Hang Ho

Alex Lk Leung

Thierry Ng

Ryo Durand

Camille Kawasaki

Laurence Oger

Jean-Marie Balas

Ian Yh Galano

Jetty Chung-Yung Wong

Lee

Ho Hang Leung

Alex Lk Ng

Thierry Durand

Ryo Kawasaki

Camille Oger

Laurence Balas

Jean-Marie Galano

Ian Yh Wong

Jetty Chung-Yung Lee

à la diffusion de documents scientifiques de niveau

Keywords:

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

Increase in omega-6 and decrease in omega-3 polyunsaturated fatty acid oxidation elevates the risk of exudative AMD development in adults with Chinese diet

Introduction

Age-related macular degeneration (AMD) is an acquired disease of the macula and depicted by progressive visual impairment due to deterioration of the photoreceptor-retinal pigment epithelial complex. It is one of the leading causes of blindness in elderlies of both Western and Asian countries, and the number of patients is expected to increase significantly [START_REF] Al-Zamil | Recent developments in age-related macular degeneration: a review[END_REF]. The exact pathophysiology of AMD is not fully elucidated, but it is believed that the pathogenesis of AMD is the result of complex multi-factor interactions between metabolic, genetic and environmental factors [START_REF] Kijlstra | Age-related macular degeneration: a complementopathy?[END_REF].

Of all the environmental factors, nutrition plays an important role in the prevention and progression of AMD. In the AREDS I and AREDS 2 studies, the only 2 large prospective randomized trials on AMD nutritional interventions showed that supplementation of an antioxidant combination (beta-carotene or lutein/zeaxanthin plus vitamin C, E, zinc and copper) potentially reduced the progression of AMD to advanced level in some patients by 25% within five years [START_REF]Age-Related Eye Disease Study Research, A randomized, placebo-controlled, clinical trial of high-dose supplementation with vitamins C and E, beta carotene, and zinc for age-related macular degeneration and vision loss: AREDS report no. 8[END_REF][START_REF]Age-Related Eye Disease Study 2 Research, Lutein + zeaxanthin and omega-3 fatty acids for age-related macular degeneration: the Age-Related Eye Disease Study 2 (AREDS2) randomized clinical trial[END_REF]. It is also proposed lutein/zeaxanthin supplementation may be beneficial to some individuals [START_REF]Age-Related Eye Disease Study 2 Research, Lutein + zeaxanthin and omega-3 fatty acids for age-related macular degeneration: the Age-Related Eye Disease Study 2 (AREDS2) randomized clinical trial[END_REF] however, the addition of omega-3 polyunsaturated fatty acid (omega-3 PUFA), namely eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) had no effect in lowing the risk of developing AMD. These findings contradict to the conclusions of epidemiological studies [START_REF]Age-Related Eye Disease Study 2 Research, Lutein + zeaxanthin and omega-3 fatty acids for age-related macular degeneration: the Age-Related Eye Disease Study 2 (AREDS2) randomized clinical trial[END_REF][START_REF] Souied | Omega-3 fatty acids and age-related macular degeneration[END_REF], in which DHA is associated with a lower incidence of advanced AMD.

The early sign of AMD onset is characterized by the presence of extracellular deposition, known as drusen, which accumulates between the Burch membrane and retinal pigment epithelium. Previous studies isolated the metabolic debris from AMD patients and detected advanced glycation end products and carboxyethylpyrrole adducts formed from oxidation of fatty acids in the photoreceptor. Indeed, these components [START_REF] Crabb | Drusen proteome analysis: an approach to the etiology of age-related macular degeneration[END_REF][START_REF] Farboud | Development of a polyclonal antibody with broad epitope specificity for advanced glycation endproducts and localization of these epitopes in Bruch's membrane of the aging eye[END_REF] are PUFA metabolites that are associated to oxidative damage therefore suggests oxidative stress takes part in the pathogenesis and progression of AMD [START_REF] Beatty | The role of oxidative stress in the pathogenesis of age-related macular degeneration[END_REF].

Oxidative stress generally refers to cellular damage caused by reactive oxygen species (ROS) [START_REF] Muller | Trends in oxidative aging theories[END_REF]. Macromolecules in particular lipids, are targets of the damage. Absorbed dietary lipids will be differentially transported to the tissue site to perform their function. Under oxidative stress and inflammation, these lipids become dysfunctional and generate a mix of lipid oxidation products (LOP). Depending on the type of product generated, it may be harmful or beneficial to human health and disease.

It has been found that LOPs derived from PUFA peroxidation is biologically active by participating in various cellular signalling pathways related to pathophysiological conditions [START_REF] Galano | Non-enzymatic cyclic oxygenated metabolites of adrenic, docosahexaenoic, eicosapentaenoic and alpha-linolenic acids; bioactivities and potential use as biomarkers[END_REF]. In particular, 15-F 2t -isoprostane (15-F 2t -IsoP) derived from arachidonic acid (ARA) is not only a hallmark for oxidative stress in vivo, but also known to trigger an excitatory response that are mediated by the thromboxane A2 receptors in a large number of tissues and vascular systems, including the aorta, carotid, coronary, cerebral, pial, and retinal vasculatures [START_REF] Galano | Non-enzymatic cyclic oxygenated metabolites of adrenic, docosahexaenoic, eicosapentaenoic and alpha-linolenic acids; bioactivities and potential use as biomarkers[END_REF][START_REF] Roberts | Measurement of F(2)-isoprostanes as an index of oxidative stress in vivo[END_REF][START_REF] Bauer | Pathophysiology of isoprostanes in the cardiovascular system: implications of isoprostane-mediated thromboxane A2 receptor activation[END_REF][START_REF] Janssen | The pulmonary biology of isoprostanes[END_REF].

Isoprostanes are also derived from other PUFA, namely F 2 -dihomoisoprostanes (F 2 -dihomo-IsoPs) from adrenic acid (AdA), F 3 -isoprostanes (F 3 -IsoPs) from eicosapentaenoic acid (EPA) and F 4 -neuroprostanes (F 4 -NeuroPs) from docosahexaenoic acid (DHA). Although it is not widely investigated, they are considered to be tissue-specific markers of oxidative stress related to neurological diseases such as Rett Syndrome and epilepsy, but also have potential benefits for cardiovascular health [START_REF] Galano | Non-enzymatic cyclic oxygenated metabolites of adrenic, docosahexaenoic, eicosapentaenoic and alpha-linolenic acids; bioactivities and potential use as biomarkers[END_REF][START_REF] Roy | Non-enzymatic oxidized metabolite of DHA, 4(RS)-4-F4tneuroprostane protects the heart against reperfusion injury, Free Radic[END_REF][START_REF] Signorini | Relevance of 4-F4t-neuroprostane and 10-F4t-neuroprostane to neurological diseases[END_REF]. In addition, LOPs originating from lipoxygenase-mediated PUFA oxidation are largely involved in inflammatory response where those from ARA namely hydroxyeicosatetraenoic acids (HETE) are pro-inflammatory and stimulates platelet activation, while DHA-derived hydroxydocosahexaenoic acids (HDHA) are anti-inflammatory with pro-resolving actions [START_REF] Ikei | Investigations of human platelet-type 12-lipoxygenase: role of lipoxygenase products in platelet activation[END_REF].

The breakdown of lipid and formation of an array of LOPs in retinal cells thereby may contribute to the pathophysiology of AMD, while antioxidants like lutein/zeaxanthin and omega-3 PUFAs may modify the oxidation process. However, despite the prompted interest in the role of lipid abnormalities in AMD, previous studies have focused specifically on the role of malondialdehyde and 4-hydroxynonenal (4-HNE) [START_REF] Kopitz | Lipids and lipid peroxidation products in the pathogenesis of age-related macular degeneration[END_REF]. These aldehydes are toxic and potentially form adducts by cross-linking with protein [START_REF] Shoeb | 4-Hydroxynonenal in the pathogenesis and progression of human diseases[END_REF]. Nonetheless, information on the role of other LOPs such as isoprostanes of AdA, DPA, DHA and EPA, and different isomers of HETE and HDHA on AMD are insofar not investigated in AMD.

Although AREDS showed the importance of antioxidants and DHA plus EPA in the progression of AMD, the studies were based on non-Asian community [START_REF]Age-Related Eye Disease Study Research, A randomized, placebo-controlled, clinical trial of high-dose supplementation with vitamins C and E, beta carotene, and zinc for age-related macular degeneration and vision loss: AREDS report no. 8[END_REF][START_REF]Age-Related Eye Disease Study 2 Research, Lutein + zeaxanthin and omega-3 fatty acids for age-related macular degeneration: the Age-Related Eye Disease Study 2 (AREDS2) randomized clinical trial[END_REF]. Furthermore, oxidative stress status of the patients depended on lipid peroxidation products related to ARA only and not EPA, DHA or other PUFA, that are suggested to have pivotal roles in disease development. We hypothesize that the poor DHA and antioxidant carotenoid diet as reported in our previous study [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF] increased the risk of AMD development in adults in part, attributed by certain oxidized products derived from PUFA. In support, we determined (1) the antioxidant status (antioxidant enzymes, carotenoid levels), ( 2) plasma omega-6 PUFA to omega-3 PUFA ratio status and (3) plasma and urine lipid oxidation products (isoprostanes, HETE, HDHA, specialized pro-resolving mediators, aldehydes) and 8-hydroxy-2'deoxyguanosisne (a DNA adduct of oxidative damage) of AMD patients compared to healthy age-matched controls in adults with Chinese diet.

Materials and methods

Chemicals and materials

All organic solvents were at least analytical grade and purchased from Sigma-Aldrich (St. Louis, Mo, USA). All isoprostanoids, HDHAs, HETEs, 4-hydroxyhexenal (4-HHE), 4-HNE, and resolvin standards were purchased from Cayman Chemical Co. (Ann Arbor, MI, USA); 8-Hydroxy-2′-deoxyguanosine (8-OHdG) was purchased from Sigma-Aldrich; neuroprotectin (NPD1), dihomo-IsoPs, dihomo-isofurans (dihomo-IsoFs) and NeuroPs were synthesized by the Institut des Biomolécules Max Mousseron (IBMM, Montpellier, France) as previously described [START_REF] De La Torre | Total syntheses and in vivo quantitation of novel neurofuran and dihomo-isofuran derived from docosahexaenoic acid and adrenic acid[END_REF][START_REF] De La Torre | Synthesis, discovery, and quantitation of dihomo-isofurans: biomarkers for in vivo adrenic acid peroxidation[END_REF][START_REF] Oger | The handy use of Brown's P2-Ni catalyst for a skipped diyne deuteration: application to the synthesis of a [D4]-labeled F4t-neuroprostane[END_REF][START_REF] Guy | Oxygenated metabolites of n-3 polyunsaturated fatty acids as potential oxidative stress biomarkers: total synthesis of 8-F3t-IsoP, 10-F4t-NeuroP and [D4]-10-F4t-NeuroP[END_REF][START_REF] Dayaker | Total synthesis of neuroprotectin D1 analogues derived from omega-6 docosapentaenoic acid (DPA) and adrenic acid (AdA) from a common pivotal, late-stage intermediate[END_REF]. Glutathione (GSH), superoxide dismutase (SOD), catalase (CAT) and bicinchoninic acid (BCA) assay kits were all purchased from Cayman Chemical Co.

Study design

This is a cross-sectional case-control study conducted at the Queen Mary Hospital in Hong Kong and the Grantham Hospital in Hong Kong from December 2015 to December 2017 for a period of two years. The study was approved by the University of Hong Kong Institutional Review Board/Hospital Authority Hong Kong West Cluster (No. UW 14-405) and complies with the principles of the Helsinki Declaration. All patients signed informed consent before joining the study. Detailed recruitment criteria and baseline characteristics of the subjects are described in previous study [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF]. Plasma fatty acids and carotenoid levels were measured [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF] and reported as total carotenoids and omega-6/omega-3 PUFA ratios in this study. These values were not calculated previously [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF].

Measurement of PUFA oxidation products

The plasma and urine samples were prepared and quantified using liquid chromatography quadrupole time-of-flight tandem mass spectrometry (LC-QTOF-MS/MS) method described [START_REF] Lee | LC-MS/MS analysis of lipid oxidation products in blood and tissue samples[END_REF] with modification. In brief, plasma samples were thawed on ice and centrifuged at 3000 x g for 10 min at 4 °C to remove denatured proteins. A volume of 200 μL plasma was added to 10 mL Folch solution for lipid extraction. After phase separation, the lipid-containing lower layer of chloroform was taken and dried under nitrogen gas. The sample was resuspended with deuterated internal standard mixture (0.1 ng/μL in methanol). Afterwards, 1 N potassium hydroxide in methanol was added for alkaline hydrolysis, and then neutralized with 5 N hydrochloric acid. The oxidized PUFA products were extracted by mixed anion solid phase extraction (SPE, MAX Waters, USA). The SPE column was first conditioned with methanol and 20 mM formic acid. After loading the sample, it was washed with 2% ammonium hydroxide and 20 mM formic acid. Then, the final elute hexane/ethanol/acetic acid (v/v/v; 70/29.4/0.6) was collected. The elute was dried at 37°C under nitrogen gas and resuspended in 100 μL methanol for immediate analysis. Sciex X500R QTOF system (Sciex Applied Biosystems, MA, USA) coupled to an Exion LC AC liquid chromatograph with C18 column (150 x 2.1 mm, 2.6 μm particle size, Phenomenex, USA) maintained at 40°C was used in the analysis. The flow rate was set to 300 μL/min and the injection volume was 10 μL. A mobile phase consisting of 0.1% aqueous acetic acid (A) and 0.1% acetic acid/methanol (B) was used for LC. The gradient conditions were set at 20% of solvent B for 2 min, increased from 20% to 98% within 8 min and then held for 5 min. Thereafter, solvent B was returned to 20% in 1 min and held for another 5 min. The QTOF system was operated in negative electrospray ionization (ESI) mode. The spray voltage was set to -4500V and nitrogen was used as the curtain gas. The temperature of the ionization chamber was set at 350°C, and the ion source gases 1 and 2 were maintained at 35 and 45 psi, respectively. For TOF MS, the declustering potential (DP) was -80eV and the collision energy (CE) was -10eV. The scan mode was set as multiple reaction monitoring (MRM).

A total of 39 LOPs (chromatogram, m/z of the precursor and daughter, DP and CE ions are found in Tables S1 andS2 in the Supplementary Material 1) were evaluated in this study, of which 37 products were within the quantitative range. The quantification of each analyte was determined by correlating the peak area to its corresponding deuterated internal standard peak. For analytes without corresponding deuterated internal standard, the following deuterated internal standard, 15-F 2t -IsoP-d 4 and 4-(RS)-4-F 4t -NeuroP-d 4 , 5(S)-HETE-d 8 and DHA-d 5 , were used for quantification. Calibration curves were set up over a concentration range of 0.01-10 ng/μL. Each curve point was determined in triplicate and the regression coefficient ranged between 0.933 -0.998. For urine samples, the LOP concentrations were normalized with creatinine levels and expressed as μg/mg creatinine.

Measurement of 4-hydoxynonenal and 4-hydroxyhexenal

To 100 μL plasma or 1 mL urine samples, 200 μL of BHT solution (1 mg/mL in ethanol), 100 μL of internal standard mix containing d 3 -HHE and d 3 -HNE (0.25 ng/μL in ethanol) and 2200 μL of ethanol-water (50:50 v/v) were added. The sample vortexed for 1 min before centrifuged at 2100 x g for 10 min. A volume of 2 mL of the supernatant was collected, filtered with a hydrophilic filter, and transferred to a new test-tube with a cap. Dinitrophenylhydrazone (DNPH) derivatization was performed by adding 2 mL 0.05 M DNPH solution in acetonitrile/acetic acid (9:1 v/v) into 2 mL of the extract. The tubes were incubated in 60 °C water bath for 2 h. After derivatization, 2 mL of water and 2 mL of hexane were added to the solution to collect the derivatives. The extraction was repeated with another 2 mL hexane. The solvents were pooled together and evaporated at 37 °C under nitrogen. The dried extract was resuspended with 100 μL of 0.1% acetic acid/acetonitrile (60:40 v/v).

The separation and quantification of 4-HHE and 4-HNE were performed using an ExionLC™ AC analytical HPLC, coupled with a Sciex X500R QTOF System. The HPLC system was connected with Kinetex C18 column (150 x 2.1mm, 2.6um, Phenomenex, USA) and warmed to 40 °C to achieve stable separation. The mobile phases are 0.1 % acetic acid in water (A) and 0.1% acetic acid in methanol (B), with a flow rate of 0.3 mL/min. A gradient elution was developed as follow: 40% to 65% B for the first 8.5 min, gradual increase to 100%B in 4 min and maintain for 7.5 min, then decrease to 40%B in 2 min and maintain for 6 min reconditioning. The analysis with the mass spectrometer was performed in MRM HR mode, with negative electrospray ionization at a source temperature of 500 °C. Quantification was made by by correlating the peak area to its corresponding deuterated internal standard peak.

Measurement of DNA oxidation products

A DNA adduct, 8-hydroxy-2'-deoxyguanosine (8OHdG) was determined in plasma and urine samples. The samples were prepared and determined as described by Hu et al. method [START_REF] Hu | Comparison of analyses of urinary 8-hydroxy-2'-deoxyguanosine by isotope-dilution liquid chromatography with electrospray tandem mass spectrometry and by enzyme-linked immunosorbent assay[END_REF] using LC-QTOF-MS/ MS (X500R Sciex Applied Biosystems, MA, USA).

GSH, SOD, CAT, BCA and creatinine assay kits

The plasma samples were prepared and measured as described in the user manuals. BCA levels in plasma samples were used for normalization of GSH, SOD and CAT values (n = 20 for both control and AMD group); creatinine levels were used for normalization of LOPs in urine samples.

Statistical analysis

All statistical methods were performed by IBM® SPSS® Statistics (version 25.0 for Windows, Armonk, NY USA) and the data is expressed as mean (standard deviation). The concentration of LOPs between AMD group and healthy control group was compared using an independent ttest. Conditional logistic regression model was also used to adjust for the multiple variables to calculate the odds ratios (ORs) of having AMD and the level of each LOP in the plasma. Pearson correlation was performed to study the relationship between total carotenoids, omega-6/ omega-3 PUFA ratio and LOP. A p-value of 0.05 or less were considered statistically significant.

Results

As previously described [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF], a total of 99 exudative AMD patients and 198 age-and gender-matched controls were required to meet the number of sampling criteria for the study. The mean age was 73.7 (10.2) years in AMD and the control group was 67.1 (9.3). There was no significant difference in BMI between the healthy control group (24.48 ± 3.81 kg/m 2 ) and AMD (24.17 ± 3.22 kg/m 2 ).

As shown, patients with exudative AMD had lower total carotenoid (lutein + zeaxanthin + lycopene + beta-carotene) levels compared to healthy control. The omega-6/omega-3 PUFA ratio in the healthy control group was approximately 8 to 1, but significantly higher in the AMD group where a ratio of 11 to 1 approximately was recorded (Table 1).

Due to limited number of plasma samples, only 20 samples from each AMD group and healthy control group were tested for antioxidant related enzymes (GSH, SOD, catalase). The group size was based on our pilot study as described in the previous report [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF]. There was no difference in GSH levels between the groups (Fig. 1A). On the other hand, SOD (Fig. 1B) and catalase (Fig. 1C) were both significantly higher in the AMD group.

A range of LOPs derived from PUFAs were measured in plasma and urine samples. The profiles were noticeably different between AMD and healthy control groups. Especially in the AMD group, the plasma levels of ARA-derived 5-F 2t -IsoP, 15-F 2t -IsoP, 5-HETE and 11-HETE were significantly higher (Table 2). The concentration of their urinary 15-F 2t -IsoP and its down-stream metabolite 2,3-dinor-5,6-dihydro-15-F 2t -IsoP also showed to be elevated. Consistently, levels of 4-F 3t -IsoP from omega-6 docosapentaenoic acid (DPA) and 17-dihomo-F 2t -IsoP from AdA was significantly higher in both plasma and urine for AMD compared to control, while 17-dihomo-F 2t -IsoF from AdA was elevated only in the urine. Other omega-6 PUFA oxidation products were similar levels between AMD group and the control group.

On the other hand, the urinary level of EPA-derived 8-F 3t -IsoP and resolvin E1 (RvE1) in plasma were lower in the AMD group compared to healthy control group (Table 3). Significantly lower plasma concentrations of DHA-derived 14-HDHA, 17-HDHA, resolvin D1 (RvD1), NPD1, and NeuroPs including 4-F 4t -NeuroP, 10-F 4t -NeuroP, 13-F 4t -NeuroP and 20-F 4t -NeuroP were also observed. The urinary levels of 4- Other omega-3 PUFA oxidation products levels were similar between the AMD group and the healthy control group. Aside from PUFA oxidation, oxidative damage of the DNA as measured by 8-OHdG was significantly higher in the AMD group compared to healthy control group in both plasma and urine (Fig. 2).

Using logistic regression analysis, specific LOPs in both plasma and urine have been identified as protective and risk factors of exudative AMD. It was found that 5-F 2t -IsoP, 17-dihomo-F 2t -IsoP, 4-F 3t -IsoP and particularly, 4-HNE and 8-OHdG in plasma were associated with higher risk of developing exudative AMD (Table 4) whereas 2,3-dinor-5,6-dihydro-15-F 2t -IsoP, 4-HNE, 4-F 3t -IsoP, 17-dihomo-F 2t -IsoF, 8-OHdG, and particularly, 11-HETE in urine indicated higher AMD risk (Table 5). In contrast, higher levels of 10-F 4t -NeuroP, 20-F 4t -NeuroP, 14-HDHA, 17-HDHA, NPD1 and RvE1 in plasma and urinary 4-F 4t -NeuroP, 10-F 4t -NeuroP, 8-F 3t -IsoP are associated to lower AMD risk.

Pearson correlation was tested to determine the relationship between nutrition status i.e., total carotenoids and omega-6/omega-3 PUFA ratio, and antioxidant status (GSH, SOD, CAT) of the AMD patients. There was a strong, positive correlation between omega-6/ omega-3 PUFA ratio and SOD (r = 0.45, n = 40, p = 0.004), while a negative correlation between omega-6/omega-3 PUFA ratio and GSH (r = -0.38, n = 40, p = 0.036) was also observed. Interestingly, there was a negative correlation between omega-6/omega-3 PUFA ratio and total carotenoid levels (r = -0.18, n = 286, p = 0.002).

When applying the Pearson correlation to determine relationship between nutrition status and oxidative damage (lipid oxidation, DNA oxidation), it showed that levels of 5-HETE, 4-HNE and 8-OHdG were negatively correlated to total carotenoid levels (Table 6). There was also a strong, negative correlation between NeuroP and RvD1 concentrations with omega-6/omega-3 PUFA ratio. Moreover, levels of 15-F 2t -IsoP, 2,3-dinor-15-F 2t -IsoP, 17-dihomo-F 2t -IsoP and 4-F 3t -IsoP were found to be positively related with omega-6/omega-3 PUFA ratio.

Discussion

It is acknowledged that lifestyle factors, especially eating habits, have a significant effect on the pathogenesis of AMD [START_REF] Kijlstra | Age-related macular degeneration: a complementopathy?[END_REF]. However, most of the research to date, emphasized on the effect of supplements, and few studies have focused on the relationship between circulating nutrients (including fatty acids and carotenoids) and AMD. Furthermore, most observations on PUFA and carotenoid supplements are based in Caucasian population with western diet [START_REF] Souied | Oral docosahexaenoic acid in the prevention of exudative age-related macular degeneration: the Nutritional AMD Treatment 2 study[END_REF][START_REF] Merle | Circulating omega-3 Fatty acids and neovascular age-related macular degeneration[END_REF]. There is little data on whether these supplements are equally relevant in reducing AMD risks in Asian populations, where dietary lipid intake is different from Western populations.

In our previous study [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF], patients with exudative AMD in Hong Kong Chinese population were found to have lower plasma carotenoids and omega-3 PUFA, in particular EPA, DHA and α-linolenic acid (ALA) levels, but high omega-6 PUFA and saturated fatty acids. The results are consistent with other epidemiological study [START_REF] Souied | Omega-3 fatty acids and age-related macular degeneration[END_REF]. Of note, patients who sustained high levels of erythrocyte DHA after DHA supplementation were reported to be significantly protected against AMD compared to those who continued to have low DHA levels (HR 0.32, 95% CI 0.10-0.99) [START_REF] Souied | Oral docosahexaenoic acid in the prevention of exudative age-related macular degeneration: the Nutritional AMD Treatment 2 study[END_REF]. The omega-6/omega-3 PUFA ratio of AMD subjects were approximately 35% higher than the healthy controls in this study. The difference is moderate and likely to be influenced by the dietary habit of the healthy controls who consumed less omega-6 PUFA food (e.g. red meat) without increasing omega-3 PUFA rich food (e.g. fatty fish) and in AMD subjects, more omega-6 PUFA food and similar amount of omega-3 PUFA food as healthy controls in their meals [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF]. Nonetheless, higher plasma omega-6 PUFA levels and lower plasma levels of omega-3 PUFA were found to be risk factors for the development of exudative AMD [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF].

Unexpectedly, carotenoid levels were not associated to the disease risk of exudative AMD in the previous study. Although carotenoids (provitamin A) are not potent antioxidants in repressing cellular PUFA Odds ratio adjusted for age and gender. CI: confidence interval; HETE: hydroxyeicosatetraenoic acid; IsoP: isoprostane; Dihydro-15-F 2t -IsoP: 2,3-dinor-5,6dihydro-15-F 2t -IsoP; IsoF: isofuran; NeuroP: neuroprostane; HNE: hydroxynonenal; 8-OHdG: 8-hydroxy-2'-deoxy-guanosine.

oxidation compared to vitamin C and vitamin E, high levels of carotenoids have been constantly reported to be associated with the development of AMD [START_REF] Wu | Intakes of lutein, zeaxanthin, and other carotenoids and age-related macular degeneration during 2 decades of prospective follow-up[END_REF], suggesting carotenoids may also take part in regulating in vivo oxidation, including lipid peroxidation, in AMD pathogenesis. Taken together, it is regarded the pathophysiology of AMD risk is due to the resulting metabolites of PUFA oxidation and hence, poor dietary omega-3 PUFA and high omega-6 PUFA may greatly affect the consequence. Therefore, we hypothesized that oxidative damage and low grade inflammation from PUFA oxidation, are partly caused by nutritional status and involved in the development of AMD.

In this study, we determined general in vivo antioxidant-oxidative stress status of the subjects by measuring systemic SOD and GSH and CAT activity. It should be noted, circulating SOD level and catalase activity are not indicators of antioxidant-PUFA oxidation relationship at cellular level. Upregulation of SOD and catalase in this study suggests the AMD patients were under oxidative stress, which matches with a study by Nita and Grzybowski [START_REF] Nita | The role of the reactive oxygen species and oxidative stress in the pathomechanism of the age-related ocular diseases and other pathologies of the anterior and posterior eye segments in adults[END_REF]. In addition, plasma levels of GSH were unchanged between groups which also coincides with other report [START_REF] Yildirim | The role of oxidative stress and antioxidants in the pathogenesis of age-related macular degeneration[END_REF]. A positive correlation between SOD and omega-6/omega-3 PUFA ratio was observed in this study and supports the suggestion that high omega-6/omega-3 PUFA ratio to be associated to greater oxidative stress in inflammation [START_REF] Dinicolantonio | Importance of maintaining a low omega-6/ omega-3 ratio for reducing inflammation[END_REF].

We evaluated a wide panel of PUFA oxidation products from omega-6 and omega-3 PUFA, and a DNA oxidation product. The results showed that the plasma levels of omega-6 PUFA oxidation products in the AMD group were significantly higher, including 5-F 2t -IsoP, 15-F 2t -IsoP, 4-F 3t -IsoP, 17-dihomo-F 2t -IsoP, and proinflammatory products 5-HETE and 11-HETE. In addition, elevated urinary 15-F 2t -IsoP and its downstream metabolite 2,3-dinor-5,6-dihydro-15-F 2t -IsoP were also observed. These findings are consistent with other report but provide additional details on changes in omega-6 PUFA metabolism of AMD [START_REF] Sabanayagam | Urinary isoprostane levels and age-related macular degeneration[END_REF]. Notably, elevation of urinary 11-HETE strongly increased the odds of AMD development. Although not much is known about 11-HETE in AMD, high levels in circulation was associated with coronary syndrome [START_REF] Zu | Relationship between metabolites of arachidonic acid and prognosis in patients with acute coronary syndrome[END_REF] and lung cancer [START_REF] Liu | Serum free fatty acid biomarkers of lung cancer[END_REF], and take part in neuroinflammation [START_REF] Hayashi | Frataxin deficiency increases cyclooxygenase 2 and prostaglandins in cell and animal models of Friedreich's ataxia[END_REF] and neurotransmission [START_REF] Yu | Glutamate affects the CYP1B1-and CYP2U1-mediated hydroxylation of arachidonic acid metabolism via astrocytic mGlu5 receptor[END_REF]. In recent report, free 11-HETE was five times higher in obese individuals who had high BMI and waist circumference [START_REF] Pickens | Obesity is positively associated with arachidonic acid-derived 5-and 11-hydroxyeicosatetraenoic acid (HETE)[END_REF].

Oxidative damage has been implicated as a major contributor in the pathogenesis of AMD. ROS burden in the eye, particularly in the macula, suggests DNA damage to take place. A recent report showed that mitochondrial DNA damage in the retina was associated to AMD. It was found to be localized in the retinal pigment epithelium layer, and in both the macula and peripheral parts [START_REF] Terluk | Investigating mitochondria as a target for treating age-related macular degeneration[END_REF]. Further, 8-OHdG is one of the most abundant forms of oxidative damage products of DNA and has been associated with ocular diseases such as Grave's ophthalmopathy [START_REF] Tsai | Oxidative stress change by systemic corticosteroid treatment among patients having active graves ophthalmopathy[END_REF] and found in trabecular meshwork of glaucoma patients [START_REF] Kau | Increased oxidative DNA damage, 8-hydroxydeoxy-guanosine, in human pterygium[END_REF] and also, augmented in plasma and urine of the AMD patients in this study. Furthermore, it was detected in aqueous humor of exudative AMD patients aged > 55 years old of Chinese ethnic and were significantly elevated compared to healthy controls [START_REF] Lau | Increase of 8-hydroxy-2'-deoxyguanosine in aqueous humor of patients with exudative age-related macular degeneration[END_REF]. Moreover, in the logistic regression analysis, the oxidation products derived from omega-6 PUFA (5-F 2t -IsoP, 2,3-dinor-5,6-dihydro-15-F 2t -IsoP, 17-dihomo-F 2t -IsoP, 17dihomo-F 2t -IsoF, 4-F 3t -IsoP, 4-HNE) and 8-OHdG were associated with a high risk of exudative AMD. Similar to this study, the association between omega-6 PUFA IsoPs and chronic diseases were related to oxidative stress [START_REF] Garcia-Blanco | Reliable determination of new lipid peroxidation compounds as potential early Alzheimer Disease biomarkers[END_REF]. Nevertheless, 17-dihomo-F 2t -IsoP is considered to be a potential biomarker for neurogenic diseases such as Alzheimer's disease and perhaps, could be the same for exudative AMD [START_REF] Vanrollins | F2-dihomo-isoprostanes arise from free radical attack on adrenic acid[END_REF].

In elevated in vivo oxygen tension (> 21%, 760 mmHg), certain PUFAs are oxidized and form isofuranoids [START_REF] Galano | Non-enzymatic cyclic oxygenated metabolites of adrenic, docosahexaenoic, eicosapentaenoic and alpha-linolenic acids; bioactivities and potential use as biomarkers[END_REF]. It was found urinary 17-dihomo-F 2t -IsoF derived from AdA showed high odds in developing AMD. To date, dihomo-IsoF has been identified to be one of the most elevated isofuranoids in the brain tissues of pig and rats [START_REF] Taber | 2nd, A nomenclature system for isofurans[END_REF]. Due to the high content of AdA in the white matter, dihomo-IsoF is considered to be one of the potential biomarkers for lipid peroxidation in the nervous system [START_REF] De La Torre | Synthesis, discovery, and quantitation of dihomo-isofurans: biomarkers for in vivo adrenic acid peroxidation[END_REF]. It is the very first time that an association has been found between 17-dihomo-F 2t -IsoF and neurodegenerative diseases like exudative AMD.

The results from Pearson correlation further explain the relevance of PUFA oxidation in AMD, where high omega-6/omega-3 PUFA ratio and low carotenoids levels were related to the prevalence of omega-6 PUFA oxidation products (mainly 4-HNE, 15-F 2t -IsoP, 2,3-dinor-15-F 2t -IsoP, 17-dihomo-F 2t -IsoP, 4-F 3t -IsoP). In particular, total carotenoid levels were inversely correlated with plasma 4-HNE (r = -0.291, n = 269, p < 0.0001). This matches with previous findings in which elevated 4-HNE was found in patients with neurodegenerative diseases [START_REF] Csala | On the role of 4-hydroxynonenal in health and disease[END_REF]. The report also points to the role of 4-HNE in pathophysiology of AMD, which has the ability to modify and inactivate retinal proteins by forming adducts [START_REF] Shoeb | 4-Hydroxynonenal in the pathogenesis and progression of human diseases[END_REF][START_REF] Ethen | Age-related macular degeneration and retinal protein modification by 4-hydroxy-2-nonenal[END_REF]. However, it is unknown if circulating 4-HNE and retinal 4-HNE are proportional, and the likelihood of it being transported to the macula remains unclear.

On the other hand, it was primarily found that plasma F 4t -NeuroPs, 14-HDHA, 17-HDHA, RvD1 and NPD1 were down-regulated in the AMD group. To the best of our knowledge, this study is the first attempt to measure metabolism of circulatory omega-3 PUFA in exudative AMD. Increased ROS or oxidative stress typically result in non-specific oxidation of omega-6 and omega-3 PUFA. Lesser dietary intake of omega-3 PUFA in the AMD subjects compared to healthy controls reported in previous study [START_REF] Ng | Dietary habits, fatty acids and carotenoid levels are associated with neovascular age-related macular degeneration in Chinese[END_REF] may partly explain the unexpected reduction of DHA peroxidation products in exudative AMD. Products from the oxidation of omega-3 PUFA (8-F 3t -IsoP, RvE1, 4-F 4t -NeuroP, 10-F 4t -NeuroP, 20-F 4t -NeuroP, 14-HDHA, 17-HDHA, NPD1) was also found to be correlated with a lower AMD risk in the regression model. This observation suggests the resolvins and protectins from EPA and DHA take part in the anti-inflammatory and pro-resolving response in AMD [START_REF] Serhan | Resolving inflammation: dual anti-inflammatory and pro-resolution lipid mediators[END_REF], thus regulating the low grade inflammation involved in the development of exudative AMD. Furthermore, it is known that NPD1 promotes homeostatic regulation of retinal epithelial cells and photoreceptor cell integrity, particularly during oxidative stress [START_REF] Bazan | Neuroprotectin D1-mediated anti-inflammatory and survival signaling in stroke, retinal degenerations, and Alzheimer's disease[END_REF]. The reduction in NPD1 level may implicate a decrease in oxidative protection in the macula of AMD patients.

Although the effects of F 4t -NeuroP on AMD have not previously been studied, recent reports have revealed an increase in 4-F 4t -NeuroP and 10-F 4t -NeuroP in patients with other neurological disorders such as Rett Syndrome [START_REF] Signorini | Isoprostanoids in clinical and experimental neurological disease models[END_REF]. The inconsistency between previous report and our current findings can be explained by the nutritional difference between healthy and AMD groups, in which high omega-6/omega-3 PUFA ratio is associated with the down-regulation of omega-3 PUFA oxidation products. Also, previous research reported the level of free F 4 -NeuroPs while we reported the total (free + esterified) plasma levels. In summary, a low plasma concentration of omega-3 PUFA resulted in decreased bioactive omega-3 PUFA oxidation products level, thus inherently promoted the development of exudative AMD. Together, results from lipid oxidation product profile indicate patients with exudative AMD have greater oxidative stress in vivo, which leads to oxidative damages, altered lipid metabolism and a higher risk of systemic inflammation. A high omega-6/omega-3 PUFA ratio and low carotenoid levels lead to the prevalence of omega-6 PUFA metabolites, as well as down-regulation of anti-inflammatory and pro-resolving omega-3 PUFA oxidation products. All of these factors contribute to the development of exudative AMD. Notably, the measurement of circulating PUFA oxidation products only relatively reflects changes in systemic PUFA metabolism and the condition may differ from that of target tissue in AMD (i.e. retinal pigment epithelial cells). Nonetheless, current research enhances our understanding of how nutrition regulates the pathogenesis of exudative AMD in the molecular level through lipid metabolism modulation. It also provides a context for future prospective studies on the use of carotenoids and omega-3 PUFA in AMD prevention and regulation.

F

  4t -NeuroP, 10-F 4t -NeuroP and 14-F 3t -IsoP from omega-3 DPA were consistently lower in AMD group compared to healthy control group.

Fig. 1 .

 1 Fig. 1. Antioxidant enzymes determined in plasma. (A) glutathione (GSH) concentration, (B) superoxide dismutase (SOD) concentration and (C) catalase (CAT) activity of AMD group and healthy control (Ctrl) group. Column bars are mean (S.D.). *p < 0.05, ***p < 0.001 AMD vs Ctrl.

) 3 .

 3 61 (2.09) IsoP: isoprostane; NeuroP: neuroprostane; RvD: resolvin D; RvE: resolvin E; HDHA: hydroxy-docosahexaenoic acid; HHE: hydroxy-hexenal; Cr: creatinine. Values are expressed as mean (S.D.). *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001 statistical significance between AMD patients and healthy control subjects.

Fig. 2 .

 2 Fig. 2. Concentration of urinary and plasma 8-hydroxy-2'-deoxyguanosine (8-OHdG) of AMD group and healthy control group. Column bars are mean (S.D.) ***p < 0.001, ****p < 0.0001 statistical significance between AMD patients (n = 99) and healthy control subjects (n = 198).

Table 1

 1 Plasma carotenoids levels and omega-6/omega-3 PUFA ratios of AMD group and control group.

	Mean (SD)	95% Confidence
		Interval

Total carotenoids is the sum of lutein + zeaxanthin + lycopene + beta-carotene. PUFA: polyunsaturated fatty acid. ***p < 0.001 and ****p < 0.0001 indicates statistical significance between AMD patients (n = 99) and healthy control subjects (n = 198).

Table 2

 2 Concentration of omega-6 PUFA oxidation products in plasma and urine samples of AMD group and healthy control group.

		Healthy control (n = 198)		AMD (n = 99)
		Plasma (ng/mL)	Urine (μg/mg Cr)	Plasma (ng/mL)	Urine (μg/mg Cr)
	Arachidonic acid derived
	5-F 2t -IsoP	29.63 (17.07)	3.96 (2.77)	39.61 (24.46) ***	3.92 (3.47)
	15-F 2t -IsoP	69.55 (45.78)	39.80 (26.52)	108.8 (66.74) ***	47.09 (20.65) **
	Dinor-15-F 2t -IsoP		3.58 (2.56)		3.62 (2.45)
	Dihydro-15-F 2t -IsoP		3.76 (2.34)		4.63 (2.23) **
	Tetranor-15-F 2t -IsoP		3.83 (2.57)		3.75 (2.56)
	PGF 2α	44.41 (30.85)	3.68 (2.59)	43.56 (30.5)	3.82 (2.85)
	5-HETE	24.28 (13.47)	1.02 (0.59)	30.39 (27.74) **	0.91 (0.60)
	8-HETE	40.77 (9.58)	4.02 (2.56)	42.28 (31.74)	3.87 (1.86)
	9-HETE	37.66 (20.91)	0.50 (0.30)	33.72 (23.28)	0.44 (0.24)
	11-HETE	33.45 (23.29)	0.89 (0.51)	40.98 (29.60) *	1.01 (0.67)
	12-HETE	30.66 (21.70)	0.70 (0.44)	35.54 (24.05)	0.64 (0.38)
	15-HETE	45.38 (26.52)	0.82 (0.49)	49.04 (23.24)	0.72 (0.51)
	20-HETE	44.41 (27.34)	2.86 (2.41)	41.08 (23.95)	2.35 (1.44)
	Adrenic acid derived
	7-dihomo-F 2t -IsoP	45.59 (40.53)	0.38 (0.24)	44.30 (27.16)	0.39 (0.24)
	17-dihomo-F 2t -IsoP	21.41 (13.37)	0.89 (0.69)	34.73 (25.45) ***	1.15 (0.82) *
	7-dihomo-F 2t -IsoF	7.01 (4.84)	0.77 (0.60)	6.77 (3.86)	0.73 (0.45)
	17-dihomo-F 2t -IsoF	36.84 (26.09)	0.66 (0.45)	41.48 (22.83)	0.81 (0.49) *
	Omega-6 Docosapentaenoic acid derived
	4-F 3t -IsoP	39.83 (32.66)	0.73 (0.58)	50.96 (35.77) **	1.02 (0.78) **
	Peroxidation end product
	4-HNE	5.62 (2.71)	2.67 (1.94)	12.76 (4.55)	3.78 (2.75)
	PGF

2α : prostaglandin F 2α ; IsoP: isoprostane; Dinor-15-F 2t -IsoP: 2,3-dinor-15-F 2t -IsoP; Dihydro-15-F 2t -IsoP: 2,3-dinor-5,6-dihydro-15-F 2t -IsoP; HETE: hydroxyeicosatetraenoic acid; IsoF: isofuran; HNE: hydroxy-nonenal; Cr: creatinine. Values are expressed as mean (S.D.). p < 0.05, **p < 0.01, ***p < 0.001 statistical significance between AMD patients and healthy control subjects.

Table 3

 3 Concentration of omega-3 PUFA oxidation products in plasma and urine samples of AMD group and control group.

		Healthy control (n = 198)	AMD (n = 99)
		Plasma (ng/	Urine (μg/mg	Plasma (ng/	Urine (μg/mg
		mL)	Cr)	mL)	Cr)
	Eicosapentaenoic acid derived
	5-F 3t -IsoP	38.02 (24.31) 3.52 (2.35)	40.74 (23.17)	3.24 (2.40)
	8-F 3t -IsoP	38.10 (21.97) 0.56 (0.39)	39.75 (23.26)	0.35 (0.26)
					***
	18-F 3t -IsoP	96.15 (83.31) 3.94 (2.62)	100.20 (82.15)	4.03 (2.38)
	RvE1	4.36 (2.63)		3.45 (2.93) **
	Docosahexaenoic acid derived
	4-F 4t -NeuroP	41.61 (22.88) 1.98 (1.62)	34.35 (18.47)	1.01 (0.79)
				**	****
	10-F 4t -NeuroP 45.26 (24.30) 1.38 (1.14)	32.45 (15.18)	0.43 (0.31)
				***	****
	13-F 4t -NeuroP 4.29 (2.82)		2.36 (1.70)
				****
	14-F 4t -NeuroP 21.03 (10.48)		18.79 (12.16)
	20-F 4t -NeuroP 9.76 (6.76)		5.53 (2.79)
				****
	RvD1	6.89 (4.52)		5.07 (2.33) ***
	NPD1	34.54 (23.39)		28.60 (17.95) *
	4-HDHA	28.14 (23.75)		23.92 (13.94)
	7-HDHA	29.76 (21.80)		34.77 (21.57)
	8-HDHA	40.01 (26.62)		36.81 (23.96)
	11-HDHA	3.36 (2.45)		3.25 (1.87)
	14-HDHA	40.65 (28.04)		24.95 (15.26)

**** 17-HDHA 37.47 (18.25) 30.83 (15.6) ** Omega-3 Docosapentaenoic acid derived 14-F 3t -IsoP 4.63 (2.55) 0.95 (0.69) 3.50 (2.57) 0.71 (0.49) * Peroxidation end product 4-HHE 50.44 (11.72) 3.95 (2.94) 49.84 (12.48

Table 4

 4 Odds ratio of plasma PUFA oxidation products of being an AMD in the sample population. Only those with statistical significance are shown.

		Odds Ratio	Standard error	p-value	95% CI
	Arachidonic acid-derived
	5-F 2t -IsoP	1.041	0.020	0.046	1.001-1.083
	Adrenic acid-derived
	17-dihomo-F 2t -IsoP	1.163	0.043	< 0.0001	1.070-1.265
	Docosapentaenoic acid-derived
	4-F 3t -IsoP	1.042	0.015	0.006	1.012-1.072
	Eicosapentaenoic acid-derived
	RvE1	0.810	0.095	0.026	0.672-0.975
	Docosahexaenoic acid-derived
	10-F 4t -NeuroP	0.962	0.014	0.007	0.936-0.989
	20-F 4t -NeuroP	0.756	0.082	0.001	0.644-0.887
	14-HDHA	0.969	0.012	0.006	0.947-0.991
	17-HDHA	0.955	0.016	0.003	0.926-0.985
	NPD1	0.969	0.013	0.018	0.944-0.995
	Peroxidation end product
	4-HNE	2.509	0.185	< 0.0001	1.747-3.604
	DNA oxidation
	8-OHdG	2.732	0.172	< 0.0001	1.929-3.830
	Odds ratio adjusted for age and gender. CI: confidence interval; IsoP: iso-
	prostane; NeuroP: neuroprostane; HDHA: hydroxy-docosahexaenoic acid; RvE:
	resolvin E; NPD: neuroprotectin; HNE: hydroxy-nonenal; 8-OHdG: 8-hydroxy-
	2'deoxy-guanosine.

Table 5

 5 Odds ratio of urinary oxidation products of being an AMD in the sample population. Only those with statistical significance are shown.

		Odds Ratio	Standard error	p-value	95% CI
	Arachidonic acid-derived
	Dihydro-15-F 2t -IsoP	1.024	0.009	0.011	1.005-1.043
	11-HETE	9.551	0.816	0.006	1.930-47.275
	Adrenic acid-derived
	17-dihomo-F 2t -IsoF	4.725	0.479	0.001	1.850-12.069
	Docosapentaenoic acid-derived
	4-F 3t -IsoP	2.430	0.327	0.007	1.281-4.610
	Eicosapentaenoic acid-derived
	8-F 3t -IsoP	0.159	0.748	0.014	0.037-0.688
	Docosahexaenoic acid-derived
	4-F 4t -NeuroP	0.554	0.224	0.008	0.357-0.859
	10-F 4t -NeuroP	0.102	0.540	< 0.0001	0.035-0.294
	Peroxidation end product
	4-HNE	1.314	0.096	0.004	1.089-1.586
	DNA oxidation
	8-OHdG	1.072	0.020	< 0.0001	1.031-1.115

Table 6

 6 Correlation of nutrition status (total carotenoids and omega-6/omega-3 PUFA ratio) with oxidative damage in vivo (lipid oxidation, DNA oxidation). Only those with statistical significance are shown. :hydroxyeicosatetraenoic acid; HNE: hydroxy-nonenal; 8-OHdG: 8-hydroxy-2'-deoxyguanosine; IsoP: isoprostane: NeuroP: neuroprostane; Dinor-15-F 2t -IsoP: 2,3-dinor-5,6-dihydro-15-F 2t -IsoP; RvD: resolvin D.

		Plasma			Urine
		r	n	p-value	r	n	p-value
	Total Carotenoids
	5-HETE	-0.123	286	0.038
	4-HNE	-0.291	269	< 0.0001
	8-OHdG	-0.161	286	0.006
	Omega-6/omega-3 PUFA ratio
	15-F 2t -IsoP	0.138	255	0.028
	Dinor-15-F 2t -IsoP				0.125	259	0.044
	17-dihomo-F 2t -IsoP	0.120	286	0.043
	4-F 3t -IsoP	0.214	286	< 0.0001	0.161	245	0.011
	4-F 4t -NeuroP	-0.136	286	0.022
	10-F 4t -NeuroP				-0.123	256	0.049
	13-F 4t -NeuroP	-0.166	275	0.006
	14-F 4t -NeuroP	-0.154	275	0.010
	20-F 4t -NeuroP	-0.205	275	0.001
	RvD1	-0.140	275	0.021
	8-OHdG	0.121	286	0.041

HETE

Funding

This work was supported by Health and Medical Research Fund (Ref: 13142301), Food and Health Bureau, Hong Kong.

Appendix A. Supplementary data

Supplementary data to this article can be found online at .


Thibault Liard

Enrique Zuazua

Initial data identification for the one-dimensional Burgers equation

, the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn't hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.

I. INTRODUCTION A. Sonic-boom minimization

A supersonic airplane, flying above the speed of sound, creates a near-field pressure disturbance. This near-field pressure propagates via an augmented Burgers equation [START_REF] Olav | Propagation of sonic booms through a real, stratified atmosphere[END_REF], [START_REF] Sriram | Advanced sonic boom prediction using the augmented burgers equation[END_REF] (1) Above, P = P (σ, τ ) is the pressure disturbance, σ is the distance from the airplane, τ is the time and the nearfield pressure is localized at σ 0 . For a description of other parameters, we refer to [START_REF] Olav | Propagation of sonic booms through a real, stratified atmosphere[END_REF], [START_REF] Sriram | Advanced sonic boom prediction using the augmented burgers equation[END_REF], [START_REF] Allahverdi | Numerical aspects of sonic-boom minimization[END_REF]. When the pressure disturbance reaches the ground level, it creates two subsequent loud bangs with a short time lapse in between, so-called sonicboom [START_REF] Juan | Multidisciplinary optimization with applications to sonic-boom minimization[END_REF], see Figure 1.

The sonic-boom effects could be minimized by tailoring the shape of the airplane [START_REF] Pawlowski | Origins and overview of the shaped sonic boom demonstration program[END_REF]. The near-field pressure is approximated using F-functions theory [START_REF] Beresford | The flow pattern of a supersonic projectile[END_REF] from the shape of the airplane. We confine our study to the propagation of the sonic-boom from the near-field of the plane down to a desired ground target. More precisely, given a desired ground target P T and the distance of the propagation Σ, our aim 1 Thibault Liard is with LS2N, Ecole Centrale de Nantes & CNRS UMR 6004, F-44000 Nantes, France., Thibault.Liard@ls2n.fr. 2 Enrique Zuazua is with Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics Friedrich-Alexander-Universitat, Erlangen-Nurnberg, 91058 Erlangen, Germany, Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain and Chair of Computational Mathematics, Fundación Deusto Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, enrique.zuazua@fau.de.

Fig. 1: Evolution of pressure disturbances. The near-field disturbance propagates away from the aircraft to the far-field generating a classic N-wave consisting only of a leading and trailing shock. Extract from [START_REF] Juan | Multidisciplinary optimization with applications to sonic-boom minimization[END_REF]. is to identify the set of near-field pressure disturbances P 0 leading to P T at σ = Σ along augmented Burgers evolution at the ground level, i.e P (Σ, •) = P T where P solution of (1) with initial data P 0 . We refer the reader to [START_REF] Juan | Multidisciplinary optimization with applications to sonic-boom minimization[END_REF] for a detailed review on the state of the art related to sonic-boom minimization problem.

In this paper, we take into account the non linear hyperbolic dynamic of (1) disregarding the non-local term and the diffusion effects. Hence, the augmented Burgers equation is reduced to a one-dimensional Burgers equation (2) using the change of variable u(t, x) = P (t, -x). In Section III, we construct multiple approximate initial data P 0 leading to ground-boom signature P T optimized for minimum pressure rise, see [3, Figure 2].

B. Presentation of the problem

We consider the one-dimensional scalar conservation laws

∂ t u(t, x) + ∂ x f (u(t, x)) = 0, (t, x) ∈ R + × R, u(0, x) = u 0 (x), ( 2
)
where u is the state, u 0 is the initial state and the flux function f is defined by f (u) = u 2 2 . Kruzkov's theory [START_REF] Stanislav | First order quasilinear equations in several independent variables[END_REF] provides existence and uniqueness of a solution of (2) with bounded initial datum u 0 ∈ L ∞ (R).

Let T > 0 a final time and u T a target function. As ( 2) is time-irreversibility, some conditions on u T need to be imposed for it to be attainable. This is shown in [12, Theorem 3.1, Corollary 3.2], [START_REF] Gosse | Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation[END_REF]Corollary 1] or [START_REF] Sundar Ghoshal | Exact controllability of scalar conservation laws with strict convex flux[END_REF] where they prove that u T is truly attainable in an exact manner by a solution of (2) if and only if u T satisfies the one-sided Lipschitz condition [START_REF] Bressan | Decay of positive waves in nonlinear systems of conservation laws[END_REF], [START_REF] Glass | An extension of oleinik's inequality for general 1d scalar conservation laws[END_REF], [START_REF] Arsen | Discontinuous solutions of non-linear differential equations[END_REF], [START_REF] Constantine M Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of grundlehren der mathematischen wissenschaften[END_REF], i.e

∂ x u T ≤ 1 T in D (R). (3)
Due to the property of non-backward uniqueness of (2), there may exist multiple initial data leading to the same attainable target u T , as seen in Figure 3. In [START_REF] Gosse | Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation[END_REF], the authors prove that the set of initial data evolving to an attainable target u T is a convex set. Later on, the aforementioned set was fully characterized in [START_REF] Colombo | Initial data identification in conservation laws and hamilton-jacobi equations[END_REF], [START_REF] Esteve | The inverse problem for hamiltonjacobi equations and semiconcave envelopes[END_REF] using the classical Lax-Hopf formula [START_REF] Peter | Hyperbolic systems of conservation laws ii[END_REF]Theorem 2.1].

In [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF], [START_REF] Castro | Optimal control and vanishing viscosity for the burgers equation[END_REF], [START_REF] Allahverdi | Numerical aspects of large-time optimal control of burgers equation[END_REF], [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF], initial data identification of Burgers equation is regarded as an optimal control problem

min u0 J 0 (u 0 ) := u T (•) -u(T, •) L 2 (R) (4)
To solve the optimal control problem (4), some difficulties arise from a theoretical and numerical point of view. Because of the presence of discontinuities (called shocks) in the solution u of (2), the derivative of the cost function J 0 in ( 4) is regarded in a weak sense by requiring strong conditions on the set of initial data [START_REF] Bressan | A maximum principle for optimally controlled systems of conservation laws[END_REF], [START_REF] Bressan | A variational calculus for discontinuous solutions of systems of conservation laws[END_REF], [START_REF] Franc | One-dimensional transport equations with discontinuous coefficients[END_REF], [START_REF] Franc | Differentiability with respect to initial data for a scalar conservation law[END_REF]. This leads to require that entropy solutions of (2) have a finite number of non-interacting jumps. When J 0 is weakly differentiable, gradient descent methods have been implemented in [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF], [START_REF] Castro | Optimal control and vanishing viscosity for the burgers equation[END_REF], [START_REF] Allahverdi | Numerical aspects of large-time optimal control of burgers equation[END_REF] to solve numerically the optimal problem (4). In the cases where it was applied successfully, only one possible initial datum emerges, namely the backward entropy solution, see Section I-C. To find some multiple minimizers, the authors in [START_REF] Gosse | Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation[END_REF] use a filtering step in the backward adjoint solution.

In our paper, an alternative proof of the characterization of the set of initial data in [START_REF] Colombo | Initial data identification in conservation laws and hamilton-jacobi equations[END_REF], [START_REF] Esteve | The inverse problem for hamiltonjacobi equations and semiconcave envelopes[END_REF] is given using only backward generalized characteristics. This leads to the hope of investigate systems of conservation laws in one dimension. More precisely, [START_REF] Colombo | Initial data identification in conservation laws and hamilton-jacobi equations[END_REF]Theorem 4.1] cannot be extend to systems as Euler equations, Saint-Venant equations or Aw-Rascle-Zhang model since Lax-Hopf formula does not hold anymore. Moreover, a wave-front tracking method is implemented to construct numerically all of them. In the simulations, the target function is optimized for minimum pressure rise in the context of sonic-boom minimization problems.

C. The backward-forward method

For a sake of completeness, we recall the definition of a weak-entropy solution of (2).

Definition 1.

• We say that u

∈ L ∞ (R + × R) ∩ C 0 (R + , L 1 loc (R)) is a weak solution if for all ϕ ∈ C 1 c (R 2 , R), R + R (u∂ t ϕ + f (u)∂ x ϕ)dxdt + R u 0 (x)ϕ(0, x)dx = 0. • We say that u ∈ L ∞ (R + × R) ∩ C 0 (R + , L 1 loc (R)) is a weak-entropy solution if u is a weak solution and for every k ∈ R, for all ϕ ∈ C 1 c (R 2 , R + ), R + R (|u -k|∂ t ϕ + sgn(u -k)(f (ρ) -f (k))∂ x ϕ)dxdt + R |u 0 -k|ϕ(0, x)dx ≥ 0.
Kruzkov's theory [START_REF] Stanislav | First order quasilinear equations in several independent variables[END_REF] provides existence and uniqueness of a weak-entropy solution (t, x) → S + t (u 0 )(x) of ( 2) with initial datum u 0 ∈ L ∞ (R). For a given function u T , we introduce the backward function (t, x) → S - t (u T )(x) as follows: for every t ∈ [0, T ], for a.e x ∈ R,

S - t (u T )(x) = S + t (x → u T (-x))(-x). (5)
Remark 1. The solutions S + t (u 0 ) and S - t (u T ) may be regarded as the zero viscosity limit of the solutions S +,ε t (u 0 ) and S -,ε T (u T ) respectively where S +,ε t (u 0 ) and S -,ε t (u T ) are defined as follows: S +,ε t (u 0 ) is the solution of the following viscous Burgers equation

∂ t u(t, x) + ∂ x f (u(t, x)) = ε∂ 2 xx u(t, x), (t, x) ∈ R + × R, u(0, •) = u 0 (x), x ∈ R,
and S -,ε t (u 0 ) is the solution of the following backward equation

∂ t u(t, x) + ∂ x f (u(t, x)) = -ε∂ 2 xx u(t, x), (t, x) ∈ R + × R, u(T, •) = u T (x), x ∈ R.
Using the change of variable (t, x) → (Tt, -x), we notice that the backward equation above is well-defined. Thus, S - T (u T ) is called the backward entropy solution.

•

The backward-forward method consists in solving backward in time the PDE (2) with final target u T and then solving it forward in time with initial datum S - T (u T ), the solution of the backward PDE.

For any attainable target u T , we have

S + T (S - T (u T )) = u T as seen in [12, Theorem 3.1, Corollary 3.2] and [19, Corollary 1]. II. MAIN RESULTS Fix u T ∈ L ∞ (R), we introduce the set I(u T ) = {u 0 ∈ L ∞ (R; R) : S + T (u 0 ) = u T }. (6)
From [12, Corollary 3.2], I(u T ) = ∅ if and only if a suitable representative of u T satisfies the Oleinik condition [START_REF] Bressan | Decay of positive waves in nonlinear systems of conservation laws[END_REF], [START_REF] Glass | An extension of oleinik's inequality for general 1d scalar conservation laws[END_REF], [START_REF] Arsen | Discontinuous solutions of non-linear differential equations[END_REF], [START_REF] Constantine M Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of grundlehren der mathematischen wissenschaften[END_REF], i.e for every x ∈ R and y ∈ R + \{0},

f (u T (x + y) -f (u T (x))) ≤ y T . (7)
When f ∈ BV loc (R), the limits

lim x→x0,x>x0 f (x) := f (x 0 +) and lim x→x0,x<x0 f (x) := f (x 0 -) exist.
Theorem 1. Let T > 0 and a suitable representation of u T ∈ L ∞ (R) satisfies the Oleinik condition [START_REF] Bressan | A maximum principle for optimally controlled systems of conservation laws[END_REF]. Then the initial data u 0 ∈ L ∞ (R) verifies S + T (u 0 ) = u T if and only if the following statements holds. For any

(x, y) ∈ X(u T ) × R y x-T f (u T (x)) S - T (u T )(s) ds ≤ y x-T f (u T (x)) u 0 (s) ds, (8)
For any (x, y) ∈ X(u T ) 2 , y-T f (u T (y))

x-T f (u T (x))

S - T (u T )(s) ds = y-T f (u T (y)) x-T f (u T (x)) u 0 (s) ds, (9) where X(u T ) = {x ∈ R, u T (x-) = u T (x+)} ) and S - T (u T ) is defined in (5). Remark 2. When u T ∈ L ∞ (R) satisfies the Oleinik condition (7), then u T ∈ BV loc (R). Thus, X(u T ) is well-defined. • III. SIMULATIONS
Let u T a given target defined by

u T (x) = min(1, max(-1, 1 2 (x -1) -1))1 (0,6) (x). ( 10
)
Note that P T : x → u T (-x) is of the form of the groundboom signature optimized for minimum pressure rise, see [3, Figure 2].

To solve (2) with initial datum u 0 ∈ BV (R), we have implemented a wave-front tracking algorithm, proposed by Dafermos [START_REF] Constantine M Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF], in the software Matlab. If u ≤ u 0 ≤ u, we introduce the state mesh

M n := u + (u -u)(2 -n N ∩ [0, 1])
and the approximate wave-front tracking solution of S + t (u 0 ) is denoted by S +,n t (u 0 ) : R → M n with n ∈ N the discretization parameter.

Step 1. We construct an approximate backward entropy solution

S -,n t (u T ) : x → S +,n t (x → u T (-x))(-x),
with u = -1, u = 1 and n = 6. An illustration of S -,n t (u T )(x) in the plane (x, t) is given in Figure 2a. The solution S -,n t (u T ) at time t = T is plotted with respect to x in Figure 2b. In figure 2c, the approximate solution (t, x) → S +,n t (S -,n T (u T ))(x) of ( 2) with initial data S -,n T (u T ) is plotted in the plane (x, t). Note that S +,n t (S -,n T (u T )) at time t = T is an approximate function of u T , see Figure 2d.

Step 2. From ( 8) and ( 9), we construct the set of initial data u 0 leading to u T along forward entropy evolution of Burgers equation (2). In Figure 3, four of them are plotted in 3. As a consequence, S +,n T (u 0 ) is an approximate function of u T . Note that, from (9), for any

u 0 satisfying S + T (u 0 ) = u T , for any x ∈ (-∞, 0) ∪ (2, 4) ∪ (6, +∞), u 0 (x) = S - T (u T )(x), Solution u(t, x) Position x Time t (a) (x, t) → -,n t (u T )(-x) (b) x → S -,n T (u T )(x) Solution u(t, x) Position x Time t (c) (x, t) → S t -,n T (u T ))(x) (d) u T (--) and x → S +,n T (S -,n T (u T ))(x) (--)
Fig. 2: T = 2, n = 6. Construction of an approximate backward entropy solution of S - t (u T ) and an approximate solution of S + T (S - T (u T )) of u T defined in [START_REF] Castro | Optimal control and vanishing viscosity for the burgers equation[END_REF] see Figure 2b and Figure 3.

Back to the sonic-boom minimization problem, using the change of variable P (t, x) = u(t, -x), we construct theoretically and numerically the set of near-pressure disturbances P 0 leading to ground bound signature P T optimized for minimum pressure rise. Among this set, a selection of admissible P 0 could be imposed to ensure feasible aircraft design (for instance aerodynamic lift) [3, Section 3.3], [START_REF] Minelli | Advanced optimization approach for supersonic low-boom design[END_REF], [START_REF] Rallabhandi | Sonic boom adjoint methodology and its applications[END_REF].

IV. PROOF OF THEOREM 1

The proof of Theorem 1 is based on the two following lemma and the notion of generalized backward characteristics. This latter notion is an important tool for the study of analytical and geometric properties of weak solutions of (2). We refer to [15, Section 10, Section 11] for more details.

Lemma 1. For every (γ 1 , γ 2 ) ∈ C 0,1 ([0, T ]; R) 2 , for every t 1 < t 2 , γ2(t1) γ1(t1) u(t 1 , x) dx + t2 t1 F (u(t, γ 1 (t)-)) dt = γ2(t2) γ1(t2) u(t 2 , x) dx + t2 t1 F (u(t, γ 2 (t)+)) dt
with u a weak-entropy solution of (2) and for every t ∈ R + , for every γ ∈ C 0,1 ([0, T ]; R),

F (u(t, γ(t)±)) = f (u(t, γ(t)±) -γ(t)u(t, γ(t)±).
The proof of Lemma 1 is given in [13, Lemma 3.2]. We introduce the function Q : R 3 → R defined by

Q(u, v, w) = f (v) -f (u) -f (u)-f (w) u-w (v -u), if u = w, f (v) -f (u) -f (u)(v -u), if u = w. ( 11
)
Since f is strictly convex, we deduce the following Lemma Lemma 2. Let u, w ∈ R.

• If v ∈ [u, w] with u ≤ w, Q(u, v, w) ≤ 0. • If v / ∈ [u, w], Q(u, v, w) ≥ 0. • If u = v, Q(u, v, w) = 0.
Proof of Theorem 1. Let u 0 ∈ L ∞ (R) and assuming that a suitable representation of u T ∈ L ∞ (R) satisfies the Oleinik condition [START_REF] Bressan | A maximum principle for optimally controlled systems of conservation laws[END_REF]. We denote by u the weak-entropy solution of (2) with initial data u 0 and u * the weak-entropy solution of (2) with initial data u * 0 := S - T (u T ) where S - T (u T ) is defined in [START_REF] Franc | Differentiability with respect to initial data for a scalar conservation law[END_REF].

Assuming that u(T, •) = u T (•). Let (x, y) ∈ X(u T ) × R with x -T f (u T (x)) < y. Since u T ∈ L ∞ (R) satisfies the one-sided Lipschitz condition, from [12, Theorem 3.1], we have u * (T, •) = u T (•).
From Lemma 1 with t 1 = 0 and t 2 = T applied with u 0 and u * 0 and using that u(T,

•) = u * (T, •) = u T , we have γ2(0) γ1(0) (u 0 (s) -u * 0 (s)) ds = T 0 F (u(t, γ 2 (t)+)) -F (u * (t, γ 2 (t)+)) dt - T 0 (F (u(t, γ 1 (t)-)) -F (u * (t, γ 1 (t)-))) dt (12)
Fig. 3: T = 2, n = 6 and u T defined in [START_REF] Castro | Optimal control and vanishing viscosity for the burgers equation[END_REF]. Plotting of four different u 0 with respect to x yielding solutions of Burgers equation that coincide with u T at time t = T .

for every (γ 1 , γ 2 ) ∈ C 0,1 ([0, T ]; R).

• Since x ∈ X(u T ), we have u T (x-) = u T (x+) = u(T, x-) = u(T, x+). Thus, from [15, Theorem 11.1.3], the minimal backward generalized characteristic ξ -and the maximal backward generalized characteristic ξ + , associated to u (or u T or u * ), emanating from (T, x) coincide over [0, T ], i.e ξ -(t) = ξ + (t) for every t ∈ [0, T ]. Moreover, ξ -is a straight line with slope f (u

T (x-)) = f (u T (x+)) and ξ -(T ) = x -T f (u T (x)). The function ξ : [0, T ] → R defined by ξ(t) = ξ -(T -t
) is a forward characteristic associated with u (or u T or u * ), issuing from (0, x -T f (u T (x)). • There exists a forward generalized characteristic ξ * ∈ C 0,1 ([0, T ]; R), associated with u * , issuing from (0, y) 1 .

From [12, Lemma 7.2], (t, x) → S - t (u T )(x) is Lipschitz continuous on any compact (0, T )×R, thus ξ * is a straight line with slope f (u * ( T 2 , ξ * ( T 2 ))). Thus, for every s ∈ (0, T ), u * (s, ξ * (s)-) = u * (s, ξ * (s)+). From [15, Corollary 11.1.2], since ξ and ξ * are distinct generalized characteristics for (2) associated with the admissible weak solution u * , which are shock-free on the time interval [0, T ]. Then ξ and ξ * cannot intersect for any t ∈ (0, T ) as seen in Figure 4. From [15, Theorem 10.2.3] and ( 12) with ), emanating from (T, x) and plotting of ξ * , the forward generalized characteristic, associated to u * , emanating from (0, y) with (x, y) ∈ X(u T ) × R and x -T f (u T (x)) < y.

x t t = T × x x -T f (u T (x)) y ξ(•) ξ * (•)
γ 1 = ξ and γ 2 = ξ * , y x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds = T 0 Q(u * (s, ξ * (s)+), u(s, ξ * (s)+), u * (s, ξ * (s)-)) ds + T 0 Q(u(s, ξ(s)-), u * (s, ξ(s)-), u(s, ξ(s)-)) ds, (13)
where Q is defined in [START_REF] Olav | Propagation of sonic booms through a real, stratified atmosphere[END_REF]. From Lemma 2 and ( 13), for every

(x, y) ∈ X(u T ) × R with x -T f (u T (x)) < y, y x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds ≥ 0. ( 14
)
Let (x, y) ∈ I(u T ) 2 with x < y.

• There exists a forward generalized characteristic ξ * ∈ C 0,1 ([0, T ]; R), associated with u * , issuing from (0, x -T f (u T (x))). From [12, Lemma 7.2], (t, x) → S - t (u T )(x) is Lipschitz continuous on any compact (0, T ) × R, thus ξ * is a straight line with slope f (u * ( T 2 , ξ * ( T 2 ))). Thus, for every s ∈ (0, T ), u * (s, ξ * (s)-) = u * (s, ξ * (s)+).

• From [START_REF] Constantine M Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of grundlehren der mathematischen wissenschaften[END_REF]11.1.3 Theorem], since u T (y-) = u T (y+) = u(T, y-) = u(T, y+) the minimal backward generalized characteristic ξ -and the maximal backward generalized characteristic ξ + , associated to u (or u T or u * ), emanating from (T, y) coincide over [0, T ], i.e ξ -(t) = ξ + (t)

for every t ∈ [0, T ]. Moreover, ξ -is a straight line with slope f (u T (y-)) = f (u T (y+)) and ξ -(T ) = y -T f (u T (y)). The function ξ : [0, T ] → R defined by ξ(t) = ξ -(T -t)
is a forward characteristic associated with u, issuing from (0, y -T f (u T (y)). From [START_REF] Constantine M Dafermos | Hyperbolic conservation laws in continuum physics, volume 325 of grundlehren der mathematischen wissenschaften[END_REF]Corollary 11.1.2], ξ and ξ * cannot intersect for any t ∈ (0, T ) as seen in Figure 5. Thus, from [15, Theorem 10.2.3] and ( 12) with γ 1 = ξ * and γ 2 = ξ,

x t t = T × x × y x -T f (u T (x)) y -T f (u T (y)) ξ * (•) ξ(•)
y-T f (u T (y)) x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds = - T 0 Q(u(s, ξ(s)+), u * (s, ξ(s)+), u(s, ξ(s)-)) ds - T 0 Q(u * (s, ξ * (s)-), u(s, ξ * (s)-), u * (s, ξ * (s)-)) ds, (15)
From Lemma 2 and ( 13), for every (x, y) ∈ X(u T ) 2 with x < y,

y-T f (u T (y)) x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds ≤ 0. ( 16
)
From [15, Corollary 11.1.2], we have x -T f (u T (x)) < y -T f (u T (y)). Thus, combining ( 14) with ( 16), we deduce that, for every (x, y) ∈ X(u T ) 2 with x < y y-T f (u T (y))

x-T f (u T (x))

(u 0 (s) -u * 0 (s)) ds = 0. (17)
The inequality (17) also holds for every (x, y) ∈ X(u T ) 2 with x < y, whence [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF]. Assuming that y < x -T f (u T (x)). There exists a z ∈ X(u T ) such that z -T f (u T (z)) < y since a suitable representative of u T ∈ L ∞ (R) satisfies the Oleinik condition [START_REF] Bressan | A maximum principle for optimally controlled systems of conservation laws[END_REF]. Using z -T f (u T (z)) < y and z ∈ X(u T ), from ( 14)

y z-T f (u T (z)) (u 0 (s) -u * 0 (s)) ds ≥ 0. (18)
Since (z, x) ∈ X(u T ), from (8), we have

x-T f (u T (x)) z-T f (u T (z)) (u 0 (s) -u * 0 (s)) ds = 0. (19)
Combining ( 18) and ( 19), the inequality ( 14) holds also for y < x -T f (u T (x)), whence [START_REF] Bressan | A variational calculus for discontinuous solutions of systems of conservation laws[END_REF].

Since u T ∈ L ∞ (R) satisfies the one-sided Lipschitz condition, from [12, Theorem 3.1], we have u * (T, •) = u T (•). From Lemma 1 applied with u 0 and u * 0 , we have

γ2(t1) γ1(t1) (u(t 1 , s) -u * (t 1 , s)) ds - γ2(t2) γ1(t2) (u(t 2 , s) -u * (t 2 , s)) ds = t2 t1 F (u(t, γ 2 (t)+)) -F (u * (t, γ 2 (t)+)) dt - t2 t1 (F (u(t, γ 1 (t)-)) -F (u * (t, γ 1 (t)-))) dt (20) for every (γ 1 , γ 2 ) ∈ C 0,1 ([0, T ]; R). Fix (x, y) ∈ X(u T ) 2 with x < y,
Step 1. Fix (x, y) ∈ R × X(u T ) with x < y.

• Let ξ -the minimal backward generalized characteristic, associated to u, emanating from (T, x). Them, ξ -is a straight line with slope f (u(x-)) and ξ

-(T ) = x - T f (u(x-)). The function ξ : [0, T ] → R defined by ξ(t) = ξ -(T -t) is a forward characteristic associated with u, issuing from (0, x -T f (u(x-)). • Since y ∈ X(u T ), we have u T (x-) = u T (x+) = u * (T, x-) = u * (T, x+).
Thus, the minimal backward generalized characteristic ξ * -and the maximal backward generalized characteristic ξ * + , associated to u * , emanating from (T, y) coincide over

[0, T ], i.e ξ * -(t) = ξ * + (t) for every t ∈ [0, T ]. Moreover, ξ * + is a straight line with slope f (u T (x-)) = f (u T (x+)) and ξ * + (T ) = y -T f (u T (y)). The function ξ : [0, T ] → R defined by ξ * (t) = ξ * + (T -t)
is a forward characteristic associated with u * , issuing from (0, y -T f (u T (y)).

We now distinguish if ξ doesn't interact with ξ * or not as seen in Figure 6.

• Assuming that ξ doesn't interact with ξ * over (0, T ). In this case, we have x -T f (u(x-)) ≤ y -T f (u T (y)).

where Q is defined in [START_REF] Olav | Propagation of sonic booms through a real, stratified atmosphere[END_REF]. Applying (8) with x = y and y = x -T f (u(x-)), we have

y-T f (u T (y)) x-T f (u(x-)) (u 0 (s) -u * 0 (s)) ds ≤ 0. ( 22
)
From Lemma 2, ( 21) and ( 22), (x, y) ∈ R × X(u T ) with x < y,

y x (u * (T, s) -u(T, s))ds ≥ 0 (23)
• Assuming that ξ interacts with ξ * at time t I ∈ (0, T ).

From ( 23) and ( 25), we conclude that for every (x, y) ∈ R × X(u T ) with x < y,

y x (u * (T, s) -u(T, s))ds ≥ 0 (26)
Step 2. Fix (x, y) ∈ X(u T ) 2 with x < y.

• Since x ∈ X(u T ), we have u T (x-) = u T (x+) = u * (T, x-) = u * (T, x+). Thus, the minimal backward generalized characteristic ξ * -and the maximal backward generalized characteristic ξ * + , associated to u * , emanating from (T, x) coincide over [0, T ], i.e ξ * -(t

) = ξ * + (t) for every t ∈ [0, T ]. Moreover, ξ * -is a straight line with slope f (u T (x-)) = f (u T (x+)) and ξ * -(T ) = x -T f (u T (x)). The function ξ * : [0, T ] → R defined by ξ * (t) = ξ * -(T -t) is a forward characteristic associated with u * , issuing from (0, x -T f (u T (x)).
• Let ξ + the maximal backward generalized characteristic, associated to u, emanating from (T, y). Them, ξ + is a straight line with slope f (u(y+)) and ξ

+ (T ) = y - T f (u(y+)). The function ξ : [0, T ] → R defined by ξ(t) = ξ + (T -t)
is a forward characteristic associated with u, issuing from (0, y -T f (u(y+)). We now distinguish if ξ doesn't interact with ξ * or not as seen in Figure 7.

• Assuming that ξ * doesn't interacts with ξ over (0, T ). In this case, we have x -T f (u T (x)) ≤ y -T f (u(y)+).

From [15, Theorem 10.2.3] and [START_REF] Stanislav | First order quasilinear equations in several independent variables[END_REF] with

t 1 = 0, t 2 = T , γ 1 = ξ * and γ 2 = ξ, y x (u * (T, s) -u(T, s)) ds + y-T f (u(y+)) x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds = - T 0 Q(u(s, ξ(s)+), u * (s, ξ(s)+), u(s, ξ(s)-)) ds - T 0 Q(u * (s, ξ * (s)-), u(s, ξ * (s)-), u * (s, ξ * (s)-)) ds, (27)
where Q is defined in [START_REF] Olav | Propagation of sonic booms through a real, stratified atmosphere[END_REF]. Applying (8) with x = x and y = y -T f (u(y+)), we have

y-T f (u(y+)) x-T f (u T (x)) (u 0 (s) -u * 0 (s)) ds ≥ 0. (28)
From Lemma 2, ( 27) and (28), for every (x, y) ∈ X(u T ) 2 with x < y, y x (u * (T, s)u(T, s))ds ≤ 0 (29)

x t t = T × x × y x -T f (u T (x)) y -T f (u(y+)) ξ * (•) ξ(•) ξ doesn't interact with ξ * x t t = T × x × y t = t I ξ * (•) ξ(•) ξ interacts with ξ * at time t = t Fig. 7: Plotting of ξ * = ξ * -(•) and ξ = ξ + (T -•)
where ξ -is the minimal backward generalized characteristic, associated to u * , emanating from (T, x) and ξ + (•) is the maximal backward generalized characteristic, associated to u, emanating from (T, y) with (x, y) ∈ X(u T ) 2 and x < y.

• Assuming that ξ * interacts with ξ at time t I ∈ (0, T ).

For every t ∈ (t I , T ), we have ξ * (t) ≤ ξ(t). From [15, Theorem 10.2.3] and (20) with t 1 = t I , t 2 = T , γ 1 = ξ * and γ 2 = ξ, y x (u * (T, s)u(T, s)) ds = -T t I Q(u(s, ξ(s)+), u * (s, ξ(s)+), u(s, ξ(s)-)) ds -T t I Q(u * (s, ξ * (s)-), u(s, ξ * (s)-), u * (s, ξ * (s)-)) ds, (30) From Lemma 2, (30), for every (x, y) ∈ X(u T ) 2 with x < y, From ( 29) and (31), we conclude that for every (x, y) ∈ X(u T ) 2 with x < y, Combining ( 26) with (32), we deduce that, for every (x, y) ∈ X(u T ) with x < y, y x (u * (T, s)u(T, s))ds = 0.

(33)

Step 3. From [15, Theorem 11.2.2], u(T, •) ∈ BV loc (R) and u * (T, •) ∈ BV loc (R). Thus, X(u T ) has Lebesgue measure 0. The equality (33) holds for a.e (x, y) ∈ R with x < y. Since y → y x u(T, s) ds and y → y x u * (T, s) ds are absolute continuous functions, (33) holds for every y ∈ R. Using the fundamental theorem of calculus, we deduce that, for a.e y ∈ R, u(T, y) = u * (T, y).

V. CONCLUSION

In this paper, we construct theoretically and numerically the set of initial data yielding solutions of Burgers equation (2) that coincide with a given target. When the augmented Burgers equation ( 1) is reduced to Burgers equation [START_REF] Allahverdi | Numerical aspects of sonic-boom minimization[END_REF], this allows to find the set of near-pressure disturbances P 0 leading to the bound signature P T optimized for minimum pressure rise. The proof is based on a backward-forward method and generalized backward characteristics.

Let us address some related open questions and possible extensions of this work.

In the development of low-boom supersonic airplane, it would be interesting to understand the impact of the nonlocal term and the zero-order term in (1) for the initial data identification problem. Moreover, some constraints on the admissible set of initial data could be imposed to ensure feasible aircraft design.

Since we don't need Lax-Hopf formula as in [12, Theorem 4.1], we may extend our approach to systems of conservation laws in one dimension (Euler equations, Saint-Venant equations, Aw-Rascle-Zhang model). Note that, as soon as the backward-forward operator S + T (S - T ) is welldefined, S + T (S - T )(u T ) may give a good candidate to solve the initial data identification of systems of conservation laws.

  0 , •) = P 0 (•).

Fig. 4 :

 4 Fig.4: Plotting of ξ(•) = -(T -•) with ξ -the minimal backward generalized characteristic, associated to u (or u T or u * ), emanating from (T, x) and plotting of ξ * , the forward generalized characteristic, associated to u * , emanating from (0, y) with (x, y) ∈ X(u T ) × R and x -T f (u T (x)) < y.

Fig. 5 :

 5 Fig.5: Plotting of ξ , the forward generalized characteristic, associated to u * , emanating from (0, x) and plotting of ξ(•) = ξ + (T -•) with ξ -the maximal backward generalized characteristic, associated to u (or u T or u * ), emanating from (T, y) with (x, y) ∈ X(u T ) 2 and x < y.

ξ

  Fig. Plotting of ξ(•) = ξ -(T -•) and ξ * (•) = ξ * + (•) where ξ -is the minimal backward generalized characteristic, associated to u, emanating from (T, x) and ξ * + (•) is the maximal backward generalized characteristic, associated to u * , emanating from (T, y) with (x, y) ∈ R × X(u T ) and x < y.

  For every t ∈ (t I , T ), we have ξ(t) ≤ ξ * (t). From [15, Theorem 10.2.3] and[START_REF] Stanislav | First order quasilinear equations in several independent variables[END_REF] with t 1 = t I , t 2 = T , γ 1 = ξ and γ 2 = ξ * , y x (u * (T, s)u(T, s)) ds = T t I Q(u * (s, ξ * (s)+), u(s, ξ * (s)+), u * (s, ξ * (s)-)) ds + T t I Q(u(s, ξ(s)-), u * (s, ξ(s)-), u(s, ξ(s)-)) ds.(24) From Lemma 2, (24), for every (x, y) ∈ R × X(u T ) with x < y, y x (u * (T, s)u(T, s))ds ≥ 0

  (T, s)u(T, s))ds ≤ 0 (31)

  (T, s)u(T, s))ds ≤ 0 (32)

If a rarefaction wave is created at t = 0, multiple forward generalized characteristics exist.