Antoine Rauch

Annabelle Dupont

Mickael Rosa

Maximilien Desvages

Christina Le Tanno

Johan Abdoul

Mélusine Didelot

Alexandre Ung

Richard Ruez

Emmanuelle Jeanpierre

Mélanie Daniel

Delphine Corseaux

Hugues Spillemaeker

Julien Labreuche

Bénédicte Pradines

Natacha Rousse

Peter J Lenting

Mouhamed D Moussa

André Vincentelli

Jean-Claude Bordet

Bart Staels

Flavien Vincent

Cécile V Denis

Eric Van Belle

Caterina Casari
email: caterina.casari@inserm.fr

Sophie Susen
email: sophiesusen@aol.com

Shear forces induced platelet clearance is a new mechanism of thrombocytopenia

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.

Conclusion

ECMO-associated shear forces induce GPIbα-shedding and thrombocytopenia due to faster clearance of GPIbα-negative platelets. Inhibiting GPIbα-shedding could represent an approach to reduce thrombocytopenia during ECMO.

Keywords: Extracorporeal Membrane Oxygenation, thrombocytopenia, bleeding, platelet clearance, glycoprotein Ibα, shedding, shear forces

INTRODUCTION

The use of extracorporeal membrane oxygenation (ECMO) devices has largely increased over the last two decades, and even more in recent years due to the COVID-19 pandemic. Despite major improvements in the design and engineering of the devices and circuit components, severe hemostatic complications (i.e. bleeding or thromboembolic events) [START_REF] Chung | Hemocompatibility-Related Adverse Events and Survival on Venoarterial Extracorporeal Life Support: An ELSO Registry Analysis[END_REF][START_REF] Aubron | Predictive factors of bleeding events in adults undergoing extracorporeal membrane oxygenation[END_REF] often occur in patients undergoing ECMO and are associated with increased morbidity, mortality and hospitalization costs.

Thrombocytopenia, a major risk-factor for bleeding, has been consistently described during ECMO [START_REF] Aubron | Predictive factors of bleeding events in adults undergoing extracorporeal membrane oxygenation[END_REF][START_REF] Jiritano | Platelets and extra-corporeal membrane oxygenation in adult patients: a systematic review and metaanalysis[END_REF] but its real prevalence is poorly established. In a recent meta-analysis, Jiritano F. et al analyzed fourteen studies documenting thrombocytopenia in ECMO patients and reported high heterogeneity, including prevalence as high as 80% [START_REF] Aubron | Predictive factors of bleeding events in adults undergoing extracorporeal membrane oxygenation[END_REF] . Importantly, only few studies until now have reported the kinetics of platelet count changes after ECMO implantation and its progression while on ECMO [START_REF] Jiritano | Platelets and extra-corporeal membrane oxygenation in adult patients: a systematic review and metaanalysis[END_REF] . The origin of thrombocytopenia in ECMO patients is thought to be multifactorial, due to infections, heparin treatment or specific patient-related factors, although its risk factors, prevalence and evolution are yet to be determined. However, it is well established that thrombocytopenia exposes to lifethreatening bleeding, requiring blood transfusions, use of pro-hemostatics, and various invasive procedures. Ultimately, these bleedings may also lead to interruption of some of the antithrombotic medications, which is associated with an increased rate of thrombotic complications.

ECMO support is associated with multiple procedure-related modifications that can affect platelets, for example, the contact with artificial surfaces, blood cell fragmentation or release of inflammatory mediators [START_REF] Mazzeffi | Thrombopoietin, Soluble CD40 Ligand, and Platelet Count During Veno-arterial Extracorporeal Membrane Oxygenation[END_REF][START_REF] Doyle | Current Understanding of How Extracorporeal Membrane Oxygenators Activate Haemostasis and Blood Components[END_REF] . However, one of the major hemodynamic changes induced by the mechanical device is an increase in the shear forces applied to the blood components.

Elevated shear stress induces von Willebrand factor (VWF) conformational changes and binding to platelet glycoprotein Ib (GPIb) [START_REF] Deng | Platelet clearance via shear-induced unfolding of a membrane mechanoreceptor[END_REF] . Recently, GPIb ectodomain shedding has been reported in ECMO patients [START_REF] Lukito | Mechanical circulatory support is associated with loss of platelet receptors glycoprotein Ibα and glycoprotein VI[END_REF][START_REF] Van Den Helm | Platelet Phenotype and Function Changes With Increasing Duration of Extracorporeal Membrane Oxygenation[END_REF] and it has been hypothesized that mechanical circulatory support enhances VWF-GPIb interaction [START_REF] Nascimbene | Acquired von Willebrand syndrome associated with left ventricular assist device[END_REF] . These observations suggest that GPIb shedding requires ligand-receptor engagement. Although shear force-induced receptor shedding independent of agonist receptor engagement has been described for GPVI [START_REF] Al-Tamimi | Pathologic shear triggers shedding of vascular receptors: a novel mechanism for down-regulation of platelet glycoprotein VI in stenosed coronary vessels[END_REF] , to the best of our knowledge, this has not yet been reported for GPIb.

A limited number of studies have investigated the direct effects of ECMO and the mechanisms underlying the onset and progression of the thrombocytopenia and the modifications of platelet functions through a systematic temporal analysis of platelet counts and functions.

To address these questions, we prospectively studied platelet count, desialylation, activation and GPIbα shedding in patients included in the WITECMO-H study. Ex vivo, we assessed platelet responses to low-and high-flow rates in a mock circulatory ECMO loop and at pathologic shear rates in a microfluidic device. The impact of red blood cells (RBCs), anticoagulation, VWF-GPIbα interactions and VWF supraphysiological concentrations were also evaluated. Finally, we studied the clearance of human platelets subjected or not to ex vivo low-and high-flow rates, using an in vivo transfusion model in NOD/SCID mice.

MATERIAL AND METHODS

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Please see the Major Resources

Patients on ECMO

Consecutive adult patients on ECMO were prospectively included in the WITECMO-H trial (https://clinicaltrails.gov/study/NTC03070912) from February 2018 to September 2020. Details of the protocol are described in the Supplemental material and methods. All patients were sampled at least three times: at admission immediately before ECMO implantation (Ti), 1-hour and 24-hours postimplantation (Ti+1h and Ti+24h, respectively). All patients, except for ten who died early on ECMO, were also sampled 48-hour post-implantation (Ti+48h). Survivors (73 patients) were subsequently sampled immediately before ECMO explantation (Te) and 48-hour post-explantation (Te+48h).

Mock circulatory ECMO loop model and microfluidic device

We used a standardized mock circulatory ECMO loop in accordance with the American Society of the international association for Testing and Materials guidelines [START_REF]Standard practice for assessment of hemolysis in continuous flow blood pumps[END_REF] . Perfusions with whole blood, plasma with normal platelet concentration (NPP) or poor platelet plasma (PPP) were performed ex vivo for 180 minutes (min) at 37°C in a thermostated tubing system using the ECMO Rotaflow (Maquet Cardiopulmonary AG, Hirrlingen, Germany). Experimental ECMO settings (see supplemental material and methods) were applied to achieve a range of outflow rates close to those obtained in ECMO patients [START_REF] Lehle | Flow dynamics of different adult ECMO systems: a clinical evaluation[END_REF] . Time points for sample collection and analysis were chosen to mimic the number of passages of the blood within ECMO in vivo (Supplementary Figure 1). Additional ex vivo perfusions were performed in presence of unfractionated heparin (UFH), tirofiban, apyrase or EDTA.

Separate experiments were performed with inhibitors of VWF degradation, VWF-GPIbα interaction or after addition of recombinant VWF concentrate to mimic the supra-physiological VWF plasma concentrations observed in ECMO-patients [START_REF] Lukito | Mechanical circulatory support is associated with loss of platelet receptors glycoprotein Ibα and glycoprotein VI[END_REF] . We also used a microfluidic device allowing to circulate fluids at a constant and controlled pathologic shear rate (see supplemental material and methods).

Human platelet transfusion in NOD/SCID mice

To assess the impact of ECMO rheology on platelet survival in vivo, we used a NOD/SCID mouse model infused with human platelets isolated from NPP exposed for 180 min to low-or high-flow rates in the mock circulatory ECMO loop. Details are provided in Supplemental material and methods.

Statistical analysis

Quantitative variables are expressed as means (standard deviation), or medians (interquartile range) in case of non-Gaussian distribution. Categorical variables are expressed as frequencies (percentages).

Normality of distribution was assessed graphically and by using the Shapiro-Wilk test. Between group comparisons were performed using Student-test (or ANOVA) in case of normal distribution and Mann-Whitney test or Kruskal-Wallis in case of non-normal distribution or in case of small sample size (<10). Within group comparisons according to different time points were performed using the paired Student-test given no deviation to normality was observed or Wilcoxon test in case of nonnormal distribution. No correction for multiplicity was done and results are considered as hypothesis generating. Statistical testing was conducted at the two-tailed α-level of 0.05. All statistical analyses were performed using GraphPad Prism software version 8.0 (GraphPad Software, CA, USA).

RESULTS

Patients

Patients' characteristics are summarized in Table 1. One hundred and seven consecutive patients were recruited, 90 patients were treated with veno-arterial (VA)-ECMO and seventeen with veno-venous (VV)-ECMO.

Thrombocytopenia occurs early after ECMO implantation

A time-dependent decrease in peripheral platelet count was observed after ECMO implantation (Figure 1A). Platelet count dropped in 90% (96/107) of patients 1-hour post-ECMO initiation (177±97 at Ti+1h versus 191±104 G.L -1 at Ti, p=0.00013). Platelet count continued to decrease with a mean drop of 35% 24 hours later (125±72 G.L -1 at Ti+24h, p<0.0001) that further decreased to 41% of the initial count two days post-ECMO implantation (112±61 G.L -1 at Ti+48h, p<0.0001). At this later time point, 75% of survivors (n=85) still under ECMO presented thrombocytopenia (platelet count<150 G.L -1 ). As expected, thrombocytopenic patients received more units of platelet and packed RBCs (1 versus 0, p=0.00022, 7 concentrates versus 4, p=0.0084, respectively) during ECMO support, compared to patients with normal platelet count. No other outcomes (major bleeding, thrombosis, death during ECMO or within 30 days after ECMO implantation) were associated with platelet count in our study (data not shown). A similar time course of platelet drop was observed in VA-and VV-ECMO patients (Supplementary Figure 2) suggesting that the drop in platelet count is dependent on ECMO but independent of the type of support. Platelets were also quantified immediately prior to ECMO explantation (Te; time range 1 to 37 days post-ECMO initiation, n=73) and two days later (Te+48h). Thrombocytopenia was present in 78% of patients before ECMO removal (Te=107±48 G.L -1 ). A remarkable 51% increase in platelet count was observed two days after device removal (162±80 G.L -1 at Te+48h, p<0.0001), resulting in 22% of patients that resolved their thrombocytopenia at this early post-removal time point.

As a potential cause of thrombocytopenia, we evaluated platelet desialylation. β-galactose exposure was only modestly increased on platelets from ECMO patients (and after ex vivo perfusion in the mock circulatory ECMO loop) to levels largely below the threshold associated with thrombocytopenia [START_REF] Dupont | Relevance of platelet desialylation and thrombocytopenia in type 2B von Willebrand disease: preclinical and clinical evidence[END_REF] (Supplementary Figures 3A and3B), indicating that platelet desialylation does not explain ECMO-induced platelet count drop.

Overall, these data confirm that ECMO support is associated with a rapid drop in platelet count that corrects quickly after device removal, suggesting a causal link between ECMO and thrombocytopenia, independent of platelet desialylation and of the type of support.

ECMO promotes platelet GPIbα shedding in patients

We investigated platelet functions in the first 42 patients, representative of the entire cohort (Table 1).

Surface and soluble GPIb levels were similar in patients before ECMO implantation (Ti) and in healthy controls (47.8±9.1 at Ti vs 50.3±7.2 MFI in controls and 5.6±3.3 at Ti vs 3.4±0.8 µg.mL -1 in controls respectively, not-significant), suggesting that pre-existing illness conditions do not significantly affect platelet GPIbα expression.

However, ECMO implantation coincided with a significant: 1) reduction in platelet GPIbα surface expression that decreased over time (47.8±9.1 at Ti versus 44.7±10.3 MFI at Ti+1h, p=0.0011 and 42.3±8.9 at Ti+24h, p=0.0020; Figure 1B) and 2) progressive increase in soluble GPIbα levels in plasma (5.6±3.3 at Ti versus 6.7±4.7 µg.mL -1 at Ti+1h, p=0.083, and 10.8±4.1 at Ti+24h, p<0.0001, Figure 1C), indicating that GPIbα shedding rather than receptor internalization occurs under ECMO. Similar results were observed in platelets and plasma from VA-and VV-ECMO patients (Supplementary Figures 4A and4B) pointing to a mechanism of shedding independent of the type of ECMO support. Interestingly, at Ti+24h we observed a negative correlation between sGPIbα levels and platelet count (r=-0.29, p=0.044; Figure 1D) suggesting a link between GPIbα shedding and platelet clearance.

We also monitored platelet GPVI surface expression (Figure 1E). In contrast to GPIb and in accordance with the literature [START_REF] Montague | Mechanisms of receptor shedding in platelets[END_REF] , GPVI expression was already lower in ECMO patients before support initiation (Ti) compared to healthy controls (19.9±4.6 at Ti versus 26.0±2.7 MFI in controls, p<0.0001), with only 3% additional reduction 24 hours post-ECMO onset (19.1±4.3 at Ti+24h, p=0.00052) independently of the type of ECMO (Supplementary Figure 4C).

ECMO promotes platelet GPIbα shedding ex vivo

To validate the results obtained in patients and exclude all confounding variables, except the direct effect of ECMO on platelets, experiments were performed ex vivo in a mock circulatory ECMO loop at different flow rates. Because VWF degradation is highly dependent on shear conditions and happens rapidly [START_REF] Zhang | Mechanoenzymatic cleavage of the ultralarge vascular protein von Willebrand factor[END_REF] , HMWMs disappearance was evaluated in every ex vivo experiment, to validate the perfusion conditions (i.e. elevated shear forces). HMWMs disappearance was almost identical in NPP versus PPP and in NPP versus whole blood (Supplementary Figure 5), suggesting that very high shear is generated within the ECMO loop model regardless of RBCs-induced changes in viscosity.

A time-dependent decrease of platelet GPIbα surface expression was observed at both low and high flow rates (Figure 2A). At high flow rate, a loss of GPIbα expression rapidly occurred 30 min after turning on the pump (p=0.0098), that further increased 60 (p=0.0019) and 180 min later (p=0.0019), with a maximum drop of 77% at the latest time point. At low flow rate, the loss of GPIbα expression was slower, with a drop of 35% at the latest time point. At the end of the ex vivo perfusions western blot analysis of platelet lysates revealed a decrease in ectodomain levels of GPIbα post-perfusion (Supplementary Figure 6) supporting that ECMO induces ectodomain shedding of GPIbα rather than receptor internalization.

In accordance with time-and flow rate-dependent shedding of GPIbα from the platelet surface, we observed, the appearance and expansion of a sub-population of GPIbα-negative platelets (Figure 2B).

Detection of GPIbα-negative platelets increased over-time at both flow rates but these platelets were consistently 2.6-to 2.8-fold more abundant at high compared to low flow rates, at every time point (30 min, 17.4±12.1 versus 6.2±3.3%, p=0.011, 60 min, 23.8±9.4 versus 8.6±4.1%, p=0.00025, 180 min, 37.2±7.4 versus 14.3±4.6%, p=0.00025), whereas almost no GPIbα-negative platelets (3%) were detected after 3 hours in samples not exposed to shear (data not shown). Strikingly, in the subset of 42 patients analyzed by flow cytometry, we observed only few GPIbα-negative platelets (3.1±2.9% at Ti versus 3.9±3.6%, at Ti+1h, p=0.0069 and 2.5±2.3% at Ti+24h, not-significant, Supplementary Figure 7A). Similar data were observed for GPVI-negative platelets (Supplementary Figure 7B).

Therefore, we hypothesized that, while GPIbα-negative platelets accumulate in the closed circuit, they are rapidly eliminated in vivo, which would explain why they are barely detectable in circulation in ECMO patients.

GPVI shedding also occurred in the ECMO loop model (Figure 2C), was more pronounced at high flow rate (maximum drop of 69%) and was associated with the appearance of GPVI negative platelets (Figure 2D). The time course of GPIbα and GPVI shedding (Figures 2A -2D) was not modified by the presence or absence of RBCs.

In a microfluidic device (supplemental material and methods and Supplementary Figure 1), GPIbα and GPVI receptor cleavage similarly occurred at pathologic shear rates of 10 000s -1 (Figure 2E), concomitant with the appearance of GPIbαand GPVI negative platelets (Figure 2F).

Overall, these data suggest that GPIbα and GPVI shedding are dependent on blood flow rate ex vivo and that this parameter can be used as proxy of more complex shear conditions in vivo.

We further hypothesized that GPIbα receptor shedding could be mediated by activation of the ADAM10 and ADAM17 proteases and performed perfusion experiments in presence of EDTA, a broad metalloprotease inhibitor. Pre-treatment with EDTA abrogated GPIbα shedding from the platelet surface (Figure 2G) and the appearance of a GPIbα-negative subpopulation of platelets (Figure 2H), providing evidence for the involvement of metalloproteases in ECMO-induced platelet receptor shedding ex vivo.

ECMO does not induce major platelet activation

Because GPIbα shedding is often associated with platelet activation [START_REF] Mazzeffi | Thrombopoietin, Soluble CD40 Ligand, and Platelet Count During Veno-arterial Extracorporeal Membrane Oxygenation[END_REF][START_REF] Doyle | Current Understanding of How Extracorporeal Membrane Oxygenators Activate Haemostasis and Blood Components[END_REF][START_REF] Van Den Helm | Platelet Phenotype and Function Changes With Increasing Duration of Extracorporeal Membrane Oxygenation[END_REF] , we measured soluble Pselectin (sP-selectin) and soluble CD40 ligand (sCD40L) in plasma from ECMO patients. Before ECMO initiation, both markers were elevated compared to healthy controls (Figures 3A and3B), suggesting that the underlying diseases induce a mild platelet activation. However, when these markers were evaluated 24 hours post-ECMO onset, discordant and minor changes were observed.

While sP-selectin levels modestly increased (42.6±12.5 at Ti versus 49.7±11.2 ng.ml -1 at Ti+24h, p=0.035, Figure 3A), sCD40L decreased (184±36 at Ti versus 64±13 ng.ml -1 at Ti+24h, p=0.0038, Figure 3B), suggesting that, if any, ECMO has only a minor effect on platelet activation.

To further evaluate the platelet activation state, TRAP-induced activation was assessed one day after ECMO implantation in 5 patients (3 on, and 2 without aspirin, none on a P2Y12 inhibitor). Upon in vitro stimulation, GPIIb and P-selectin surface expression significantly increased in all patients (43.7±4.2 versus 75.5±9.9, p<0.0001, 0.6±0.2 versus 9.5±0.6, p<0.0001, respectively; Figures 3C and3D) and in the same range as platelets from healthy subjects (50.7±7.3 versus 82.2±11 p<0.0001, 0.5±0.2 versus 9.5±3, p<0.0001, respectively). These observations support the hypothesis that platelets remaining in circulation during ECMO are in a minor pre-activated state that does not prevent a subsequent strong agonist-induced response.

Platelet morphology one day after ECMO implantation assessed in 6 patients appeared normal and similar to that of platelets collected pre-ECMO implantation, except for a reduction in -granule content (2.2±0.3 versus 1.8±0.2 -granules per µm 2 , p=0.031, Wilcoxon test) (Figure 3E). Although statistically significant, the difference was only negligible, which is consistent with minor platelet preactivation results.

To exclude platelet activation-dependent shedding of GPIbα (ADP-mediated), we next measured platelet surface expression of GPIbα in patients treated or not with a P2Y12 inhibitor and found no significant difference (Figure 3F). Moreover, no associations were found between platelet activation markers and GPIbα shedding (sCD40L and GPIbα MFI, r=-0.12 p=0.73, sP-selectin and GPIbα MFI,r=-0.053 p=0.81), further supporting that ECMO-associated receptor shedding occurs independently of platelet activation.

When NPP was perfused ex vivo in the mock circulatory ECMO loop, platelet activation profiles were similar to those obtained on patients' samples. Both P-selectin expression at the platelet surface and sCD40L levels did not significantly change after 180 min (Figures 4A and4B).

We also observed similar decreased GPIbα expression (and accumulation of GPIbα-negative platelets) in NPP perfused at high flow rate, in presence or absence of the platelet inhibitors tirofiban or apyrase (Figures 4C-F).

Overall, concordant ex vivo and in vivo results show that ECMO does not induce major platelet activation nor is ECMO-dependent GPIbα-shedding affected by platelet activation.

ECMO-induced GPIbα shedding is independent of anticoagulation regimen

We investigated whether anticoagulation could affect receptor cleavage on ECMO. At 24 hours post-ECMO implantation, three out of 36 alive patients still under ECMO had undetectable anti-FXa activity levels. However, GPIbα expression, at the platelet surface and in plasma (sGPIbα), was similar between patients with or without detectable anti-FXa activity levels (Figure 5A and5B).

Moreover, thrombin generation was evaluated in 18 patients one day after ECMO implantation (Ti+24h) and, as expected, very low thrombin generation was observed in patients with detectable anticoagulation (data not shown).

Mirroring the results observed in patients, when ex vivo perfusions were performed in the mock circulatory ECMO loop in presence of heparin (0.5 IU.ml -1 , mimicking the patients' regimen), thrombin generation was nearly completely inhibited (not shown). An almost identical time course was observed for both the loss of GPIbα ectodomain from the platelet surface (Figure 5C) and the accumulation of GPIbα-negative platelets (Figure 5D) in presence or absence of heparin. These data indicate that the ECMO effects on platelets are independent of coagulation activation both in patients and ex vivo.

ECMO-induced platelet GPIbα-shedding is independent of VWF-GPIbα interaction

Because elevated shear forces under ECMO promote VWF conformational changes and disappearance of HMWMs, we investigated whether ECMO-associated GPIbα shedding relies on VWF multimerization or VWF-GPIbα binding. Inhibiting VWF degradation using an antibody against the VWF-D4 domain prevented the loss of VWF HMWMs in the mock circulatory ECMO loop (Figure 6A), but did not prevent GPIb shedding at a high flow rate (Figure 6B). Inhibition of VWF-GPIbα binding using an antibody against the VWF-A1 domain also failed to inhibit GPIb shedding (Figure 6B). These data indicate that GPIb ectodomain release happens independently of VWF degradation and VWF-GPIb interaction. Furthermore, shedding was still observed after perfusion of NPP from a von Willebrand disease-type 3 patient with undetectable VWF (Supplementary Figure 8A and8B). Finally, we found that GPIbα shedding over time was similar at physiological or supra-physiological VWF concentrations (Figure 6C and6D).

These data confirm that ECMO support promotes platelet-GPIbα shedding independently of ECMOassociated effects on VWF.

ECMO drives the clearance of human platelets transfused in NOD/SCID mice through a shear forces dependent mechanism

As GPIbα-shedding has been reported to be associated with increased platelet clearance from the circulation, we aimed to evaluate the in vivo kinetics of platelets exposed to low or high flow rates in the mock circulatory ECMO loop. First, NPP was perfused in the ECMO loop for 180 min and a control aliquot was kept at 37°C for the same time. We subsequently transfused similar amounts of human platelets into NOD/SCID immune deficient mice and measured human platelet counts over time (Figure 7A). ECMO-exposed platelets were eliminated more rapidly than control platelets. At 6 hours (360 minutes), 81±13, 60±26 and 54±16% of platelets exposed to high-, low-or no flow, respectively, had disappeared (Figure 7B). Interestingly, we also observed that GPIbα-negative platelets disappeared more rapidly than platelets still bearing intact receptors at their surface (Figure 7C) and thus had a shorter half-life (1.30±0.2 hours for GPIbα-positive platelets versus 0.41±0.1 hours for GPIbα-negative platelets, Mann Whitney test: p=0.028), indicating that GPIbα-negative platelets are cleared more rapidly from the circulation.

DISCUSSION

Using a translational approach, we demonstrate that the thrombocytopenia observed during ECMO support: 1) has an early onset and offset in close relation to ECMO implantation and withdrawal, 2) is associated with shear-rate induced GPIbα shedding, and 3) is the result of faster clearance of GPIbαnegative platelets. Importantly, our data also show that ECMO-associated decline in platelet count is independent of platelet activation or desialylation, presence of RBCs, activation of coagulation and VWF-GPIb interaction.

Consistent with other studies [START_REF] Jiritano | Platelets and extra-corporeal membrane oxygenation in adult patients: a systematic review and metaanalysis[END_REF][START_REF] Mazzeffi | Thrombopoietin, Soluble CD40 Ligand, and Platelet Count During Veno-arterial Extracorporeal Membrane Oxygenation[END_REF] , in all but eleven patients of the cohort (>90%), we observed a progressive decrease in platelet count over time following ECMO initiation, which resolved rapidly post-device explantation, suggesting a causal link between thrombocytopenia and ECMO. Previous studies have reported that thrombocytopenia is not dependent on the type of ECMO, cannulation nor on treatment regimen of the ECMO patients. [START_REF] Kohs | Severe thrombocytopenia in adults undergoing extracorporeal membrane oxygenation is predictive of thrombosis[END_REF] Our data confirmed that platelet count drop was similar in VA-and VV-ECMO patients, with/without antiplatelet therapy and despite anticoagulation regimen. Overall, these data indicate an accelerated clearance of platelets due to the high shear induced by the device, independent of underlying medical conditions and treatments, as a systematic and specific mechanism of thrombocytopenia under ECMO support.

Thrombocytopenia and GPIb-shedding have been both associated with desialylation [START_REF] Grozovsky | Novel mechanisms of platelet clearance and thrombopoietin regulation[END_REF] , therefore, we investigated if ECMO-associated decrease in platelet count could be due to the loss of platelet sialic acid content. In our previous study, we established a threshold above which thrombocytopenia is likely to be desialylation-dependent [START_REF] Dupont | Relevance of platelet desialylation and thrombocytopenia in type 2B von Willebrand disease: preclinical and clinical evidence[END_REF] . Our results show that both in ECMO patients and in the ex vivo ECMO model, platelet desialylation is significantly increased, but remains below this threshold indicating that mechanisms other than platelet desialylation are probably responsible for the thrombocytopenia.

VWF interaction with its GPIbα platelet receptor can trigger various platelet responses and mediate

GPIbα-ectodomain shedding, which has been reported to promote rapid platelet clearance, possibly preventing aggregation and thrombus formation [START_REF] Deng | Platelet clearance via shear-induced unfolding of a membrane mechanoreceptor[END_REF] . Lukito et al. [START_REF] Lukito | Mechanical circulatory support is associated with loss of platelet receptors glycoprotein Ibα and glycoprotein VI[END_REF] reported that GPIbα and GPVI surface levels are significantly reduced in patients undergoing ECMO-or ventricular assist device-support compared to healthy controls, but the time-course of these events post-ECMO implantation was not described in their paper. In our study, GPIbα ectodomain shedding occurs rapidly, within 24 hours post-ECMO and concomitant with an increase of sGPIbα plasma levels, demonstrating that ECMO favors rapid GPIbα proteolysis. At the same time, GPVI shedding was also observed. However, while sGPIbα measured immediately before ECMO implantation was modestly increased compared to levels in healthy individuals as previously reported [START_REF] Montague | Mechanisms of receptor shedding in platelets[END_REF] , GPVI surface expression was severely decreased in patients before ECMO initiation compared to controls, suggesting that pulmonary and cardiac distress are associated with substantial GPVI-and minor GPIbα shedding. Conversely, shedding post-ECMO onset was pronounced for GPIbα but minor for GPVI. These results largely confirm data from the literature showing that GPIbα and GPVI shedding can be regulated by independent, simultaneously occurring mechanisms [START_REF] Montague | Mechanisms of receptor shedding in platelets[END_REF] .

Mock circulatory loop experiments were instrumental to better understand the relationship between blood flow rates as proxy of shear stress and GPIbα shedding. We first carefully validated that flow dynamics within the mock circulatory ECMO loop were consistent with blood hemodynamics in ECMO patients by 1) using a microfluidic device allowing to circulate fluids at a constant and controlled pathological shear rate; 2) assessing ADAMTS13-mediated VWF degradation as internal control of shear conditions at every perfusion and 3) demonstrating that ECMO-associated effects on platelets were independent of the presence of RBCs during ex vivo perfusions. While low flow rates promoted significant but minor reduction of GPIbα surface levels of circulating platelets, a rapid shedding of ≈90% of the receptors from the platelet surface was observed within 3 hours at high flow rates. At these high flow rates, two distinct platelet populations were present: one expressing variable amounts of GPIbα, which decreased over time, and one without any detectable surface GPIbα.

Similarly, a small, but consistent increase in GPIbα shedding and accumulation of GPIbα-negative platelets has been reported in platelet storage lesion studies [START_REF] Chen | Inhibiting GPIbα Shedding Preserves Post-Transfusion Recovery and Hemostatic Function of Platelets After Prolonged Storage[END_REF][START_REF] Baaten | Platelet heterogeneity in activation-induced glycoprotein shedding: functional effects[END_REF][START_REF] Rijkers | Evaluation of the role of the GPIb-IX-V receptor complex in development of the platelet storage lesion[END_REF] . Therefore, it seems reasonable to hypothesize that shear stress induced by high flow rates promotes efficient GPIbα shedding.

Since ECMO patients suffer from acquired von Willebrand syndrome and since all the aforementioned platelet abnormalities can be associated with signaling defects downstream of VWF-GPIbα engagement, a direct role of VWF in this clinical picture seemed possible. However, we demonstrated that ECMO-associated platelet abnormalities are independent of either VWF proteolytic degradation or its interaction with GPIbα and are, instead, directly triggered by the elevated shear stress secondary to high blood flow rates.

We also demonstrated that shear-dependent ECMO-associated receptor shedding was independent of platelet-or coagulation activation, while it was abrogated by EDTA, similarly to what was previously demonstrated in similar conditions for VWF degradation by ADAMTS13 [START_REF] Rauch | Antibody-based prevention of von Willebrand factor degradation mediated by circulatory assist devices[END_REF] . Recently, Six et al.

reported that strong and sustained dual agonist-induced platelet activation induces ADAM17mediated shedding of an intracellular subpopulation of GPIb receptors [START_REF] Six | GPIbα shedding in platelets is controlled by strict intracellular containment of both enzyme and substrate[END_REF] . Interestingly, these authors also demonstrated that the degree and kinetics of GPIb cleavage depend on the agonist and receptor(s) engaged and thus on the platelet activation response, further supporting the notion of heterogeneity in platelet glycoprotein shedding that can happen in association with different events.

The putative mechanisms of release of platelet protein's ectodomains can differ significantly, as shown by the results of perfusion experiments with EDTA in the ECMO loop model. A mild increase in P-selectin expression was still observed in platelets exposed to high flow rate (data not shown) whereas the loss of GPIbα from the platelet surface was completely inhibited.

Only few single-center observational studies have evaluated the impact of ECMO on platelet function reporting discrepant results regarding platelet activation markers and agonist-induced platelet aggregation [START_REF] Kalbhenn | Acquired von Willebrand syndrome and impaired platelet function during venovenous extracorporeal membrane oxygenation: Rapid onset and fast recovery[END_REF][START_REF] Tauber | Predicting Transfusion Requirements During Extracorporeal Membrane Oxygenation[END_REF] . We observed only a minor platelet pre-activation before and during ECMO, which does not prevent agonist-induced responses, nor corresponds to the strong activation evoked by Six and colleagues [START_REF] Six | GPIbα shedding in platelets is controlled by strict intracellular containment of both enzyme and substrate[END_REF] . We also demonstrated that receptor shedding was similar in patients with/without platelet inhibitors, confirming that platelet activation plays a minor role in GPIbα shedding and thrombocytopenia under ECMO, and that ECMO support does not induce major platelet activation. However, while we don't observe major changes in platelet surface p-selectin, we observed a moderate reduction in -granule content in ECMO patient's platelets. This discrepancy could be due to platelet microparticle release, known to occur with mechanical shear [START_REF] Wang | Device-Induced Hemostatic Disorders in Mechanically Assisted Circulation[END_REF] . Although parent platelets may retain normal or slightly reduced reactivity, platelet microparticles could still promote thrombosis in ECMO setting.

We performed ex vivo perfusions in presence of heparin to mimic full-dose anticoagulation. Receptor shedding was similar in patients with normal or undetectable thrombin generation potential i.e. inefficient or efficient anticoagulation at the time of blood collection, confirming that the anticoagulation regimen does not affect shear-induced platelet shedding.

When human platelets perfused within the mock circulatory ECMO loop were infused into immunedeficient mice, we noticed a rapid drop in platelet count in the first 6 hours post-transfusion, followed by a more gradual disappearance at later time points, suggesting a two-phase elimination kinetic. It is tempting to speculate that GPIbα-negative platelets are rapidly eliminated post-infusion in mice (calculated half-life of 0.4 hours), while platelets bearing residual receptors at their surface are eliminated more slowly (half-life of 1.30 hours). This hypothesis is supported by the finding that only very small amounts of GPIbα-negative platelets are present in the circulation of the patients, without any significant difference between levels measured before and 24h post-ECMO implantation. To the best of our knowledge, this is the first study comparing clearance of GPIbα-positive and -negative platelets following ECMO support. It remains to be determined whether GPIbα-negative platelets are being sequestered in a specific organ such as the liver or the spleen and the molecular mechanism by which GPIbα shedding leads to platelet clearance in the context of elevated shear forces.

We propose here a new mechanism of thrombocytopenia, in which elevated shear forces directly induce only minor platelet activation, but a pronounced GPIbα-shedding resulting in a rapid clearance of platelets expressing low or undetectable GPIbα levels at their surface. ADAM17 has been shown to cleave the extracellular portion of GPIbα in many different conditions secondary to VWF-GPIbα engagement [START_REF] Bergmeier | Tumor necrosis factor-alpha-converting enzyme (ADAM17) mediates GPIbalpha shedding from platelets in vitro and in vivo[END_REF] or profound platelet activation [START_REF] Six | GPIbα shedding in platelets is controlled by strict intracellular containment of both enzyme and substrate[END_REF] . However, we have previously demonstrated that ADAM17-independent GPIbα-shedding can happen even in presence of active GPIbα-binding VWF bearing the p.V1316M/VWD-type 2B mutation [START_REF] Casari | Protein kinase C signaling dysfunction in von Willebrand disease (p.V1316M) type 2B platelets[END_REF] , suggesting that several proteases can proteolyze GPIbα. Whether shear stress induced-, VWF independent-GPIbα-shedding is mediated by ADAM17 or by other EDTA-sensitive proteases require further investigation.

Our study has a number of limitations. First it is a single-center study that wasn't designed to assess association between outcomes and platelet count. However, we found an association between early decrease of platelet count and transfusion of RBCs suggesting an increased risk of bleeding.

Adherence to antiplatelet medications was not assessed. Thrombocytopenia is probably multifactorial in heterogeneous and critically ill patients requiring ECMO. Although the ECMO loop model cannot recapitulate all the aspects of ECMO support and patient heterogeneity, we aimed to model as accurate as possible both the shear conditions and the anticoagulation regimen present in vivo.

Despite these limitations, our concordant results clearly indicate that platelet GPIbα shedding is directly dependent on elevated shear stress.

In conclusion, our results demonstrate that ECMO-associated GPIbα shedding from the platelet surface is directly dependent on blood flow rates but independent of VWF-GPIbα interactions and coagulation potential and promotes platelet clearance. We also demonstrate that the occurrence of thrombocytopenia is inherent to ECMO support and to the high-shear induced by the device, which should be taken into account for the management of these patients. Inhibiting platelet-GPIbα shedding could represent an innovative approach to reduce thrombocytopenia during ECMO, which could potentially be more efficient than platelet transfusion which is the only current therapeutic option. This unique and previously undescribed mechanism of shear-induced thrombocytopenia appears a key feature of ECMO support. Whether a similar mechanism may happen in other conditions, is still unknown and requires further investigations.

Figure 1 :Figure 2 :

 12 Figure 1: Platelet count timeline, GPIbα and GPVI levels in patients undergoing ECMO

Figure 3 :

 3 Figure 3: Platelet activation and morphology prior-and post-ECMO implantation.

Figure 4 :

 4 Figure 4: ECMO does not induce major platelet activation in vitro and platelet GPIbα shedding

Figure 5 :

 5 Figure 5: ECMO-associated platelet GPIbα shedding is independent of anticoagulation regimen

Figure 6 :

 6 Figure 6: GPIbα shedding observed during ECMO support is independent of VWF-GPIbα

Figure 7 :

 7 Figure 7: Platelet perfused in a mock circulatory loop model are rapidly eliminated in vivo in

  Table and the Supplemental Methods included in the Supplemental Material.

Table 1 : Characteristics of the studied population

 1 Values are numbers(percentage), mean(standard deviation) or median[interquartile]. ARDS: acute respiratory distress syndrome, BMI: body mass index, CHF: congestive heart failure, ECMO: extracorporeal membrane oxygenation; NA: non applicable, STEMI: ST-Elevation Myocardial Infarction, VA: veno-arterial, VV: veno-venous; *none associated with heparin-induced thrombocytopenia. Categorical variables were compared using chi-squared test. Continuous variables were compared using the Student-test in case of normal distribution or Mann-Whitney test otherwise.

	1
	2
				ECMO
			ECMO patients (n=107)	patients shedding analyzed for GPIb	P
				(n=42)
	Age (years), mean(SD)		55(14)	53(13)	0.68
	Male gender, n(%)		69(64)	27(64)	>0.99
	BMI (kg.m -2 ) mean(SD)		27.2(5.8)	26.4(5.9)	0.51
	Cardiac risk factors, n(%)
	Hypertension		38(36)	14(33)	0.85
	Dyslipidemia		25(23)	9(21)	>0.99
	Smoking		37(36)	12(28)	0.56
	Diabetes mellitus		33(31)	13(31)	>0.99
	Platelet count at admission (G/L), mean(SD)	191(104)	205(110)	0.45
	Type of ECMO, n(%)
	VA-ECMO		90(84)	32(77)	0.34
	VV-ECMO		17(16)	10(23)
	ECMO indication, n(%) Post-cardiotomy shock Cardiac arrest Pulmonary embolism Decompensated CHF STEMI ARDS Others	23(22) 17(16) 2(2) 16(15) 26(24) 13(12) 10(9)	5(12) 7(17) 1(2) 7(17) 10(24) 8(19) 4(9)	0.24 >0.99 >0.99 0.80 >0.99 0.3 >0.99
	Pump type, n(%)
	Rotaflow (Maquet)		56(52)	24(57)	0.72
	Revolution (Livanova)		44(41)	16(38)	0.85
	Centrimag (Thoratec)		7(7)	2(5)	>0.99
	ECMO implantation (L/min), mean(SD) outflow 1h after	3.8(1.1)	3.9(1.1)	0.69
	ECMO outflow 24h after implantation (L/min), mean(SD)	3.7(0.8)	3.8(0.8)	0.97
	ECMO median(IQR) duration,	days,	6[3-12]	5[2-9]	0.56
	Antiplatelet drug before and/or
	during ECMO, n(%)
	Aspirin		61(57)	23(55)	>0.99
	P2Y12 inhibitors		16(15)	6(14)	>0.99
	Antifibrinolytic during ECMO, n(%)	37(35)	8(19)	0.08
	Packed red blood cells during ECMO, units, median(IQR)	5[2-13]	4[0-10]	0.39
	Fresh frozen plasma during ECMO, units, median(IQR)	0[0-4]	NA	NA
	Platelet concentrates during ECMO, units, median(IQR)	0[0-3]	0[0-2]	0.29
	Bleeding complication during ECMO, n(%)	75(70)	27(64)	0.56

ACKNOWLEDGMENTS

We thank our research study nurses and hemostasis team of the national reference center for von Willebrand disease at the Lille University Hospital.

SOURCES OF FUNDING

This study was funded by the Conseil Régional Nord-Pas-de-Calais, the French national Research Agency (Programme d'Investissement d'Avenir) with the Hospital-University Research in Health program (Recherche Hospitalo-Universitaire, WILL-ASSIST-HEART ANR-17-RHUS-0011).

CONFLICT OF INTEREST DISCLOSURES

None

SUPPLEMENTAL MATERIALS

Expanded Materials and Methods

Major Resources Table

What is Known?

-Thrombocytopenia is frequently observed in patients assisted with extracorporeal membrane oxygenation (ECMO) for pulmonary and/or cardiac support.

-Thrombocytopenia is a major risk-factor for severe, life-threatening complications in ECMO patients -Shedding of the platelet receptor GPIbα has been reported in ECMO patients and is associated with increased platelet clearance, but the mechanisms contributing to receptor cleavage are poorly understood.

What New Information Does This Article Contribute

-Thrombocytopenia appears rapidly after ECMO support initiation, corrects quickly after support withdrawal and is associated with a parallel platelet GPIbα shedding.

-Platelet count is inversely correlated with GPIbα shedding.

-Rapid elimination of GPIbα-negative platelets from the circulation is directly dependent on shear forces pertinent to the mechanical support. Patients under mechanical pulmonary and/or cardiac support often display low peripheral platelet count and hemostatic complications.

Thrombocytopenia could be due to platelet clearance secondary to GPIbα shedding but the mechanisms responsible for cleavage of the platelet receptor in ECMO settings are currently unknown.

Patients under mechanical pulmonary and/or cardiac support often display low peripheral platelet count and hemostatic complications. Thrombocytopenia could be due to platelet clearance secondary to GPIbα shedding but the mechanisms responsible for cleavage of the platelet receptor in ECMO settings are currently unknown.

To investigate the causes of ECMO-associated thrombocytopenia we studied platelet count, modifications and functions in patients, and developed in vitro & animal models mimicking those conditions. Our study demonstrates that thrombocytopenia has a high prevalence and an early onset/offset closely related to ECMO implantation and withdrawal in assisted patients. Our translational models were instrumental to demonstrate that shear forces generated during ECMO support directly induce GPIbα shedding from the platelet surface but do not inhibit platelet functions.

GPIbα-negative platelets are rapidly cleared from circulation in NOD/SCID mice.

Overall, we describe a new mechanism of thrombocytopenia pertinent to ECMO patients that could be targeted by new therapeutic approaches aiming to decrease and counteract life-threatening complications.


J M Bonnisseau
email: jean-marc.bonnisseau@univ-paris1.fr

A Jamin

The Survival Assumption in Intertemporal Economies

Keywords: General Equilibrium Theory, Increasing Returns, Survival Assumption, Marginal Pricing, General Pricing Rules. JEL Classification: C62, C67, D21, D51

In an economy with a non-convex production sector, we provide an assumption on each individual producer, which implies that the survival assumption holds true at the aggregate level for general pricing rules. For the marginal pricing rule, we derive this assumption from the bounded marginal productivity of inputs. We apply this approach to intertemporal economies and we show how our assumption fits well with the time structure leading to a tractable existence result of equilibria, which could be apply to discrete dynamical growth models.

Introduction

The presence of increasing returns to scale, of fixed costs, and, more generally the lack of convexity in the production sector, are recognized as a failure of the competitive mechanism. [START_REF] Walras | Elements d'Economie Politique Pure[END_REF] first proposed that the non convex firms should be set to follow an average cost pricing behaviour. Later, the theory of marginal cost pricing has been developed, with the works of [START_REF] Pigou | The Economics of Welfare[END_REF], [START_REF] Lange | On the economic theory of socialism (parts one and two)[END_REF][START_REF] Lange | On the economic theory of socialism (parts one and two)[END_REF], [START_REF] Lerner | A note on socialist economics[END_REF], [START_REF] Hotelling | The general welfare in relation to problems of taxation and of railway and utility rates[END_REF], and later of [START_REF] Allais | Traité d'Economie Pure[END_REF]. Since [START_REF] Guesnerie | Pareto optimality in non-convex economies[END_REF], this theory has been formulated in the general framework of general equilibrium theory, and many results, addressing as well the existence problem of marginal cost pricing equilibria as their optimality, has been published since. In parallel, another approach was developed, with the emergence of existence results for general pricing rules with bounded losses. For a global survey, see the special issue of the Journal of Mathematical Economics on General Equilibrium and Increasing Returns (See, Cornet 1988, Brown's chapter in the Handbook of Mathematical Economics 1992, or [START_REF] Villar | Equilibrium and Efficiency in Production Economies[END_REF].

Beside standard assumptions on the consumers and producers, the existence results require a crucial assumption, namely the survival assumption, which is sufficient for the marginal pricing rule and for the bounded losses general pricing rules. The purpose of this hypothesis is to solve the aggregate survival problem. In the classical Arrow-Debreu framework, it is sufficient to assume that the total initial endowments vector is positive and the inactivity is feasible for each firms. Then, the Survival Assumption is satisfied when the firms are price-taker and profit maximizer. In models incorporating increasing returns, where firms may exhibit losses, this kind of assumption is no longer enough.

Let us present more precisely the Survival Assumption. In an economy with finite numbers, of commodities, and n of firms, we denote by Y j the production set of firm j and by ϕ j its pricing rule, that associates to any production y j ∈ Y j a set ϕ j (y j ) of price systems compatible with y j , according to the pricing behaviour of firm j. Firm j is thus said to be at equilibrium for a pair (p, y j ) if the price system p is acceptable for the firm j at y j ∈ Y j , i.e. p ∈ ϕ j (y j ). A production equilibrium is then a state (p, (y j )) of the economy in which the price system p is acceptable for each firm, according to its pricing behaviour, for the given productions (y j ). [START_REF] Bonnisseau | General equilibrium theory with increasing returns: the existence problem[END_REF] propose the following version of the survival assumption: at production equilibrium, if a positive amount of a reference commodity bundle is added to the total initial endowments, so that the production equilibrium becomes attainable, then the total wealth in the economy is above the consumers aggregate subsistence level. This kind of assumption is unsatisfactory in the sense that it states a condition on the set of production equilibria, an endogenous variable of the model. It would be worth preferable to state a condition on the individual primitive data of the economy, notably on the firms characteristics, which does not involve any coordination among the producers.

In the case of marginal pricing, [START_REF] Vohra | Marginal cost pricing under bounded marginal returns[END_REF] proposes a conditions, called bounded marginal return, which generalizes a result in [START_REF] Bonnisseau | Existence of marginal cost pricing equilibria: the nonsmooth case[END_REF]. The intuition comes from the single consumer, single producer, two commodities economy. In that case, the Survival Assumption may not be satisfied if the production set exhibits a vertical tangent, that is when the marginal productivity of the input is infinite. In [START_REF] Vohra | Marginal cost pricing under bounded marginal returns[END_REF], this idea is formalized in a general setting but whith the same a priori given sets of inputs and outputs for all firms.

The aim of this article is to provide a condition on the firms individual pricing behaviour under which the aggregate survival is ensured for the marginal pricing rule as well as for general pricing rules. We first states this condition in an abstract framework with a partition of the set of commodities. We provide a first interpretation in terms of degree of sophistication of the commodities.

The most interesting application of our result concerns intertemporal economies. Indeed, we then consider the the economically meaningful partition given by the time structure of the economy. Our result holds true with an arbitrary number of periods and it opens the way to infinite horizon models through a limit argument. Furthermore, we now have a tractable existence result, which opens the possibility to apply it in dynamical macroeconomic models.

The amazing ascertainment is how mild is our condition in this context. Indeed, for a general pricing rule, it merely requires that if a costly investment is done at one period, there exists a positive return at a future period, that is a least an output with a positive price. For the marginal pricing rule, we consider the case where the production possibilities are defined recursively by a sequence of one-period production mappings at it is done in most of the dynamical models. This just means that production is time consuming. Then, the global production sector satisfies the Survival Assumption if, for each period, the marginal productivity of inputs is bounded above. So, it appears that it is economically not restrictive.

The paper is organized as follows: in Section 2, we briefly present the model of an Arrow-Debreu economy and the representation of the behavior of non-convex producers by pricing rules; Section 3 is devoted to the statement of Assumption (CH) on each producer and we show that it is sufficient to obtain the Survival Assumption at the aggregate level; Section 4 deals with the marginal pricing rule and we state the generalization of the Bounded Marginal Return Assumption associated to a partition of the commodity set; finally, Section 5 studies the case of intertemporal economies and provides the adaptation of Assumptions (CH) and (BMR) to the natural partition

We consider an economy with positive finite numbers of commodities and n of firms. We take R for commodity space, and we consider normalized price vectors in S, the unit simplex of R .

The technological possibilities of firm j, (j = 1, ..., n) are represented by a subset Y j ⊂ R satisfying the following assumption: Assumption (P) For every j, Y j is a closed subset of R satisfying the free-disposal property: Y j -R + ⊂ Y j and the possibility of inaction: 0 ∈ Y j .

Note that, under Assumption (P), for every j, the boundary ∂Y j of the production set Y j exactly coincides with the set of (weakly) efficient productions of firm j. For every j, the behavior of firm j is described by its pricing rule ϕ j , a correspondence which associates with each (weakly) efficient production y j a subset ϕ j (y j ) of S of admissible price vectors. This formalization is compatible with various behaviors considered in the economic literature, notably with the profit maximization, the average pricing or the marginal pricing, defined for producer j, respectively, by:

P M j (y j ) = {p ∈ S | p • y j ≥ p • y j for every y j ∈ Y j }, AP j (y j ) = {p ∈ S | p • y j = 0}, M P j (y j ) = N Y j (y j ) ∩ S,
where N Y j (y j ) denotes the Clarke's normal cone to Y j at y j (See [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Cornet | General equilibrium theory and increasing returns: Presentation[END_REF]). For every j, firm j is said to be in equilibrium at the pair (p, y j ) ∈ S × ∂Y j if p is an admissible price vector for firm j given the (weakly) efficient production y j , i.e. if p ∈ ϕ j (y j ). Given the productions (y j ) ∈ n j=1 ∂Y j , if there exists a price vector p ∈ S admissible for every firm j, then we say that the collection (p, (y j )) is a production equilibrium. We shall denote by P E the set of production equilibria of the economy, that is:

P E = (p, (y j )) ∈ S × n j=1
∂Y j p ∈ ϕ j (y j ) for every j} .

Since our main focus is on the production sector of the economy, we just consider for the consumption sector the agregate consumption set X ⊂ R and the agregate initial endowments ω ∈ R . We make the following assumption on the consumption side:

Assumption (C) X = R + and ω h > 0 for all h = 1, . . . , .

An alternative to the survival assumption

It is quite justified to wonder about the appropriateness of the survival assumption since it poses a condition on endogenous variables, the production equilibria. In [START_REF] Bonnisseau | General equilibrium theory with increasing returns: the existence problem[END_REF], existence of equilibria is proven under the following version of the survival assumption :

Assumption (SA) For every (p, (y j )) ∈ P E, p • n j=1 y j + ω > min 0, min n j=1 y jh + ω h | h = 1, . . . , .
Since p ∈ S, one always has

p • n j=1 y j + ω ≥ min n j=1 y jh + ω h | h = 1, . . . , .
Hence Assumption (SA) is satisfied when the productions (y j ) is strictly feasible in the sense that n j=1 y jh + ω h > 0 for all h. Otherwise, Assumption (SA) only requires to have a strict inequality p

• n j=1 y j + ω > min n j=1 y jh + ω h | h = 1, . . . ,
instead of a weak inequality. So, Assumption (SA) means that the firms following independently their pricing rules do not select the worse possible joint price p in terms of global wealth. Note that this assumption is satisfied if the firms follow loss-free pricing rules, that is if q • y j ≥ 0 for all q ∈ ϕ j (y j ).

Before stating the sufficient conditions on the general pricing rules, we shall introduce several notations. For every j and every (weakly) efficient production y j ∈ ∂Y j , we let O(y j ) be the set of y j -outputs, that is the subset of commodities h, (h = 1, . . . , ) satisfying y jh ≥ 0, and I(y j ) = {1, . . . , } \ O(y j ) the set of y j -inputs. Note that the possibility of inaction and the freedisposal assumption imply that 0(y j ) is nonempty for all weakly-efficient production y j .

Our hierarchical condition involves a partition of the set of commodities. So, for any H ⊂ {1, . . . , }, z H is the vector of R H with the coordinates of the vector z for the commodities h ∈ H. For any partition H = (H 1 , . . . , H k ) of the commodity set {1, . . . , }, for every commodity h, we let ν(h) be the unique integer in {1, . . . , k} satisfying h ∈ H ν(h) .

Assumption (CH) There exists a partition H = (H 1 , . . . , H k ) of the commodity set {1, . . . , }, such that, for every j, every y j ∈ ∂Y j , every q j ∈ ϕ j (y j ) and every h ∈ I(y j ), if q jh > 0, then there exists a commodity k such that ν(k) > ν(h) and q jk > 0.

For a pair (q j , y j ), if it does not exist a commodity h ∈ I(y j ) with q jh > 0, then Assumption (CH) is obviously satisfied.

We remark that Assumption (CH) involves only fundamentals of the economy, namely production sets and pricing rules and it does not involve the set of production equilibria, which is endogenous. Assumption (CH) can be checked independently for each producer. This is the major improvement with respect to the standard survival assumption.

To interpret this assumption, we remark that the partition H induces a hierarchy among the commodities. The higher is ν(h), the higher is the commodity h in the hierarchy. Then, Assumption (CH) means that if an input has a positive price, then there exists another commodity with a positive price, which is higher in the hierarchy. By a recursive argument, this implies that there exists an output with a positive price, which is higher in the hierarchy than an input with a positive price. In particular, this implies that the commodities in the highest class H k are either outputs or free inputs in the sense that the price is equal to 0.

A first example of a hierarchy given by a partition H of the commodity set is associated to a degree of sophistication. We can rank the commodities from the raw materials to the consumption goods, which are not used in the production of other commodities. In that case, Assumption (CH) means that if the firm chooses a price q j associated to the production y j according to the pricing rule ϕ j , then if the price of an input is positive, there exists an output with a positive price and a degree of sophistication higher than the one of the input.

Globally, the production is oriented towards more sophisticated commodities. This leads to a weak coordination of the production sector which avoids the worse possible choice of the price when each firm follows its pricing rule.

Proposition 1 Assumption (SA) is satisfied if Assumptions (C), (P) and (CH) hold true.

A fundamental consequence of this proposition is the fact that we can get the existence of an equilibrium with general pricing rules under standard assumptions on the consumer side of the economy, Assumption (P) on the production side, the bounded losses Assumption and Assumption (CH) replacing Assumption (SA). So, we have an existence result under assumptions only on the fundamentals of the economy like for a competitive equilibrium in the Arrow-Debreu theory.

Proof: Let us assume by contraposition that there exists (p, (y j )) ∈ P E such that p • ( n j=1 y j + ω) ≤ min{0, min h { n j=1 y jh + ω h }}. Since p ∈ S, p • ( n j=1 y j + ω) ≥ min h { n j=1 y jh + ω h }. Hence, one gets

p • ( n j=1 y j + ω) = min h { n j=1 y jh + ω h } ≤ 0
This implies that n j=1 y jh +ω h = min h { n j=1 y jh +ω h } for every commodity h such that p h > 0. From Assumption (C), ω h > 0 for all h, hence if p h > 0 then n j=1 y jh < 0. Since p ∈ S, there exists h 0 such that p h 0 > 0. Since n j=1 y jh 0 < 0, there exists j 0 such that y j 0 h 0 < 0. Let h0 ∈ {1, . . . , } such that ph 0 > 0, y j 0 h0 < 0 and ν( h0 ) ≥ ν(h) for all h satisfying p h > 0 and y j 0 h < 0. Such h0 exists since the set of commodities is finite. Assumption (CH) implies that there exists h 1 such that p h 1 > 0 and ν(h 1 ) > ν( h0 ). The choice of h0 implies that y j 0 h 1 ≥ 0. Consequently, since n j=1 y jh 1 < 0, there exists j 1 such that y j 1 h 1 < 0. With the same argument, we find a commodity h1 such that ν( h1 ) ≥ ν(h 1 ) > ν( h0 ). By iterating this procedure, one gets a contradiction since the number of commodities is finite.

With the marginal pricing rule, we can be more precise and give a condition on the production sets under which Assumption (CH) holds true. This more primitive condition is a condition of bounded marginal returns. Indeed, in a single firm and two commodity economy, the Survival Assumption may not be satisfied if the boundary of the production set has a vertical tangent. Such a tangency implies that the marginal productivity is infinite. We show that, even if they are increasing, marginal returns must be bounded above for the Survival Assumption to be satisfied.

If the production set Y j is defined by a smooth transformation function g j on a neighborhood of a weakly efficient production y j that is, for a neighborhood U of y j , Y j ∩ U = {z j ∈ U | g j (z j ) ≤ 0}, then the marginal return of an input h 1 for an output h 2 is

∂g j ∂y h 1 (y j ) ∂g j ∂y h 2 (y j )
So, the marginal return is bounded if ∂g j ∂y h 2 (y j ) is positive. This means that, locally, the level of production in commodity h 2 is a smooth function of quantities of inputs.

To capture this idea without smoothness assumption on the production set, we can remark that the above condition implies that there exists an inward direction2 z at y j , satisfying z h > 0 for all h = h 2 . So, we will put our assumption on the set of inward directions, that is on the tangent cone.

We recall that the Clarke's tangent cone to Y j at y j is defined by:

T Y j (y j ) =    v ∈ R ∀(y ν j ) ⊂ Y j , (y ν j ) → y j , ∀(t ν ) ⊂ R + , (t ν ) → 0, ∃(v ν ) ⊂ R such that (v ν ) → v and y ν j + t ν v ν ∈ Y j ∀ν    .
From the free-disposal assumption, we may have some useless inputs for the production set. These inputs are not relevant in the production process and they have a zero price when the firm follows the marginal pricing rule.

Definition 1 Let us consider a production set Y j satisfying Assumption (P).

A commodity h is a useless input if for all y j ∈ ∂Y j , y j = ((y jk ) k =h , 0) belongs to Y j whenever y jh < 0.

The following lemma recall that a useless input has a zero price when the firm follows the marginal pricing rule. The proof is given in Appendix.

Lemma 1 Let us consider a production set Y j satisfying Assumption (P). Let h be a useless input. Then, for all y j ∈ ∂Y j and q ∈ M P j (y j ), if y jh < 0, then q h = 0.

In the following, Ĩ(y j ) is the set of non-useless input and, for any partition

H = (H 1 , . . . , H k ) of the commodity set, ν(y j ) = max{ν(h) | h ∈ Ĩ(y j )}.
We can now state our assumption on the bounded marginal return of Y j , which implies that Assumption (CH) holds true for the marginal pricing rule.

Assumption (BMR) There exists a partition H = (H 1 , . . . , H k ) of the commodity set {1, . . . , }, such that, for every j, every y j ∈ ∂Y j , there exists z ∈ T Y j (y j ) such that z h > 0 for all h ∈ Ĩ(y j ) and z h ≥ 0 for all h ∈ ∪ ν≤ν(y j ) H ν .

The interpretation of Assumption (BMR) is the following. From the freedisposal assumption, a vector in the tangent cone cannot be positive. So, if we consider the vector z given by Assumption (BMR), the exists an output k, satisfying ν(k) > ν(y j ) and z k < 0. If z is an inward direction, then -z is an outward direction. So, for t small enough, y j -tz does not belongs to Y j . This means that the increase of the quantities of inputs (tz h ) h∈ Ĩ(y j ) is not enough to produce the additional quantity of output -tz k > 0. Hence, the productivity of the inputs is bounded above.

To compare with Vohra (1992), we just have to note that the assumption in this paper involves only a two set partition with the upper class being the one of "final" outputs, i.e., commodities that are never used as an input by any firm. Vohra's assumption rules out the possibility of an intermediate producer, in the sense that it only produces commodities that will be inputs for others firms since his Bounded Marginal Return Assumption implies that each firm is able to produce at least one final output.

Proposition 2 Under Assumptions (C) and (P), Assumption (BMR) implies that Assumption (CH) holds for the marginal pricing rule.

An immediate corollary of this proposition and Proposition 1 is that the survival assumption is a consequence of Assumptions (C), (P) and (BMR) for the marginal pricing rule. So, as for general pricing rules, we obtain the existence of a marginal pricing equilibrium under assumptions only on the fundamentals of the economy.

Proof: Let y j ∈ ∂Y j and q j ∈ M P j (y j ). Since the normal cone is the negative polar cone of the tangent cone, one has q j • z ≤ 0 for the vector z given by Assumption (BMR). Furthermore, for all useless inputs k, if y jk < 0, then the vector with zero components but the kth one equal to 1 belongs to the tangent cone T Y j (y j ). This implies that q jk = 0.

Let a commodity h such that y jh < 0 and q jh > 0. Consequently, h is not a useless input and ν(h) ≤ ν(y j ). Let z as given by Assumption (BMR). Then,

h |ν(h )≤ν(y j ) q jh z h > 0.
Since q j •z ≤ 0, there exists h such that ν( h) > ν(y j ) ≥ ν(h) such that zh < 0 and q j h > 0. Hence Assumption (CH) holds true.

The survival assumption in an intertemporal economy

We now consider the most important application of the general result given above, namely, an intertemporal economy. Indeed, in this case, the partition of the set of commodities is natural: it is defined by the date at which the commodity is available. This case is fundamental since it allows us to consider the existence of equilibrium in a dynamical model with a finite horizon and in an overlapping generation model. We can then expect to consider an infinite horizon model through a limit argument.

We are considering an economy with T periods and commodities at each period. Then the commodity space is R T and each commodity is represented by a pair (h, t) in {1, . . . , } × {1, . . . , T }. In this setting, the set of commodities is naturally partitioned by the date at which the commodity is available. So we have a natural partition H = (H 1 , . . . , H T ). For each commodity (h, t), the index ν(h, t) defined in the previous section is merely equal to t, that is the date at which the commodity h is available.

In this framework, we can rewrite Assumption (CH) as follows:

Assumption (CT)For every j, every y j ∈ ∂Y j , every q j ∈ ϕ j (y j ) and every commodity (h, t) ∈ I(y j ), if q j(h,t) > 0, then there exists a commodity (k, τ ) such that τ > t and q j(k,τ ) > 0.

This assumption is really mild since it simply claims that a costly investment q j(h,t) y j(h,t) < 0 at a period t is done only if there is a possibility of a non negative return in a subsequent period q j(k,τ ) y j(k,τ ) ≥ 0. If the production is time consuming in the sense that an investment at one period produces some outputs only in the next periods, the statement is totally intuitive since no return can be expected at the period of investment. An economically reasonable pricing rule should satisfy this requirement, which is much more lighter than a no-loss condition, which requires that the return is larger than the investment.

So, in a multi-period economy with increasing returns, the existence of a general economic equilibrium with pricing rules can be obtained under the standard assumptions as the ones presented in [START_REF] Bonnisseau | General equilibrium theory with increasing returns: the existence problem[END_REF] or [START_REF] Villar | Equilibrium and Efficiency in Production Economies[END_REF] but Assumption (CT) replacing the Survival Assumption. Hence, the class of pricing rules compatible with an equilibrium is very large and far beyond the no-loss pricing rules.

We now turn to the case of the marginal pricing rule with the production possibilities represented by production mappings as it is done in most of the dynamical economic models.

A production mapping is a set-valued mapping F from -R + to R . For a given vector of inputs z ∈ -R + , F (z) is the set of possible outputs. We posit the following assumption on F . the sense that the consumption of an input at a period t has no influence on the output at this period. To a production mapping F , we associate an elementary production set Z in (R ) 2 defined by:

Assumption (F) F has a closed graph, nonempty values, F (0) = -R + , for all z ∈ -R + , F (z) ∩ R + is bounded, F (z) = (F (z) ∩ R + ) -R + and for all (z, z ) ∈ (-R + ) 2 , if z ≤ z , then F (z ) ⊂ F (z).
Z = {(z, ζ) ∈ -R + × R | ζ ∈ F (z)}
For the intertemporal production, we assume that the production possibilities of a producer j at a period t is represented by a production mapping F jt or by the associated production set Z jt . To describe the global production possibilities on the T periods, we use the following linear mapping Γ from (R × R ) T -1 to (R ) T defined by:

Γ (z t , ζ t+1 ) T -1 t=1 = z 1 , (z t + ζ t ) T -1 t=2 , ζ T
The production set is then :

Y j = Γ T -1 t=1 Z jt
One checks that Assumption (P) is satisfied by Y j when the production mappings F jt satisfy Assumption (F). Now, we proved that if the production mappings F jt satisfy the bounded marginal return condition then the global production set Y j satisfies Assumption (CT) for the marginal pricing rule . We can now state our result.

Proposition 3 Under Assumptions (C) and (F), if the elementary production sets Z jt satisfy Assumption (BMR) for the natural partition associated to the two periods, then the production set Y j satisfies Assumption (CT) for the marginal pricing rule.

Using this result, one remarks that we can check directly on the elementary production sets Z jt if the whole production set Y j satisfies Assumption (CT) and the result is independent of the number of periods. On Z jt , since the partition has only two elements, we are in the same setting as Vohra (1992). Our contribution is to show that we can extend the result from a two-period economy to a T -period economy. Furthermore, Assumption (BMR) has a simpler formulation for an elementary production set Z jt . Indeed, the commodities in the second period are either outputs or useless inputs. Hence Assumption (BMR) holds true if for every (z j , ζ j ) ∈ ∂Z j , there exists (z, ζ) ∈ T Z j (z j , ζ j ) such that z ≥ 0 and z h > 0 if z jh < 0.

Proof of Proposition 3. We first recall some definitions of normal cones that we will use in the following.

Definition 2 Let X be a subset of R n and x an element of X. Let x be the canonic Euclidean norm of x.

(a) The proximal normal cone to X at x, denoted N P X (x), is defined by:

N P X (x) = {v ∈ R n | ∃ α > 0, B(x + αv, α v ) ∩ X = ∅} (b)
The limiting normal cone to X at x, denoted N L X (x), is defined by: N X (x) = y ∈ R n ∃ (x ν , y ν ) ⊂ X × R n , (x ν , y ν ) → (x, y), y ν ∈ N P X (x ν ), ∀ν ∈ N (c) The Clarke's normal to X at x, denoted N X (x) is closed convex hull of N L X (x).

We now state a lemma, the proof of which is given in Appendix, on the relation between the limiting normal cone of Y j and the limiting normal cones of the elementary production set Z j .

Lemma 2 Let q an element of N L Y j (y j ). Then, under Assumption (F), there exists (z j,t , ζ j,t+1 ) T -1 t=1 ∈ T -1 t=1 Z jt such that (i) y j = Γ (z j,t , ζ j,t+1 ) T -1 t=1 ) ;

(ii) for all t = 1, . . . , T -2, ζ j,t+1 ≥ 0;

(iii) for all t = 1, . . . , T -1, (q j,t , q j,t+1 ) ∈ N L Z jt (z j,t , ζ j,t+1 ).

We now come to the proof of Proposition 3. Let y j ∈ ∂Y j , q j ∈ M P j (y j ) and (h, τ ) such that y j(h,τ ) < 0 and q j(h,τ ) > 0.

  Assumption (F) is standard. It incorporates the free-disposability in the production process and the fact that the production process takes time in 11 Document de Travail du Centre d'Economie de la Sorbonne -2009.76

Document de Travail du Centre d'Economie de la Sorbonne -2009.76

Document de Travail du Centre d'Economie de la Sorbonne -2009.76induced by the sequence of periods.

Document de Travail du Centre d'Economie de la Sorbonne -2009.76

That is, y j + tz ∈ Y j for all t > 0 small enough. In the smooth case, if ∇g j (y j ) • z < 0, then z is inward. 8 Document de Travail du Centre d'Economie de la Sorbonne -2009.76

Under Assumption (P), the limiting normal cone is a closed cone included in R T + . Hence its closed convex hull is merely the convex hull. So, there exists (q 1 j , . . . , q k j ) ∈ (N L Y j (y j )) k and β ∈ R k ++ such that q j = k κ=1 β κ q κ j . Consequently, there exists at least an index κ such that q κ j(h,τ ) > 0. From Lemma 2, there exists (z j,t , ζ j,t+1

(ii) for all t = 1, . . . , T -2, ζ j,t+1 ≥ 0;

(iii) for all t = 1, . . . , T -1, (q κ j,t , q κ j,t+1 ) ∈ N L Z jt (z j,t , ζ j,t+1 ).

Since q κ j(h,τ ) > 0 and y j(h,t) < 0, one first remarks that τ < T . Indeed, if

From the definition of Z j,T -1 through the production mapping F j,T -1 satisfying Assumption (F), one deduces that the commodity (T, h) is a useless input of Z j,T -1 , which satisfies Assumption (P). So, from Lemma 1, q κ j(h,T ) = 0, which contradicts q κ j(h,T ) = q κ j(h,τ ) > 0. Since y j(h,τ ) < 0 and ζ j,τ ≥ 0, one deduces that z j,(h,τ ) < 0. Since, Z j,τ satisfies Assumption (BMR), Proposition 2 shows that Z j,τ satisfies Assumption (CH) for the marginal pricing rule. Consequently, since (q κ j,τ , q κ j,τ

), there exists a commodity h such that q κ j,(h ,τ +1) > 0. Since for all κ , q κ j,(h ,τ +1) ≥ 0, one concludes that q j,(h ,τ +1) > 0. So, Y j satisfies Assumption (CT) for the marginal pricing rule.

Appendix

Proof of Lemma 1. Let y j ∈ ∂Y j and q ∈ M P j (y j ) such that y jh < 0. From the definitions of a useless input and the tangent normal cone, one easily shows that the vector ε h , the vector with 0 components but the hth one equal to 1, belongs to the tangent cone T Y j (y j ). Since the Clarke's normal cone is the negative polar cone of the tangent cone, one deduces that q h ≤ 0. Since Y j satisfies the free-disposal Assumption, then q h ≥ 0. So, q h = 0.

Proof of Lemma 2. Let y j ∈ ∂Y j and q ∈ N L Y j (y j ). From the definition of the limiting normal cone, there exists a sequence (y ν j ) in ∂Y j , which converges to y j and a sequence (q ν ), which converges to q such that q ν ∈ N P Y j (y ν j ) for all ν.

The definition of Y j implies that there exists (z ν j,t , ζ ν j,t+1

) for all ν. We can assume that ζ ν j,t+1 ≥ 0 for all t = 1, . . We now prove that the sequence (z ν j,t , ζ ν j,t+1 ) T -1 t=1 ) is bounded. We first remark that z ν j,1 = y ν j,1 and it converges to y j,1 , hence it is bounded. Let z 1 be such that z 1 ≤ z ν j,1 for all ν. If T + 2, then (ζ ν j,2 = y ν j,2 ) and it is bounded since (y ν j,2 ) converges to y j,2 . If T > 2, for all ν, ζ ν j,2 ∈ F j,1 (z ν j,1 ) ⊂ F j,1 (z 1 ). So, from Assumption (F), the sequence (ζ ν j,2 ) is bounded since

) converges to y j,2 , one deduces that the sequence (z ν j,2 ) is bounded. Using recursively the same argument, we finally show that the sequence (z

) is bounded, it has a converging subsequence that we denote again by (z ν j,t , ζ ν j,t+1 ) T -1 t=1 ) for the sake of simpler notations. The limit is denoted (z j,t , ζ j,t+1 ) T -1 t=1 ) . Since the elementary production sets Z j,t are closed and the linear mapping Γ is continuous, we immediately show that (z j,t , ζ j,t+1 ) T -1 t=1 ∈ T -1 t=1 Z jt , y j = Γ (z j,t , ζ j,t+1 ) T -1 t=1 ) and for all t = 1, . . . , T -2, ζ j,t+1 ≥ 0.

We now show that for all ν, for all t = 1, . . . , T -1, (q ν j,t , q ν j,t+1 ) ∈ N P Z jt (z ν j,t , ζ ν j,t+1 ). Since q ν ∈ N P Y j (y ν j ), there exists α ν > 0 such that B(y ν j + α ν q ν , α ν q ν ) ∩ Y j = ∅. Let us assume by contraposition, that there exists (z j,t , ζ j,t+1 ) ∈ B((z ν j,t , ζ ν j,t+1 ) + α ν (q ν t , q ν t+1 ), α ν (q ν t , q ν t+1 ) ) ∩ Z j,t . Then,

) belongs to Y j . Furthermore, y j -y ν j -α ν q ν = (-α ν q ν τ ) τ =t,t+1 , (z ν j,t , ζ ν j,t+1 ) + α ν (q ν t , q ν t+1 )

So, y j -y ν j -α ν q ν 2 = (α ν ) 2 τ =t,t+1 q ν τ 2 + (z ν j,t , ζ ν j,t+1 ) + α ν (q ν t , q ν t+1 ) 2 < (α ν ) 2 T τ =1 q ν τ 2

Consequently, y j belongs to B(y ν j +α ν q ν , α ν q ν )∩Y j , which is a contradiction with B(y ν j + α ν q ν , α ν q ν ) ∩ Y j = ∅.