C Heurteaux
email: heurteau@ipmc.cnrs.fr

C Laigle

N Blondeau

G Jarretou

M Lazdunski

ALPHA-LINOLENIC ACID AND RILUZOLE TREATMENT CONFER CEREBRAL PROTECTION AND IMPROVE SURVIVAL AFTER FOCAL BRAIN ISCHEMIA

Keywords: neuroprotection, polyunsaturated fatty acids, riluzole, focal ischemia, long-term survival, 2P-domain K ؉ channels ALA, alpha-linolenic acid; ALA250

We investigated here the effects of alpha-linolenic acid and riluzole, both activators of the 2P-domain K ؉ channel family TREK/TRAAK, in a model of focal ischemia clinically relevant to stroke, not only assessing neuronal protection, but also long term survival. Moreover, all the drug treatments were initiated post-ischemia.

Mice were subjected to transient middle cerebral artery occlusion (1 h) and reperfusion according to the intraluminal filament model. Drugs were injected into the jugular vein according to three protocols: (i) a single dose of 4 mg/kg riluzole or 500 nmol/kg alpha-linolenic acid at different reperfusion time; (ii) a three-day therapy (a single dose of 2 mg/kg riluzole and 250 nmol/kg alpha-linolenic acid given 1-2, 48 and 72 h after reperfusion); (iii) a three-week therapy (a single dose of 2 mg/kg riluzole and 250 nmol/kg alpha-linolenic acid given once a week during three weeks after reperfusion. A combined treatment with 2 mg/kg riluzole؉250 nmol/kg alpha-linolenic acid injected 2 h after reperfusion was also tested.

A single dose of riluzole (4 mg/kg) or alpha-linolenic acid (500 nmol/kg) injected up to 3 h after reperfusion reduced drastically the stroke volume by 75% and 86%, respectively. Neurological deficits 24 h after ischemia were significantly improved by alpha-linolenic acid500 or riluzole4 with a neurological score of 1.8 as compared with 2.5 observed in vehicle-treated mice. Alpha-linolenic acid-and riluzole treatment were associated with a reduction in cytopathological features of cell injury, including DNA fragmentation and Bax expression in the cortex and the caudate putamen. With regard to the survival rate at 30 days, the best protections were obtained with the alpha-linolenic acid-injection in the threeweek therapy as well as with a single dose of the combined treatment (2 mg/kg riluzole؉250 nmol/kg alpha-linolenic acid). Palmitic acid, a saturated fatty acid that does not activate the 2P-domain K-channel TREK/TRAAK family, did not provide any neuroprotection.

Taken together, these data suggest that the TREK/TRAAK K-channel family may be a promising target for neuroprotection, and that riluzole and alpha-linolenic acid could be of therapeutic value against focal ischemia/reperfusion injury to the brain.

Stroke remains the third leading cause of death and the most common cause of adult disability in industrialized countries. Until now, the only treatment for acute ischemic stroke that has a proven efficacy is i.v. thrombolysis within 3 h of onset. However, only a small percentage of the patients are eligible for this therapy. Anticoagulant therapy is also performed for the prevention of stroke in patients with high risk factors and for halting the evolution of a thrombotic stroke [START_REF] Hankey | Treatment and secondary prevention of stroke: evidence, costs, and effects on individuals and populations[END_REF]. It remains therefore of foremost importance to develop efficacious neuroprotective agents.

The neuroprotective effects of riluzole (RLZ, 2-amino-6-trifluoromethoxy benzothiazole) and alpha-linolenic acid (ALA) have been well established in different models of neurodegeneration. RLZ currently in clinical use among patients with amyotrophic lateral sclerosis [START_REF] Bensimon | A controlled trial of riluzole in amyotrophic lateral sclerosis. ALS/Riluzole Study Group[END_REF] induces major protective effects against brain, spinal cord and retina injuries [START_REF] Malgouris | Riluzole, a novel antiglutamate prevents memory loss and hippocampal neuronal damage in ischemic gerbils[END_REF][START_REF] Pratt | Neuroprotective actions of Riluzole in rodent models of global and focal cerebral ischemia[END_REF][START_REF] Martin | The neuroprotective agent riluzole inhibits release of glutamate and aspartate from slices of hippocampal area[END_REF][START_REF] Barneoud | Neuroprotective effects of riluzole on a model of Parkinson's disease in the rat[END_REF][START_REF] Ettaiche | Riluzole improves functional recovery following ischemia in the rat retina[END_REF][START_REF] Lang-Lazdunski | Riluzole improves neurological function following severe spinal cord ischemia[END_REF].

Both in vitro and in vivo studies have shown alphalinolenic acid-benefit for neurons in hyperexcitability-induced neuronal death such as epileptic seizures [START_REF] Voskuyl | Anticonvulsant effect of polyunsaturated fatty acids in rats, using the cortical stimulation model[END_REF][START_REF] Leaf | The antiarrhythmic and anticonvulsant effects of dietary N-3 fatty acids[END_REF][START_REF] Lauritzen | Polyunsaturated fatty acids are potent neuroprotectors[END_REF][START_REF] Blondeau | Polyunsaturated fatty acids induce ischemic and epileptic tolerance[END_REF]. It also prevents neuronal cell death and paraplegia after transient spinal cord ischemia [START_REF] Lang-Lazdunski | Linolenic acid prevents neuronal cell death and paraplegia after transient spinal cord ischemia in rats[END_REF].

The middle cerebral artery occlusion (MCAO) using the intraluminal suture technique has been shown to reliably and rapidly induce neurological deficits and hemispheric infarcts restricted to the territory of the MCA 24 h after the onset of ischemia [START_REF] Ginsberg | Rodent models of cerebral ischemia[END_REF][START_REF] Sharp | Multiple molecular penumbras after focal cerebral ischemia[END_REF]. Using this model described as the closest to human stroke and monitoring the long-term recovery, the aim of the present work is to demonstrate the relevance of the protective effect of alpha-linolenic acid and RLZ against stroke for future clinical study. Both drug treatments were initiated post-ischemia, to determine their therapeutic windows. Three-day and three-week therapies effects were then analyzed to characterize a potential improvement in the survival of animals and their cerebral protection. The drug combination was also tested to evaluate whether the association of these drug actions could enhance the protective effects. To test the link between alpha-linolenic acid, riluzole and the 2P-domain K ϩ channel TREK/ TRAAK family, palmitic acid (PALM), a saturated fatty acid, which does not activate these channels (Lesage andLazdunski, 1999, 2000;[START_REF] Patel | Properties and modulation of mammalian 2P domain K ϩ channels[END_REF] was injected as negative control and his effects analyzed. Moreover, we paid a particular attention to the comparison of the shortterm and the long-term effects of these agents administered in single or combined doses and in repeated singledrug treatment over 3 days and 3 weeks.

EXPERIMENTAL PROCEDURES Animals

All experiments were conducted according to the NIH guide and the Society of Neurosciences guidelines for the Care and Use of Laboratory Animals (NIH publications No. 80-23, 1985). All efforts were made to minimize the number of animals used and their suffering. Adult male C57/Bl6 mice, weighing 22-26 g were used in this study. The animals housed under controlled laboratory conditions with a 12-h light/dark cycle, a temperature of 21Ϯ2 °C, and a humidity of 60 -70% for at least one week prior to drug treatment or surgery. The mice had free access to standard rodent diet and tap water.

Physiological parameters

General anesthesia was induced with 3% isoflurane and maintained with 1% isoflurane by means of an open facemask for each mouse. Mice were allowed to breathe spontaneously. A subset of animals (nϭ5 per group) was monitored for physiological parameters including mean arterial blood pressure (MABP), rectal temperature, arterial blood gases and pH before, during and after ischemia. The right femoral artery was catheterized with PE-10 polyethylene tubing and connected to a blood pressure transducer (Harvard Apparatus, Les Ulis, France) for continuous monitoring of MABP (mm Hg). A heparinized blood sample (75 l) was then obtained from the catheterized femoral artery and blood partial pressure of oxygen (pO 2 ), partial pressure of carbon dioxide (pCO 2 ) and pH were measured using an Acid-Base Laboratory system (ABL 555, Radiometer-Copenhagen, Neuilly/Plaisance, France). Core temperature was monitored continuously with a thermometer (3-mm probe diameter, Harvard Apparatus), inserted into the rectum and maintained at physiological temperatures using a thermostatically controlled heating blanket (Harvard Apparatus).

Induction of transient focal cerebral ischemia

Focal ischemia was induced by occlusion of the left middle cerebral artery (MCA) using an intraluminal filament technique [START_REF] Ding-Zhou | L-NAME reduces infarction, neurological deficit and bloodbrain barrier disruption following cerebral ischemia in mice[END_REF] based on modifications of the original rat model [START_REF] Longa | Reversible middle cerebral artery occlusion without craniectomy in rats[END_REF]. After a midline neck incision was made, the left common and external carotid arteries were isolated and ligated with a 4-0 silk suture (Ethicon, Johnson and Johnson Intl, Brussels, Belgium). A temporary Yasargil aneurysm clip (BMH31, Aesculap, Tuttlingen, Germany) was temporarily placed on the internal carotid artery. A 6-0 nylon monofilament (Ethicon), blunted at tip with an open flame, was introduced through a small incision into the common carotid artery and 13 mm distal to the carotid bifurcation for occlusion of the origin of the MCA. Animals were kept at 37 °C for one hour, after which time the thread was carefully withdrawn to allow reperfusion of the MCA territory. To control the MCAO severity regional CBF (rCBF) was determined by laser-Doppler flowmetry (Perimed, Craponne, France) using a flexible 0.5-mm fiber optic extension to the master probe fixed on the intact skull over the ischemic cortex. Sham-operation was performed inserting the thread into the common carotid artery without advancing it to occlude the MCA. The animals were allowed to regain full consciousness on a heating pad before returning to the cage.

Drug treatments

The experimental protocol is shown in Fig. 1. It includes one study at short term, where the animals were analyzed at 24 h postischemia (Fig. 1A) and three studies at long term (Fig. 1B,C,D) where the mice were evaluated one month following MCA occlusion. In study A, mice received a single dose of 4 mg/kg riluzole (RLZ4) or 500 nmol/kg alpha-linolenic acid (ALA500), or 500 nmol/kg palmitic acid (PAL500) at different times after MCAO to determine the therapeutic window. The doses used were selected based on our previous studies in different animal models [START_REF] Lang-Lazdunski | Riluzole improves neurological function following severe spinal cord ischemia[END_REF][START_REF] Lauritzen | Polyunsaturated fatty acids are potent neuroprotectors[END_REF][START_REF] Blondeau | Polyunsaturated fatty acids induce ischemic and epileptic tolerance[END_REF] and from pilot studies on mice searching for the best protection (data not shown). Single doses of RLZ and ALA were tested at different time points to establish therapeutic windows. It is based on these results that we chose as reference treatment a fixed dose for a determined time point of injection after reperfusion.

Next, the efficiency of a single dose of RLZ4 or ALA500 injected one (RLZ4) or two (ALA500) hours after reperfusion was compared with a dose of 2 mg/kg riluzole (RLZ2) or 250 nmol/kg alpha-linolenic acid (ALA250) given 1-2, 48 and 72 h after reperfusion (Fig. 1B) or 1-2 h after reperfusion following MCAO and once a week during the next 2 weeks (Fig. 1C). In study C, mice were treated with a combined treatment of RLZ2ϩALA250 injected 2 h after reperfusion following MCAO (Fig. 1D). RLZ (RBI, USA), ALA and PALM (Biomol, USA) were injected as a bolus directly in the jugular vein. Sham-operated animals and mice injected with vehicle were used as controls.

Neurological deficits

Neurological deficits of mice were assessed 24 h post-ischemia in a blinded fashion according to a scoring scale in a postural reflex test developed by [START_REF] Bederson | Rat middle cerebral artery occlusion: evaluation of the model and development of a neurologic examination[END_REF]: Grade 0: no visible deficits; Grade 1: forelimb flexion; Grade 2: unidirectional circling when the animal is pulled by the tail; Grade 3: circling and rolling movement; Grade 4: decreased level of consciousness. Mice with a neurological deficit above grade 3 were excluded from the study.

Determination of infarct volume

To assess the infarct volume in the short-term study (single injection of ALA, RLZ or PALM), mice were killed at 24 h after reperfusion. Their brains were removed and sectioned into six 1 mmthick coronal slices using a tissue chopper (Phymep, France). Coronal brain slices were immediately immersed into 2% 2,3,5triphenyltetrazolium chloride (TTC, Sigma, France) for 20 min at room temperature in the dark followed by fixation in a 4% paraformaldehyde solution overnight prior to analysis as described previously [START_REF] Ding-Zhou | L-NAME reduces infarction, neurological deficit and bloodbrain barrier disruption following cerebral ischemia in mice[END_REF]. The striatal and cortical areas of infarction, outlined in light were measured on each section using a computer image analysis system and corrected for brain edema according to [START_REF] Golanov | Contribution of cerebral edema to the neuronal salvage elicited by stimulation of cerebellar fastigial nucleus after occlusion of the middle cerebral artery in rat[END_REF]. Infarct volume, expressed in mm 3 was calculated by a linear integration of the corrected lesions areas. The same TTC staining at 24 h postischemia was also applied to assess the first effect on infarct size of the combined administration used in the long-term study.

In addition Cresyl Violet, a dye that stains the Nissl bodies in the stellate somas of viable neurons was used to confirm the evolution and the extent of infarct volume and analyze cell survival in ischemic mice. Coronal frozen sections of brain (10 m-thick) were added to a solution of 1% Cresyl Violet in 0.25% acetic acid for 3 min, rinsed, dehydrated and mounted with Entellan. Sections were analyzed under light microscopy. Cell counting using standard methods [START_REF] Blondeau | A potent protective role of lysophospholipids against global cerebral ischemia and glutamate excitotoxicity in neuronal cultures[END_REF] were performed in a blind fashion in striatal and cortical penumbra at 30 days to determine the level of cerebral damage. Staining coronal brain sections with TTC and Cresyl Violet after 24 h and 30 days of recovery, respectively has been shown to provide a reliable estimate of the infarct margins and the boundary between penumbra and core, which was estimated by the difference between the size of both infarcts studied at 24 and 3 weeks post-ischemia. Histologically, the boundaries between core and penumbra can be well evaluated. Unlike the core region where many neurons are dead, the penumbra region appears with intact neurons and cells presenting pyknosis.

TUNEL staining

Terminal deoxynucleotidyl transferase (TdT)-mediated 2=-deoxyuridine 5=-triphosphate-biotin nick-end labeling (TUNEL) staining was performed by using the In Situ Cell Death Detection Kit (Roche, Basel, Switzerland), which allows to demonstrate apoptotic cells in ischemic brain tissue as described previously [START_REF] Blondeau | A potent protective role of lysophospholipids against global cerebral ischemia and glutamate excitotoxicity in neuronal cultures[END_REF]. Coronal 10 m frozen sections were rehydrated in ethanol (95, 70 and 50%) followed by PBS and bathed in 0.3% hydrogen peroxide/methanol to inactivate endogenous peroxidase. Sections were then permeabilized in 0.3% Tween/PBS and washed twice in PBS. Labeling of 3=-OH terminal DNA fragments was then performed at 37 °C for 2 h by using the TUNEL reaction mixture according to the manufacturer's protocol. Positive control was obtained by pre-incubation of a section with DNAse I (20 g/ml) for 15 min at 37 °C before incubation with biotinylated dUTP. Quantitative (number of TUNEL-positive cells/0.5 mm 2 ) analyses were performed in the ipsilateral cortical penumbra. TUNEL-pos- itive cells displayed a brown staining within the nucleus in the apoptotic cells.

Western blot analysis

Whole-cell protein extraction was performed using standard procedures. Samples were obtained from the MCA territory cortex on the ischemic sides and from nonischemic controls. (nϭ3 per group). Fresh brain tissue was cut into pieces after 24 h of reperfusion and homogenized in four volumes of cold lysis buffer (20 nmol/l Tris pH: 7.5, 137 mmol/l NaCl, 2 mmol/l EDTA, 1% Triton X-100, 10% glycerol, and protease inhibitor cocktail) on ice. The homogenates were centrifuged at 12,000ϫg for 30 min at 4 °C. The supernatant was stored at Ϫ70 °C until further use. Protein concentrations were measured using conventional Bradford's method. Fifty microgram proteins from each experimental group were applied to 10% SDS PAGE gels and electrophoresed for 1 h at 100 mA. Proteins were transferred onto nitrocellulose membrane (Hybond) in blotting buffer (156 mmol/l Tris, 1 mol/l glycine, PBS) for 90 min at 80 mA and blocked with 5% skim milk (Regilait) in PBS for 2 h at room temperature. The blotted membrane was incubated with the rabbit polyclonal Bax antibody (Santa Cruz, SC493, diluted 1/1000) for 2 h at room temperature. Western blots were incubated with horseradish peroxidaseconjugated anti-rabbit IgG (Jackson ImmunoResearch, diluted 1/15,000) for 1 h at room temperature and revealed using enhanced chemiluminescence Western blotting detection reagents (Pierce). To control for sample loading, stripped membranes were rehybridized with a ␣-tubulin antibody as internal control (data not shown). Films with specific bands were scanned and quantified using an imaging densitometer.

Immunohistochemistry

Frozen brain sections 25 m thick were immersed in 0.3% H 2 O 2 / methanol for 10 min, permeabilized in 0.1% Triton/PBS for 10 min and blocked with 3% goat serum/PBS for 2 h at room temperature followed by a single rinse in PBS. Sections were incubated then with the primary antibody overnight. The antiserum used was the rabbit polyclonal Bax antibody (Santa Cruz, SC493, diluted 1/300). After the primary incubation and three rinses in PBS, sections were then incubated in biotinylated horse anti-rabbit IgG (Jackson ImmunoResearch, diluted 1/15,000) for 2 h at room temperature. Bax expression was visualized by 3,3=-diaminobenzidine (DAB) staining using the VectaStain ABC kit (Biovalley). All sections were washed in PBS and then in distilled water, and mounted with Entellan.

Statistical analysis

Data were expressed as meanϮS.D. of n observations, where n represents the number of animals or samples. Comparison between multiple groups was evaluated by one-way analysis of variance (ANOVA) followed by a PLSD (protected least significant difference) Fisher's test. When measures were repeated (physiological variables) data were analyzed by a two-factorial ANOVA for repeated measures. Significance was assumed at a value of PϽ0.05.

RESULTS

A single dose of ALA500 or RLZ4 induced cerebral protection at 24 h and 30 days after MCAO

To determine whether acute administration of ALA or RLZ protects against ischemic stroke, mice were subjected to transient (1 h) MCAO and injected intraveniously with a single dose of ALA500 or RLZ4 at different times after reperfusion. This paradigm corresponds to Fig. 1A. Administration of RLZ or fatty acids (ALA or PAL) did not affect MABP, arterial pO 2 or pCO 2 and arterial blood pH (Table 1). The intraluminal suture method resulted in an infarct in the right MCA perfused region. Representative photographs of brain slices stained with TTC at 24 h after MCAO are shown in Fig. 2A. As indicated by the white area, cerebral infarcts in ALA500-and RLZ4-treated mice were reduced. The infarct spread into the dorsomedial cortex in the region of the anterior cerebral artery/MCA and the caudate-putamen was particularly inhibited by both treatments. Quantitative assessment of infarct volume 24 h after MCAO revealed the therapeutic window of ALA500 or RLZ4 treatment (Fig. 2B). Administration of ALA500 or RLZ4 up to 6 h after reperfusion drastically decreased cerebral infarction. Maximal ALA-induced protection was achieved with a dose of 500 nmol/kg injected 2 h postischemia with a 86% reduction as compared with vehiclegroup (PϽ0.001), an effect that was lost when treatment was initiated 12 h after reperfusion. Administration of RLZ up to 3 h after reperfusion at a dose of 4 mg/kg reduced the stroke volume by approximately 75% (PϽ0.001) as compared with control mice, with no significant difference between reperfusion times. In contrast, PALM500, a saturated fatty acid, had no beneficial effect on the infarct size when injected 2 h after reperfusion, a time corresponding to the best cerebral protection obtained with ALA, a polyunsaturated fatty acid (Fig. 2B).

Based on these results, we chose as reference treatment the dose of RLZ4 or ALA500, injected 1 or 2 h after reperfusion, respectively. In this context, neurological deficits 24 h after ischemia were also significantly improved by ALA500 or RLZ4 (Table 1).

The protection induced by ALA500-or RLZ4-injection was also associated with a reduction in cytopathological features of cell injury, including cell shrinkage (cresyl violet staining, not shown), DNA-fragmentation (Fig. 3A), and Bax expression (Fig. 3B-D). TUNEL method revealed massive DNA fragmentation by 24 h in the ipsilateral cortex of vehicle-treated mice, which underwent 1 h-MCAO as compared with sham-operated animals. Fig. 3A shows the number of TUNEL-positive cells, characterized by shrunken cytoplasm and dark nuclei in the ipsilateral cortex of different groups. ALA500-or RLZ4-injection signifi- cantly reduced the percentage of positive cells with a better protection with RLZ (13.9% in RLZ4-and 36.5% in ALA-groups vs 49% in vehicle-group, PϽ0.001). Expression of the protein Bax, known to have a role in promoting apoptosis pathways, is regarded as a diagnostic marker of neurons undergoing cell death [START_REF] Tsujimoto | Bcl-2 family: life-or-death switch[END_REF]. The induction of the Bax protein in the cortical penumbra of vehicle-and treated-mice has been assessed using Western blotting (Fig. 3B and3C) and immunochemistry (Fig. 3D) at 3 h after reperfusion following MCAO. The Western blotting analysis, using the polyclonal anti-Bax antibody revealed a single major band at 21 kDa, the size predicted for the Bax-␣ protein (Fig. 3B). Fig. 3C summarizes the quantification of Bax expression in ipsilateral cortical extracts from the complete set of mice. Bax induction remained undetectable in protein extracts from cortex of sham-operated groups. One hour MCAO induced strong Bax expression in cortical extracts of vehicle-treated mice 3 h after reperfusion. In contrast, injection of a dose of ALA500 or RLZ4 after MCAO led to a significant decrease in Bax expression in the cortical extracts of treated mice. While Bax expression increased 5.9-fold in vehicle-treated mice, injection of RLZ4 or ALA500 significantly prevented this increase. Immunostaining on brain sections (Fig. 3D) confirmed the findings obtained with Western blot analysis. Weak Bax immunoreactivity decorated the cytoplasm and cell processes of neurons in the cortex of sham-operated mice, but increased Bax immunoreactivity occurred in the cytoplasm of penumbra neurons of the ipsilateral cortex at 3 h following reperfusion. Post-treatment with ALA500 or RLZ4 strongly reduced Bax immunoreactivity in penumbra cortical neurons of ischemic brains (Fig. 3D). Importantly, the reduction of the infarct volume induced by ALA500-or RLZ4-injection, previously observed by TTC staining, was still occurring at 30 days after MCAO with Nissl staining. The lesion volume, revealed by the lack of Nissl staining, was drastically reduced in ALA500-or RLZ4-treated mice, as compared with vehicle-treated animals (data not shown). The infarct size reduction was associated to better cell preservation after ALA500 or RLZ4 administration that lasted for at least one month as determined by cell counting in well-known structures injured by MCAO (i.e. dorsomedial cortex and caudate-putamen) at 30 days postischemia (Fig. 4A). As expected, one hour MCAO produced severe cell loss. Compared with the sham-operated group, only 57.9 and 43% penumbra neurons of cerebral cortex and striatum were preserved, respectively. This cellular loss was almost completely inhibited by ALA500-or RLZ4-injection in both structures. Cortical cell survival was around 95% in ALA500-and RLZ4-treated groups (PϽ0.001). Administration of ALA500 or RLZ4 limited striatal cell destruction to 5.9 and 7.9% of neurons, respectively (PϽ0.001). In contrast, injection of the saturated fatty acid PALM500 failed to protect the brain and no rescue of the cell number was observed in both structures (Fig. 4A).

Three-day and three-week therapies reduced mortality and increased cell survival following ischemia at 30 days after MCAO

As indicated in the paradigms of Fig. 1B and1C, the efficiency of a single dose of RLZ4 or ALA500 injected 1 (RLZ4) or 2 h (ALA500) after reperfusion was compared with a dose of RLZ2 or ALA250 given 1-2, 48 and 72 h after reperfusion or 1-2 h after reperfusion following MCAO and once a week during the next two weeks of reperfusion. The protection was assessed by neuronal counting and mortality rate one month after MCAO. In terms of cell survival, the weekly injection of ALA250 or RLZ2 during three weeks also induced a potent protection (Fig. 4B and4C). It limited the striatal cell destruction to 4.6 and 5.7% of neurons (Fig. 4B), and induced a cortical cell survival of 96.5 and 95.1%, respectively (PϽ0.001) (Fig. 4C). In terms of mortality (Fig. 5), the best protection against ischemia was obtained with a weekly treatment of ALA250 during three weeks (70% survival rate) followed by RLZ-treatment at a dose of 2 mg/kg with the same periodicity (60% survival rate). Compared with the reference treatment (a single dose of ALA500 or RLZ4), the three-day therapy did not significantly improve the survival rate of ischemic mice (Fig. 5).

Combined doses of RLZ2 and ALA250 induced the best brain resistance to ischemia

As indicated in paradigm D, we next tested the hypothesis that combined administration of a dose of RLZ2ϩALA250 given 2 h after MCAO could improve the protection against the deleterious effects of ischemia. Representative photographs of brain slices stained with TTC at 24 h after MCAO with the corresponding quantitative assessment of infarct volume are shown in Fig. 6. As indicated by the white area, the cerebral infarct in mice treated with a combined injection of RLZ2ϩALA250 was significantly reduced. Cell counting in the peri-infarct zone 30 days after MCAO shows that a combined injection of RLZ2 and ALA250 induced a penumbra cortical and striatal cell survival of Fig. 5. Comparative effects between three-day or three-week therapies and administration of a single or combined dose of ALA and RLZ on survival rate of mice one month after 1-hour reversible MCAO. Single dose: a dose of RLZ4 or ALA500 injected 1 or 2 h after reperfusion. Days 1-3: a dose of RLZ2 or ALA250 given 1-2, 48 and 72 h after reperfusion. Weeks 1-3: a dose of RLZ2 or ALA250 given 1-2 h after reperfusion and once each week during the next two weeks of reperfusion. Combined: a single dose of ALA250ϩRLZ4 injected once 2 h after reperfusion (nϭ15 per group). 98.7 and 98%, respectively (PϽ0.001) (Fig. 4B and4C). One month after MCAO, the mortality rate was reduced to 20% (Fig. 5) compared with 75% in the sham-operated group.

DISCUSSION

The purpose of the current study was to investigate the protective effects of ALA and RLZ in a clinically relevant model associating transient focal ischemia and posttreatment. After 24 h of MCA occlusion, histological changes indicating ischemic brain damage have been observed in caudate putamen, pallidum, basolateral cortex, hippocampus and parts of thalamus, which is consistent with previous studies of intraluminal MCA occlusion [START_REF] Hata | Evolution of brain infarction after transient focal cerebral ischemia in mice[END_REF][START_REF] Ding-Zhou | L-NAME reduces infarction, neurological deficit and bloodbrain barrier disruption following cerebral ischemia in mice[END_REF]. We have tested a doserange of both drugs (ALA500 or RLZ4) based on previously published data in other models of neurodegeneration [START_REF] Pratt | Neuroprotective actions of Riluzole in rodent models of global and focal cerebral ischemia[END_REF][START_REF] Lang-Lazdunski | Riluzole improves neurological function following severe spinal cord ischemia[END_REF][START_REF] Lauritzen | Polyunsaturated fatty acids are potent neuroprotectors[END_REF]. In the first step of the study, we showed that an i.v. injection with a single dose of ALA500 or RLZ4 initiated after the onset of MCAO significantly reduced the infarct size and improved the neurological score, both examined at 24h post-ischemia. ALA500 and RLZ4 were both efficient up to 6 h following reperfusion, i.e. within a therapeutic window, which is relevant clinically. Ischemic injury is associated with progressive DNA fragmentation in both striatal and cortical neurons within the infarct area including the core and penumbra. At 24 h post-reperfusion, TUNEL labeling appeared intense and abundant in these damaged structures. Ischemia-induced DNA fragmentation contains not only internucleosomal fragmentation but also random breakdown products often associated with necrosis. The TUNEL procedure labels all types of DNA fragments. It cannot discriminate accurately between apoptotic and necrotic cells [START_REF] Charriaut-Marlangue | Apoptosis and necrosis after reversible focal ischemia: an in situ DNA fragmentation analysis[END_REF][START_REF] Unal-Cevik | Apoptotic and necrotic death mechanisms are concomitantly activated in the same cell after cerebral ischemia[END_REF], and therefore is mostly considered as a marker of damaged cells. In that context, we observed that a single dose of ALA500 or RLZ4 injected 1-2 h following MCAO reduced dramatically the number of TUNEL-positive cells in cortex after 24 h reperfusion. Programmed cell death and apoptosis are regulated by members of the Bcl-2 family, including Bcl-2, which promotes cell survival and Bax, which promotes cell death. Several studies in vivo have demonstrated modifications in the expression levels of members of the Bcl-2 family. Bax upregulation has been found following ischemia [START_REF] Krajewski | Upregulation of bax protein levels in neurons following cerebral ischemia[END_REF][START_REF] Ferrer | Caspasedependent and caspase-independent signalling of apoptosis in the penumbra following middle cerebral artery occlusion in the adult rat[END_REF] in association with Bcl-2 reduction in the infarct core [START_REF] Matsushita | Alterations of Bcl-2 family proteins precede cytoskeletal proteolysis in the penumbra, but not in infarct centres following focal cerebral ischemia in mice[END_REF]. Bax is a potent proapoptotic molecule, which triggers activation of terminal caspases and subsequently promotes neuronal cell death and has been described to increased in the penumbra 3 h following the reperfusion [START_REF] Ferrer | Caspasedependent and caspase-independent signalling of apoptosis in the penumbra following middle cerebral artery occlusion in the adult rat[END_REF]. Using Western blotting and immunohistochemistry, this work confirms the upregulation of the Bax protein in the penumbra cortical neurons of ischemic vehicle-treated mice at 3 h following reperfusion and demonstrated that a single dose of ALA500 or RLZ4 strongly reduced Bax expression. Taken together with the fact that penumbra is the privileged site for apoptosis, both reduction of TUNEL labeling and Bax reactivity suggest that the neuroprotection induced by ALA500 and RLZ4 involves a decrease of apoptotic pathways occurring in this area. Nevertheless, we cannot exclude a direct effect of these drugs on apoptosis.

In the second step, to design the best type of treatment we compared neuronal protection, but also long term survival induced by a single dose of ALA500 or RLZ4 to three day or three week therapies with ALA250 or RLZ2 or a combination therapy of ALA250ϩRLZ2 injected 2 h after reperfusion. The results show that the greatest differences between paradigms appeared in the survival rate of the mice. The model of focal ischemia used in this study is very severe, since more than 75% of the ischemic mice treated with vehicle died within the month of reperfusion. Interestingly, three-week therapies with ALA250 or RLZ2 improved the survival rate (at one month) as compared with a single dose of ALA500 or RLZ4. The best protection in terms of survival rate (83.6%) was obtained with the cocktail therapy of ALA250ϩRLZ2 injected 2 h after reperfusion, suggesting a combined effect of both drugs. When we compare the different paradigms and analyze the results obtained in terms of brain protection and mice survival, it seems that salvaging cortical and caudate neurons did not necessarily improve survival. This point can be explained by the development of the ischemic edema (vasogenic and cytotoxic), that we have not measured in this study. As mentioned in the Experimental Procedures, neurological deficit was assessed at 24 h post-ischemia. Mice with a neurological deficit above grade 4 (decreased level of consciousness), were excluded from the study to exclude ischemic edema, one of the causes of clinical deterioration within the first 24 h. The clinical manifestations of ischemic brain swelling generally appear between 2 and 5 days after stroke and this propensity of ischemic brain tissue to develop edema and swelling remains the major cause of death within a week. Then, cresyl violet staining, used to count viable neurons and analyze cell survival in ischemic mice, was performed at three weeks on the animals that had survived, probably overcoming the deleterious effects of edema. In this context, we cannot exclude that the single combined therapy (ALA250ϩRLZ2) as well as the three daily injections (ALA250 or RLZ2) reduced the development of edema and consequently reduced brain and animal death.

Taken together, this study demonstrates that alphalinolenic acid and riluzole induce a beneficial effect on the infarct size, neurological score, neuronal survival and mortality rate of ischemic mice in a clinically relevant model of stroke. The neuroprotective properties of RLZ and PUFAs have already been described in other models of neurodegeneration [START_REF] Malgouris | Riluzole, a novel antiglutamate prevents memory loss and hippocampal neuronal damage in ischemic gerbils[END_REF][START_REF] Pratt | Neuroprotective actions of Riluzole in rodent models of global and focal cerebral ischemia[END_REF][START_REF] Barneoud | Neuroprotective effects of riluzole on a model of Parkinson's disease in the rat[END_REF][START_REF] Voskuyl | Anticonvulsant effect of polyunsaturated fatty acids in rats, using the cortical stimulation model[END_REF][START_REF] Ettaiche | Riluzole improves functional recovery following ischemia in the rat retina[END_REF][START_REF] Lang-Lazdunski | Riluzole improves neurological function following severe spinal cord ischemia[END_REF][START_REF] Leaf | The antiarrhythmic and anticonvulsant effects of dietary N-3 fatty acids[END_REF][START_REF] Lauritzen | Polyunsaturated fatty acids are potent neuroprotectors[END_REF]. The main interest of this work has been to compare the neuroprotective effects of RLZ or/and ALA, both at short (24 h) and long terms (one month). Most studies using the MCAO model describe the beneficial effect of a drug by the reduction of the infarction and the improvement of the neurological score at 24 h. As for the rapid ischemic preconditioning protection, which disappears when the reperfusion is extended to 7 days [START_REF] Perez-Pinzon | Rapid ischemic preconditioning protects rats from cerebral anoxia/ischemia[END_REF], it is known that neuronal protection can only be temporary with some therapies [START_REF] Dietrich | Intraischemic but not postischemic brain hypothermia protects chronically following global forebrain ischemia in rats[END_REF]. In our study, we analyzed not only the effectiveness of RLZ or/ and ALA on the infarct size and the neurological outcome at 24 h post-ischemia, but also the cell survival in the damaged cerebral structures and the death rate of mice one month after ischemia. Looking at cell survival shortly after stroke, it appears that the various paradigms (single dose of ALA or RLZ, 3 days or three weeks therapy, or combined treatment) equally induce a remarkable protection of the neurons. On the other hand, for animal survival, only the 3 week therapy or combined treatment provides a real improvement. This observation is interesting for later clinical applications, since it shows that one protocol effective in rescuing cellular populations does not necessarily improve survival.

It is clear that multiple and complex pathways involving excitotoxicity, oxidative and nitrosative stress, periinfarct depolarization, inflammation and apoptosis-like mechanisms cause cell death in stroke [START_REF] Dirnagl | Pathobiology of ischaemic stroke: an integrated view[END_REF][START_REF] White | Brain ischemia and reperfusion: molecular mechanisms of neuronal injury[END_REF]. In this context, it is reasonable to develop combination therapies targeting several of deleterious mechanisms, simultaneously. In this work, we show that a combined injection of RLZ and ALA not only induced effective neuronal survival but more importantly an improvement of the animals' survival. Several research groups have analyzed the different targets of RLZ and ALA. Both molecules are known to reduce hyperexcitability and excitatory neurotransmission [START_REF] Cheramy | Riluzole inhibits the release of glutamate in the caudate nucleus of the cat in vivo[END_REF][START_REF] Martin | The neuroprotective agent riluzole inhibits release of glutamate and aspartate from slices of hippocampal area[END_REF][START_REF] Doble | The pharmacology and mechanism of action of riluzole[END_REF][START_REF] Lauritzen | Polyunsaturated fatty acids are potent neuroprotectors[END_REF]. They both partially inhibit voltage-sensitive Na ϩ channels and voltage-sensitive Ca ϩϩ channels [START_REF] Benoit | Riluzole specifically blocks inactivated Na channels in myelinated nerve fibre[END_REF][START_REF] Hebert | Block of the rat brain IIA sodium channel alpha subunit by the neuroprotective drug riluzole[END_REF][START_REF] Huang | Effects of the neuroprotective agent riluzole on the high voltage-activated calcium channels of rat dorsal root ganglion neurons[END_REF]. More importantly both activate the background TREK-1, TREK-2 and TRAAK potassium channels (Lesage andLazdunski, 1999, 2000;[START_REF] Duprat | The neuroprotective agent riluzole activates the two P-domain Kϩ channels TREK-1 and TRAAK[END_REF][START_REF] Patel | Properties and modulation of mammalian 2P domain K ϩ channels[END_REF]. On the other hand recent work has shown enhanced expression of the lipid-sensitive 2P-domain K ϩ channels (the TREK/ TRAAK channel family) in cortex and hippocampus after MCAO and suggested a correlation between TREK channels and acute cerebral ischemia [START_REF] Li | Enhanced expressions of arachidonic acid-sensitive tandem-pore domain potassium channels in rat experimental acute cerebral ischemia[END_REF]. This idea was reinforced by recent data demonstrating that TREK-1 channels were required for the protective properties of polyunsaturated fatty acids against global ischemia and kainate-induced seizures [START_REF] Heurteaux | TREK-1, a K(ϩ) channel involved in neuroprotection and general anesthesia[END_REF].

Our data on the neuroprotective effects of alpha-linolenic acid and riluzole in a model of focal ischemia are supportive of the idea that a combined therapy with these molecules may have important clinical implications for a successful therapy of stroke in the future. They also strengthen the general opinion that polyunsaturated fatty acids have beneficial effects in human health.

Fig. 1 .

 1 Fig. 1. Flowchart illustrating the different paradigms. Timing of MCAO/reperfusion, ALA and RLZ treatment at different doses, neurological score, mortality rate, TTC-staining, Bax expression, TUNEL staining and cell counting. (A) Single dose paradigm (RLZ4 or ALA500 or PAL500). (B) Three-day therapy paradigm (RLZ2 or ALA250). (C) Three-week therapy paradigm (RLZ2 or ALA250). (D) Combined therapy paradigm (ALA250ϩRLZ2).

Fig. 2 .

 2 Fig. 2. Effects of a single injection of ALA, RLZ or PALM on infarct size in brain sections after 1-hour reversible MCAO. (A) Representative samples of TTC-stained from mice killed 24 h after ischemia. One-hour MCAO/reperfusion animals treated with vehicle or PALM showed severe infarctions. The white areas represent the infarct regions in these sections. In contrast, ALA500 RLZ4 treatment had a significant reduction in infarction. (B) Therapeutic window of ALA500 and RLZ4 on infarct volume 24 h after MCAO. ALA, RLZ or PALM was i.v. injected (nϭ15 per experimental group). Data are expressed as meansϮS.E.M. * PϽ0.05; ** PϽ0.001 versus vehicle-treated ischemic mice.

Fig. 3 .

 3 Fig. 3. Decrease of TUNEL-positive cells and Bax expression following the acute injection of ALA500 or RLZ4 after MCAO. (A) Histograms representing percentage of TUNEL-positive cells/total in the ipsilateral penumbral cortex (per 0.5 mm 2 ) 24 h after reperfusion (nϭ10 per group, 10 slides per mouse). * PϽ0.05, ** PϽ0.001 versus vehicle (VEH)-treated ischemic mice. (B) Western blotting analysis of Bax protein induction in cortical extracts isolated from sham-operated (Sham), VEH-injected brains or brains treated with a dose of RLZ4, ALA500 or PAL500 (PALM). RLZ4, ALA500 or PALM500 was i.v. injected 1 (for RLZ4) or 2 h (for ALA500 or PALM500) after reperfusion. Brains were removed from mice 3 h after reperfusion. (C) Quantification of Bax expression obtained in Western blotting. The specific bands were quantified using an imaging system as described in Experimental Procedures. Values are expressed as a percentage of sham-operated mice. Data are expressed as meanϮS.E.M. Data are representative of three separate experiments (nϭ4 per experimental group). * PϽ0.05, ** PϽ0.001 versus sham-operated mice. (D) Decreased expression of Bax protein in cortical penumbra of RLZ4-or ALA500-treated ischemic mice as compared with VEH-or PALM-injected ischemic mice at 3 h after reperfusion following MCAO. Data are representative of three separate experiments (nϭ5 per experimental group). Scale barϭ100 m.

Fig. 4 .

 4 Fig. 4. Comparative long-term protective effects between single, three-day or three-week therapy and administration of a combined dose of ALA or RLZ on cell death one month after 1-hour reversible MCAO. Histograms represent the neuronal density per 0.5 mm 2 brain structure (cortex or caudate-putamen in (i) ipsi-and (c) contralateral sides) assessed in cresyl-violet-stained sections (cortex and caudate-putamen) one month after ischemia. (A) Single dose paradigm: RLZ4 or ALA500 was i.v. injected 1 or 2 h after reperfusion, respectively. (B, C) Three-day, three-week and combined therapies paradigms. Days 1-3: a dose of RLZ2 or ALA250 given 1-2, 48 and 72 h after reperfusion. Weeks 1-3: a dose of RLZ2 or ALA250 given 1-2 h after reperfusion and once each week during the next two weeks of reperfusion. Combined: a single dose of RLZ2ϩALA250 injected once 2 h after reperfusion. Data are expressed as meansϮS.E.M. (nϭ15 per experimental group, 10 slides per animal). ** PϽ0.001 versus vehicle-treated (VEH) ischemic mice.

Fig. 6 .

 6 Fig. 6. Effects of combined administration of ALA250 and RLZ2 on infarct size in brain sections after 1-hour MCAO. (A) Representative samples of TTC-stained from mice killed 24 h after ischemia. The white areas represent the infarct regions in these sections. (B) Histograms representing the corresponding infarction volumes 24 h after ischemia. A single dose of ALA250ϩRLZ2 was i.v. injected 2 h after reperfusion (nϭ15 per group). Data are expressed as meansϮS.E.M. (nϭ15 per experimental group). ** PϽ0.001 versus vehicle-treated (VEH) ischemic mice.

Table 1 .

 1 Physiological parameters 30 minutes after injection of drugs (nϭ5 per group) and neurological deficit score after 1-hour MCAO in drug-and vehicle-treated mice (24 hours following reperfusion)

		VEH	LIN	RLZ	PALM
	MABP (mm Hg)	90Ϯ2	9 2 Ϯ3	9 4 Ϯ6	9 2 Ϯ4
	paCO 2 (mm Hg)	39Ϯ6	4 4 Ϯ3	4 6 Ϯ2	4 1 Ϯ3
	paO 2 (mm Hg)	444Ϯ41	420Ϯ9	462Ϯ29	452Ϯ27
	pH (arterial)	7.30Ϯ0.02 7.35Ϯ0.01 7.31Ϯ0.02 7.33Ϯ0.03
	RT (°C)	37.4Ϯ0.3 37.2Ϯ0.4	37.2Ϯ0.4	37.4Ϯ0.3
	Neurological	2.48Ϯ0.11 1.79Ϯ0.14* 1.78Ϯ0.13* 2.59Ϯ0.15
	score
	* PϽ0.05 vs. vehicle-treated ischemic mice.

Acknowledgments-This work was supported by the Centre National de la Recherche Scientifique (CNRS) and the Paul Hamel

Foundation. We are grateful to the Fondation de la Recherche Médicale for a fellowship to C. Laigle, to Professor M. Plotkine for training with the focal ischemia model, and to Dr. Lochlan Rash for careful reading of the manuscript.


Luc Biasiori-Poulanges

Hazem El-Rabii
email: hazem.elrabii@cnrs.pprime.fr

Shock-induced cavitation in a water droplet

The objective of the present study is to develop a basic understanding of the interaction of shock waves with density inhomogeneities. We consider here the particular instance of a planar air shock impinging on a spherical water-droplet and discuss to what extent this interaction can lead to the inception of cavitation inside the droplet. The effort centers on the early phases of the interaction process during which the geometry and amplitude of the propagating wavefront is modified by refraction and the subsequent internal reflections at the droplet interface. The problem is analysed using both simple ray theory and a 2-D multiphase, compressible hydrodynamic code (ECOGEN). Within the context of ray theory, the occurrence of focusing is examined in details and parametric equations are derived for the transmitted wavefront and its multiple internal reflections. It is found that wave patterns predicted by ray calculations compare extremely well with the more accurate numerical solutions from simulations. In particular, it is shown that the internal wavefront assumes a complex time-dependent shape whose dominant feature is the existence of cusp singularities. These singular points are shown to trace out surfaces that are the caustics of the associated system of rays. From the singularities of the energy flux density of the refracted wave, the parametric equations of the caustic surface associated to the k-th reflected wavefront are deduced. As a consequence of the focusing process, the simulations show the formation of negative-pressure regions in the internal flow field. These low-pressure zones are identified as possible spots where cavitation may occur, depending on the magnitude of the pressure reached. Finally, the numerical results provide quantitative information on the dependence of negative pressure peak upon incident-shock-wave strength.

I. INTRODUCTION

The fundamental mechanisms governing aerobreakup have been addressed in numerous studies where the fragmentation of a single spherical drop suddenly exposed to a uniform high-speed gas flow was considered [START_REF] Guildenbecher | Secondary atomization[END_REF] . The relative velocity of the drop with respect to the ambient flow field has often been realized by its injection into the uniform flow field behind a shock wave. Curiously enough, the question whether the shock wave itself may have any effect on the deformation and breakup process has not received much attention. The underlying reasons likely are twofold: first, the time it takes for the shock wave to transit the drop is too short to cause any significant drop response during the interaction [START_REF] Aalburg | Deformation and Drag Properties of Round Drops Subjected to Shock-Wave Disturbances[END_REF] . Secondly, the large difference in the shock impedance between the ambient gas and the liquid results in a poor energy transfer into the liquid (e.g., transmission coefficient from air to water ≈ 0.1%). While it is tempting to conclude from the above that the shock wave has no direct effect on the droplet evolution, a closer consideration of the matter shows that the answer is not that straightforward. Indeed, because of the large shock impedance contrast between air and water, the interface bounding the liquid medium acts as a perfect mirror trapping the transmitted wave energy within the droplet. As a result, the confined shock wave experiences nearly-total reflections and focusing that amplifies its local interaction with the liquid on short time scales. The important point here is that the reflected wave is a focused expansion wave that can, under some conditions, expose regions of the liquid to a pulling force. This suggests the possibility for the liquid to cavitate. Water, for instance, cannot withstand significant tension and starts to cavitate whenever pressure falls below some critical value. Given that the presence of vapour cavities inside liquid droplets alters the interfacial dynamics [START_REF] Liang | Interaction of a planar shock wave and a water droplet embedded with a vapour cavity[END_REF][START_REF] Biasiori-Poulanges | Multimodal imaging for intra-droplet gas-cavity observation during droplet fragmentation[END_REF] , changes in the fragmentation process are to be expected, especially if high-speed jets develop during cavity collapsing [START_REF] Liang | Interaction of a planar shock wave and a water droplet embedded with a vapour cavity[END_REF] .

In this paper, the question we are concerned with is under what conditions low-enough negative pressure to cause cavitation can be reached inside a shock impacted water droplet. We use numerical simulations to identify these conditions by considering that cavitation starts whenever pressure falls below some critical value. The simulations' results and, in particular, the complex wavefront patterns generated inside the droplet are interpreted qualitatively using the classical ray-tracing approach to geometrical acoustics.

II. CONFINED WAVEFRONT PROPAGATION

We consider a planar shock wave propagating through air and impinging on a spherical water droplet. The shock-droplet interaction results in a shock that is transmitted through the interior of the water droplet while a portion of the incident shock diffracts around the edge of the droplet.

Here, we focus on the transmitted shock and analyse the wavefront evolution following a dynamic ray-tracing method [START_REF] Cerveny | Seismic ray theory[END_REF] . This approach consists in studying shock propagation on the basis of the concept of rays considered as orthogonal trajectories along which wavefronts travel. Both media are assumed homogeneous so that rays are straight lines along which wavefronts propagate at constant speed. In accordance to the ray formalism, the incident shock wave propagates along a family of parallel rays incident from the right onto the droplet. Figure 1 illustrates the geometry of the problem. The radius of the spherical droplet is denoted by a and its center is at the origin of the coordinate system. We conveniently choose the x-axis in the direction of the incoming parallel bundle of rays and the time origin as the instant at which the shock wave reaches x = -a. Consider now an arbitrary ray AB striking the droplet surface at point B with an incident angle α. The major part of the ray amplitude is reflected at B, while the remaining part of the intensity is transmitted into the droplet. The refracted ray makes an angle θ with the interface normal at B. The incident and refracted angles are related by the fundamental law of refraction 6 : sin θ = n sin α.

(1)

Here, n is the ratio of wave velocity in water to that of air. The transmitted ray is then internally reflected at each interaction with the droplet surface at angle to normal of θ . The typical ray path of the refracted wavefront consists therefore of many successive segments separated by reflection points, P k , at the droplet boundary. Following usage in geometrical optics, we define a k-ray family to be rays within the droplet (for all α) that have undergone k -1 internal reflections [START_REF] Adler | High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder[END_REF] . According to this definition, the transmitted rays that have not suffered internal reflection belong to the 1-ray family. They become the 2-rays after their first reflection, and so on. The wavefront travelling along a k-ray family will be denoted by F k .

We shall now derive the parametric equations for F k . To that end, let us consider an arbitrary point M belonging to a k-ray. The location of this point is expressed by its position vector as

r M = r P k-1 + e k P k-1 M , (2)
where e k are the unit vectors in the k-ray direction, and P k-1 M is the length of the segment P k-1 M.

Elementary geometrical considerations (Fig. 1) yield the coordinates of the k-th internal reflection point, P k ,

(x P k , y P k ) = (a cos γ k , a sin γ k ), (3)
where γ k = 2kθα -(k -1)π. P 0 corresponds to the entry point (B) of ray AB into the droplet.

Using (3) we can then write down at once that

e k = e x cos(γ k -θ ) + e y sin(γ k -θ ), k ≥ 1, (4)
where e x and e y are the unit vectors in xand y-direction, respectively.

To determine P k-1 M , we note that the time t required for point M on the k-ray to be reached by the wavefront

F k is t = a u a (1 -cos α) + 2(k -1) a u w cos θ + P k-1 M u w . (5)
The first term in (5) is the time for the front to travel along the ray AB from x = -a to B, whilst the second term represents the total travel time it takes for the front to go from B to P k-1 along the segment-rays in between. Consequently, Eq. ( 5) yields indicates the physical time in tenth of microsecond. Wavefronts F k=1,2 are plotted from parametric equations (7). The case k = 3 only shown for t = 1.8 µs is intended to be illustrative.

P k-1 M = u w t -n a (1 -cos α) -2(k -1)a cos θ . (6)
After substituting ( 6) into (2), and expressing the result in Cartesian components we find

x M = (u w t -n a (1 -cos α) -2(k -1)a cos θ ) cos(γ k -θ ) -a cos(γ k -2θ ), (7a)
y M = (u w t -n a (1 -cos α) -2(k -1)a cos θ ) sin(γ k -θ ) -a sin(γ k -2θ ). (7b)
Given that the wavefront shape is defined as the locus of points reached by a disturbance in a

given time along all possible ray paths, Eqs (7a)-(7b) represent the parametric equations of the wavefront F k , with α as parameter.

Since our interest is the determination of the refracted wavefront and its internal reflections, we need consider only the incident rays that meet the upstream droplet surface at an angle lower than the critical angle for total reflection, that is, |α| < α c = arcsin(1/n). It is to be noted that the range of α values is not restricted solely by α c . Indeed, for a k-ray family, two specific rays (R l and R u ) bound the region in which the k-rays lie. The ray R l corresponds to the ray reaching the point P k-1 at time t, whereas R u is the ray that hits, at the same instant, the inner droplet surface at P k .

The associated incident angles, α l and α u , are obtained by setting P k-1 M equals to 0 and 2a cos θ , respectively, into (6):

2 (k -1)a cos θ l = u w t -n a (1 -cos α l ), (8)
2 ka cos θ u = u w t -n a (1 -cos α u ), (9)
where θ l and θ u are the refraction angles corresponding to α l and α u , respectively. Equations ( 8) and ( 9) can be solved exactly, as they are quadratic in cos α. The values of α that should be considered for k = 1 are such that |α| is bounded by the lowest value between α c , α l , and α u . For k greater than one, the absolute value of α is between α l and α u . that the transmitted front appears as originating from an external point source located on the symmetry axis. It does not exhibit any singular point during early times (t 0.6 µs), i.e., the front shape is smooth. Closer examination of the front shape at t = 0.6 µs reveals that the wavefront folds itself where it is in contact with the droplet boundary. The fold moves along the boundary as F propagates and splits F into two sub-fronts, F 1 and F 2 . The segment of the front ahead of the fold (F 1 ) remains smooth all along its propagation and corresponds to the rays that have experienced only a single refraction. The front segment F 2 starts to develop simultaneously with the appearence of the fold, near the droplet surface (see Fig. 2(a), t = 0.6 µs). Contrary to F 1 , F 2 exhibits a singular point (cusp) as it is particularly apparent at t 0.9 µs and grows as the front travels through the droplet. On reaching the downstream droplet surface, F is completely A striking feature of the cusp motion is evidenced by superposing successive fronts F 2 , as shown in Fig. 3. Indeed, as the wavefront advances, we see that the cusp of F 2 traces out a curve, the so-called caustic, which is shown by a dashed line in Fig. 3. We notice that the caustic has a cusp singularity at C, where it is clearly seen that the front's cusp cancels.

III. CAUSTICS INSIDE THE LIQUID DROPLET

It is well known that a caustic corresponds to regions where several rays bunch together to form discontinuities at which the intensity diverges. This means that the front's cusp is a moving focus, and the caustic is the surface traced by it.

To determine the parametric equation of the caustic, we note that the loci of points of high ray concentration can be obtained from the singularities of the energy flux density of the transmitted wave. If σ denotes the flux density associated with the plane incident wave, the flux incident upon an element of area dS d on the droplet surface is dΦ = σ cos α dS d . Due to the axial symmetry about the x-axis, we have that dS d = 2πa 2 sin α dα. The fraction of dΦ that survives k -1 internal reflections is thus given by

dΦ M = T (α) R k-1 (α) dΦ = 2πa 2 σ T (α) R k-1 (α) sin α cos α dα, (10)
where T (α) and R(α) are the transmission and reflection coefficient. The flux density over the transmitted wavefront is equal to dΦ M divided by the element of area dS mapped out by the rays that have crossed dS d :

dΦ M dS = a 2 σ T (α) R k-1 (α) sin α cos α y M ( ẋ2 M + ẏ2 M ) 1/2 . ( 11
)
The superposed dot indicates derivative with respect to α. Inserting Eqs.(7a)-(7b) in (11), we find

that dΦ M dS = a 2 σ T (α) R k-1 (α) sin α cos α ( P k-1 M sin(γ k -θ ) + a sin γ k-1 ) ( γk -θ ) P k-1 M -a γk-1 cos θ . ( 12
)
Shock-induced cavitation in a water droplet

The parametric equations for the caustic surfaces are obtained from the condition that the denominator of expression (12) be zero. Thus,

( γk -θ ) P k-1 M -a γk-1 cos θ = 0, (13)
P k-1 M sin(γ k -θ ) + a sin γ k-1 = 0. ( 14
)
The condition (13) relates the angle value corresponding to the front cusp at time t, which after some algebra can be recast as

2n sin 2 α 2 + 2( f (α) + k -1) cos θ = u w t a , (15)
where

2 f (α) = 2n 2 (k -1) sin 2α -sin 2θ n 2 (2k -1) sin 2α -sin 2θ .
If we eliminate t from Eqs. (7a)-(7b) by means of the condition (15), we obtain for the caustic of

order k x C = a f (α) cos γ k + a( f (α) -1) cos(γ k -2θ ), (16a)
y C = a f (α) sin γ k + a( f (α) -1) sin(γ k -2θ ). (16b)
Since the flux density (12) becomes infinite at the caustic C it cannot be used to quantify the density of rays at the caustic. Following Burkhard and Shealy 8 , we therefore compute the density of rays tangent to the caustic, which gives a relative measure of the focusing strength over the caustic. This quantity is obtained by dividing an element of incident flux by the area of the caustic formed by the associated rays, dS C . For the caustic of singly-reflected rays, we find

dΦ dS C = 2aσ T (α) R(α) cos θ (cos θ -3n cos α) 2 3n 2 (6 + 5n 2 + 11n 2 cos 2α -18n cos α cos θ ) |y C | . ( 17
)
It is apparent from this expression that the concentration of tangent rays is the highest at the intersection of the caustic and the symmetry axis (y C = 0), that is, at the caustic's cusp. This is because the degree of focusing at the caustic's cusp is higher than over a small element of the caustic's surface area. As exemplified in Fig. 4 (for k = 2), the concentration increases when α decreases, becoming infinite for α = 0. The location of the cuspidal point of the caustic can readily be found by setting α = 0 in (16a), which gives for the horizontal coordinate Equation (18) shows that the sign of the abscissa x cusp is determined by the parity of k, and the position of caustic's cusp gradually approaches the point O with increasing k.

x cusp = (-1) k n (2k -1)n -1 a k=2 = n 3n -1 a. (18)
It is also interesting to consider the time, t f , at which focusing at this point occurs. As shown in Fig. 3(b), t f coincides with the instant when the two cusps of the once-reflected front merge. This time can be determined from the condition (15). As the left-hand side of the latter is symmetric in α with a maximum for α equals to zero, the sought time is obtained by evaluating (15) at α = 0, yielding

t f = 4k(k -1) n + 1 -2k (2k -1) n -1 a u w k=2 = 8n -3 3n -1 a u w . (19)
Beside the cusp trajectory described by Eqs. (16a)-(16b), there exists another region where the flux density ( 12) is singular. The equations of the loci of these singularities can be determined by using (14) to eliminate the time t from Eqs. (7a).-(7b). In doing so, we find

x A = - a sin θ sin(γ k -θ ) , y A = 0, (20)
which are parametric equations of a straight line segment along the x-axis. This high density region results from the focusing on the x-axis of incident rays that enter the droplet over a ring of constant α. It is straightforward to see that the highest degree of focusing in this case is also achieved at the point of coordinates (x cusp , 0) and time t f .

The determination of the pressure amplitude on caustic surfaces will be described in the next section.

IV. CAVITATION INSIDE A WATER DROPLET

As we have seen in the preceding section, ray theory provides a direct physical interpretation of the wave patterns observed within the droplet. Furthermore, it enables to determine regions of the pressure field where the wave is focused (i.e. caustics). The quantitative prediction of pressures at caustics is, however, beyond the scope of ray calculations, which indicate infinite pressure in caustic regions, see Eq. ( 12). To identify conditions inducing cavitation inside a droplet hitted by a shock wave, it is then necessary to complement the ray approach with numerical simulations to determine the pressure on caustic surfaces. Before proceeding, however, it is useful to say a few words on the cavitation threshold.

It is well-known that liquids rupture (or cavitate) when subjected to tensions in excess to some critical tensile that depends on the nature of the liquid and its purity. For pure liquids, cavitation arises from microscopic voids caused by random thermal motions of the molecules. [START_REF] Balibar | Metastable liquids[END_REF] The process of vapor bubble formation by this mechanism is referred in the literature to as homogeneous nucleation. In contrast, when liquids contain impurities, the maximum tensile they can withstand drastically decreases. This process, termed heterogeneous nucleation, results from the expansion of submicroscopic gas pockets trapped at the solid/liquid interface on the wall of the container or on particles present in the liquid. Water, in particular, has a wide range of measured tensile limits.

The maximum tensile that pure water can withstand is 134 MPa at 300 K, according to vapor nucleation theory. [START_REF] Fisher | The fracture of liquids[END_REF] Such a high tensile value has been achieved experimentally. [START_REF] Zheng | Liquids at large negative pressures: water at the homogeneous nucleation limit[END_REF] For not-purified water, the tensile limit becomes less specific and is found to be a few orders of magnitude lower, 0.1-1 MPa. [START_REF] Caupin | Cavitation in water: a review[END_REF] Given such a disparity in tensile limits for water, a pressure cavitation threshold has to be chosen, somewhat arbitrarily, within the range of data reported in the scientific literature. We have opted to consider two different values. As far as pure water is concerned, a natural choice is the above-mentioned theoretical limit of -134 MPa, which will be denoted by p c,1 . To address the case of not-purified water it is also necessary to consider a higher value as a threshold, p c,2 . We set this value to -2.3 MPa on the basis of the experimental results reported by Sembian et al. [START_REF] Sembian | Plane shock wave interaction with a cylindrical water column[END_REF] We simulate the interaction of a planar air shock wave with a spherical water drop using the open-source hydrodynamics code ECOGEN [START_REF] Schmidmayer | Ecogen: An open-source tool for multiphase, compressible, multiphysics flows[END_REF] . In this code, the dynamics of water and air are modelled using compressible multicomponent flows in which fluid components are assumed immiscible [START_REF] Saurel | Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures[END_REF] . The water obeys the stiffened gas equation of state with the parameters given in Ref. [START_REF] Dorschner | On the formation and recurrent shedding of ligaments in droplet aerobreakup[END_REF] , whereas air follows the ideal gas law. Viscous and capillary effects are accounted for ac-cording to Schmidmayer et al. [START_REF] Schmidmayer | A model and numerical method for compressible flows with capillary effects[END_REF] , while phase changes are not modelled. An interface-capturing scheme is used, combining the flow model with a shock-capturing finite-volume method. Additionally, three levels of refinement were used in order to resolve the flow discontinuities. We refer the reader to Refs. [START_REF] Schmidmayer | Ecogen: An open-source tool for multiphase, compressible, multiphysics flows[END_REF][START_REF] Dorschner | On the formation and recurrent shedding of ligaments in droplet aerobreakup[END_REF] , and references therein for more details. The problem at hand is treated with a two-dimensional, axisymmetric formulation. A schematic diagram of the computational domain is illustrated in Fig. 5, where the x-axis is the symmetry axis on which the center of the spherical droplet of radius a is located. A symmetric boundary condition is applied to the bottom side of the computational domain and non-reflective boundary conditions are used for the remaining boundaries to avoid contamination of calculations from the reflected outgoing waves. [START_REF] Thompson | Time dependent boundary conditions for hyperbolic systems[END_REF][START_REF] Thompson | Time-dependent boundary conditions for hyperbolic systems, ii[END_REF][START_REF] Meng | Numerical simulations of the early stages of high-speed droplet breakup[END_REF] The shock is initialized inside the domain and travels from left to right. For a given incident shock Mach number M s , the initial flow field is determined from the Rankine-Hugoniot jump relations for an ideal gas using a downstream density of 1.204 kg • m -3 , pressure of 1 atm, and water density of 10 3 kg • m -3 . The surface tension between air and water is 72 mN • m -1 . The water drop is assumed to be in mechanical equilibrium with the surrounding air.

The excess of pressure inside the droplet over the ambient pressure was computed by employing the Laplace-Young equation. Theoretical wavefronts are given by Eq. ( 7) for n = 2.12 and displayed in red dotted line when they overlap the schlieren visualization. Times in µs are given in the sub-captions and the arrow between parentheses indicates the direction of wave propagation.

the theoretical and numerical results. We point out that such an excellent agreement is obtained with no adjustable parameters.

In Fig. 7, we plot the caustic associated with the rays reflected once at the boundary of the droplet (red line in the figure), as expressed by Eqs. (16a)-(16b). As we have already mentioned above, this surface is the locus of points where the ray intensity is the highest. Since upon reflection at the droplet interface the compressive wave is transformed into an expansion wave, it means that this caustic corresponds to the region of lowest pressure. The color-filled circles represent the lowest pressure as obtained from the simulation: the darker the color, the lower the pressure (see the color bar to the right of the figure). Each circle corresponds to the position of the front's cusp at different time instants, which are indicated by the same color code as for pressure, from white (1.07 µs) to black (1.75 µs). We note, in accordance with ray calculations, that the caustic's cusp is the point of lowest pressure. A slight shift between the caustic and the trajectory described by the front's cusp is observed. The shift is the largest on the x-axis. Equation (18) gives for x cusp = 0.40a, which reasonably agrees with the simulation result of 0.51a. The time at which the caustic's cusp is reached is 1.78 µs, Eq. (19), which is very close to the value of 1.75 µs obtained from the simulation.

As already mentioned, our simulations do not take into account phase changes and interactions.

This implies that we ignore the effects relevant to the dynamics of bubble formation and their feedbacks on the droplet evolution. Because these effects are expected to significantly alter the droplet dynamics, our simulation results can only be considered as valid up to the instant the first inception of cavitation is observed. Our concern here nevertheless is in determining incident shock conditions leading to cavitation inside a spherical droplet. For such a purpose, it seems reasonable to consider cavitation event is occurred in regions where pressure has dropped below some threshold value.

In order to identify conditions prone to the advent of cavitation zones, we have performed simulations for incident-shock Mach numbers, M s , varying from 1.1 to 6.0. In Fig. 8a, we have plotted the lowest pressure reached inside the droplet, P min , downstream of the first-reflected wave against M s . The location of the minimum pressure corresponds approximately to that of the caustic's cusp. A few trends stand out from Fig. 8a. One is that P min is negative for all M s , with the exception at M s = 1.1 for which it is nearly zero. Additionally, P min is a decreasing function of M s , as should be expected. We note that P min decreases slowly from 0 at M s = 1.1 to -5 MPa at M s = 2.0. On increasing M s beyond this latter point, P min decreases at a higher pace. The most interesting aspect of this graph is the linear dependence of P min on M s , which is observed over the range 3.5 M s ≤ 6.0. At this stage, we cannot offer any explanation for this behavior. If we compare P min with the pressure threshold p c,1 , whose location is displayed in Fig. 8a as the horizontal red line, we see that the regimes in which a liquid gas phase transition is likely to develop correspond to M s above 5. Figure 8b is a zoom-in of Fig. 8a covering a much smaller M s -range times smaller than the one obtained for pure water. As a final word, it can be mentioned that, in their study of shock-water column interaction, Sembian et al. [START_REF] Sembian | Plane shock wave interaction with a cylindrical water column[END_REF] found that cavitation may arise for an incident shock wave Mach number greater than 2.4. This value is 50% higher than what we found. The difference with our value of 1.7 can obviously be attributed to the higher degree of the rarefaction wave focusing achieved in a spherical droplet. In Table I, we report the shock wave Mach numbers for which homogeneous and heterogeneous cavitation is likely to occur according to our simulations.

V. CONCLUDING REMARKS

In this paper, we have examined the initial phases of the interaction between a planar shock wave in air and a spherical water-droplet. The analysis was conducted using ray theory, which provides analytical results that were compared with and complemented by numerical simulations.

There are several remarks and conclusions that we consider of particular relevance concerning the results reported herein. First, we saw that the wavefront inside the droplet assumes a complex time-dependent shape whose dominant feature is the existence of cusp singularities. From ray calculations, it was clearly shown that these singular points result from the focusing process. Second, we derived parametric equations for the surface of confined wavefront. Comparisons with simulations showed that the front shape and its evolution are perfectly well described by these equations.

This result offers a simple description of the geometry and the process of focusing of the wavefront during the interaction. Third, it was proved that each wavefront cusp traces out a surface, which is the caustic of the associated system of rays. The energy flux density turns out to be singular over these caustic surfaces. Although physically unrealistic, this singular behaviour may be interpreted as revealing regions of highest ray density. Furthermore, we showed that caustics exhibit cusps where the concentration of rays forming the caustics is the strongest. Finally, as a consequence of the wave impedance, the compression wave inside the water droplet reflects at the interface as an expansion wave, thereby forming low-pressure regions in the internal flow field. On the basis of cavitation pressure thresholds from the literature, we obtained the incident-shock-strength conditions under which a planar shock wave can cause cavitation within a droplet.

FIG. 1 :

 1 FIG. 1: Ray diagram showing the refraction and the multiple internal reflection of initially parallel rays incident crossing the droplet boundary.

FIG. 2 :

 2 FIG.2: Propagation of the confined wavefront within the droplet (n = 2.12). Labels on contour

Figure 2

 2 Figure 2 shows the wavefront pattern generated inside the droplet, at different instants, as calculated from Eqs. (7a)-(7b) for k = 1 and 2. To avoid overloading the figure with crossed and/or juxtaposed fronts, the successive wavefront positions are displayed in different panels: (a) 50 ns ≤ t ≤ 1.3 µs, (b) t = 1.5 µs, (c) t = 1.7 µs, (d) 1.8 µs ≤ t ≤ 2.5 µs. We observe from Fig. 2(a)

FIG. 3 :

 3 FIG. 3: (Left) Illustration of the k-ray family (gray) and the corresponding F k=1,2 wavefronts (red). Four rays are highlighted in red solid and dashed lines for readability purposes. (Right) Superposition of successive fronts F 2 that evidences the cusp motion and the caustic traced out (black dashed line). The black-to-red color scale is representative of the physical time.

4 FIG. 4 :

 44 FIG.4: Variation of the density of rays tangent to the caustic (for k = 2 and n = 2.12) over the incident angle α. The concentration increases when α decreases, becoming infinite for α = 0.

FIG. 5 : 2 -

 52 FIG. 5: 2-D axisymmetric computational domain setup. States 1 and 2 refer to the pre-shock and post-shock conditions, respectively.

Figure 6

 6 Figure 6 displays the time evolution of the wavefront during a few round-trips. The upper half of each panel in the figure shows numerical Schlieren images 21 (magnitude of density gradient)from the simulations, whereas the lower half displays the corresponding fronts as predicted by ray theory, see Eqs. (7a)-(7b). The sequence runs from left to right and then top to bottom, and is not uniformly spaced in time (see caption for details). The time steps are selected so as to exhibit the principal features of the front evolution. It is clear from the comparison offered here that, with regard to both the shape and location of the wavefront, we have a remarkable agreement between

FIG. 7 :

 7 FIG. 7: The black solid line displays the droplet boundary (n = 2.12). The red solid line is the caustic traced out by F 2 wavefront cusp. Circle markers are extracted from numerical simulations and refer to the spatial location of the peak negative pressure over time, colored with the pressure magnitude.

FIG. 8 :

 8 FIG.8: Peak minimum pressure compared to (a) the cavitation pressure threshold p c,1 as predicted by the classical nucleation theory and (b) the pressure thresholds p c,2 experimentally reported by Sembian et al.[START_REF] Sembian | Plane shock wave interaction with a cylindrical water column[END_REF]

TABLE I :

 I Shock wave Mach numbers for which homogeneous and heterogeneous cavitation is likely to occur. The horizontal red line indicates the location of p c,2 . In this case, we see that cavitation is likely to occur whenever M s exceeds 1.7. Such a critical value is almost three

		Shock wave Mach number, M s
	Cavitation threshold	Sembian et al. 13	Present simulations
	p c,1	-	≈ 5.0
	p c,2	2.4	1.7
	between 1.0 and 2.5.

(1996).23 E. Johnsen and T. Colonius, "Numerical simulations of non-spherical bubble collapse," J. Fluid Mech. 629, 231-262 (2009).

ACKNOWLEDGMENTS

The authors gratefully acknowledge fruitful discussions with Tim Colonius from California Institute of Technology. This work was partially supported by the Région Nouvelle-Aquitaine as part of the SEIGLE project (grant number 2017-1R50115). The fisrt author acknowledges the support received by an ETH Zurich Postdoctoral Fellowship.