. conversely, a positive value (below the red line) meant that the normalized ERP amplitude in the right hemisphere was lower to that of the homologous site and that the asymmetry exceeded that observed in healthy subjects. ERP event-related potential, L + RH -CS left + right hand with the central stimulus (arrow) 1 3 11.5 cm 3 , in the present study) and only local connectivity through "U" fibres is altered. For P5, this observation was still valid more than 12 months after surgery. In the latter case, some local connectivity may still remain-enabling local or perilesional remodelling, rather than migration of the original function to more remote areas of the brain (as is suspected for more extensive lesions [START_REF] Bonnetblanc | low grade gliomas and cerebral plasticity: fundamental and clinical implications[END_REF][START_REF] Desmurget | contrasting acute and slow-growing lesions: a new door to brain plasticity[END_REF]). as a consequence, surgery may induce lesions in a slightly reorganized network in which the local connectivity is less affected; in turn, this would decrease the degree of local inhibition. similarly, Jacobs and Donoghue (1991) used the pharmacological blockade of cortical inhibition to demonstrate the role of inhibitory circuits in the shaping of cortical motor maps and unmasking of latent, intracortical connections.

In the present study, the electrophysiological imbalance in patients with small lesions did not appear to have a detrimental effect on behavioural performance levels. For extensive lesions, local connectivity is much more impaired and local disinhibition may be prevented. Increases in cortical excitability in the injured hemisphere may be limited by the lesion's extent and boundaries. We were surprised to observe that for patients with small lesions, there were more ERP increases in the right (injured) hemisphere in the condition that (in theory) would have minimized interhemispheric imbalance (l + Rh -CS). Indeed, since both hands were involved, the reciprocal interhemispheric inhibitions would supposedly be symmetrical. Given that the stimulus was presented in the sagittal axis, symmetrical visual processing would have also been expected. at the EEG level, bimanual coordination seems particularly challenging for interhemispheric balance in these patients. More exactly, simultaneously coordination of the two hands would maximize interhemispheric transfer. In such a case, the motor network in the injured hemisphere may saturate in response to increased input and because of the reorganization in functional connectivity. this saturation may induce an increase in ERP amplitude and cortical excitability. In summary, one can consider that increased ERP amplitude reflects difficulties in separating and Fig. 6 asymmetry indexes computed for P2 (with a large frontal lesion: green line) at each electrode site, relative to the upper bound (red line) and lower bound (blue line) of the 95 % cI in the Rh -RS condition. sites at which the aI exceeded the upper bound of the 95 % cI are indicated by the symbol "+". Increased cortical excitability in the contralesional frontal homologous electrode was observed for this patient. For the sake of clarity, aIs are shown in the right hemisphere only. however, note that aIs represent a direct comparison of homologue pairs of electrodes between the right and left hemisphere. a negative value (above the blue line, since the scales are inverted) meant that the normalized ERP amplitude in the right (injured) hemisphere was higher than that at the homologous site and that the asymmetry exceeded that observed in healthy subjects. conversely, a positive value (below the red line) meant that the normalized ERP amplitude in the right hemisphere was lower to that of the homologous site and that the asymmetry exceeded that observed in healthy subjects. ERP event-related potential, RH-RS right hand with the left stimulus (arrow) 1 3 processing the sources of information that arise in each of the two hemispheres and converge towards a smaller network. More specifically, local inhibition may become less effective. this hypothesis could be tested by using pairedpulse tMs paradigms to assess intracortical inhibition.

We were intrigued to read that in patients with extensive lesions of the right temporoparietal junction (that overlap with lesions observed in our patients), bilateral stimulation (whether visual or tactile) induced the extinction of the visual stimuli applied to the impaired hemifield or hemibody (but not those applied to the healthy ones) [START_REF] Driver | Perceptual awareness and its loss in unilateral neglect and extinction[END_REF]. similarly, we found the bilateral and symmetrical stimulation was the most challenging paradigm in our two patients with small lesions (in the right middle and inferior temporal area and in the temporal pole, respectively)-despite the fact that these patients did not exhibit any sign of visual extinction. as shown previously (sallard et al. 2012b) and in the present study, major lesions in the temporoparietal junction induce a general increase in reaction times for both hands; however, the increase is greatest for the healthy (right) hand when both hands are used synchronously in bilateral conditions. the opposite results are observed under unilateral conditions. hence, one can hypothesize that extensive lesions to the right temporoparietal junction or to a distributed network involving the arcuate fasciculus and its ramifications in the temporal lobe result in desynchronization of both the right and left hands. More exactly, it suggests that this right distributed temporal network (especially the deeper part) set the pace for the left hemisphere and synchronizes the right hand with the left hand during simultaneous bimanual movements. Right hand movement may depend on that of the left hand. We did not observe these effects in patients with more frontal lesions (i.e. the left hand was more impaired in all conditions). It is also intriguing to note that for patients with smaller lesions in the right middle and inferior temporal area and in the temporal pole, respectively, hand movements on both body sides were fast and well synchronized-even though excitability was greater in the right hemisphere only. the latter increase may have been required to maintain bimanual synchronization.

We observed a marked increase in excitability in the contralesional homologue for a patient with an extensive frontal lesion (P2). Importantly, these increases did not appear to be artifactual and may evidence compensation by the contralesional frontal hemisphere. however, the reasons for this increase in cortical excitability are more difficult to interpret, since this phenomenon was not observed Fig. 7 Data on all ERPs for P2 in the Rh -RS condition, showing that ERP amplitudes were already greater in frontal sites than in dorsal sites before the aIs were computed. the value of this control experiment was further strengthened by the fact that increases in ERP amplitude were never observed in the contralesional frontal homologue under any of the experimental conditions. We therefore concluded that the observed increases in frontal sites were not artifactual. ERP event-related potential, RH -RS right hand with the right stimulus (arrow) 1 3 in the other patient with a frontal lesion. these increases in the contralesional frontal hemisphere were observed for two conditions that (in theory) were not supposed to be particularly challenging for the right (injured) hemisphere (Rh -RS and Rh -L + RS) because they should have involved the left (healthy) perceptual-motor network.

the present study had several limitations. the sample size was small because the pre-and post-operative assessments were lengthy and burdensome. Furthermore, the tumour size and site varied markedly from one patient to another. to circumvent these limitations, we used a differential approach in which individual performance levels were compared with a baseline. For each condition, each patient's performance was compared with the 95 % cI derived from a control group. Differential approaches are particularly appropriate when between-subjects variability is large (as in many clinical studies) and so are fully justified in the present case. this approach was even more justified when the variability in the lesion extent and size is considered. It is important to bear in mind that lesion variability is always present in studies of patients with brain lesions. Even though researchers try to control for lesion size and locations, lesion reconstruction (when performed) is not usually accurate enough to determine which white matter pathways are injured, for instance. Furthermore, as demonstrated by functional mapping of the brain, there is also huge between-subjects variability in anatomical landmarks and functional performance. In many cases, datasets from individual patients are averaged-constituting a major limitation. Given that white matter pathways link remote brain areas with various functions, small differences in the state of damage may lead to complete different functional states. accordingly, we consider that our differential approach was justified for determining whether it was possible to measure changes in interhemispheric electrophysiological balance after wide-awake surgery.

With our method, the ERP amplitudes in the two hemispheres are symmetrical before the stimulus, but are modified afterwards; we can therefore consider that the interhemispheric excitability has been modified. For a given stimulus intensity and baseline EEG signal, increases in excitability are defined as an increase in the amplitude of response. classically, changes in cortical excitability in the motor domain and in sensory domains (visual and auditory domains, etc.) are studied by measuring evoked potentials. Furthermore, evoked potentials are always a combination of multiple sources and generators. EEG signals will always have several sources. lastly, the presence of a preparatory period (due to the fixation cross prior to the appearance of the visual stimulus) does not invalid our methodology and results. For each patient and each experimental condition, we determined whether the ERP values were abnormal (i.e. outside the 95 % cI derived from a control group) after the stimulus onset. conversely, we checked that the ERP values were normal (i.e. within the 95 % cI) before the stimulus onset. In other words, any pre-motor potential that may have occurred was symmetrical in the two hemispheres. In contrast, we observed asymmetrical ERP values when comparing the hemispheres after the "go" signal.

the type of surgical procedure (wide awake vs. anesthetized) should not have any influence on the results. however, the wide-awake procedure allows a functional mapping of cortical and subcortical areas around the tumour using direct electrical stimulation. as such, the resection is more precise and can preserve the connectivity online much more than classical procedures with anaesthesia. Using our method, the hypothesis that identical results and changes in local excitability could also be observed, with more classical surgical procedures (and for different nature of the lesion), is appealing, but it needs to be demonstrated.

to the best of our knowledge, this is the first study to have compared interhemispheric ERP amplitudes with a view to detecting asymmetries in cortical excitability. Our simple method may have value as a clinical tool for assessing and then monitoring interhemispheric electrophysiological imbalance in these patients with a slow-growing tumour. More particularly, this measurement method may have clinical value for determining which environmental and visuo-manual task constraints induce interhemispheric electrophysiological asymmetries (especially if behavioural measurements are not enough accurate). this simple approach may help to fine-tune rehabilitation and occupational therapy by identifying the situations that perturb the patient's interhemispheric balance. Our results also allowed strengthening a number of hypotheses concerning recovery and plasticity in patients with slow-growing lesions [START_REF] Bonnetblanc | low grade gliomas and cerebral plasticity: fundamental and clinical implications[END_REF][START_REF] Desmurget | contrasting acute and slow-growing lesions: a new door to brain plasticity[END_REF]). We found that the smaller the lesion, the more the ipsilesional hemisphere was involved in the recovery. conversely, the greater the lesion, the more the contralesional hemisphere was involved. For patients with smaller lesions, the interhemispheric imbalance in excitability is the opposite of what is observed in stroke patients (especially during the chronic phase of stroke) (hummel and cohen 2005, 2006). With a view to better understanding brain plasticity and neuromodulation, there is a continued need to study how slowgrowing lesions induce recovery processes and influence brain dynamics.

Fig. 2

 2 Fig.2Illustration of the aIs computed for two patients (P4: cyan line; P3: green line) for a single pair of homologous electrodes (e.g. c3-c4), in comparison with the upper bound (red line) and lower bound (blue line) of the 95 % cI determined for the same pair of homologous electrodes in control subjects. Negative values (i.e. below the lower bound of the 95 % cI) after the "go" signal (t = 0 ms) therefore correspond to a greater ERP amplitude in the right (injured) hemisphere than in the left (healthy) hemisphere. con-

Fig. 3

 3 Fig. 3 Reaction times for control subjects (histograms: means ± 95 % cI) and the patients (geometric forms: individual means) in each of the experimental conditions. CI confidence interval, CS central stimulus,

Fig. 4

 4 Fig. 4 Number of electrode sites with an aI outside the 95 % cI in the right hemisphere (positive aI, black) and left hemisphere (negative aI, grey) for each patient and under all experimental conditions.

Fig. 5

 5 Fig. 5 asymmetry indices computed for the two patients with the smallest lesions (P4: cyan line; P5: green line) and for each all electrode site, in comparison with the upper bound (red line) and lower bound (blue line) of the 95 % cI in the l + Rh -CS condition. For each patient, sites for which the aI was below the lower bound of the 95 % cI are indicated by the symbol "+". Increased cortical excitability was observed at several sites in the right (injured) hemisphere in these two patients with smaller lesions. For the sake of clarity, aIs are shown in the right hemisphere only. however, note that aIs represent a direct comparison of homologue pairs of electrodes between

Table 1

 1 when the lesion is extensive(andersen et al. 2010). During the registration process, cost function masking avoids the bias caused by abnormal, lesion-induced radiological signals. Briefly, the lesion was contoured (using MRIcron software (http://www.mccauslandcenter.sc.edu/mric ro/mricron) and transformed into a binarized image. this image was inserted during the normalization processes performed here with sPM 8 (http://www.fil.ion.ucl.ac.uk/spm/ software/spm8). the lesion is then redrawn on the normalized scan, in order to generate an individual volume of interest (illustrated in Fig.1for each patient). clinical characteristics of the study population Patients were thoroughly examined by a neuropsychologist before and after surgery. None had clinically evident neurological disorders. Data on surgical resection volumes are summarized in table 1. three patients had extensive resections(79.2, 119.1 and 116.8 cm 3

		socio-demographic and clinical data
	socio-demographic				clinical			time of assessment after
	Id.	age (years)	Gender	Edu. level	handed ness	Grade of	surgery	Volume of resection	the surgery (months)
				(years)		the tumour		(cm 3 )
	P1	38	M	15	R	II	awake	79.2	3
	P2	44	F	12	R	II	awake	119.1	3
	P3	33	M	17	R	II	awake	116.8	9
	P4	35	M	17	R	II	awake	7.1	3
	P5	45	M	17	R	II	awake	11.5	12
	Fig. 1 lesion mapping results
	for each patient

Acknowledgments this study was funded by the Association pour la Recherche sur le Cancer charity (Grant 3184). We thank the patients for their participation in the study. We also thank the three anonymous reviewers for their constructive and stimulating comments about the present work.

Conflict of interest None.


Nicolas Bertheuil

Laurent Sulpice

Giovanni Battista

Levi Sandri

Vincent Lavoué

Eric Watier

Bernard Meunier

Inguinal Lymphadenectomy For Stage Iii Melanoma: A Comparative Study Of Two Surgical Approaches At The Onset Of Lymphoedema

Keywords: inguinal lymphadenectomy, lymphedema, stage III melanoma, surgery

Introduction: When sentinel lymph node is positive for metastasis (exclusion for micro-metastasis) and in cases of palpable adenopathy, a lymphadenectomy should be performed. Many incisional surgical approach have been described in literature. We perform two type of incision (vertical with skin excision and transversal) for inguinal lymphadenectomy. The aim of this study was to compare post-operative morbidity between these two approach in cases of Stage III Melanoma. We analyzed chronic lymphoedeme, skin necrosis, wound dehiscence, wound infection and seroma rates between the two techniques.

Methods: From April 2000 to February 2012 fifty-three patients underwent to inguinal lymphadenectomy for Stage III melanoma at CHU of Rennes. Patients were stratified in 2 groups according to the surgical approach, group 1 with a vertical incision with skin excision and group 2 with a transverse incision.

Results: Chronic lymphedema rate for group 1 was 37.04% and for group 2 rate was 26.92%, this complication was lower un group 2 but no significant difference was observed (p=0.558). Skin necrosis (p=0.235), wound dehiscence (p=1.000), wound infection (p=0.236) and seroma (p=0.757) were not significantly different. Two cases of skin necrosis were observed in group 2 (7.69%) and none in group 1.

Conclusion:

We do not found significant difference for chronic lymphedema between these two approach. However, we had less lymphedema with the transversal technique which has the advantage to reduce the skin suffering when external iliac lymphadenectomy dissection is necessary in addition to the inguinal lymphadenectomy.

Introduction

The incidence of melanoma has been increasing over the past few decades [START_REF] Grange | Epidemiology of cutaneous melanoma: descriptive data in France and Europe[END_REF] and multidisciplinary treatment is recommended. A lymphadenectomy should be performed when sentinel lymph nodes are positive for metastasis (excluding micrometastases <0.1 mm [START_REF] Van Akkooi | Clinical relevance of melanoma micrometastases (<0.1 mm) in sentinel nodes: are these nodes to be considered negative?[END_REF]) and in cases of palpable lymphadenopathy [START_REF] Négrier | Clinical practice guideline: 2005 update of recommendations for the management of patients with cutaneous melanoma without distant metastases (summary report)[END_REF]. Many incisional surgical approaches have been described in the literature. To prevent the skin necrosis that is observed with vertical incisions, a modified technique has been described using an ellipse-shaped incision [START_REF] Baas | Groin dissection in the treatment of lower-extremity melanoma. Short-term and long-term morbidity[END_REF][START_REF] Poos | Therapeutic groin dissection for melanoma: risk factors for short term morbidity[END_REF][START_REF] Van Der Ploeg | A modified technique of groin dissection[END_REF].

A transverse incision has been described when inguinal lymphadenectomy must be completed with external iliac lymphadenectomy. The transverse incision avoids a 'T' incision and reduces skin lesions (wound dehiscence, skin necrosis). However, a few recent series have reported morbidity rates ranging from 10% to 50% with this technique [START_REF] Baas | Groin dissection in the treatment of lower-extremity melanoma. Short-term and long-term morbidity[END_REF][START_REF] Van Akkooi | Morbidity and prognosis after therapeutic lymph node dissections for malignant melanoma[END_REF][START_REF] Dardarian | Saphenous vein sparing during inguinal lymphadenectomy to reduce morbidity in patients with vulvar carcinoma[END_REF]. The aim of this study was to compare post-operative morbidity between the transverse and vertical incision approaches in cases of stage III melanoma. We compared chronic lymphoedema, skin necrosis, wound dehiscence, wound infection and seroma rates between the two techniques.

Materials and Methods

From April 2000 to February 2012, 53 patients underwent inguinal lymphadenectomy for stage III melanoma at CHU Rennes. Patients were stratified into two groups according to the surgical approach that was used as follows: group 1 had vertical incisions and group 2 had transverse incisions. Clinical variables were compared between the two groups, including age, sex, height, weight, body mass index, smoking habits before surgery, histological type of melanoma, Breslow depth, Clark level, presence of ulceration and regression. Information regarding surgical complications, such as chronic lymphoedema, skin necrosis, wound dehiscence, wound infection, seroma and 1-year survival, was collected from all patients. Due to the retrospective nature of the study, lymphoedema was considered as 'present' or 'not present', and it was not possible to confirm the stage of lymphoedema in all patients. This study was approved by our Institutional Review Board, performed in accordance with

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

the principles of the Declaration of Helsinki (1964) and was in agreement with French bioethics laws (7 July 2011). We obtained informed consent from all patients.

Surgical Procedures

The two procedures are described in Figure 1. In group 1 (Figure 1-A), a vertical incision with skin excision was made. Drawings were made with the patient in the supine position; an upper point was marked at the femoral pulse under the inguinal ligament and a lower point was marked at the junction between the adductor longus and the sartorius muscle. An ellipse was marked on the skin between these two points, the amount of skin resection is determined by pitching the skin. An incision was made on the superficial fascia, including the skin in the specimen. In group 2 (Figure 1-B), a transverse incision was made at a three-finger distance below the inguinal ligament. The decision regarding which incision to use was made by the surgeon in the operating room. The remaining steps of the surgical procedure were identical for both groups. Briefly, the great saphenous vein was systematically ligated, followed by lymphadenectomy of the tissue under the superficial fascial layer, over the aponeurosis of the adductor longus and over the aponeurosis of the sartorius up to the inguinal ligament deep to the femoral pedicle. Finally, two suction drains were placed and skin closure was achieved in three planes without tension using 2/0 Vicryl® for the subcutaneous plane, 3/0 Monocryl® for the intradermal plane and staples for the skin. A compression bandage was maintained until drain removal.

Statistical Analysis

Data analyses were performed using the SPSS statistical software (version 16.0 for Windows, SPSS, Inc., Chicago, IL, USA). The Student's t-test, Mann-Whitney U-test and Fisher test were used where appropriate. Statistical significance was defined by p 0,5.

Results

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

We retrospectively analysed 53 patients, 27 in group 1 (G1: vertical incision with skin excision) and 26 in group 2 (G2: transverse incision). The results are summarized in Table 1. No significant differences were observed in patient characteristics between the two groups. The mean age was 48.85 years (G1) vs. 56.73 years (G2) (p = 0.071), the mean height was 167.58 cm (G1) vs. 169.35 cm (G2) (p = 0.562) and the mean body mass index was 24.54 kg/m 2 (G1) vs. 25.24 kg/m 2 (G2) (p = 0.527).

Similarly, no significant differences were found in sex (p = 0.586), histological type (p = 0.293), Breslow depth (G1 = 3.0 mm vs. G2 = 3.1 mm, p = 0.516), Clark level (p = 0.218), ulceration (p = 1.000) and regression (p = 0.146).

No significant difference was observed in the rate of chronic lymphoedema between groups 1 (37.04%) and 2 (26.92%) (p = 0.558). There were no significant differences in skin necrosis (p = 0.235), wound dehiscence (p = 1.000), wound infection (p = 0.236) and seroma (p = 0.757) between the two groups. Furthermore, there was no significant difference in the 1-year survival (p = 1.000) between the two groups. The post-operative morbidity rates between the two groups are summarized in Table 2.

Discussion

Many previous studies have described modifications of the incisional techniques for inguinal lymphadenectomy because morbidity rates associated with this procedure remain high [START_REF] Baas | Groin dissection in the treatment of lower-extremity melanoma. Short-term and long-term morbidity[END_REF][START_REF] Poos | Therapeutic groin dissection for melanoma: risk factors for short term morbidity[END_REF][START_REF] Van Der Ploeg | A modified technique of groin dissection[END_REF][START_REF] Van Akkooi | Morbidity and prognosis after therapeutic lymph node dissections for malignant melanoma[END_REF][START_REF] Fraley | Radical ilio-inguinal node dissection: the skin bridge technique. A new procedure[END_REF]. A prospective randomized study of two transverse incisions-one above, and one below, the inguinal ligament-was performed on 62 patients [START_REF] Manci | Inguinofemoral lymphadenectomy: randomized trial comparing inguinal skin access above or below the inguinal ligament[END_REF]. A higher complication rate was observed in the group whose incision was below the inguinal ligament (the same procedure described in our study) but no significant difference in chronic lymphoedema or survival was observed between the two groups.

In our study, the transverse incision was favoured because it avoided a « T » incision in cases where crural or external iliac lymphadenectomy was required. Transverse incisions should reduce the risk of skin suffering. Although statistical significance was not achieved, less lymphoedema was observed in

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

G2, which may be explained by the reduction in the 'tourniquet effect' observed in cases with a vertical incision, which is caused by healing in the member axis (Figure 2).

The vertical incision (as used in group 1) offers better exposure for lymphadenectomy but may increase skin suffering; thus, many surgeons perform an ellipse-shaped incision to reduce this risk (Figure 1-A).

We observed a higher complication rate of chronic lymphoedema in group 1 compared with group 2 (37.04%vs.26.92% , p = 0.558), though the difference was not statistically significant. Two cases of skin necrosis were observed in group 2 (7.69%) vs. none in group 1. This may underline the utility of an ellipse-shaped incision to reduce skin suffering when a vertical incision is chosen.

No significant difference in lymphoedema was observed between the two groups. Previous authors have described procedures to decrease the incidence of lymphoedema, including preservation of the great saphenous vein [START_REF] Dardarian | Saphenous vein sparing during inguinal lymphadenectomy to reduce morbidity in patients with vulvar carcinoma[END_REF][START_REF] Abbas | Systematic review and meta-analysis of the used surgical techniques to reduce leg lymphedema following radical inguinal nodes dissection[END_REF]12]. Zhang et al. described a 50% decrease in lymphoedema with this method in a study of 64 patients with vulvar carcinoma, with no difference in survival was observed [12]. Dardarian et al. demonstrated a reduction in chronic lymphoedema from 39% to 11% (p < 0.05) associated with a reduction in cellulitis from 45% to 0% (p < 0.001) and a reduction in wound dehiscence from 25% to 0% (p < 0.02) [START_REF] Dardarian | Saphenous vein sparing during inguinal lymphadenectomy to reduce morbidity in patients with vulvar carcinoma[END_REF]. A recent meta-analysis confirmed Dardarian's results regarding lymphoedema reduction with an odds ratio of 0.24 and a confidence interval of 0.11-0.53 without local recurrence [START_REF] Abbas | Systematic review and meta-analysis of the used surgical techniques to reduce leg lymphedema following radical inguinal nodes dissection[END_REF].

New solutions to reduce lymphoedema have been proposed. Preservation of muscular fascia was described with an associated incidence of chronic lymphoedema of 14% [13], which is lower than both our series and other publications [START_REF] Van Akkooi | Morbidity and prognosis after therapeutic lymph node dissections for malignant melanoma[END_REF]. Sartorius transposition was evaluated and did not reduce postoperative wound morbidity [14]. Transplantation of the greater omentum in the treatment of chronic lymphoedema has been described with positive results, but it was a small case series and an abdominal approach was necessary [START_REF] Abbas | Systematic review and meta-analysis of the used surgical techniques to reduce leg lymphedema following radical inguinal nodes dissection[END_REF]15].

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Although our study was a retrospective cohort study, to the best of our knowledge, it is the first comparative study of these two incisional approaches for inguinal lymphadenectomy for stage III melanoma.

Conclusions

We did not find a significant difference in the incidence of chronic lymphoedema between patients who had transverse and vertical incisions. With regard to the complication rates of inguinal lymphadenectomy for stage III melanoma, we plan to conduct a prospective study comparing the two incisions to validate the present results.

Acknowledgment: Authors like to thank Camille Buscail, MD (service d'épidémiologie et de santé publique, Hopital Pontchaillou, CHU Rennes, Rennes, France) for statistic analysis.

Conflict of interest disclosure: All Authors do not have conflict of interest. This paper has not been funded.

Figure 1 :

 1 Figure 1: The two-incision procedure. A: A vertical incision with an ellipse-shaped incision (Group 1) B: A transverse incision made at a three-finger distance below the inguinal ligament (Group 2).

Figure 2 :

 2 Figure 2: A: A vertical incision. Blue arrows indicate the healing process; red arrows indicate the 'tourniquet effect'. B: A transverse incision. Blue arrows indicate the healing process that may avoid the 'tourniquet effect'.

  Figure 2. A: A vertical incision. Blue arrows indicate the healing process; red arrows indicate the 'tourniquet effect'. B: A transverse incision. Blue arrows indicate the healing process that may avoid the 'tourniquet effect'.

Table 1 .

 1 Patients characteristics Student test, ** Mann Whitney test, *** Fisher test. SSM: spreading superficial melanoma. Quantitative data value are express in mean (except Breslow express in median) Qualitative data are express in number and percentage.

					p value
	Variable		Group 1	Group 2	(*,**,***)
	Number		27	26
	Age		48,85 +/-15,89	56,73 +/-15,29	0,071	*
	Size		167,58 +/-9,75	169,35 +/-9,70	0,562	*
	Weigh		63,37 +/-13,19	72,16 +/-12,36	0,326	*
	Gender	Men Women	11 (40,74%) 16 (59,26%)	13 (50,00%) 13 (50,00%)	0,586	***
	Smoke	non-smoker	18 (66,67%)	11 (73,33%)
		smoker	9 (33,33%)	4 (26,67%)	0,739	***
		unknown	.	11
	Histological type of melanoma
		SSM	17 (73,91%)	8 (53,33%)
		Nodular	3 (13,04%)	4 (26,67%)
		Acral Acromic	1 (4,35%) 1 (4,35%)	3 (20,00%) .	0,293	***
		Spitz	1 (4,35%)	.
		unknown	4	11
	Breslow	(mm)	3,00 (1,56-5,00)	3,10 (1,15-4,75)	0,516	**
	Clark	III	8 (33,33%)	7 (33,33%)
		IV V	13 (54,17%) 3 (12,50%)	7 (33,33%) 7 (33,33%)	0,218	***
		unknown	3	5
	Ulceration	No	17 (68,00%)	11 (64,71%)
		Yes	8 (32,00%)	6 (35,29%)	1,000	***
		unknown	2	9
	Regression No	25 (100,00%)	14 (87,50%)
		Yes	.	2 (12,50%)	0,146	***
		unknown	2	10
	* t


V Hovelaque
email: vincent.hovelaque@univ-rennes1.fr

L Bironneau
email: laurent.bironneau@univ-rennes1.fr

The carbon-constrained EOQ model with carbon emission dependent demand

Keywords: Inventory, EOQ, environmental awareness, carbon emission, carbon tax

Currently companies are looking for solutions to reduce carbon emissions associated with their operations. Operational adjustments, such as modifications in batch sizes or order quantities, have proven to be an effective way to decrease emissions. In this paper, a novel model is proposed that takes into account the link between an inventory policy (EOQ), total carbon emissions, and both price and environmental dependent demand. In the case of an exogenous price, two optimal quantities are determined which maximize a retailer's profit and which minimize carbon emissions. Conditions that allow a company to maximize profit while minimizing emissions and mechanisms that allow a firm to maximize its profit and to decrease its carbon emissions are determined. In the case of an endogenous price, some empirical results are also discussed. When a firm optimizes its profit through both its selling price and its order quantity, some experiments match empirical observations. On the one hand, an environmental strategy is more significant for cheaper and green-labeled products.

On the other hand, a public mechanism such as a carbon tax will decrease total and marginal emissions.

Introduction

Currently companies are increasingly sensitive and responsive to the carbon emissions (emissions of carbon dioxide and other greenhouse gases) associated with their operations [START_REF] Cholette | The energy and carbon intensity of wine distribution: a study of logistical options for delivering wine to consumers[END_REF][START_REF] Stock | Research opportunities in supply chain management[END_REF]. Under the influence of their customers, who increase their socially responsible consumption practices [START_REF] Gonzalez | How do socially responsible consumers consider consumption? An approach with free associations method[END_REF], and also of governments and other pressure groups, they are undertaking initiatives to reduce their carbon footprint. In a recent paper [START_REF] Chen | The carbon-constrained EOQ[END_REF] note that firms have focused for the most part on reducing emissions through innovations of the physical processes involved, for example by redesigning products and packaging, deployment and use of less polluting sources of energy, or replacing energy inefficient equipment and facilities… For example, determining how frequently supply deliveries are made could be as important in mitigating carbon emissions as the energy efficiency of the vehicles used to make these deliveries [START_REF] Chen | The carbon-constrained EOQ[END_REF].

Operational adjustments, such as modifications in batch sizes or order quantities, have proven to be an effective way to reduce carbon emissions impact [START_REF] Cholette | The energy and carbon intensity of wine distribution: a study of logistical options for delivering wine to consumers[END_REF][START_REF] Hua | Managing carbon footprints in inventory management[END_REF][START_REF] Bouchery | Including sustainability criteria into inventory models[END_REF][START_REF] Chen | The carbon-constrained EOQ[END_REF]. Thus, in recent years, green and sustainable supply chain management have been studied thoroughly, as evidenced by the literature review of [START_REF] Seuring | A review of modeling approaches for sustainable supply chain management[END_REF]. In the following state of the art, we will focus on papers dealing with sustainability modeled in inventory management, demand and vertical coordination. [START_REF] Hua | Managing carbon footprints in inventory management[END_REF] propose an environmental approach to inventory management that integrates the impacts of a carbon cost associated with both order placement (mainly transport) and holding stock (storage). The proposed model extends the Economic Order Quantity model [START_REF] Harris | How many parts to make at once. Factory[END_REF][START_REF] Wilson | A scientific routine for stock control[END_REF] to take into account carbon emissions under the cap and trade system. They conclude that the optimal order quantity is between the classically calculated economic quantity (without taking into account the CO2 cost) and the quantity that minimizes carbon emissions. They also prove that there exist some conditions to decrease both economic costs and carbon emissions. [START_REF] Wahab | EOQ models for a coordinated twolevel international supply chain considering imperfect items and environmental impact[END_REF], meanwhile, offer an approach to optimally define the delivery / production policy to minimize the total cost of supply in a global supply chain. They show that taking into account the environmental impact leads to reducing the frequency of deliveries. [START_REF] Bonney | Environmentally responsible inventory models: nonclassical models for a non-classical era[END_REF] show that increasing the amount of products transported and reducing the frequency of delivery compared to those proposed by the traditional EOQ model gives better results in terms of ordering costs and carbon emissions.

However the latter two models are more limited than that of [START_REF] Hua | Managing carbon footprints in inventory management[END_REF]; they do not take into account the "cap and trade" system, and are also limited to the study of the reduction of CO2 emissions in transport, (emissions associated with storage are not considered). [START_REF] Bouchery | Including sustainability criteria into inventory models[END_REF] propose to integrate all the parameters of sustainable development (SD), including the social aspect missing from other works, into inventory management models. They reformulate the classic economic order quantity model as a multi-objective problem (called the sustainable order quantity model -SOQ). It follows two main points of their work. Firstly, that it enables reducing any SD impact by requiring a small increase in cost, which is in accord with the results in [START_REF] Hua | Managing carbon footprints in inventory management[END_REF]. Secondly, it is sometimes not possible to have overall environmental, economic and social efficiency, which led them to propose a trade off between various parameters. [START_REF] Chen | The carbon-constrained EOQ[END_REF] provide conditions under which, first, it is possible to reduce emissions by modifying order quantities and, second, the relative reduction in emissions is greater than the relative increase in cost. This paper is also interesting because the authors discuss the applicability of their results to systems under a variety of environmental regulations, for example settings with a carbon tax, cap-and-offset, and cap and price. [START_REF] Battini | A sustainable EOQ model: Theoretical formulation and applications[END_REF] propose a sustainable EOQ model by integrating environmental impacts of transportation and storage. They assess the impact of sustainability considerations on purchasing decisions in comparison to a traditional EOQ model. In particular, for the same transportation modality, there is no significant difference between the traditional and the sustainable model, but this difference increases with an increasing product price.

El [START_REF] El Saadany | Environmental performance measures for supply chains[END_REF] studied a simple two-echelon supply chain model in which demand depends on the environmental quality of the systems (measured using thirty criteria) and the associated costs. [START_REF] Glock | Sustainability strategies in an EPQ model with priceand quality-sensitive demand[END_REF] present a model that illustrates the trade-offs between sustainability, demand, costs and profit in a supply chain. They show that stakeholders can attract more customers by controlling emissions and scrap. Moreover, the total profit is higher but the product quality is lower in the case of cooperation. Nouira et al. (2013) propose a nonlinear profit function that takes into account a greenness dependent demand. The selection of manufacturing processes (technology) and input items allows optimizing a profit function in the case of an environmental dependent demand. In particular, they show that a firm increases its profit and decreases its carbon footprints if its faces to both an ordinary and green market. [START_REF] Liu | Consumer environmental awareness and competition in two-stage supply chains[END_REF] also develop a model that includes consumer environmental awareness and competition (three supply chain scenarii). They find that stakeholders with superior ecofriendly operations will benefit from an environmental demand increase. Some other results are proposed based on the eco-friendly level of manufacturers and retailers. [START_REF] Ghosh | A comparative analysis of greening policies across supply chain structures[END_REF] examine some supply chain coordinations with players initiating product "greening".

Cooperation between stakeholders does lead to higher greening levels but also to higher retail prices. In some coordination cases, the retailer has to provide suitable incentives to the manufacturer form him to participate in the bargaining process. Swami and Shah (2013) develop a model with a manufacturer and a retailer that coordinate their operations (wholesale price and green effort for the manufacturer, market price and green effort for the retailer).

They propose a two-part tariff contract to produce channel coordination and 'greener' efforts.

The majority of articles have focused on the minimization of the total cost supported by the manufacturer/retailer by introducing in the EOQ model some costs parameters linked to environment considerations. To take into account consumer's environmental awareness, some articles introduce an independent environmental parameter in a deterministic demand function (defined as the manufacturer's effort to be less polluting). To our knowledge, no model proposes to introduce the carbon emissions directly in the demand function, complementary to price sensitivity. Such a model is of interest because (i) some retailers developed a carbon labeling on their products linked with the emissions (Casino in France and Tesco in Great Britain), (ii) stock policy will impact carbon emission and then both economic profit and consumer's attractiveness and (iii) the retailer's environmental strategy should affect the price of the product. As proved in several surveys, the consumer is increasingly attracted by less polluting products, even if the price factor remains the first criteria of purchase.

The purpose of this paper is to address this gap by proposing a novel model that takes into account the link between inventory policy, total carbon emissions and both price and environmental dependent demand. We study the structure in two steps: first, the retailer maximizes its profit through the economic order quantity solely; secondly, through EOQ and price. In the first step, the mathematical model addresses the conditions that allow a firm to maximize its profit and to minimize its carbon emissions. We find that a carbon tax is always beneficial to the environment (lower emissions), to the consumer (higher demand) but unfavorable to the retailer (lower profit). Furthermore, an increase in the carbon price (cap and trade mechanism) has the same effects excepted on the retailer's profit that should increase or decrease depending on the parameters. Depending on certain conditions, total emissions should increase as one cost parameter increases (ordering or holding cost). This result shows that, for instance, a higher price of oil (picked up by the ordering cost) could generate more emissions. Finally, above a specific threshold, the retailer may increase its profit with an increasing consumer's awareness but may be penalized if the awareness becomes too high. For this first step, we also analyze the behaviour of the marginal rate of carbon emissions. To our knowledge, this performance indicator was not studied in literature.

Its evolution is not necessarily similar to that of CO2. Thus, we show that, in some situations, the amount of CO2 emitted decreases while the marginal rate increases. This phenomenon is related to the independent evolution between demand and emissions. In a second step, some empirical results are discussed in case of an endogenous price. If price elasticity increases, the retailer will reduce its price to increase its demand. For some parameters, this generates a higher CO2 amount. Moreover, an increase of the carbon tax leads to a decrease in the emissions but entails a higher price and a higher marginal rate. Thus, this work highlights some ambiguities of environmental strategies: a decrease of the carbon emissions could lead to a higher price supported by consumers. Moreover, the marginal rate decreases in some cases because of a lower demand. This paper is organized as follows. Section 2 introduces the mathematical model. In section 3, the optimal quantities that maximize the profit and minimize the emissions are derived in the case of an exogenous selling price. Then, conditions are determined to both a profitmaximization and an environmental minimization. We also analyze behaviors of profit and emissions with some parameters evolutions. In section 4, some results with an endogenous selling price are presented. Finally, this leads over the discussion of the findings and future research opportunities.

The model

In this work, a retailer buys a single product at a wholesale price r and sells it at a unit price w on a single channel defined by a deterministic demand D uniformly distributed in time (no hazard). The classical inventory management policy determines the economic order quantity (EOQ) that maximizes the profit function:

P(Q) = (w -r)D -k D Q + h Q 2 (1)
with Q as the order size in units (a decision variable), D as the total demand, k as the fixed ordering cost per order, and h as the constant inventory holding cost per product unit and time unit. This approach has undergone many developments and research articles since the original work of [START_REF] Harris | How many parts to make at once. Factory[END_REF]

CO 2 (Q) = e D Q + g Q 2 (2)
where e is the unit carbon emission linked to a replenishment's order and g the unit variable emission in warehouse (related to inventory quantity). This function follows the classical mechanisms (trade-offs) of the EOQ model applied to carbon emissions: emissions linked to storage increase with Q whereas emissions linked to frequency (transport) decreases.

Thus, objective is to determine the optimum order quantity Q maximizing the function:

Pr(Q) = P(Q) -t.CO 2 (Q) ( 3
)
where t is the carbon tax per unit (€/ton).

In this paper, the inventory model is developed with a deterministic demand function:

D(Q) = α -βw -δCO 2 (Q) (4)
With α as the maximal market demand (end consumer demand), β as the price's sensitivity (w as unit price) and CO 2 as the amount of carbon emissions defined previously and associated with elasticity δ. This function takes into account the attractiveness of a product according to two criteria: its selling price and the amount of carbon emitted to deliver it. This second criterion is currently highlighted by retailers with eco-labeling, e.g. carbon labeling by Casino in France and Tesco in Great Britain [START_REF] Upham | Carbon labeling of grocery products: public perceptions and potential emissions reductions[END_REF] and sustainable strategies [START_REF] Chitra | In search of the green consumer: a perceptual study[END_REF][START_REF] Liu | Consumer environmental awareness and competition in two-stage supply chains[END_REF]. This kind of demand function has been widely used to incorporate quality and price impacts on the demand [START_REF] Tirole | The theory of industrial organization[END_REF]. Moreover, [START_REF] Lau | Effects of a demand-curve's shape on the optimal solutions of a multi-echelon inventory/pricing model[END_REF] show that, for a single-echelon system, any downward-sloping demand curve leads to conclusions similar to a linear price-demand relationship.

The demand function can, in turn, be rewritten using the formulas ( 2) and (4):

D(Q) = Q(2α -2wβ -2gδ -Qgδ ) 2(Q + eδ + Qeδ ) (5)
Moreover, following Hua et al ( 2011), a cap-and-trade system is modeled by CO 2 (Q)

+ X = Z
where Z is the total carbon emissions quota (ton) and X is the transfer quantity of carbon emissions (positive or negative). Introducing a unit price s of carbon (€/ton), the economic function of the retailer with the following profit function is defined as:

Pr(Q) = (w -r)D(Q) + sZ -k D(Q) Q + h Q 2 + (t + s)CO 2 (Q) (6)
Because the consumer demand is modified with some parameter evolutions, it will also be of interest to observe the impact of inventory strategies on the marginal rates of greenhouse gas emissions defined by the performance indicator:

κ (Q) = CO 2 (Q) D(Q) (7)
This model addresses (i) the conditions that allow a retailer both to maximize its profit and to minimize its carbon emissions and, (ii) mechanisms that encourage the retailer to decrease its emissions while maximizing its profit. In the first step (section 4), the model is analyzed in the case of an exogenous price w. In the second step (section 5), the retailer optimizes its profit with both the order quantity and price (endogenous).

Analysis with an exogenous price

The following property states the optimal conditions for both profit maximization and carbon emission minimization.

Proposition 1.

Order quantities maximizing profit (6) and minimizing emissions (2) are given by:

Q p * = arg max Q Pr(Q) { }= -eδ + A(k + eλ)(h + gλ) h + gλ (8) Q e * = arg max Q CO 2 (Q) { }= -eδ + egA g (9)
with A = 2α -2wβ + egδ 2 and λ = s + t + δ (wr).

Moreover:

arg min Q κ (Q) { }= Q e * ( 10
)
(all proofs are developed in Annex).

Obviously, the two optimal quantities verify the two following properties Pr(

Q p * ) ≥ Pr(Q e * )
and CO 2 (Q p * ) ≥ CO 2 (Q e * ) . The result on marginal emissions is also of interest. If some parameters behaviors generate an increasing difference between the two optimal quantities, then the system may be more polluting. There is a large consensus that the first objective of retailers is to maximize their profit. Nevertheless, the structure of the logistic costs and the carbon emissions impacts the two optimal quantities relative to each other, as stated in the following theorem.

Theorem 1.

If g / e = h / k then Q p * = Q e * (case 1) If g / e < h / k then Q p * < Q e
Notice that this result matches with theorem 1 of [START_REF] Hua | Managing carbon footprints in inventory management[END_REF]. A cost analysis allows

understanding this result. Consider C(Q) = k D Q + h Q 2 and E(Q) = e D Q + g Q 2 . Minimize these functions are equivalent to minimize D Q + h k Q 2 and D Q + g e Q 2
. Then, when the relative emission (storage / ordering) is greater than the relative cost, the optimal quantity is greater.

The relative position of Q p * and Q e * is not only dependent of inventory costs and unit levels of carbon emissions but also independent of consumer environmental awareness.

In the following subsection, impacts of the carbon tax, inventory costs and environmental sensitivity are analyzed.

Impact of carbon tax and price

The main objective of the carbon tax, considered as a public mechanism (like the cap and trade tool with the unit carbon price), is to contribute to the decrease in the amount of CO2 emitted through industrial (production and logistics) activities. As expressed in theorem 1, the policy of profit maximization is not optimal for minimizing CO2 emissions, except in the first case. Indeed, if g / e = h / k , then Q p * = Q e * and the retailer optimizes both its profit (maximization) and carbon emissions (minimization). A carbon tax is then ineffective in that case. In the two other cases, an increase of the carbon tax generates lower emissions of greenhouse gases, an increase in demand, but a decrease in the profit of the retailer. This is summarized in the next proposition.

Proposition 2.

For all parameters configuration:

∂Pr(Q p * ) ∂t ≤ 0 , ∂D(Q p * ) ∂t ≥ 0 , ∂CO 2 (Q p * ) ∂t ≤ 0. Moreover sgn ∂Q p * ∂t = sgn eh -gk ( ).
This result can be explained in Figure 1. Consider the situation where g / e < h / k , i.e.

Q p * < Q e * . First, as proved in theorem 1, Q e * is independent of t. For an initial level 0 t of ( ) Proposition 2 also proves that the performance indicator expressed in (7) will then always decrease if the tax increases. Thus, a carbon tax has a dual benefit: to improve sustainability of inventory management, and also to increase consumer demand towards more sustainable products.

p Q t * e Q 0 Pr( , ) Q t 2 ( ) CO Q * 1 ( ) p Q t 1 0 t t > Q 1 Pr( , ) Q t € Carbon emission

Proposition 3.

For all parameters configurations:

sgn ∂Q p * ∂s = sgn eh -gk ( ); ∂D(Q p * ) ∂s > 0 ; ∂CO 2 (Q p * ) ∂s < 0
Moreover, if Z > egAegδ , there exists a value of s such that ∂Pr(Q p * )

∂s = 0 . If Z ≤ egA -egδ , ∂Pr(Q p * ) ∂s < 0.
The two above propositions show that the two mechanisms (tax and carbon prices) have not the same impacts on the retailer. Both mechanisms reduce the total amount of carbon emissions and increase the final demand. Obviously, a carbon tax always reduces the retailer's profit. This is not the case with the cap-and-trade mechanism that should advantage or penalize the retailer in function of the carbon quota Z and the unit price of carbon s. That is interesting to note that the threshold at which profit should increase with a sufficient high s is equal to CO 2 (Q e * ) . Then, if the quota is less than CO 2 (Q e * ) , the retailer has to buy carbon credit and this purchase increases with s, and has a negative impact on the profit. Otherwise, if the quota and the carbon price are relatively high, the retailer can improve its profit by selling carbon credit on the market without deteriorating its carbon emissions.

Impact of ordering and holding costs

In a supply chain, the strategy of lower cost storage generally requires a search of lower costs of holding inventory (h) and/or of ordering (k). The decrease of these costs generates a change in inventory policy (favoring either more possession or more delivery frequency). In our model, such a policy change will impact the amount of greenhouse gas emissions, and therefore demand. The following proposition shows that an increase or decrease of inventory costs can generate an increase in GHG emissions.

Proposition 4.

The carbon function is not monotonous with the ordering cost and the holding cost. In particular, CO 2 (Q p * , k) and CO 2 (Q p * ,h) respectively have an extremum for: k = h e g and ĥ = k g e .

This results comes directly from the first order condition of each derivative function 2 illustrates this result for a given set of parameters. = 0 and then κ (Q p * ) increases. This result once again is consistent with property 1 and theorem 1. It confirms that the evolution of a parameter is not a sufficient condition for an overall decrease in carbon emissions. The parameter k can be associated with the price of oil. Thus, an increase in the price (all other parameters remaining equal) will change the ordering policy of the retailer by an increase in the order quantity. The weight of the storage will be more important and this can generate an increase in emissions if the emissions unit structure is unfavorable to storage.

∂CO 2 (Q p * , k) ∂k = 0 and ∂CO 2 (Q p increase further. Figure
k k * ( ( )) p D Q k * ( ) p Q k unit k k * Pr( ( )) p Q k * 2( ( )) p CO Q k € Carbon emission unit

Impact of CO2 elasticity

In formula (4), a modification of consumer environmental awareness is captured by the elasticity indicator . Thus, an increase of corresponds to consumers' expected increased sustainability, which induces a decrease in the demand related to CO2 emissions. If e / g = k / h , then the retailer optimizes both environmental and economic functions for all parameters. Otherwise, there exists no elasticity level to reach this twofold objective (complex

solution to Q p * (δ ) = Q e * (δ ))
. For e / g ≠ k / h , one obtains:

∂Pr(Q p * ) ∂δ = 0 for δ = 2(w -r) 2 (α -βw) -(k + eλ)(h + gλ) (w -r)(e(h + 2g(s + t)) + gk) .
The convexity of the profit function cannot be proved due to the complexity of its form.

Nevertheless, for all simulations, either δ ≤ 0 and the profit function decreases with δ , or δ > 0 and the profit function decreases then increases with δ .

This results illustrates that, above some threshold, the retailer may increase its profit with an increasing consumer's awareness. It is interesting to note that for an initial low elasticity δ , the retailer will be penalized with higher consumer's awareness. However, if the consumer becomes increasingly sensitive to sustainability, retailer will decrease sufficiently emissions to compensate an increase of δ .

The sign of ∂Q p * / ∂δ depends on the parameters but the complexity of the equations does not allow to obtain overall demonstrable analytical results.

Consider the following set of parameters: α = 10;w = 10;r = 0;β = 0.1;t = 0; s = 0; g = 1; e = 1; k = 1;h = 0.5 . Figure 3 is obtained by varying the environmental sensitivity with an exogenous price. So, even if emissions decrease overall with an increase in δ , the marginal importance of emissions increases because the overall demand decreases. This important point highlights some ambiguous mechanisms linked to sustainable strategies: under parameters conditions, increasing carbon sensitivity encourages the retailer to reduce its emissions but this reduction does not compensate the decrease in demand. In the case of commodities (food products, etc.), there may be a substitution from eco-friendly products to "classical" products, and then a total raise of carbon emissions.

Figure 3: Marginal carbon emissions behavior with consumer environmental awareness Furthermore, excess δ could lead a firm to not produce (under this threshold value, the optimal quantity and/or the profit would be negative). So, all other factors being equal, an oversensitivity to CO2 emissions could totally block a company's strategy of holding and ordering, the strategy of a CO2 decrease not being sufficient to outweigh the increase of δ . In the case of excess environmental sensitivity, the retailer will probably adjust its price to generate profit. This notion will be discussed in the next section.

Analysis with an endogenous price

Previously, the stock management policy was discussed with an exogenous price. Some mechanisms are also observed independently of a retailer's pricing behavior. In this section, the retailer optimizes its profits through both inventory management policy ( Q p ) and sales price policy ( w p ). Thus: Max Q p ,w p Pr(Q p ,w p ) and Min Q e ,w e Pr(Q e ,w e ) are investigated.

In fact, the CO2 minimization has not an economic sense in this case. To minimize its emissions, the firm will hardly increase its price so that the demand will tend to 0 (and Q e will tend to 0 too).

In the profit's maximization case, the first order conditions lead to: The second order conditions cannot be proved formally due to the complexity of the Hessian matrix. Nevertheless, it is proved that ∂ 2 Pr/ ∂Q p 2 < 0 and ∂ 2 Pr/ ∂w p 2 < 0 for all (Q p ,w p ).

For all following simulations, it is verified that the determinant of the Hessian matrix is positive for (Q p * ,w p * ) and thus that the above solutions are optimum.

Several simulations are described afterwards, based on the following parameter values: α = 10;r = 0; Z = 10.

Considering a lack of carbon tax and cap-and-trade mechanism ( s = t = 0), optimal solutions are computed for different values of price and carbon sensitivities in the three cases are presented in Table 1. In order to evaluate the tax utility, case 2 ( g / e < h / k ) is closely examined in Table 2.

β = 0.1 case 2 β = 0.1 β = 0.2 β = 0.3 case 3 β = 0.1 β = 0.2 β = 0.3 g=1 e=1 h=0.7 k=1
An increased sensitivity to CO2 (via δ ) generates a decrease in price, quantity, CO2 emissions, and profit as well as an increase in the marginal emission indicator. This result tends to confirm an empirical analysis showing that the overall change in purchasing is more significant for cheaper and green-labeled products [START_REF] Vanclay | Customer response to carbon labeling of groceries[END_REF]. An inventory policy linked to CO2 and price also allows a more sustainable strategy.

Moreover, increased sensitivity to price generates a decrease in optimal price, profit and marginal emission indicators. However, certain situations generate an increase in quantity (case 3) and emissions (case 2 and 3 with δ = 0.1 ). When consumers are more price dependent, retailers lower their price and capture an increased demand D(Q p * ) . When environmental sensitivity is low, the influence of inventory costs is stronger than the influence of carbon emission costs.

β = 0.1 β = 0.2 β = 0.3 case 2 t = 0 β = 0.1 β = 0.2 β = 0.3 case 2 t = 1 β = 0.1 β = 0.2 β = 0.3
The first observation is that a tax leads to a decrease in total emissions and therefore the mechanism reaches its objective. However, a tax also leads to an increase in the optimal price, and therefore to a decrease in attractiveness for all consumers. This example highlights a major point of discussion about the introduction of a carbon tax and its sharing between all the actors of the supply chain (stakeholders and consumers).

Concerning an ordering policy, the optimal quantity increases if environmental sensitivity is weak, but decreases in the opposite case. This has been explained above by the relative cost of an order compared to the inventory cost (30% lower).

Concluding remarks

Currently companies are looking for solutions to reduce carbon emissions associated with their operations. Operational adjustments, like modifying batch sizes or order quantities, have proven to be an effective way to decrease emissions. In this paper, a model is proposed that takes into account the link between an inventory policy (EOQ), total carbon emissions, and both price and environmental dependent demand. In the case of an exogenous price, optimal quantities (EOQ) that maximize a retailer's profit and minimize carbon emissions are determined. First, the increase of a carbon tax generates lower emissions, except when

Annexes

For all proofs, we note A = 2(α -βw) + egδ 2 and λ = s + t + δ (wr).

Proof of proposition 1

The second derivative of the profit function and of the carbon function are as follows:

∂ 2 Pr(Q) ∂Q 2 = - A(k + eλ) (Q + eδ ) 3 and ∂ 2 CO 2 (Q) ∂Q 2 = eA (Q + eδ ) 3
These functions are respectively concave and convex in Q.

Then, the optimal values Q p * and Q e * are determined by the first order conditions:

Figure 1 .

 1 Figure 1. Impact of an increase of the carbon tax

Figure 2 :

 2 Figure 2: Evolution of order quantity, demand, emissions and profit with ordering cost.

  + 2βr -gδQ p * ) + 2β(k + e(s + t)) 4βQ p * with A = 2α -2w p * β + egδ 2 and λ = s + t + δ (w -r).

Proof

  + gλ) g(k + eλ) + (h + gλ)eThe first results are derived from the shape of first derivates:with A = 2(α -βw) + egδ 2 A ≥ δ eg B = g(k + eλ) + e(h + gλ) (k + eλ)(h + gλ) = eg φ + 1 φ ≥ 2 eg , with φ = k / e + λ h / g + λ gk)(k + eλ)A 2 2(A(k + eλ)(h + gλ)) gk) 2 A 2 4(A(k + eλ)(h + gλ)) 3/2 ≤ 0 egλ -A g(k + eλ) + e(h + gλ) s = (ehgk)(Z + egδ ) D -D g(k + eλ) + e(h + gλ) 2 -2eg(α -βw) + 2egZδ . s exists if: D ≥ 0 and (ehgk)(Z + egδ ) D -D g(k + eλ) + e(h + gλ) gk)(k + eλ)A 2 2 A(k + eλ)(h + gλ) gk) 2 A 2 4 A(k + eλ)(h + gλ) ( ) 3/2 ≤ 0

  . For instance,[START_REF] Andriolo | A century of evolution from Harris's basic lot size model: survey and research agenda[END_REF] and[START_REF] Glock | The lot sizing problem: a tertiary study[END_REF] produced literature reviews on the EOQ model and its many adaptations. It is also possible to consider the many published papers on this model, it's evolution and potential research perspectives in the special issue "Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris" published by the International Journal of Production Economics in 2014[START_REF] Cárdenas-Barrón | Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris[END_REF].[START_REF] Stock | Research opportunities in supply chain management[END_REF] show that transport and warehouse operations generate large amounts of carbon emissions. Like the cost function above, the emission function models a trade-off

between the frequency of delivery (increasing emissions of transport and decreasing emission of storage) and the storage amount (decreasing emissions of transport and increasing emission of storage). Among the methodologies, which calculate carbon emissions, the Greenhouse Gas protocol (2012) developed a linear function that depends on a fixed coefficient and a variable part. Thus, referring in particular to

[START_REF] Hua | Managing carbon footprints in inventory management[END_REF]

and

[START_REF] Bouchery | Including sustainability criteria into inventory models[END_REF]

, the total carbon emission function (in tons) linked to delivering and storing products is then defined by the formula:

Table 1 :

 1 Order quantity and price optimum without carbon tax and cap-and-trade

	κ

Table 2 :

 2 Impact of a carbon tax on the optimal order quantity and price

	κ

Acknowledgements: authors thank the anonymous referee for his useful comments.

g / e = h / k . In this case, the optimal quantity does not depend on the carbon tax, and both economic and environmental optimality exist. Growing consumer expectation for sustainable products has a positive impact on the environment, but a negative impact on a firm's profit (in constant prices). Finally, an increase in ordering costs (due to increased transport costs: increased taxes on gas or transportation) may, in some cases, have an adverse effect on the environment. When the retailer optimizes its profit with quantity and price, an increase of environmental sensitivity leads to a decrease in price and carbon emission in some numerical examples. Moreover, a carbon tax allows an emission decrease but generates an increase of price.

In the case of an endogenous price, when a firm optimizes its profit through both its selling price and its order quantity, some experiments match with empirical observations. On the one hand, an environmental strategy is more significant for cheaper and green-labeled products.

On the other hand, a public mechanism such as a carbon tax allows to decrease total emissions but to increase marginal emissions.

Further research directions may be looked into. This work could be extended to multi-level systems with manufacturer-retailer cooperation. In particular, it may be of interest to analyze a multi-level system with a wholesale price (r) depending on the emission costs. For example, all transport costs could be assigned to the manufacturer, which determines its strategy by maximizing its profit with the wholesale price.

All results of this paper and the marginal analysis, with parameters defined in (7), show the complexity of environmental issues. Thus, some strategies may be effective in a given sector but implicitly, can cause an increase in emissions in other sectors. For example, if standard goods (food, habitation, health…) are considered, there will necessarily be a coexistence of a chain of green products (often more expensive) and a chain of so-called standard products (cheaper but more polluting). Finally, the strategy of a green supply chain should be compared to high quality food chains (e.g. without GMOs, non-allergenic products...). It therefore seems interesting to analyze the research on the quality of products and to develop green supply chain management insights.