Jérémy Ortillon

Nathalie Hézard

Karim Belmokhtar

Charlotte Kawecki

Christine Terryn

Guenter Fritz

Alexandre Kauskot

Ann Marie Schmidt

Philippe Rieu

Philippe Nguyen

Marie

Nathalie Hezard

Dr Pascal Maurice
email: pascal.maurice@univ-reims.fr

Dr Fatouma Touré
email: fatouma.toure@chu-limoges.fr

Receptor for Advanced Glycation End Products is Involved in Platelet Hyperactivation and Arterial Thrombosis during Chronic Kidney Disease

Keywords: Chronic Kidney Disease, RAGE, Platelets, Arterial thrombosis, Uremic toxins

 Chronic kidney disease (CKD) is associated with an increased frequency of thrombotic events and accumulation of uremic toxins.

 Some of these uremic toxins can interact with the Receptor for Advanced Glycation End products (RAGE).

What does this paper add?

 CKD induces platelet hyperactivation and accelerates arterial thrombus formation.

 RAGE deletion has a protective role in CKD-induced platelet hyperactivation and arterial thrombosis.

ABSTRACT

Background -Chronic kidney disease (CKD) is associated with a high cardiovascular mortality due to increased rates of vascular lesions and thrombotic events, as well as serum accumulation of uremic toxins. A subgroup of these toxins (AGEs and S100 proteins) can interact with the receptor for advanced glycation end-products (RAGE). Objective -In this study, we analyzed the impact of CKD on platelet function and arterial thrombosis, and the potential role of RAGE in this process. Materials and methods -Twelve weeks after induction of CKD in mice, platelet function and time to complete carotid artery occlusion were analyzed in four groups of animals (sham-operated, CKD, Apoe -/-, and Apoe -/-/Ager -/-mice). Results -Analysis of platelet function from whole blood and platelet-rich plasma showed hyperactivation of platelets only in CKD Apoe -/-mice. There was no difference when experiments were done on washed platelets. However, pre-incubation of such platelets with AGEs or S100 proteins induced RAGE-mediated platelet hyperactivation. In vivo, CKD significantly reduced carotid occlusion times of Apoe -/- mice (9.2±1.1 vs 11.1±0.6 min for sham, p<0.01). In contrast, CKD had no effect on occlusion times in Apoe -/-/Ager /-mice. Moreover, carotid occlusion in Apoe -/-CKD mice occurred significantly faster than in Apoe -/-/Ager -/-CKD mice (p<0.0001). Conclusion -Our results show that CKD induces platelet hyperactivation, accelerates thrombus formation in a murine model of arterial thrombosis, and that RAGE deletion has a protective role. We propose that RAGE ligands binding to RAGE is involved in CKD-induced arterial thrombosis.

INTRODUCTION

Individuals with chronic kidney disease (CKD) have a 10-to 20-fold higher risk of death due to cardiovascular complications than the general population [START_REF] Parfrey | The clinical epidemiology of cardiac disease in chronic renal failure[END_REF][START_REF] Go | Chronic kidney disease and the risks of death, cardiovascular events, and hospitalization[END_REF] . The risk of cardiovascular events increases steadily with declining glomerular filtration rate [START_REF] Go | Chronic kidney disease and the risks of death, cardiovascular events, and hospitalization[END_REF] . Underlying factors include accelerated atherosclerosis, increased arterial stiffness and pathological arterial thrombosis, with related increases in the risk of myocardial infarction and stroke [START_REF] Vanholder | Chronic kidney disease as cause of cardiovascular morbidity and mortality[END_REF][START_REF] Drueke | Atherosclerosis in CKD: differences from the general population[END_REF][START_REF] Fort | Chronic renal failure: a cardiovascular risk factor[END_REF][START_REF] London | Arteriosclerosis, vascular calcifications and cardiovascular disease in uremia[END_REF] . Arterial thrombosis is one of the key features of uremic vasculopathy but the pathophysiology is not completely understood.

Indeed, the modifications of the vascular wall cannot totally account for this increased risk of arterial thrombosis. In addition, platelet dysfunction has been described in CKD patients, including defects in adhesion, secretion of alpha granule content, arachidonic acid and prostaglandin metabolism, and calcium mobilization [START_REF] Landray | Inflammation, endothelial dysfunction, and platelet activation in patients with chronic kidney disease: the chronic renal impairment in Birmingham (CRIB) study[END_REF] . Furthermore, platelets from patients with CKD have been reported to be hyperreactive, with increased expression of P-selectin and  IIb  3 integrin [START_REF] Landray | Inflammation, endothelial dysfunction, and platelet activation in patients with chronic kidney disease: the chronic renal impairment in Birmingham (CRIB) study[END_REF] . The uremic milieu may play a role in CKD-related platelet dysfunction.

Aside from increased cardiovascular risk, CKD is also characterized by the retention of various solutes that are normally excreted by the kidneys [START_REF] Vanholder | Review on uremic toxins: classification, concentration, and interindividual variability[END_REF] . Those solutes have emerged as key factors in CKD-related cardiovascular disease and are collectively named uremic toxins [START_REF] Brunet | Does uremia cause vascular dysfunction?[END_REF][START_REF] Jourde-Chiche | Vascular incompetence in dialysis patients--protein-bound uremic toxins and endothelial dysfunction[END_REF] . Recent findings suggest that uremic toxins may play a role in CKD-related platelet hyperactivation and arterial thrombosis [START_REF] Karbowska | Indoxyl Sulfate Promotes Arterial Thrombosis in Rat Model via Increased Levels of Complex TF/VII, PAI-1, Platelet Activation as Well as Decreased Contents of SIRT1 and SIRT3[END_REF][START_REF] Holy | Carbamylated Low-Density Lipoproteins Induce a Prothrombotic State Via LOX-1: Impact on Arterial Thrombus Formation In Vivo[END_REF] . For instance, indoxyl sulfate, a typical uremic toxin derived from tryptophan, has been reported to enhance platelet response and arterial thrombosis in uremic mice [START_REF] Karbowska | Indoxyl Sulfate Promotes Arterial Thrombosis in Rat Model via Increased Levels of Complex TF/VII, PAI-1, Platelet Activation as Well as Decreased Contents of SIRT1 and SIRT3[END_REF] . Similarly, carbamylated low-density lipoproteins (cLDL), resulting from post-translational modification of LDL by carbamylation, is particularly increased in uremic conditions.

Intravenous injection of cLDL in animal models accelerated platelet aggregation and arterial thrombus formation [START_REF] Holy | Carbamylated Low-Density Lipoproteins Induce a Prothrombotic State Via LOX-1: Impact on Arterial Thrombus Formation In Vivo[END_REF] . Among the uremic toxins, Advanced Glycation End products (AGEs) are long-lived heterogeneous protein adducts produced from a non-enzymatic chemical reaction between sugars and the amino groups of proteins (Maillard reaction) [START_REF] Maillard | Action des acides aminés sur les sucres ; formation des mélanoïdes par voie méthodiques[END_REF] . AGEs are of particular interest because they have established cardiovascular toxicity and are binding partners for the Receptor for Advanced Glycation End products (RAGE) [START_REF] Schmidt | Advanced glycation endproducts interacting with their endothelial receptor induce expression of vascular cell adhesion molecule-1 (VCAM-1) in cultured human endothelial cells and in mice. A potential mechanism for the accelerated vasculopathy of diabetes[END_REF][START_REF] Schmidt | The multiligand receptor RAGE as a progression factor amplifying immune and inflammatory responses[END_REF] . RAGE is a multiligand, cell surface receptor of the immunoglobulin superfamily involved in various processes, such as cell migration, adhesion, and proliferation [START_REF] Schmidt | The multiligand receptor RAGE as a progression factor amplifying immune and inflammatory responses[END_REF][START_REF] Koch | Structural basis for ligand recognition and activation of RAGE[END_REF][START_REF] Goury | Deletion of receptor for advanced glycation end products exacerbates lymphoproliferative syndrome and lupus nephritis in B6-MRL Fas lpr/j mice[END_REF][START_REF] Rai | Lysophosphatidic acid targets vascular and oncogenic pathways via RAGE signaling[END_REF] . Other RAGE ligands are also elevated in CKD, such as S100 family proteins and serum amyloid A (SAA) [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF][START_REF] Gawdzik | Vascular remodeling and arterial calcification are directly mediated by S100A12 (EN-RAGE) in chronic kidney disease[END_REF] . RAGE engagement by its ligands initiates a pro-inflammatory response [START_REF] Schmidt | Advanced glycation endproducts interacting with their endothelial receptor induce expression of vascular cell adhesion molecule-1 (VCAM-1) in cultured human endothelial cells and in mice. A potential mechanism for the accelerated vasculopathy of diabetes[END_REF][START_REF] Xu | Impairment of vascular endothelial nitric oxide synthase activity by advanced glycation end products[END_REF][START_REF] Toure | Formin mDia1 mediates vascular remodeling via integration of oxidative and signal transduction pathways[END_REF] , and RAGE signaling is also involved in uremiainduced atherosclerosis and atherosclerotic plaque vulnerability 23 . We and others have previously reported that RAGE is a key player in CKD-related atherosclerosis and vascular calcification [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF][START_REF] Gawdzik | Vascular remodeling and arterial calcification are directly mediated by S100A12 (EN-RAGE) in chronic kidney disease[END_REF][START_REF] Harja | Vascular and inflammatory stresses mediate atherosclerosis via RAGE and its ligands in apoE-/-mice[END_REF][START_REF] Bro | A neutralizing antibody against receptor for advanced glycation end products (RAGE) reduces atherosclerosis in uremic mice[END_REF] . However, the role of RAGE in platelet behavior and arterial thrombus formation has not been widely explored. Here, we used ex vivo and in vivo approaches to investigate the role of CKD and RAGE in modulating platelet function and the formation of arterial thrombus.

METHODS

Reagents and antibodies

Horm collagen was purchased from Nycomed. ADP was obtained as Multiplate reagents from Roche Diagnostics. Rhodamine 6G, apyrase, prostaglandin E 1 (PGE 1 ), ferric chloride (FeCl 3 ), and thrombin were obtained from Sigma. D-Phe-Pro-Arg chloromethyl ketone dihydrochloride (PPACK) was obtained from Calbiochem. PE-coupled anti-mouse activated GPIIb/IIIa (clone JON/A), GPIb  (clone Xia.B2), and CD62P (clone Wug.E9) antibodies, and FITC-coupled antimouse  2 (clone Sam.G4), GPIb  (clone Xia.C3), and GPVI (clone JAQ1) antibodies were obtained from Emfret Analytics. PE-coupled anti-CD36 (clone CRF D-2712) and FITC-coupled anti- IIb (clone MWReg30) antibodies were obtained from BD Biosciences. Glycated-human serum albumin (HSA) and non-glycated-HSA were produced as previously described [START_REF] Ikeda | N (epsilon)-(carboxymethyl)lysine protein adduct is a major immunological epitope in proteins modified with advanced glycation end products of the Maillard reaction[END_REF] .

Animals

Apolipoprotein E knockout (Apoe -/-) mice on a C57BL/6 background were purchased from Charles River Laboratories. Apoe -/-/Ager -/-(RAGE knockout) mice were generated using a backcross-intercross scheme in the animal facility of New York University (Prof AM Schmidt) [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF][START_REF] Harja | Vascular and inflammatory stresses mediate atherosclerosis via RAGE and its ligands in apoE-/-mice[END_REF][START_REF] Belmokhtar | Receptor for advanced glycation end products: a key molecule in the genesis of chronic kidney disease vascular calcification and a potential modulator of sodium phosphate co-transporter PIT-1 expression[END_REF] . Embryos from the F3 generation were rederived and mice arising from the rederivation process transferred to the animal facility of Reims France (IFR53 Pôle Santé, Reims, France). Animals were housed in a barrier-and pathogen-free animal facility with a 12-h lightdark cycle and fed a regular chow diet. All experiments were approved by the ethics committee for the use and care of animals of Reims Champagne Ardenne (CEEA-RCA-56). Twelve-week-old Apoe -/-or Apoe -/-/Ager -/-mice were randomly assigned to CKD or shamoperated groups (Sham). A two-step surgical procedure was used to induce CKD [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF] . Surgery was performed under isoflurane (Axience). The two poles of the left kidney were excised, and one week later the right kidney was removed, resulting in a 75% reduction of total renal mass.

Induction of uremia

Control animals underwent sham operations involving decapsulation of both kidneys. All mice were sacrificed 12 weeks after surgery. All surgeries were done by the same operator blinded to the genotype of the mice.

Biochemical and hematological analysis

The serum concentration of urea, total cholesterol, triglycerides, calcium, and phosphorus were measured by automated assays using a Modular® analyzer (Roche Diagnostics). Complete blood counts, hematocrit, and hemoglobin were determined with a hematology cell counter ADVIA 2120 (Siemens).

Whole blood platelet aggregation

Mice were anesthetized with 25% urethane (Sigma). Whole blood was collected from the inferior vena cava in 0.129 M sodium citrate (1/6, v/v). Whole blood aggregation was monitored by impedance aggregometry at 37°C under stirring using the Multiplate analyzer ® . Whole blood was diluted in physiological saline (1:1) and platelet aggregation was induced by adding Horm collagen (2.5 μg/mL), ADP (5 μM) or thrombin (10 mU/mL). Curves were recorded for 6 min and platelet aggregation was determined as the area under the curve (AUC) in arbitrary units (AU*min).

Platelet-rich plasma aggregation

Blood was collected from mice as described above. Whole blood (700 L) was mixed with Tyrode buffer (500 L). Diluted platelet-rich plasma (PRP) was separated by centrifugation at 100 x g for 10 min at 22°C. Diluted platelet-poor plasma (PPP) was prepared by further centrifugation at 800 x g for 2 min. Platelets were counted and adjusted to 2.10 8 /mL. Platelet aggregation was monitored by measuring light transmission through the stirred suspension of diluted PRP at 37°C with a Soderel ® aggregometer. Thrombin was used as platelet agonist for homogeneity with whole blood and washed platelet aggregation experiments. Aggregations were triggered by 50 and 100 mU/mL thrombin since low concentrations of thrombin (< 10 mU/mL) were without effect.

Washed platelet aggregation

Mouse platelets were prepared as described previously [START_REF] Adam | Apelin: an antithrombotic factor that inhibits platelet function[END_REF] . Using this protocol, platelet suspensions were estimated to be more than 99% pure [START_REF] Adam | Apelin: an antithrombotic factor that inhibits platelet function[END_REF] . Briefly, whole blood was collected from the inferior vena cava and mixed with 80 M PPACK and 10% (v/v) ACD-C buffer (124 mM sodium citrate, 130 mM citric acid, 110 mM dextrose, pH 6.5). PRP was obtained by centrifuging whole blood for 7 min at 160 x g. Platelets were obtained from PRP after centrifugation for 10 min at 670 x g and washed in the presence of apyrase (100 mU/mL) and PGE 1 (1 M) to minimize platelet activation. The number of platelets was adjusted to 2.10 8 /mL in modified Tyrode-HEPES buffer (137 mM NaCl, 2 mM KCl, 0. produce glycated albumin, enriched in carboxymethyllysine (CML), a well-characterized RAGE ligand. Briefly, 0.5 g of HSA were dissolved in 10 mL of 0.5 M sodium phosphate buffer (pH 7.4) containing 0.05% NaN 3 and 3 g of D-glucose. Non-glycated albumin was prepared similarly, except that glucose was omitted, and used as control. The solution was sterilized by ultrafiltration (0.45 µm filter) and incubated at 37 °C for 90 days. The samples were then dialyzed against PBS [START_REF] Ikeda | N (epsilon)-(carboxymethyl)lysine protein adduct is a major immunological epitope in proteins modified with advanced glycation end products of the Maillard reaction[END_REF] .

Flow cytometry

Activated GPIIb/IIIa (JON/A) and CD62P (P-selectin) were used as markers of platelet activation and secretion, respectively. Platelet surface expression of selected markers was also evaluated such as GPIb α , GPIb β , GPIV (also known as CD36), GPVI, and integrin α 2 and α IIb chains.

Diluted whole blood or washed platelets (2.10 6 /mL) were stimulated with a range of agonists, as indicated. After incubation for 5 min at room temperature without stirring, whole blood or washed platelets were incubated with appropriate fluorophore-conjugated antibodies for 10 min at room temperature. The reaction was stopped by adding 500 L of PBS, and immediately analyzed with a FC500 flow cytometer (Beckman Coulter). The platelet population was gated using their forward and side scatter characteristics. Acquisition and processing data from 10,000 platelets were collected and analyzed using CXP software (Coulter). The Mean fluorescence intensity (MFI) of the whole platelet population was used.

Carboxymethyllysine quantification by Liquid Chromatography-MS/MS

For plasma AGE measurements, CML was measured in plasma samples using mass spectrometry coupled with liquid chromatography (LC-MS/MS). Total AGEs (free adducts and protein-linked) F o r P e e r R e v i e w 10 were measured as previously described [START_REF] Gorisse | Protein carbamylation is a hallmark of aging[END_REF] . Briefly, samples were subjected to acid hydrolysis by 6 M hydrochloric acid for 18 h at 110 °C. Hydrolysates were evaporated to dryness under a nitrogen stream. Dried samples were resuspended in 100 μL of 125 mM ammonium formate containing 1 µM d7-citrulline, 1 µM d2-CML, and 65 µM d8-lysine [used as internal standards (ISs)] and filtered using Uptidisc PTFE Filters (4 mm, 0.45 μm; Interchim). Samples were then 10-fold diluted in 125 mM ammonium formate containing the three ISs. A second dilution at 1:20 (vol/vol) was performed in 5 mM ammonium formate buffer (pH 2.9) containing ISs. Diluted hydrolysates were subjected to LC-MS/MS analysis (API4000; ABSciex) to quantify CML and Lys residues. LC was performed using a Kinetex HILIC Column (100 × 4.6 mm, 2.6 µm; Phenomenex) with 5 mM ammonium formate (pH 2.9) as mobile phase A and 100% acetonitrile as mobile phase B. The flow rate was constant at 0.9 mL/min throughout separation. Parameters of HPLC and MS for CML and Lys assays have been described elsewhere [START_REF] Gorisse | Protein carbamylation is a hallmark of aging[END_REF] .

Mouse model of carotid artery thrombosis

Mice were anesthetized with 25% urethane. Rhodamine 6G (4mg/kg) was used to label endogenous platelets and leukocytes. The carotid artery was isolated from surrounding tissue by a black plastic sheet. Vascular injury was performed by applying a piece of filter paper (2 x 1 mm, 1 mm thick) saturated with 7.5% FeCl 3 onto the carotid artery for 1 min. After removing the filter paper, thrombus formation was monitored in real time with an epifluorescent microscope (Nikkon Eclipse E400) connected to a digital camera. Time to complete occlusion was defined as the arrest of blood flow for at least 1 min [START_REF] Li | Ferric chloride-induced murine carotid arterial injury: A model of redox pathology[END_REF] .

Statistical analysis

Results are expressed as the mean ± SEM. Statistical analysis was performed using GraphPad

RESULTS

We previously demonstrated that the Apoe -/-background is needed for the development of CKD induced vasculopathy [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF][START_REF] Belmokhtar | Receptor for advanced glycation end products: a key molecule in the genesis of chronic kidney disease vascular calcification and a potential modulator of sodium phosphate co-transporter PIT-1 expression[END_REF] . Therefore, this well-established uremic mouse model was used in this study to evaluate the role of CKD and RAGE in platelet aggregation and arterial thrombosis.

Apoe -/-and Apoe -/-/Ager -/-mice were subjected to a sham operation (sham) or surgically-induced chronic kidney disease (CKD). Twelve weeks after surgery, CKD induction in both genotypes was associated with significantly higher urea, creatinine, and cholesterol levels. Triglyceride, calcium, and phosphorus levels were unmodified. As expected, the number of red blood cells, hemoglobin level and hematocrit were significantly lower in the uremic versus sham-operated group (Table 1).

Uremia induces platelet hyperactivation

We first investigated the effect of CKD-induced uremia on platelet activation in whole blood by measuring platelet  IIb  3 activation using the monoclonal antibody JON/A specific for the activated conformation of the mouse integrin, and P-selectin surface expression, by flow cytometry. At resting state and relative to Apoe -/-sham mice, CKD had no effect on the activation level of  IIb  3 and P-selectin surface expression (Fig. 1A). Platelet activation by low thrombin concentration (10 mU/mL) led to significantly higher expression of activated  IIb  3 (MFI, 3.1±0.6 vs 1.9±0.5; p<0.05) and P-selectin (MFI, 7.0±1.4 vs 3.4±0.7; p<0.05) in whole blood from CKD than sham mice (Fig. 1A). We next assessed the effect of CKD on whole blood platelet aggregation induced by two different agonists, ADP and collagen. As shown in Fig. 1B, CKD-induced uremia significantly increased whole blood platelet aggregation by 2.6-fold (p<0.01) in response to 5 µM ADP and 1.7-fold (p<0.01) in response to 2.5 µg/mL collagen F o r P e e r R e v i e w 13 compared to Apoe -/-sham animals. Of note, a second wave of aggregation was systematically observed in the CKD condition during ADP-induced platelet aggregation, indicative of dense granule secretion. The effect of CKD on thrombin-induced platelet aggregation in PRP was also evaluated (Fig. 1C). Again, platelet aggregation was significantly higher in the CKD condition in response to both low (50 mU/mL) and high (100 mU/mL) thrombin concentrations. Platelet aggregation induced by low doses of thrombin was not sustained and reversible, and aggregation percentages were 41±7% for the CKD group vs 21±6% for the Apoe -/-sham group (p<0.05). In contrast, platelet aggregation induced by high doses of thrombin was stable and sustained, and the aggregation reached 82±3% for the CKD group vs 62±6% for the Apoe -/-sham group (p<0.01). Overall, these data show that CKD-induced uremia affects platelet function by increasing both platelet activation and aggregation in whole blood and PRP.

Deletion of RAGE limits uremia-induced platelet hyperactivation

The effects of RAGE deletion on both platelet activation and aggregation parameters were evaluated in whole blood and PRP. As shown in Fig. 1A, activation of  IIb  3 and P-selectin surface exposure were similar between the Apoe -/-/Ager -/-CKD and Apoe -/-/Ager -/-sham-operated groups, both in the basal state and after stimulation with low doses of thrombin (10 mU/mL).

However, stimulation with higher doses of thrombin (50 mU/mL) induces platelet hyperactivation in Apoe -/-/Ager -/-CKD compared to Apoe -/-/Ager -/-Sham mice (supplementary Fig. S1). Platelet aggregation in whole blood was significantly higher in the CKD group than in the sham-operated group in response to 5 µM ADP (575±80 vs 97±46 AU*min; p<0.01) and 2.5 µg/mL collagen (571±61 vs 241±72 AU*min; p<0.01) (Fig. 1B). Again, a second wave of aggregation was systematically observed in the CKD group during ADP-induced platelet Using PRP, we observed that CKD did not further increase aggregation of Apoe -/- /Ager -/-platelets in response to low doses (50 mU/mL) of thrombin compared to the shamoperated group (Fig. 1C), whereas CKD increased platelet aggregation at the higher concentration of thrombin (100 mU/mL) (70±1 vs 50±4%; p<0.01) (Fig. 1C). Interestingly, a spontaneous disaggregation occurring after 5 minutes was observed in the Apoe -/-/Ager -/-shamoperated group (62.5% of cases, Table 2). When comparing thrombin-induced  IIb  3 activation and P-selectin cell surface expression between the Apoe -/-/Ager -/-and the Apoe -/-groups for the CKD condition, RAGE deletion led to significantly lower values (55%, p=0.05, and 80%, p<0.05, lower, respectively) (Fig. 1A). There was no difference between the two groups under sham condition.

We next performed flow cytometry analysis and measured platelet expression levels of GPIb  , GPIb  , CD36 (GPIV), GPVI,  2 and  IIb  3 for both genotypes (Apoe -/-and Apoe -/-/Ager -/-).

Platelet surface expression of these receptors was comparable between both genotypes and CKDinduced uremia had no effect (Fig. 2). In addition, the number of circulating platelets and mean platelet volume were comparable (Table 3). Taken together these observations suggest that RAGE deletion decreases the effects of CKD on platelet activation and that RAGE participates in CKD-induced platelet hyperactivation.

Effect of CKD and RAGE deletion on washed platelet function

We further investigated whether CKD-induced uremia and/or RAGE directly affect platelet function by evaluating the impact of uremia and RAGE deletion on washed platelet activation and aggregation. Washed platelet aggregation induced by various concentrations of thrombin (Fig. 3A), as well as the levels of  IIb  3 activation (Fig. 3B) and P-selectin cell surface expression 3C), were similar for both genotypes. Moreover, there was no observable difference between the CKD and sham conditions. Overall, these data show that the effects of CKD and RAGE on platelet function are abolished when platelets are isolated from their plasma environment.

RAGE ligands accumulate in the uremic serum and are involved in platelet hyperactivation

Because our results suggested that the uremic milieu is a critical component of CKD-mediated platelet hyperactivation, we next analyzed the serum concentration of carboxymethyllysine (CML), known to be associated with CKD [START_REF] Semba | Serum carboxymethyl-lysine, a dominant advanced glycation end product, is associated with chronic kidney disease: the Baltimore longitudinal study of aging[END_REF] , in Apoe -/-sham and CKD mice. Indeed, uremia is associated with the retention of various solutes that are normally excreted by the kidneys, including potential binding partners for RAGE, such as AGEs and S100 proteins. As shown in Fig. 4A, CML concentration was significantly higher in the serum of CKD than sham mice (0.26 ± 0.02 vs 0.20 ± 0.01 mmol/mol lysine, respectively; p<0.01). In contrast, the concentration of CML in Apoe -/-/Ager -/-CKD sera was numerically, albeit non-significantly higher than that in Apoe -/-/Ager -/-sham sera, (0.21 ± 0.02 versus 0.18 ± 0.01 mmol/mol lysine, respectively; p = 0.06). There was no observable difference between the two genotypes for the sham condition.

To definitively demonstrate that RAGE ligands may directly modulate platelet function, we evaluated the effects of ex vivo incubation of two different AGEs on washed Apoe -/-and Apoe -/- /Ager -/-platelet aggregation, using glycated-HSA and S100A12, a typical calcium-binding protein belonging to the S100 protein family [START_REF] Foell | S100A12 (EN-RAGE) in monitoring Kawasaki disease[END_REF] . Incubation of washed platelets with glycated-HSA or S100A12 did not induce platelet activation or aggregation (data not shown). However, pre-incubation of washed Apoe -/-platelets with glycated-HSA (100 g/mL) for 10 minutes before stimulation by thrombin (25 mU/mL) led to a significant, 2.5-fold increase in aggregation relative to pre-incubation with non-glycated-HSA (p = 0.028) (Fig. 4B). Similar effects were observed with S100A12 (Fig. 4C). Increasing the concentration of S100A12 led to potentiation of washed platelet aggregation in response to thrombin. For instance, at 60 g/mL S100A12, platelet aggregation was 1.7-fold higher than the control. These potentiating effects of glycated-HSA and S100A12 were abolished in washed Apoe -/-/Ager -/-platelets (Fig. 4B,4C). Overall, these results strongly suggest that the potentiating effects of glycated-HSA and S100A12 on washed platelet aggregation induced by thrombin involve RAGE.

Uremia accelerates arterial thrombosis in vivo and RAGE deletion is protective

We finally explored the impact of CKD-induced uremia and RAGE on in vivo thrombosis using the FeCl 3 -induced carotid thrombosis model [START_REF] Li | Ferric chloride-induced murine carotid arterial injury: A model of redox pathology[END_REF] . Mice were injected with Rhodamine 6G and thrombus development and complete vessel occlusion were monitored in real time, under a microscope, after application of a patch containing 7.5% FeCl 3 . Time to complete arterial occlusion was significantly shorter in Apoe -/-CKD than Apoe -/-sham mice (9.2±1.1 vs 11.1±1.2 min; p<0.01; Fig. 5A,5B). Deletion of RAGE (Apoe -/-/Ager -/-sham) significantly delayed occlusion times (13.1±1.5 min) relative to Apoe -/-sham mice, and induction of CKD in these mice (Apoe -/-/Ager -/-) had no further effect (14.5±3.2 min). Interestingly, the mean arterial occlusion time of Apoe -/-/Ager -/-CKD mice was longer than that of Apoe -/-CKD mice (14.6±3.2 min vs 9.2±1.1 min; p<0.0001). In addition, the percentage of carotid arteries showing embolization before complete occlusion was 6.5 times higher in Apoe -/-/Ager -/-CKD than Apoe -/- CKD mice (54.5% vs 8.3%; Fig. 5C). These observations strongly suggest that CKD-induced uremia accelerates arterial thrombus formation and that RAGE is involved in this phenomenon.

DISCUSSION

Individuals with CKD have a much higher risk of death due to cardiovascular complications than the general population [START_REF] Go | Chronic kidney disease and the risks of death, cardiovascular events, and hospitalization[END_REF][START_REF] Fort | Chronic renal failure: a cardiovascular risk factor[END_REF] . Vascular disease in these patients is characterized by accelerated vascular ageing, including vascular calcification affecting the medial layer of vessels, and extensive atherosclerosis [START_REF] Drueke | Atherosclerosis in CKD: differences from the general population[END_REF][START_REF] Fort | Chronic renal failure: a cardiovascular risk factor[END_REF][START_REF] London | Atherosclerosis and arteriosclerosis in chronic renal failure[END_REF] . This vascular disease is also associated with thrombotic events [START_REF] Go | Chronic kidney disease and the risks of death, cardiovascular events, and hospitalization[END_REF] .

The underlying mechanisms involved in arterial thrombosis are not fully understood and therefore constitute a critical barrier for providing therapy. Recently, we and others reported that the pro-inflammatory receptor RAGE is involved in specific CKD-related vascular changes, such as uremic atherosclerosis and uremic vascular calcification [START_REF] Belmokhtar | Signaling of Serum Amyloid A Through Receptor for Advanced Glycation End Products as a Possible Mechanism for Uremia-Related Atherosclerosis[END_REF][START_REF] Gawdzik | Vascular remodeling and arterial calcification are directly mediated by S100A12 (EN-RAGE) in chronic kidney disease[END_REF][START_REF] Bro | A neutralizing antibody against receptor for advanced glycation end products (RAGE) reduces atherosclerosis in uremic mice[END_REF][START_REF] Jin | Advanced Glycation End Products Enhance Macrophages Polarization into M1 Phenotype through Activating RAGE/NF-kappaB Pathway[END_REF] . Indeed, CKD is considered to be a condition associated with increased activation of RAGE. Various solutes that are normally excreted by the kidneys accumulate in the sera with declining kidney function. Among them, AGEs and other RAGE ligands, such as S100 family proteins, accumulate during CKD, leading to increased engagement of this pro-inflammatory receptor and subsequent vascular remodeling [START_REF] Vanholder | Review on uremic toxins: classification, concentration, and interindividual variability[END_REF][START_REF] Liabeuf | Protein-bound uremic toxins: new insight from clinical studies[END_REF] .

Here, we investigated the impact of CKD on platelet function and arterial thrombus formation, and the involvement of RAGE in this phenomenon. We report that uremia induces platelet hyperactivation and accelerates arterial thrombus formation in a murine model of arterial thrombosis [START_REF] Li | Ferric chloride-induced murine carotid arterial injury: A model of redox pathology[END_REF] . We show that RAGE-ligand accumulation and RAGE engagement during CKD is involved in arterial thrombus formation, at least partially through increased platelet activation.

We analyzed platelet activation markers and platelet aggregation in whole blood and PRP. Both methods showed increased platelet aggregation in the CKD condition. This platelet hyperactivation was associated with increased  IIb  3 activation and P-selectin exposure.

Importantly, RAGE deletion inhibited CKD effects when low doses of agonists were used. Furthermore, platelet aggregation was less stable when RAGE was deleted. This suggests that 1) in CKD, RAGE modulates platelet sensitivity in the presence of small doses of agonists but this mechanism is overridden in the presence of higher doses of agonists, and 2) RAGE deletion is sufficient to inhibit the effects of CKD. In addition, isolation of platelets of either genotype from their plasma environment abolished this hyperactivation associated with CKD condition, suggesting that platelets are not intrinsically modified by CKD or RAGE deletion, but rather a priming effect of the uremic environment. We further explored this hypothesis using RAGE ligands and demonstrated that pre-incubating platelets isolated from Apoe -/-mice with RAGE ligands, such as AGEs (Glycated-BSA) or S100 proteins, induced increased platelet aggregation in response to low dose of thrombin. Moreover, deletion of RAGE prevented these effects, suggesting that this receptor is a central player in the modulatory effect in response to AGEs and S100. These data are consistent with recent studies showing that platelet aggregation in response to weak agonists, such as serotonin and ADP, increases in response to AGEs [START_REF] Gawlowski | Advanced glycation end products strongly activate platelets[END_REF][START_REF] Herczenik | Activation of human platelets by misfolded proteins[END_REF][START_REF] Zhu | Advanced glycation end products induce a prothrombotic phenotype in mice via interaction with platelet CD36[END_REF] . For instance, Gawlowski et al. reported increased platelet aggregation after stimulation with food-derived AGE or AGE isolated from human sera [START_REF] Gawlowski | Advanced glycation end products strongly activate platelets[END_REF] . This platelet hyperactivation was associated with CD62, CD63, and RAGE overexpression on the platelet surface. Similarly, Herczenik et al. reported a pro-aggregatory effect on platelets exposed to modified proteins with amyloid properties [START_REF] Herczenik | Activation of human platelets by misfolded proteins[END_REF] . They further established a potential link with RAGE using sRAGE to inhibit the interaction of amyloid proteins with RAGE. RAGE-mediated activation of platelets was also suggested by Ahrens et al. [START_REF] Ahrens | HMGB1 binds to activated platelets via the receptor for advanced glycation end products and is present in platelet rich human coronary artery thrombi[END_REF] . They reported an interaction of RAGE with HMGB1 (high mobility group box 1), involved in platelet activation and the atherosclerosis process. In the present study, we did not analyze the role of HMGB1. However, HMGB1 serum levels have been reported to be significantly increased in CKD patients and negatively correlated with eGFR, and positively correlated with Our study highlights a new role for RAGE in the modulation of platelet function and arterial thrombosis during CKD. We found that the induction of uremia in Apoe -/-mice accelerates arterial thrombosis formation in vivo. This is the first report on a specific prothrombotic effect of CKD in vivo in a mouse model of carotid-artery thrombosis. Here, we addressed the question of arterial thrombosis and focused our experiments on platelet activation and aggregation. However, platelet hyperactivation may also increase thrombogenicity in venous conditions. For instance, overexpression of P-selectin may have an impact on neutrophil and monocyte adhesion and activation which play a major part in venous thrombogenicity [START_REF] Yago | Cooperative PSGL-1 and CXCR2 signaling in neutrophils promotes deep vein thrombosis in mice[END_REF] . Therefore, further investigation is needed to test the relevance of our results on venous thrombogenicity. Moreover, we focused on a specific subclass of uremic toxins. However, our results strengthen recent findings by Holy et al. showing the prothrombotic effect of another uremic by-product, the carbamylated low density lipoprotein (cLDL) [START_REF] Holy | Carbamylated Low-Density Lipoproteins Induce a Prothrombotic State Via LOX-1: Impact on Arterial Thrombus Formation In Vivo[END_REF] , in a murine model of arterial thrombosis after carotid photochemical injury. Overall, their results suggest that the post-translational modification of proteins can induce platelet hyperaggregability, leading to arterial thrombosis. Indeed, protein carbamylation is highly elevated in CKD and considered as a hallmark of uremia. Taken together, these results suggest a critical role of the uremic milieu on platelet hyperactivation promoting arterial thrombosis. These findings are consistent with previous publications from CKD patients reporting the propensity of the uremic milieu to induce thrombosis. For instance, in the context of percutaneous coronary intervention, one-year mortality related to stent thrombosis is higher in CKD patients (15.2%) and those on dialysis (10.0%) than in non-CKD patients (1.9%) [START_REF] Naganuma | Impact of Chronic Kidney Disease on Outcomes After Percutaneous Coronary Intervention for Chronic Total Occlusions (from the Japanese Multicenter Registry)[END_REF][START_REF] Sato | Comparison of clinical outcomes of coronary artery stent implantation in patients with end-stage chronic kidney disease including hemodialysis for three everolimus eluting (EES) stent designs: Bioresorbable polymer-EES, platinum chromium-EES, and cobalt chrome-EES[END_REF] . A potential explanation has been put forward by Chitalia et al. [START_REF] Chitalia | Uremic serum and solutes increase post-vascular interventional thrombotic risk through altered stability of smooth muscle cell tissue factor[END_REF] , who reported the modulation of cell function by uremic serum. They showed increased expression of tissue factor during the incubation of primary cultures of human vascular smooth muscle cells with uremic serum and isolated uremic toxins, such as indole-3-acetic acid. Overall, these results suggest a prothrombotic state in some patients, related to CKD and mediated by the uremic milieu. Another example comes from Jain et al. who reported greater response to ADP in platelets from CKD patients compared to platelets from matched controls 45 . Furthermore, they reported higher residual platelet responsiveness in clopidrogel-treated CKD patients than control patients, again demonstrating that the uremic milieu is able to increase platelet activation and aggregation.

In summary, we report a novel mechanism by which RAGE ligand accumulation, due to uremia, increases RAGE-dependent platelet hyperactivation. This may contribute to creating a prothrombotic state in which circulating CKD platelets are primed for activation, leading to enhanced thrombus formation. These results constitute a reasonable basis for designing preclinical studies aiming at targeting the RAGE/ RAGE-ligand axis as a promising approach to preventing thrombotic complications associated with CKD.

F

 5 Number

 5 RAGE ligands binding to RAGE are involved in CKD-induced platelet hyperactivation. Keywords: Chronic Kidney Disease, RAGE, Platelets, Arterial thrombosis, Uremic toxins Number of figures:

  3 mM NaH 2 PO 4 , 5.5 mM glucose, 5 mM N-2-hydroxyethylpiperazine-N'-2-ethanesulfonic acid, 12 mM NaHCO 3 , 2 mM CaCl 2 , pH 7.3). Platelet aggregations in response to thrombin were measured with a Soderel ® aggregometer. To evaluate RAGE involvement in washed platelet aggregation, HSA was used to

11

 11 Prism software 7.0. Groups were compared using a two-tailed Mann-Whitney test. P values of less than 0.05 were considered statistically significant.

  and nutrition[START_REF] Bruchfeld | High Mobility Group Box Protein-1 correlates with renal function in chronic kidney disease (CKD)[END_REF] . Hence, our results are in agreement with these previous reports.

Figure 1 . 32 Figure 2 .

 1322 Figure 1. CKD increases platelet activation and RAGE deletion limits these effects. (A)

Figure 3 .

 3 Figure 3. Uremia and RAGE deletion have no effect on washed platelet function. (A) Platelet

Figure 4 .

 4 Figure 4. RAGE ligands accumulate during uremia and are involved in uremia-induced

Figure 5 .

 5 Figure 5. Uremia accelerates arterial thrombus formation and RAGE deletion provides

Table 1 . Biological characteristics

 1 Li X, Adams-Huet B, Sarode R, Toto RD, Banerjee S and Hedayati SS.Differences in Whole Blood Platelet Aggregation at Baseline and in Response to Aspirin andAspirin Plus Clopidogrel in Patients With Versus Without Chronic Kidney Disease. Am J

	o r
	P e
	e r
	R e
	v
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	w
	21

Hct (%) 36.8 ± 0.4 30.5 ± 0.1 * 39.4 ± 1.3 30.1 ± 0.6 ****

	F o r
	P e
	e r
	R e
	v
	i e
	w
	30

Table 2 . Effect of RAGE deletion of the stability of platelet aggregation

 2 The table shows the disaggregation observed 5 min after stimulation with 100 mU/mL of thrombin. Results are expressed as the percentage of disaggregation out of total aggregation.

Table 3 . Platelet characteristics

 3

	Data are reported as mean ± SEM, (n = 5 to 9). PLT: platelets. MPV: mean platelet volume

/Ager -/- Sham CKD Sham CKD PLT (m/mm3) 1027.2 ± 103.3 1188.8 ± 234.5 1100.8 ± 133.1 1396.6 ± 209.7 MPV (fL)

	F o r
		P e
			e r
				R e
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	6.5	± 0.1	6.4	± 0.1	6.3	± 0.1	6.1	± 0.2

Thrombosis and Haemostasis

ACKNOWLEDGEMENTS

The authors would like to thank Ms. Valerie Creuza for her appreciated technical assistance in this study and Ms. Leslie Migeon for secretarial assistance in finalizing the manuscript.

FUNDING

This work was supported by funding from CNRS, URCA. FT was recipient of a grant from the Société Francophone de Nephrologie Dialyse Transplantation (SFNDT). CK was recipient of a scholarship from the Nouvelle Société Francophone d'Athérosclérose (NSFA).

DISCLOSURE

The authors have no conflict of interest to disclose.

CONTRIBUTIONS


Keywords: assimilation de données, problème mal posés, régularisation, assimilation d'images, flot optique, mouvement

Résolution des problèmes mal posés du traitement d'image par Assimilation de Données. Application au flot optique Résumé : L'Assimilation de Données est un cadre méthodologique utilisé en sciences de l'environnement pour la prédiction des systèmes complexes, que sont les modèles météorologiques, océanographiques, ou encore de qualité de l'air. L'Assimilation de Données opère en résolvant un système à trois composantes: une première équation décrit l'évolution temporelle du vecteur d'état; une seconde équation décrit la relation entre le vecteur d'état et l'observation; enfin, une dernière équation décrit la condition initiale. Dans ce rapport, nous utilisons ce cadre mathématique pour l'étude des problèmes mal posés du traitement d'image, habituellement résolus par des techniques de régularisation. Pour cela, le problème est formulé au moyen du système à trois composantes de l'Assimilation de Données. Comme illustration, cette approche est appliquée au calcul du flot optique.

Introduction

In the research field of Image Processing, most problems are ill-posed in the sense that it is not possible to provide a unique solution [Hadamard, 1923]. A first cause of ill-posedness is that the equations used to model image properties are under-determined. An example is the famous "aperture problem" occurring in the estimation of optical flow: a further constraint is required to compute a unique field of velocity vectors. As an image processing problem is usually represented by a system of equations to be solved, the so-called Image Model, this type of ill-posedness means that this Image Model is not invertible. A second cause of ill-posedness (and a common problem in Image Processing) is that the extraction of image features is rarely obtained in a unique way. For example, the computation of image gradient is ill-posed as it requires approximating a differential operator by a discrete one among several possible finite difference formulations; each one will give a different result. In such a case, the problem is ill-posed because the uniqueness of the solution is clearly unverified.

One possible strategy to solve ill-posed problems is to provide the Image Model with additional information. Two possibilities occur. 1) Explicit information: additional images are used to enlarge the set of data, but this is generally not possible because no further acquisition is available. 2) Implicit information: either hypotheses on image properties or constraints on the solution can be used. An example of a constraint is to restrict the dimension of the space of the admissible solutions: this can be done by searching for the result among the functions which have bounded spatial variations [Tikhonov, 1963]. In the general case, image properties and constraints are expressed as equations added to the Image Model in order to obtain a new one which will be invertible. This paper focuses on the methods constraining the function space, the so-called "Tikhonov's regularizing methods", and their formulation in a new methodological framework.

Nowadays, most acquisition systems produce sequences of images which provide information on the dynamic of the objects observed. This knowledge is then used to enhance the problem modelling. Let us consider the example of image segmentation. A regularization method, such as Shah-Mumford's functional [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF], produces a segmentation which is a compromise between the regularity of the solution and the low discrepancy with the input. The dynamic estimated from the sequence of images is then exploited to manage missing data. Indeed, a result in the areas where data are not available is estimated by linear interpolation of those obtained on the adjacent frames. Linear interpolation is, however, restrictive as it is only relevant for structures displaying a linear dynamic. To overcome this difficulty, segmentation could be performed directly on the whole sequence and not on a frame-by-frame basis; following Weickert, the solution [START_REF] Weickert | Variational optic flow computation with a spatio-temporal smoothness constraint[END_REF] is searched for as a function depending on the spatial and temporal coordinates. In this case, the Image Model will remain but will depend on the time coordinate and the regularization constraint will bound the variations of the solution simultaneously in space and time. Wieckert's method has three main drawbacks. First, spatiotemporal regularization correctly describes regular dynamics but a complex configuration, such as discontinuity or non-linearity, will not be handled correctly. Second, pixels with aberrant values introduce errors in their spatio-temporal neighbourhood because the method provides a smoothed result. Third, as the solution is searched for in the spatio-temporal domain, this leads to a significant increase in complexity, which is a limiting factor for operational applications. A model of the structures' temporal dynamics, called the "evolution model", has to be stated from the a priori knowledge on the data or on the physical properties of the structures. The evolution model will be used to obtain good results even if observations are missing.

In this paper, we propose to use the Data Assimilation framework as a generic tool to solve ill-posed problems in Image Processing. Although, this approach is not new, as some algorithms have already been proposed to perform curve tracking [START_REF] Papadakis | A variational approach for object contour tracking[END_REF] and motion estimation using physical models [Papadakis andMémin, 2008, Herlin et al., 2006], we, however, propose a general framework. Data Assimilation solves a system of three components with respect to a state vector representing the solution of the problem: one is an evolution equation describing the evolution of the state vector over time using an operator called the "evolution model". The evolution model provides the description of the images' temporal dynamics.

another, called "observation equation", describes the link between the state vector and the observations included in the sequence of images, and the third describes the initial condition.

For each term, a description of the errors is stated as a covariance matrix. This matrix depicts the dependencies that exist between the components of the state vector on the one hand, and between two different locations in the space-time domain on the other hand.

Data Assimilation is a suitable approach to solve the three drawbacks of Weickert's method because: the evolution model provides an accurate description of images' temporal dynamics; the matrix associated to the observation equation overcomes the problem of missing data by weighting the importance of the observation equation in the computation of the solution; Data Assimilation is a frame-by-frame process reducing memory allocation.

This paper is organized as follow. Section 2 describes the variational Data Assimilation method known as the 4D-Var algorithm. How Data Assimilation can be used to solve ill-posed problems by assimilating image data within an appropriate evolution model is explained in Section 3. Section 4 is a direct application of Section 3 and describe a method to compute optical flow. The evolution model used in this case is the transport of velocity. Section 4 also presents and analyzes experimental results. We conclude in Section 5 and give some scientific perspectives to this work.

The Data Assimilation framework

The Data Assimilation framework aims to solve the system (1,2,3) with respect to a state vector X(x, t), depending on the spatial coordinate x and time t:

∂X ∂t (x, t) + Å(X)(x, t) = E m (x, t) (1) X(x, 0) = X b (x) + E b (x) (2) Y(x, t) = À(X)(x, t) + E O (x, t) (3)

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Equation (1) describes the temporal evolution of X. Å, called the evolution model is generally differential and possibly non linear. As Å describes approximately the evolution of the state vector, the model error E m is introduced to quantify the difference. Equation (2) states the initial condition of the state vector and the error is quantified by the background error E b . Equation (3), called the observation equation, describes the link between the observation vector, Y, and the state vector. The observation may be an incomplete measure of the state vector, in which case À is a projection operator on a discrete and finite space. In other cases, the observation may be an indirect, and sometimes complex, measure of the state vector. Equation ( 3) is the standard form of the observation equation used in the Data Assimilation literature. This formulation is quite restrictive to describe the complex link between the observation and the state vector. To be more general, the following will be used in is article:

À(Y, X)(x, t) = E O (x, t) (4)
The observation error E O represents the imperfection of À and the measurement errors. The errors E m , E b and E O are assumed to be Gaussian vectors and fully characterized by their covariance matrices Q, R and B.

Let X denote a Gaussian stochastic vector depending on a space-time coordinate (x, t), X = X(x, t) and X ′ = X(x ′ , t ′ ). As the covariance matrix Σ measures the dependency between X and X ′ , it is defined by:

Σ(x, t, x ′ , t ′ ) = (X -X) T (X ′ -X ′ )dP X,X ′ (5)
where P X,X ′ is the joint distribution of (X, X ′ ) and denotes the expectation. P X,X ′ is a Gaussian distribution depending on variables x, t, x ′ and t ′ . The inverse of a covariance matrix is formally and implicitly defined [Oliver, 1998] by:

Σ -1 (x, t, x ′′ , t ′′ )Σ(x ′′ , t ′′ , x ′ , t ′ )dx ′′ dt ′′ = δ(x -x ′ )δ(t -t ′ ) (6)
In order to solve equations ( 1), ( 2) and ( 4), the following functional is defined:

E(X) = A A ∂X ∂t + Å(X) T (x, t)Q -1 (x, t, x ′ , t ′ ) ∂X ∂t + Å(X) (x ′ , t ′ )dxdtdx ′ dt ′ + A A À(X, y) T (x, t)R -1 (x, t, x ′ , t ′ )À(X, y)(x ′ , t ′ )dxdtdx ′ dt ′ + Ω Ω X(x, 0) -X b (x) T B -1 (x, x ′ ) X(x ′ , 0) -X b (x ′ ) dxdx ′ (7)
with A = Ω×[0, T], Ω being the spatial domain and [0, T] the temporal domain.

If E m , E b and E O are assumed to be independent, the functional E represents the log-likelihood of X. The function minimizing E is therefore a maximum likelihood estimator. This minimization is carried out using a calculus of variation: the differential of E with respect to X, denoted ∂E ∂X , is calculated and the associated Euler-Lagrange equations are solved. ∂E ∂X is obtained by computing the derivative of E with respect to X in the direction ψ:

∂E ∂X (ψ) = ∂E ∂X T ψ = lim γ→0 d dγ (E(X + γψ)) (8)
and by introducing an auxiliary variable λ, called the adjoint variable in the literature of Data Assimilation:

λ(x, t) = A Q -1 (x, t, x ′ , t) ∂X ∂t + Å(X) (x ′ , t ′ )dx ′ dt (9)
The reader is referred to [Valur Hólm, 2003] for details on the determination of the differential of E and the Euler-Lagrange equations. This process leads to the following system:

λ(x, T) = 0 (10) - ∂λ ∂t + ∂Å ∂X * λ = - A ∂À ∂X * (x, t)R -1 À(X, Y)(x ′ , t ′ )dx ′ dt ′ (11) X(x, 0) = Ω Bλ(x ′ , 0)dx ′ + X b (x) (12) ∂X ∂t + Å(X) = A Qλ(x ′ , t ′ )dx ′ dt ′ (13)
Equation ( 11) makes use of adjoint operators denoted ∂• ∂X * and formally defined as the differential in the dual space of X, by:

∂Å ∂X (φ) T ψdxdt = φ T ∂Å ∂X * (ψ)dxdt (14)
for all integrable functions φ and ψ. The adjoint operator represents a compact notation for integration by parts. Riesz's theorem assumes the existence and uniqueness of the adjoint operator.

A difficulty occurs: equations (10,11) allow the state vector to determine the adjoint variable, and equations (12,13) allow the adjoint variable to determine the state vector. To break this deadlock, an incremental algorithm is used: the state vector is written as X b + δX where X b is a mean term, called the background variable in the Data Assimilation literature, and δX is an incremental term. X is then replaced by X b + δX in equations ( 11), ( 12) and ( 13). Then, the operators Å and À, not necessarily linear, have to be linearized using a first order Taylor expansion: Å(X) ≃ Å(X b ) + ∂Å ∂X b (δX). This leads to the following new system of equations:

λ(x, T) = 0 (15) - ∂λ ∂t + ∂Å ∂X b * λ = - A ∂À ∂X b * R -1 À(X b , Y) + ∂À ∂X b (δX) dx ′ dt ′ (16) X b (x, 0) = X b (x) (17) ∂X b ∂t + Å(X b ) = 0 (18) δX(x, t) = Ω Bλ(x ′ , 0)dx ′ (19) ∂δX ∂t + ∂Å ∂X b (δX) = A Qλ(x ′ , t ′ )dx ′ dt ′ (20)

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The adjoint variable is now obtained from the background variable X b using equations ( 15) and ( 16). The background variable is calculated from equations ( 17) and ( 18) independently of the adjoint variable and the state vector. The incremental variable δX is obtained from the adjoint variable using equations ( 19) and ( 20). Due to the linearization of Å and À, equations (15,16,19,20) produce an approximated solution δX + X b . The function minimizing E (equation ( 7)) is obtained using an iterative algorithm: at each iteration, the value δX is computed from the background variable X b and the value δX + X b becomes the background variable for the next iteration.

Image Assimilation

This section explains how to solve ill-posed Image Processing problems using the image assimilation. Obviously, the observations to be used in the observation equation will be images, but the nature of the state vector will depend on the problem under study. For example, segmentation, denoising and restoration use a state vector which is an image. Tracking, image registration and motion estimation use a vector field. Active contours use a curve. To solve an Image Processing problem using the 4D-Var algorithm as described in Section 2, a suitable evolution model (describing the temporal evolution of the state vector) and its error should first be chosen to constrain the state vector. Next, an observation equation (describing the link between the state vector and the image) and its error should characterize the properties of observations. Lastly, the initial condition and its background error should be defined. Although these choices generally depend on the context, we will however describe some general principles in the next subsections.

The evolution model

One first choice is to impose a temporal regularity on the state vector. This property is modeled as the transport equation dX dt = 0, depending on the spacetime coordinates. Using the chain rule, we obtain:

∂X ∂t + ∂X ∂x ∂x ∂t = 0 (21)
The evolution model is then Å(X)(x, t) = ∂X ∂x ∂x ∂t . Usually ∂x ∂t = 0 and this model becomes unsuitable. However, in the case of determining optical flow, the spatial coordinates stand for a trajectory and ∂x ∂t becomes a velocity vector. An example of using equation ( 21) as an evolution equation is given in Section 4.

Another choice of evolution model is to express the transport of the state vector as a diffusion, a physical law ruling the transport of chemical species or temperature. The general formulation is:

∂X ∂t = ∇ T (D∇X) with ∇ = ( ∂ ∂x , ∂ ∂y
) T the gradient operator of Ê 2 . The matrix D is a tensor characterizing the direction and the intensity of the diffusion. It is possible to drive the diffusion according to the image characteristics. A standard example is the Perona & Malik diffusion: the tensor matrix is equal to c( ∇X )Id with c a Gaussian function and Id the identity matrix. One of the properties of this diffusion is to smooth the image while preserving the contours. In this case, the evolution model is Å(X) = -∇ T (D∇X).

In addition to the evolution model, the matrix Q in equation ( 13) has to be specified and it is important to understand the action of Q -1 in the functional (7). Let us consider two possible choices for Q and define their impact in a functional F T (X)Q -1 F (X)dxdx ′ which has to be minimized. As a first example, let Q be the Dirac covariance defined by Q(x, x ′ ) = δ (xx ′ ). The Dirac function has the property of being its own functional inverse:

Ω δ (x ′ ) δ (x -x ′ ) dx ′ = δ (x) So, we have Q -1 (x, x ′ ) = δ (x -x ′ ) and: Ω 2 F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ (22) = Ω F (X) T (x)F (X)(x)dx = Ω F (X) 2 dx
A Dirac covariance corresponds to a L 2 regularization.

A second example is the exponential covariance defined by

Q(x, x ′ ) = exp(- x -x ′ σ ).
Let us write Q(x, x ′ ) = q(xx ′ ), then (6) becomes q -1 ⋆ q(xx ′ ) = δ (xx ′ ). The inverse of the exponential covariance is obtained from the Fourier theorem:

q -1 (ω) q(ω) = 1 q -1 (ω) = 1 q(ω) = 2σ 1 + σ 2 ω 2 q -1 (x) = 1 2σ δ (x) -σ 2 δ ′′ (x)
Let us replace the expression of Q -1 in the functional (22):

Ω 2 F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ = 1 2σ Ω F (X) T (x) F (X) -σ 2 ∂ 2 F (X) ∂x 2 dx = 1 2σ Ω F (X) 2 + σ 2 ∂F (X) ∂x 2 dx (23)
Line ( 23) is obtained from the previous line using integration by parts. The exponential covariance is then equivalent to a quadratic second-order regularization. However, the inversion of a covariance matrix is often non-trivial and usually inaccessible. Restrictive choices have to be made such as Dirac or exponential covariance. For further details, the reader is referred to [Oliver, 1998, Tarantola, 2005].

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The observation equation

As previously pointed out, the observation equation describes the link between the state vector and the observation. If the generic formulation of Tikhonov's approach is considered, it is clear that the Image Model corresponds exactly to the observation equation of the Data Assimilation framework because the regularization part of Tikhonov's method does not contain information about the input.

In addition to the observation equation, the observation error has to be stated and its matrix of covariance R must be specified. R is a good candidate to manage missing data. This matrix must also be inverted (see equation ( 16)) and, as explained before, determining the inverse is only possible in simple cases of covariance matrices. The Dirac covariance can be used to express a null interaction between two space-time locations. This choice is not restrictive because observation errors have only to be located and quantified in space-time. R is then written as:

R(x, t, x ′ , t ′ ) = r(x, t)δ (x -x ′ ) δ (t -t ′ )
where r becomes a real matrix. The advantage is that determining the inverse is trivial when r is invertible:

R -1 (x, t, x ′ , t ′ ) = δ (x -x ′ ) δ (t -t ′ ) r -1 (x, t) (24)
The function r must characterize the quality of the observation: a high value should indicate that the observation is relevant and a value close to zero indicates an irrelevant observation (like missing data), that should not be taken into account. One possible formulation of r is:

r(x, t) = r 0 (1 -f (Y(x, t)) + r 1 f (Y(x, t)) (25)
If a real function f ∈ [0, 1] characterizing the relevance of the observation Y is defined, r will be close to a "minimal covariance" r 0 on missing or irrelevant data and r will be close to a "maximal covariance" r 1 on relevant data. r 0 and r 1 are chosen to be constant and invertible matrices (in order for r to be invertible).

Estimation of optical flow

Optical flow is the apparent motion in sequences of images and is defined as the transport of brightness over time [START_REF] Horn | Determining optical flow[END_REF]. Let I be a sequence of images on a bounded domain of Ê 2 , denoted Ω. Let W(x, t) be the velocity vector of a point x ∈ Ω between frame t and frame t + δt. The optical flow is the vector W(x, t) verifying:

I(x + W(x, t)δt, t + δt) = I(x, t) ∀x ∈ Ω (26)
As this equation is non linear with respect to W, the left member of equation ( 26) is linearized using a first order Taylor development around δt = 0. This provides the optical flow constraint:

∇I T (x, t)W(x, t) + ∂I ∂t (x, t) = 0 ∀x ∈ Ω ( 27
)
RR n 6477

This is an ill-posed problem: the velocity vector has two components -W(x, t) = (U (x, t), V (x, t)) -and the optical flow constraint is not sufficient to compute this vector. A solution could be obtained in the Image Processing context using the Tikhonov's method or Weickert's method described in the Introduction and we study, in the following, the solution obtained in the Data Assimilation context.

Modelling in the Data Assimilation context

The field W of velocity vectors is now considered as the state vector and the sequence of images I constitutes the observation vector. The optical flow constraint is used as the observation equation:

À(W, I)(x, t) = ∇I(x, t) T W(x, t) + I t (x, t) (28)
Equation À = 0 means that the vector (W, 1) is orthogonal to the spatiotemporal gradient of the observation

∇ 3 I = ∂I ∂x , ∂I ∂y , ∂I ∂t
. The areas where the value of the spatio-temporal gradient are close to zero are not relevant observation because the equation ( 27) is always true. This information has to be taken into account by the observation error R. Applying the principle described in Subsection 3.2, the observation error is chosen as:

R(x, t, x ′ , t ′ ) = r 0 + (r 1 -r 0 ) exp(-∇ 3 I(x, t) 2 ) δ (x -x ′ ) δ (t -t ′ ) (29)
The transport of the velocity is chosen as the evolution equation. In this case, equation ( 21) is rewritten as the two-component system:

∂U ∂t + U U x + V U y = 0 (30) ∂V ∂t + U V x + V V y = 0 (31)
and the evolution model is

Å(W) = Å 1 (W) Å 2 (W) with Å 1 (W) = U x U + U y V and Å 2 (W) = V x U + V y V . The Q matrix is chosen as: Q(x, t, x ′ , t ′ ) = exp - 1 σ ( x -x ′ -|t -t ′ |) 1 0 0 1 (32)
to ensure a spatial second-order regularization. The initial condition of the state vector has also to be provided. We use Horn and Schunck's algorithm [START_REF] Horn | Determining optical flow[END_REF]] (Tikhonov's method with a second-order regularization) to compute the velocity field on the first image of the sequence. We assume the initial condition to be without errors and we set B(x, x ′ ) = δ(xx ′ ) as the background error.

Adjoint operators

In order to determine the adjoint operators of Å and À, the directional derivatives must first be established. Using the definition (8), we obtain: 14), integration by parts and considering the border terms to be equal to zero, the adjoint operator of Å is :

∂Å 1 ∂W (η) = ∂Å 1 ∂U (η 1 ) + ∂Å 1 ∂V (η 2 ) = U η 1 x + U x η 1 + V η 1 y + U y η 2 ∂Å 2 ∂W (η) = ∂Å 2 ∂U (η 1 ) + ∂Å 2 ∂V (η 2 ) = U η 2 x + V y η 2 + V η 2 y + V x η 1 INRIA with η = η 1 η 2 T . Using (
∂Å 1 ∂W * (λ) = -U λ 1 x -V y λ 1 -V λ 1 y + U y λ 2 ∂Å 2 ∂W * (λ) = -U x λ 2 -U λ 2 x -V λ 2 y + V x λ 1 with λ = λ 1 λ 2 T .
The directional derivative of the observation operator is:

∂À ∂W (η) = I x η 1 + I y η 2 = ∇I T η
and determining the adjoint operator is direct as it does not necessitate integration by parts:

∂À ∂W * (λ) = ∇Iλ

Discretization

Using the choices made in Subsection 4.1, the three EDPs (11,13,16) become:

∂W ∂t + W T ∇W = 0 (33) ∂λ ∂t -∇λ T W -λ T ∇ ⊥ W = - 1 2σ ∇IR ⋆ (I t + ∇I T (W + δW))(34) ∂δW ∂t + W T ∇δW + ∇W T δW = Q ⋆ λ (35) with ∇ ⊥ W = ∇ ⊥ U ∇ ⊥ V and ∇ ⊥ U = U y -U x T
. These three equations are approximated using the finite difference technique.

As W(x, t) is a vector of Ê 2 , equation ( 33) has two components. The first component combines a term of linear advection in direction y and a term of non linear advection in direction x. It can be expressed as a two-equation system using a splitting method [START_REF] Verwer | A note on operator splitting in a stiff linear case[END_REF]:

∂U ∂t + U U x = 0 (36) ∂U ∂t + V U y = 0 (37)
The equation ( 36) is efficiently approximated using the Lax-Friedrich method [Sethian, 1996] which rewrites it in the following form

∂U ∂t + ∂F (U ) ∂x = 0 with F (u) = 1 2 u 2
. This new equation is approximated by the explicit scheme:

U k+1 i,j = 1 2 (U k i+1,j + U k i-1,j ) - △t 2 (F k i+1,j -F k i-1,j )
with U k i,j = U (x i , y i , t k ), F k i,j = F (U (x i , y i , t k )) and △t the time step. The term 1 2 (U k i+1,j + U k i-1,j ) stabilizes the scheme (by adding a diffusive effect) when △t has a low value (the Courant-Friedrich-Levy condition). The linear advection (37) is approximated using a shock explicit scheme [Sethian, 1996]:

U k+1 i,j = U k i,j -△t max(V k i,j , 0) U k i,j -U k i-1,j + min(V k i,j , 0) U k i+1,j -U k i,j
Having defined the discretisation of the first component of (33), it can be seen that the second component contains a linear advection term in direction x and a non linear advection term in direction y. The same strategy is then used to perform the discretization process.

INRIA

Equation ( 34) combines a linear advection, a term of reaction and a forcing term. It also has two components. The first one is

∂λ 1 ∂t -U λ 1 x -V y λ 1 -V λ 1 y + U y λ 2 = I x B with B = -1 2σ R ⋆ (I t + ∇I T (W + δW)).
It is split into two parts. The first one contains the linear advection in direction x and the reaction term ∂λ 1 ∂t -U λ 1

x -V y λ 1 = 0 and is approximated in the same way as (37) with an explicit shock scheme. However, the equation is retrograde as the initial condition is given at time T :

(λ 1 ) k-1 i,j = 1 - 1 2 (V k i,j+1 -V k i,j-1 ) (λ 1 ) k i,j - △t max(U k i,j , 0)((λ 1 ) k i,j -(λ 1 ) k i-1,j ) + min(U k i,j , 0)((λ 1 ) k i+1,j -(λ 1 ) k i,j ))
The second part contains the linear advection term in direction y and the forcing term:

∂λ 1 ∂t -V λ 1 y + U y λ 2 = I y B.
Again, a shock explicit scheme is used:

(λ 1 ) k-1 i,j = (λ 1 ) k i,j + △t 2 U k i,j+1 -U k i,j-1 (λ 2 ) k i,j -△tB k i,j - △t max(V k i,j , 0)((λ 1 ) k i,j -(λ 1 ) k i,j-1 ) + min(V k i,j , 0)((λ 1 ) k i,j+1 -(λ 1 ) k i,j )
Having the same structure as the first component, the second component of ( 34) is approximated using the same method. The last equation, (35), is similar to equation (34): a linear advection with a reaction term and a forcing term. We therefore use the same discretization technique.

Results

We tested the algorithm on the "taxi" sequence and one a sequence provided by the Coriolis team (LEGI, France) showing a simulation (in a water tank) of the oceanic circulation (Fig 2). For these two sequences, the initial condition is given by the Horn and Schunk method. The image gradients are computed using a convolution by a derivative Gaussian kernel. The parameter σ of the matrix Q and the Gaussian kernel variance are set to 1. The number of iterations is set to 10. We also force the image gradients to zero on the fourth frame of the taxi sequence to simulate missing data. Results are displayed in Figure 1 and confirm the capability of the algorithm to manage missing data.

Conclusion

In this paper we proposed a framework to solve ill-posed problems using Data Assimilation methods. This approach is an alternative to the Weickert's method, which constrains the solution's variations in space and time.

We gave general principles to choose a suitable evolution model, an observation equation, and their matrices of covariance. The observation equation corresponds to the Image Model of the Weickert's method. The evolution equation constrains the solution in time and the model error constrains it in space. As an example, we saw that an exponential covariance is equivalent to a second order Tikhonov regularization. The three drawbacks of the Weickert's method, pointed out in the Introduction, are then overcome: the dynamic of the vector state is accurately described by the suitable evolution model; missing data are managed by the observation error: the covariance evaluates the confidence of the observation data. Depending on the covariance value, the observation will or will not be taken into account in the computation of the solution. Typically, the covariance matrix must have values close to 0 where data are missing, leaving the evolution law determining the solution; the algorithm processes a sequence of images frame by frame, which allows an implementation with low memory requirements.

The choice of the evolution model remains the most important point as it greatly depends on the applicative context. We described the process of determining it to solve the optical flow problem: the image gradients (or the image's grey level values) are assimilated in an evolution model representing a transport equation of the velocity. This transport equation has however the drawback of being non linear and we described a stable numerical scheme.

The main perspective of this work is to define a generic architecture of the evolution model which will then be applicable to specific cases. The diffusion laws, which model the transport of the state vector using the physical conservative principles, are good candidates for a number of reasons. First, their approximation in a discrete space is robust as stable numerical schemes exist. Second, the adjoint of the model operator is known explicitly. Third, several types of diffusion are possible. It may be anisotropic and driven by specific configurations of the state vector or of the image values, as the diffusion has a strong impact on spatial regularization [START_REF] Nielsen | Regularisation and scale space[END_REF]]. An adaptive diffusion can spatially regularize the state vector while preserving the discontinuities. Such a diffusion, used as an evolution equation, performs both the temporal and the spatial regularization of the state vector.

As a second perspective, we plan to apply this method to 3D reconstruction from a sequence of 2D images and one initial 3D image. The process is considered as an interpolation driven by the sequence of 2D images: these images are the observation and assimilated into an evolution model which is the transport of the voxel brightness.

Figure 1 :

 1 Figure 1: The taxi sequence. Top line: results on frame 3 for Data Assimilation and Horn & Schunk methods. Bottom line: results on frame 4 with missing data.

Figure 2 :

 2 Figure 2: The Coriolis sequence: results on frame 2 for Data Assimilation and Horn & Schunk methods.

RR n 6477