E Vollaire

M P Bellard

J Rols

J L Teissié

Muriel Coll

Golzio

V Josserand

M Kéramidas

J Lavaud

C Righini

J Vollaire

E Bellard

M P Rols

J L Coll

M Golzio

Electrochemotherapy guided by intraoperative fluorescence imaging for the treatment of inoperable peritoneal micro-metastases

Keywords:

Electrochemotherapy guided by intraoperative fluorescence imaging for the treatment of inoperable peritoneal micro-metastases

Introduction

In oncology, if tumor resection is feasible, surgery is often the first therapeutic option. For peritoneal carcinomatosis from colorectal cancer or recurrent ovarian cancer, a treatment of choice is cytoreductive surgery associated with immediate hyperthermic intraperitoneal chemotherapy. Patient survival essentially depends on the completeness of macroscopic tumor resection and on adjuvant chemotherapy for the elimination of residual tumor nodules smaller ⁎ ⁎ Correspondence to: J.-L. Coll, Institut Albert Bonniot INSERM-UJF U823, 38706 Grenoble, France. ⁎ ⁎⁎ ⁎ Correspondence to: M. Golzio, IPBS CNRS, UMR 5089, 205, Route de Narbonne, 31077 Toulouse Cedex, France.

Email addresses: jean-luc.coll@ujf-grenoble.fr (J.L. Coll); muriel.golzio@ipbs.fr (M. Golzio) 1 These authors contributed equally to this work. than 2.5 mm [START_REF] Harmon | Prognostic indicators in peritoneal carcinomatosis from gastrointestinal cancer[END_REF]. Unfortunately, the peritoneal extension, the multifocal spread of the metastases and the close contact with vital organs often lead to incomplete tumor resection. Optimizing the extensiveness of cytoreductive surgery without collateral damage is still a major challenge.

Advances in optical fluorescence imaging, and especially in intraoperative imaging devices associated with the generation of tumor-targeted contrast agents, can improve the normal vision of surgeons [START_REF] Gutowski | Intraoperative immunophotodetection for radical resection of cancers: evaluation in an experimental model[END_REF][START_REF] Keramidas | Intraoperative near-infrared image guided surgery of peritoneal carcinomatosis in a preclinical mouse model[END_REF][START_REF] Wenk | Near-infrared optical guided surgery of highly infiltrative fibrosarcomas in cats using an anti-avß3 integrin molecular probe[END_REF][START_REF] Atallah | Role of near-infrared fluorescence imaging in head and neck cancer surgery: from animal models to humans[END_REF][START_REF] Bu | Fluorescent imaging of cancerous tissues for targeted surgery[END_REF]. In addition to morphological changes and/or discoloration of the tissue that basically guide the surgeon, molecular imaging has the potential to provide additional biological information about the cancerous tissue at molecular and cellular levels [START_REF] Hussain | Molecular imaging for cancer diagnosis and surgery[END_REF]. However, despite these improvements in tumor visualization, some cancerous nodules are too dangerous to remove because of their location on sensitive organs or tissues (e.g. arteries, nerves), and cannot be excised without a major risk of causing irreparable damages. Therefore, loco-regional treatments such as cryotherapy, radiofrequency or thermal ablation

U N C O R R E C T E D P R O O F
have been developed in combination with surgery [START_REF] Hansen | Ablative technologies for hepatocellular, cholangiocarcinoma, and metastatic colorectal cancer of the liver[END_REF]. Electrochemotherapy (ECT) has emerged as a new therapy with a high objective response rate ($_amp_$gt; 80%) for treatment of cutaneous and subcutaneous tumors and new ECT protocols are being developed to treat deeper tumors [START_REF] Miklavčič | Electrochemotherapy: technological advancements for efficient electroporation-based treatment of internal tumors[END_REF][START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF]. ECT bypasses tumor removal and instead consists in the administration (intravenous or intra-tumor injections) of very low doses of non-permeant anticancer drugs, such as bleomycin or cisplatin, followed by short electric pulses applied locally to the tumor. ECT induces an antitumor effect whose main mechanism is the enhanced delivery of these low permeant cytotoxic molecules into tumor cells [START_REF] Mir | Electrochemotherapy potentiation of antitumour effect of bleomycin by local electric pulses[END_REF][START_REF] Sersa | Electrochemotherapy in treatment of tumours[END_REF]. ECT was pioneered for the treatment of equine tumors by the authors [START_REF] Tamzali | Successful treatment of equine sarcoids with cisplatin electrochemotherapy: a retrospective study of 48 cases[END_REF] and the first clinical trials in human were performed in France by the team of L.M. Mir [START_REF] Mir | Electrochemotherapy, a new antitumor treatment: first clinical trial[END_REF][START_REF] Belehradek | Electrochemotherapy, a new antitumor treatment. First clinical phase I-II trial[END_REF]. ECT was demonstrated to be a safe procedure since electric pulses used in ECT did not significantly change the expression profile of genes involved in the development of cancer in malignant melanoma cells [START_REF] Mlakar | Electric pulses used in electrochemotherapy and electrogene therapy do not significantly change the expression profile of genes involved in the development of cancer in malignant melanoma cells[END_REF], did not impact drug resistance and did not change the metastatic potential of human melanoma cells [START_REF] Todorovic | Assessment of the tumourigenic and metastatic properties of SK-MEL28 melanoma cells surviving electrochemotherapy with bleomycin[END_REF]. In clinical trials performed in Europe and the USA, ECT has been shown to be well tolerated and effective in the control of primary and metastatic solid tumors of diverse histotypes [START_REF] Miklavčič | Electrochemotherapy: technological advancements for efficient electroporation-based treatment of internal tumors[END_REF][START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF][START_REF] Tamzali | Successful treatment of equine sarcoids with cisplatin electrochemotherapy: a retrospective study of 48 cases[END_REF][START_REF] Cabula | Electrochemotherapy in the treatment of cutaneous metastases from breast cancer: a multicenter cohort analysis[END_REF]. ECT was described as a safe procedure and is now a routine practice in oncological clinics (National Institute for Health and Clinical Excellence (NICE) guidance (https://www.nice.org.uk/Guidance/IPG446; https://www.nice.org.uk/ Guidance/IPG478). The technological development of new electrodes has allowed for the treatment of deep tumors such as liver and bone metastases, brain tumors, and colorectal and esophageal tumors [START_REF] Miklavčič | Electrochemotherapy: technological advancements for efficient electroporation-based treatment of internal tumors[END_REF]. ECT was shown to be highly effective, with complete response rates between 60 and 70% and objective response rate around 80% [START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF][START_REF] Cadossi | Locally enhanced chemotherapy by electroporation: clinical experiences and perspective of use of electrochemotherapy[END_REF]. The overall effectiveness of ECT is also dependent on the involvement of the immune system. Indeed, ECT-mediated tumor regression was dramatically decreased in animals deficient in functional T lymphocytes, such as immunocompromised nude mice [START_REF] Sersa | Electrochemotherapy with CDDP on LPB sarcoma: comparison of the anti-tumor effectiveness in immunocompetent and immunodeficient mice[END_REF]. Nevertheless, ECT being a local treatment, one of the main challenge is to target the electric field onto every tumor nodule.

In the present work, we propose to evaluate, for the first time, an innovative approach of ECT guided by intraoperative fluorescence (IF) imaging for the treatment of inoperable peritoneal micro-metas tases in a preclinical model. To achieve this aim, we used a micrometastases model in which very aggressive metastases appear secondarily to primary tumor resection. No resection of metastases was performed. Instead, nodules detected by IF imaging were treated by ECT, EP alone or bleomycin alone. This experimental model offers the advantage of being finely controlled, since both primary tumor growth and metastases invasion are monitored over time with noninvasive multimodal imaging (bioluminescence and 3D fluorescence/ microCT).

Material and methods

Cell line and culture conditions

TSA-pGL3 is a cell line derived from the original TS/Apc mouse adenocarcinoma cell line and stably transfected with the pGL3-luciferase reporter gene (Promega, Charbonnieres, France). Cells were cultured at 37 °C in a humidified 95% air/5% carbon dioxide atmosphere in RPMI 1640 medium supplemented with 1% glutamine, 10% fetal bovine serum, 50 units/mL penicillin, 50 μg/mL streptomycin and 700 μg/mL Geneticin (G418 sulphate; Gibco, Paisley, UK). These cells express low levels of integrin αvβ3 [START_REF] Koenig | Fluorescence diffuse optical tomography for free-space and multifluorophore studies[END_REF].

Animal and tumor models

All animal experiments were conducted in agreement with the Principles of Laboratory Animal Care (National Institutes of Health publication no. 86-23, revised 1985) and approved by the regional ethics committee. The animal model setup is summarized in Fig. 1.

Female 8 weeks old balb/c mice (Janvier, Le Genest-Isle, France) were anesthetized (isoflurane/oxygen 3.5% for induction and 1.5% thereafter) and shaved, then a 2 cm incision was made in the left flank so that the kidney could be slightly pushed out of the cavity. By using a Hamilton syringe, 15 × 10 3 TSA-pGL3 cells, suspended in 20 μL phosphate-buffered saline, were injected under the renal capsule. Seven days later (day 8), the tumor-bearing kidney was surgically removed. After each surgery, an antibiotherapy (sulfamethoxazole 1 g/L, tetracycline 1 g/L, trimethoprim 200 mg/L) was admin istered in drinking water for 4 days. Six days after the removal of the tumor-bearing kidney (day 14), peritoneal carcinomatosis was monitored by in vivo bioluminescence imaging.

Bioluminescence imaging

Primary tumor growth and metastases development were monitored by in vivo bioluminescence imaging 6 days after tumor cell injection (D7), 6 days after tumor kidney removal (D14) and then twice a week until death. Five minutes before imaging, mice received an intraperitoneal injection of 150 μg/g of D-luciferin (Promega, France) and then were anesthetized (isoflurane 4% for induction and 1.5% thereafter) and placed in the optical imaging system (Aequoria, Hamamatsu Photonics, France). This allowed localization and quantitation of luciferase-positive tumor cells before and after treatment. Semiquantitative data were obtained from the bioluminescence images by drawing regions of interest on the abdomen with the Wasabi software (Hamamatsu Photonics, France). Results were expressed as a number of relative light units per pixel per second (RLU/pix/s) and the evolution of metastases after treatment was expressed as a percentage relative to the measure at day 14.

Fluorescence tomography/microCT imaging

For tumor and metastases imaging via integrin αvβ3 targeting [START_REF] Keramidas | Intraoperative near-infrared image guided surgery of peritoneal carcinomatosis in a preclinical mouse model[END_REF], Angiostamp™700 (Fluoptics, France) was injected intravenously (10 nmol, 200 μL) at day 7 (prior to kidney resection), and at day 14 (the day before treatment). Four hours later, mice were injected intravenously with 100 μL of Visipac 320 mg/mL iodine (GE Healthcare, USA) and were placed in a mobile animal holder compatible with the previously described fDOT (CEA-LETI, France) for 3D fluorescence imaging [START_REF] Koenig | Fluorescence diffuse optical tomography for free-space and multifluorophore studies[END_REF] and the microCT scanner (ScancoMedical, Switzerland) for whole body anatomical imaging. MicroCT acquisitions were made at 45 keV with a 177 μA intensity, a 300 ms integration time and a 80 μm isotropic voxel size. Fluorescence scanning was performed on the abdominal region only. MicroCT and 3D fluorescence data were reconstructed separately by the respective software systems and 3D images were superimposed for bimodal visualization with the open source image processing program Image J (NIH).

Intraoperative fluorescence imaging

On the day of the surgery (day 15), tracking of metastases was performed with the previously described handheld NIR 2D-Fluorescence Reflectance Imaging device (Fluobeam®700, Fluoptics, France) [START_REF] Keramidas | Intraoperative near-infrared image guided surgery of peritoneal carcinomatosis in a preclinical mouse model[END_REF][START_REF] Wenk | Near-infrared optical guided surgery of highly infiltrative fibrosarcomas in cats using an anti-avß3 integrin molecular probe[END_REF]. This system offers real time fluorescence imaging in a bright and open field environment for surgery. The surgeon can therefore operate as usual but with the assistance of a screen that dis plays the location of fluorescent tumors. In order to evaluate the Flu-obeam®700 sensitivity, a standard curve was prepared from serial Angiostamp™700 dilutions. A signal-to-noise ratio of 1.5 or greater was considered the positive signal threshold. Under these conditions, a positive signal could be detected on a 10 μL aliquot containing 0.5 nmol/L of probe (0.5 fmol/μL) with a 100 ms exposure (data not shown).

Electrochemotherapy

When metastases were detected by bioluminescence imaging (day 14), mice were randomized into 3 groups (n = 12 mice/group) -G1: ECT group (bleomycin injected intravenously followed by electric field application) G2: bleomycin alone (control) and G3: EP alone (control).

At day 15, mice from G1 and G2 received an intravenous injection of a low concentration of bleomycin (5 mg/kg in 0.15 M NaCl) whereas G3 received 0.15 M NaCl only. Five minutes later, mice were anesthetized and underwent an abdominal surgery for image-guided metastases tracking. For G1 and G3, each metastasis observed was immediately subjected to EP. Electric parameters were 8 pulses (100 μs at 10 Hz frequency) delivered with stainless steel non-invasive contact electrodes (1 mm) connected to a S20 generator from Betatech (St Orens, France). The amplitude was 110 or 130 V. These electrodes were especially designed for this study (Fig. 2B andC) and could be moved easily inside the peritoneal cavity and along the intestine to avoid puncturing the tissue. After surgical recovery, metastases progression or regression was followed by non-invasive bioluminescence imaging twice a week until mice were sacrificed for ethical considerations (more than 20% weight loss or prostration).

Statistical analyzes

Statistical analyzes (Kruskal-Wallis, Mann Whitney, 2 ways ANOVA, Mantel Cox) were performed using Graphpad prism® (La Jolla, California, USA).

Results

Animal model characterization by in vivo multimodal imaging

In vivo bioluminescence and bimodal tomographic imaging (combining 3D fluorescence imaging and microCT) were used to monitor primary tumor growth and metastasis (Fig. 3A andB). The use of non-invasive molecular imaging allowed individual longitudinal follow-up of tumor development in each animal and enabled an equal distribution of the mice within the three experimental groups before treatment (Fig. 3C). Multimodal imaging provided preoperative mol ecular diagnosis and accurate characterization of the animal model. Abdominal micro-metastases could be spotted before surgery and were then easily located during the surgery using intraoperative real time NIR fluorescence imaging (Fig. 2A). IF imaging was used to guide the application of the image-guided ECT protocol (Fig. 2C). Upon the completion of surgery, the progression of the residual nodules was measured using non-invasive bioluminescence over time. Because each animal constituted its own control, the inter-animal heterogeneity of the tumor growth was minimized.

Effect of EP and ECT treatments on tumor burden and survival

The average number of treated nodules per mouse was 14.2 ± 3.5 for EP and 14.4 ± 2.9 for ECT. Immediately after EP (with or without bleomycin), we observed that all pulsed nodules turned white. This phenomenon, which has been previously described, is due to a transient vascular lock, induced by EP, that temporary stops the blood flow [START_REF] Gehl | Vascular reactions to in vivo electroporation: characterization and consequences for drug and gene delivery[END_REF]. Mouse recovery after surgery and EP or ECT treatment was good, with less than 10% mortality.

After the treatment, mice from the bleomycin group quickly presented severe metastatic progression as seen by bioluminescence imaging (Fig. 4A). This finding is consistent with the fact that a sub-optimal bleomycin concentration was used as compared to standard bleomycin chemotherapy protocols [START_REF] Mir | Electrochemotherapy, a new antitumor treatment: first clinical trial[END_REF][START_REF] Belehradek | Electrochemotherapy, a new antitumor treatment. First clinical phase I-II trial[END_REF]. Indeed, 3 days after the bleomycin treatment (at day 18), the tumor burden had already reached 372% of the measure at the day before treatment (day 14). At the same time, the metastatic invasion was only mildly slowed in the EP group (203% of the measure at day 14). In contrast, we observed a strong tumor growth inhibition within the ECT group, represented by a tumor burden at day 18 of 109% of the measure at day 14 (Fig. 4B). These differences were strongly amplified over time (Fig. 4C) and the reduced metastatic load in the ECT group was associated with a significantly increased survival rate (median lifespan: 34 days vs 25 days; ****p $_amp_$lt; 0.0001) while there was no significant difference in life span between the mice treated with bleomycin alone and the non-operated mice (median lifespan: 27 days vs 25 days, respectively) (Fig. 4D).

Effect of the voltage intensity

We then investigated whether increasing the voltage intensity from 110 V to 130 V could improve the therapeutic effect. For both EP and ECT, we observed a two fold reduction in the tumor burden when the voltage intensity was 130 V as compared to 110 V (Fig. 5A). However, there was a sharp deterioration in the survival rate of the mice treated with ECT at 130 V in comparison with ECT at 110 V (27 days vs. 34 days), indicating a global toxicity for this higher-intensity condition (Fig. 5B).

Discussion

In this study, we developed a syngeneic immuno-competent murine model in which micro-metastases appear secondarily to primary tumor growth and resection (Fig. 1). Primary tumor development and metastases invasion were finely characterized by non-invasive multimodal imaging combining i) bioluminescence to monitor tumor progression and ii) fluorescence tomography/microCT, using a tumor-targeted probe, to allow pre-and intraoperative molecular imaging (Fig. 3). Non-invasive imaging techniques facilitate the accurate diagnosis, staging and visualization of tumors, which are prerequisites for successful surgical therapy. Moreover, the use of tumor-specific labeling allows direct translation from preoperative diagnostic to image-guided surgery. Intraoperative real time visualization of the tumor, by highlighting the molecular signature obtained from preoperative imaging, allows the surgeon to be precisely guided within the cancerous tissues (Fig. 2). Until now, many different molecular imaging modalities, either alone or in combination with a navigation system, have been used to improve surgery. Those that provide intraoperative guidance for the tumor resection, such as intraoperative nuclear imaging [START_REF] Hall | Martin EW use of intraoperative nuclear medicine imaging technology: strategy for improved patient management[END_REF], Magnetic Resonance Imaging (MRI) [START_REF] Hall | Intraoperative MR imaging[END_REF][START_REF] Black | Development and implementation of intraoperative magnetic resonance imaging and its neurosurgical applications[END_REF], Ultra Sounds (US) [START_REF] Kane | Intraoperative ultrasonography: history, current state of the art, and future directions[END_REF], and fluorescence imaging, are important innovative tools, each with its own merits and limitations [START_REF] Hussain | Molecular imaging for cancer diagnosis and surgery[END_REF]. Among them, fluorescence imaging seems to be the most suitable for intraoperative surgical guidance because i) it uses non-ionizing photons, ii) it presents high sensitivity and resolution at low depth, iii) it offers a wide field of view compatible with the operative field, iv) it displays a high speed readout allowing real time imaging and v), the absence of radioactive products in the operating room is also contributing to make it the optimal choice for intraoperative guidance [START_REF] Harlaar | Real-time near infrared fluorescence (NIRF) intra-operative imaging in ovarian cancer using an alpha(v)beta(3-)integrin targeted agent[END_REF].

In addition to the choice of the imaging modality, another issue concerns the use of a relevant molecular imaging probe. Currently, the most promising fluorescent probes for cancer imaging are based on the active targeting of tumor-specific biomarkers. Indeed they offer better tumor to healthy tissue contrast than non-targeting agents that accumulate in the tumor passively via the enhanced permeability and retention (EPR) effect [START_REF] Yu | Clearance pathways and tumor targeting of imaging nanoparticles[END_REF]. Currently, the translation of these fluorescent targeting probes to the clinic is still a major bottleneck that slows down the acceptance of optical imaging in human trials [START_REF] Rosenthal | The status of contemporary image guided modalities in oncologic surgery[END_REF]. Various cancer biomarkers that are associated with malignant phenotypes, such as growth factor receptors, matrix metallo-proteases and other markers of angiogenesis, have been considered for the design of tumor-targeting agents. Most of these biomarkers, especially those located on tumor cell-surface, are suitable for active targeting. Thus, possible targets include the folate receptor, VEGF, EGFR, HER2, ep-CAM, PSMA and αvβ3 integrin.

In this study, we focused on the αvβ3 integrin, which is overexpressed during angiogenesis and in aggressive tumor cells [START_REF] Hood | Role of integrins in cell invasion and migration[END_REF][START_REF] Hutteman | Intraoperative near-infrared fluorescence imaging of colorectal metastases targeting integrin alpha(v)beta(3) expression in a syngeneic rat model[END_REF][START_REF] Jin | Noninvasive optical imaging of ovarian metastases using Cy5-labeled RAFT-c(-RGDfK-)4[END_REF].

RGD peptides have been identified as high affinity ligands for the αvβ3 integrin and the preclinical applications of fluorescent RGD peptides in targeted surgery have been reported in ovarian cancer [START_REF] Mery | Intraoperative fluorescence imaging of peritoneal dissemination of ovarian carcinomas. A preclinical study, Gynecol[END_REF] and colorectal intra-abdominal metastases [START_REF] Dutour | Targeted imaging of α(v)β(3) expressing sarcoma tumor cells in vivo in pre-operative setting using near infrared: a potential tool to reduce incomplete surgical resection[END_REF]. Our group previously developed a fluorescent molecular scaffold presenting four copies of a cRGD called Angiostamp™. We demonstrated the high affinity and specificity of Angiostamp™ for the αvβ3 integrin, both in vitro and in vivo, in various animal models including peritoneal carcinomatosis and ovarian metastasis in mice [START_REF] Jin | Noninvasive optical imaging of ovarian metastases using Cy5-labeled RAFT-c(-RGDfK-)4[END_REF][START_REF] Mery | Intraoperative fluorescence imaging of peritoneal dissemination of ovarian carcinomas. A preclinical study, Gynecol[END_REF], orthotopic head and neck squamous cell carcinoma in mice [START_REF] Atallah | Role of near-infrared fluorescence imaging in head and neck cancer surgery: from animal models to humans[END_REF], osteosarcoma in rats [START_REF] Dutour | Targeted imaging of α(v)β(3) expressing sarcoma tumor cells in vivo in pre-operative setting using near infrared: a potential tool to reduce incomplete surgical resection[END_REF], and fibrosarcoma in cats [START_REF] Wenk | Near-infrared optical guided surgery of highly infiltrative fibrosarcomas in cats using an anti-avß3 integrin molecular probe[END_REF]. In each of these diverse conditions, Angio-stamp™ displayed a high signal in tumor tissue with low nonspecific background, generating a very good tumor contrast that allowed visualization and precise delineation of tumor margins and significantly improved the surgical resection. Moreover, the surgery duration was significantly reduced, which may also have an important impact on the patient outcome [START_REF] Keramidas | Intraoperative near-infrared image guided surgery of peritoneal carcinomatosis in a preclinical mouse model[END_REF].

In the very invasive metastatic model used in this study, surgical resection of spotted micro-metastases remained a challenge because of the potential excision-related damage that can occur on blood vessels or the intestine wall, leading to hemorrhage or septicemia respectively. Furthermore, our previous results demonstrated that the use of IF imaging led to the identification of two times more tumor nodules than the surgeon's eyes can distinguish, and particularly some very small ones (down to ~250 cells) [START_REF] Keramidas | Intraoperative near-infrared image guided surgery of peritoneal carcinomatosis in a preclinical mouse model[END_REF]. Because of its small size, the mouse is a very tricky model to perform resection surgery. Thus, improving the visualization of micro-nodules, by increasing the number of resection gestures also increases the risk of damage. Indeed, in our mouse model, the post-operative morbidity was very high in the context of an exhaustive surgical resection of the nodules (up to 85% of operative or post-operative mortality, data not shown). In the clinical situation, the surgeon will advisedly leave some tumor nodules that cannot be resected because of their delicate location and will rely on intracavital hyperthermic chemotherapy to treat nodules that are smaller than 2.5 mm. Nonetheless, if bigger nodules are left they will certainly contribute to relapse. Therefore, there is a need for improved loco-regional treatments for use in combination with surgery [START_REF] Hansen | Ablative technologies for hepatocellular, cholangiocarcinoma, and metastatic colorectal cancer of the liver[END_REF].

In this study we evaluated a new therapeutic approach combining intraoperative image-guided tumor location and ECT, using a peritoneal micro-metastases model. No resection surgery was performed, but nodules identified by IF were submitted to EP following intravenous injection of bleomycin. Under these conditions fewer than 10% of the mice died following surgeries for EP or ECT treatment.

The main mechanism of the antitumor effect of ECT is the enhanced delivery of cytotoxic molecules into tumor cells [START_REF] Sersa | Electrochemotherapy in treatment of tumours[END_REF][START_REF] Cabula | Electrochemotherapy in the treatment of cutaneous metastases from breast cancer: a multicenter cohort analysis[END_REF]. A suitable drug for ECT treatment is a hydrophilic molecule that cannot easily cross the intact plasma membrane of the cell. With ECT, a high transport across the membrane is obtained for these hydrophilic molecules. Therefore low concentrations can be administered to achieve intracellular concentrations suitable to induce efficient cytotoxic activities, thus avoiding deleterious side effects. Bleomycin and cisplatin are the two chemotherapeutic drugs currently used in the standardized ECT protocols in clinic. Bleomycin is a non-permeant antineoplastic antibiotic isolated from Streptomyces verticillus [START_REF] Mir | Bleomycin: revival of an old drug[END_REF]. Its cytotoxicity results from its ability to make single and double strand DNA breaks in mammalian cells [START_REF] Tounekti | The ratio of singleto double-strand DNA breaks and their absolute values determine cell death pathway[END_REF]. Cisplatin is a low-permeant platinum-based antineoplastic agent [START_REF] Siddik | Cisplatin: mode of cytotoxic action and molecular basis of resistance[END_REF]. Its cytotoxic mode of action is mediated by its interaction with DNA to form DNA adducts, inducing intra-strand cross-links. However, due to their relative hydrophilicity, these compounds have a very low cellular uptake [START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF][START_REF] Tamzali | Successful treatment of equine sarcoids with cisplatin electrochemotherapy: a retrospective study of 48 cases[END_REF]. The hampered cellular transmembrane transport of these molecules can be overcome by EP. The effectiveness of bleomycin and cisplatin in ECT has been demonstrated in preclinical studies using several tumor models but also for different cancer types in human and equine clinical trials [START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF][START_REF] Tamzali | Successful treatment of equine sarcoids with cisplatin electrochemotherapy: a retrospective study of 48 cases[END_REF]. ECT is highly effective, with complete response rates between 60 and 70% and objective response rate around 80% [START_REF] Mali | Antitumor effectiveness of electrochemotherapy: a systematic review and meta-analysis[END_REF][START_REF] Cabula | Electrochemotherapy in the treatment of cutaneous metastases from breast cancer: a multicenter cohort analysis[END_REF][START_REF] Cadossi | Locally enhanced chemotherapy by electroporation: clinical experiences and perspective of use of electrochemotherapy[END_REF]. ECT has already been shown to be efficient for deep tumors [START_REF] Miklavčič | Electrochemotherapy: technological advancements for efficient electroporation-based treatment of internal tumors[END_REF][START_REF] Edhemovic | Electrochemotherapy: a new technological approach in treatment of metastases in the liver[END_REF][START_REF] Girelli | Feasibility and safety of electrochemotherapy (ECT) in the pancreas: a pre-clinical investigation[END_REF]. In the case of peritoneal metastases, we believe that intraoperative image-guided tumor localization can increase the effectiveness of the ECT approach. Indeed, in our study we demonstrated the success of ECT treatment to slow down metastases progression and to increase the survival rate in a very aggressive peritoneal metastases model without resorting to tumor removal.

Beyond the cytotoxic action of bleomycin, additional mechanisms may contribute to the success of this therapy. One such mechanism involves the vascular disrupting action of ECT. While electroporation alone reduces blood flow, ECT has a delayed killing effect on tumor vasculature endothelial cells, leading to prolonged reduction of tumor blood flow and consequently to tumor hypoxia and eventually to tumor necrosis [START_REF] Sersa | Vascular disrupting action of electroporation and electrochemotherapy with bleomycin in murine sarcoma[END_REF]. In the present study, we observed that the treated metastatic nodules became white immediately after EP or ECT. This phenomenon is due to the transient vascular lock after EP and to a more prolonged blood flow reduction when combined with the killing effect on vascular endothelial cells in presence of bleomycin [START_REF] Markelc | Differential mechanisms associated with vascular disrupting action of electrochemotherapy: intravital microscopy on the level of single normal and tumor blood vessels[END_REF].

It is known that the local field strength in the tissue must be above a critical value for ECT to be effective. This local field value is under the control of the applied voltage and the geometry of the electrodes [START_REF] Corovic | Modeling of electric field distribution in tissues during electroporation[END_REF][START_REF] Marčan | Web-based tool for visualization of electric field distribution in deep-seated body structures and planning of electroporation-based treatments[END_REF]. When using higher electric voltage (130V), the overall survival of the animals was not statistically improved by ECT as compared to EP alone. Indeed, when a tissue is submitted to high voltage, irreversible electroporation (IRE) may occur [START_REF] Horowitz | In vivo results of a new focal tissue ablation technique: irreversible electroporation[END_REF][START_REF] Rubinsky | Irreversible electroporation in medicine[END_REF][START_REF] Phillips | Irreversible electroporation on the small intestine[END_REF]. In IRE, the application of high voltage across the tissue alters the transmembrane potential and disrupts the lipid bilayer of cells, allowing the exchange of intra and extra-cellular components that can contribute to cell death. Tissues under IRE ablation showed a peak apoptotic rate at 24 h after IRE ablation with viable tissues at the peripheral rim of treated tissues in histological analysis on prostate tumors [START_REF] Kim | Choi OK changes of apoptosis in tumor tissues with time after irreversible electroporation[END_REF]. The cell destruction mechanism of IRE is not based on thermal damage. IRE was not studied until the early 2000s in the context of medical applications as an ablative method but now several surgical applications are under clinical trials [START_REF] Rubinsky | Irreversible electroporation in medicine[END_REF]. In our study, we hypothesized that EP (130V) could also induce cell mortality by an IRE like process.

Another mechanism involving the immune system has been described to explain the overall efficacy of ECT. Indeed, when ECT with bleomycin or cisplatin have been applied to subcutaneous transplanted tumors in nude or immune-competent mice, the ECT treatment was less effective in nude mice, thus indicating a role for the immune system in clearing tumors [START_REF] Sersa | Electrochemotherapy with CDDP on LPB sarcoma: comparison of the anti-tumor effectiveness in immunocompetent and immunodeficient mice[END_REF][START_REF] Sedlar | Potentiation of electrochemotherapy by intramuscular IL-12 gene electrotransfer in murine sarcoma and carcinoma with different immunogenicity[END_REF]. However, it is not yet fully understood how ECT may trigger the immune system [START_REF] Calvet | Electrochemotherapy with bleomycin induces hallmarks of immunogenic cell death in murine colon cancer cells[END_REF][START_REF] Mozzillo | Assessing a novel immuno-oncology-based combination therapy: Ipilimumab plus electrochemotherapy[END_REF].

In our experiments, we used immuno-competent BALB/c mice, bearing a very aggressive tumor model that kills non-treated mice within 20 to 27 days. Due to this aggressivity, the delay between the ECT treatment and the death of the mice might not be long enough to promote a strong and efficient stimulation of the immune system and it is thus unlikely that the immune reaction participated to the antitumor effect. Indeed, we did not observe a complete regression of the disease but ECT led to a significant (****p $_amp_$lt; 0.0001) 36% increase in median lifespan (corresponding to approximately 12 days). It would be interesting to evaluate the presence and intensity of the ECT-primed antitumor immune response, but this would necessitate the use of a more adapted animal model. Finally, it has been shown that ECT can be combined with another therapeutic approach such as irradiation [START_REF] Kranjc | Radiosensitising effect of electrochemotherapy with bleomycin in LPB sarcoma cells and tumors in mice[END_REF]. Thus ECT guided by intraoperative fluorescence imaging could be an additional tool at the disposition of clinicians for improving the outcome of such patients.

Conclusion

In this study, we used a very aggressive peritoneal micro-metastases model that we finely characterized using non-invasive multimodal imaging. The use of tumor-specific fluorescent labeling allowed direct translation from preoperative molecular imaging and diagnosis to intraoperative visualization of the tumor nodules. We implemented a new therapeutic approach using tumor-targeted fluorescence guided ECT which efficiently minimized collateral damage related to resection surgery and led to a significant increase in lifespan. ECT has already been approved in several EU countries (Denmark, Germany, Greece, Italy, Poland, Portugal, Slovenia, and the UK) and we believe that intraoperative image-guided tumor localization can increase the effectiveness of this approach in treating peritoneal metastases.

Fig. 1 .

 1 Fig. 1. Animal model setup. At day 1 (D1), adenocarcinoma cells were injected under the renal capsule of the left kidney leading to the development of a primary tumor (D7). At day 8 (D8), the tumor-bearing kidney was surgically removed. At day 14 (D14), small scattered nodules resembling peritoneal carcinomatosis appeared in the peritoneal cavity.

Fig. 2 .

 2 Fig. 2. Electrochemotherapy guided by intraoperative fluorescence imaging. A) Visualization of metastases by intraoperative fluorescence imaging 24 h after intravenous injection of Angiostamp®700. B) Non-invasive 1 mm-gap contact electrodes were connected to the pulse generator. C) Each fluorescent metastasis was subjected to EP. Electric parameters were 8 pulses (100 μs at 10 Hz frequency). The amplitude was 110 or 130 V.

Fig. 3 .

 3 Fig. 3. Animal model characterization by multimodal imaging. A) In vivo bioluminescence imaging on the dorsal and ventral faces at D1 just before tumor cell implantation, at D7 just before primary tumor-bearing kidney (PTK) resection and at D14 when peritoneal metastases have developed. B) Whole-body microCT imaging 15 min after the intravenous injection of visipac and bimodal imaging combining microCT and abdominal 3D fluorescence 4 h after the intravenous injection of Angiostamp®700 at D7 and D14. 3D fluorescence imaging highlights the abdominal metastases, a local recurrence (LR) on the site of excision of the primary tumor-bearing kidney (PTK) and the renal elimination of the tracer (HK: healthy kidney). At D14, the remaining HK was hyperactive to ensure the tracer elimination. C) Abdominal bioluminescence measured at D14 and distribution of mice in the 3 experimental groups. Each black spot represents one mouse and the white spots represent the mean values with standard deviations for each group. A Kruskal-Wallis statistical analysis was performed.

Fig. 4 .

 4 Fig. 4. Effect of ECT treatment on tumor burden and survival. A) Representative images of abdominal bioluminescence at D18. B) Quantification of metastatic load measured by abdominal bioluminescence at D18. A Mann Whitney statistical analysis was performed; **p $_amp_$lt; 0.01. C) Metastatic load measured once a week by in vivo abdominal bioluminescence. Data (mean values ± SEM) in B and C are expressed as a percentage of the measure at D14. A 2 ways ANOVA statistical analysis was performed; *p $_amp_$lt; 0.05; **p $_amp_$lt; 0.01; ***p $_amp_$lt; 0.001. D) Survival rate for each experimental condition. A Mantel Cox statistical analysis was performed; **p $_amp_$lt; 0.01; ****p $_amp_$lt; 0.0001.

Fig. 5 .

 5 Fig. 5. Effect of EP intensity on tumor burden and survival. A) Quantification of metastatic load at D18 measured by abdominal bioluminescence. A Mann Whitney statistical analysis was performed. B) Survival rate for each experimental condition. A Mantel Cox statistical analysis was performed; *p $_amp_$lt; 0.05; **p $_amp_$lt; 0.01.

Journal of Controlled Release xxx (2016) xxx-xxx

Acknowledgements

Research was conducted within the scope of the EBAM European Associated Laboratory (LEA) and network of the COST action TD1104. Optical systems were acquired thanks to France Life Imaging (FLI, French program "Investissement d'Avenir"; grant "Infrastructure d'avenir en Biologie Santé", ANR-11-INBS-0006). Proof Reading Services corrected the manuscript.


Keywords: Modèle de Biot, ondes poroélastiques, ondes acoustiques, couplage acoustique/poroelastique, solution analytique, technique de Cagniard de Hoop

Solution analytique pour la propagation d'ondes en milieu stratifié hétérogène acoustique/poroélastique. Partie I : en dimension 2

Résumé : Nous nous intéressons à la modélisation de la propagation d'ondes dans les milieux infinis bicouche acoustique/poroélastique. Nous considérons le modèle bi-phasique de Biot dans la couche poroélastique. Cette première partie est consacrée au calcul de la solution analytique en dimension deux à l'aide de la technique de Cagniard-De Hoop. Nous mettons en évidence l'intérêt de la solution analytique en l'utilisant pour valider un code de calcul.

Introduction

The computation of analytical solutions for wave propagation problems is of high importance for the validation of numerical computational codes or for a better understanding of the reflexion/transmission properties of the media. Cagniard-de Hoop method [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] De Hoop | The surface line source problem[END_REF] is a useful tool to obtain such solutions and permits to compute each type of waves (P wave, S wave, head wave...) independently. Although it was originally dedicated to the solution to elastodynamic wave propagation, it can be applied to any transient wave propagation problem in stratified medium. However, as far as we know, few works have been dedicated to the application of this method to poroelastic medium. In [START_REF] Ezziani | Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques[END_REF] the analytical solution to poroelastic wave propagation in an homogeneous 2D medium is providds and in [START_REF] Feng | High-frequency acoustic properties of a fluid/porous solid interface. ii. the 2d reflection green's function[END_REF] the authors compute the analytical expression of the reflected wave at the interface between an acoustic and a poroelastic layer in two dimension but they do not explicit the expression of the transmitted waves.

In order to validate computational codes of wave propagation in poroelastic media, we have implemented the codes Gar6more 2D [START_REF] Diaz | Gar6more 2d[END_REF] and Gar6more 3D [START_REF] Diaz | Gar6more 3d[END_REF] which provide the complete solution (reflected and transmitted waves) of the propagation of wave in stratified 2D or 3D media composed of acoustic/acoustic, acoustic/elastic, acoustic/poroelastic or poroelastic/poroelastic layers. The codes are freely downloadable at http://www.spice-rtn.org/library/software/Gar6more2D and http://www.spice-rtn.org/library/software/Gar6more3D. We will focus here on the 2D acoustic/poroelastic case, the three dimensional and the poroelastic/poroelastic cases will be the object of forthcoming papers. The outline of the paper is as follows: we first present the model problem we want to solve and derive the Green problem from it (section 1). Then we present the analytical solution to the wave propagation problem in a stratified 2D medium composed of an acoustic and a poroelastic layer (section 2) and we detail the computation of the solution (section 3). Finally we show how the analytical solution can be used to validate a numerical code (section 4).

The model problem

We consider an infinite two dimensional medium (Ω = R 2 ) composed of an homogeneous acoustic layer Ω + = R×] -∞, 0] and an homogeneous poroelastic layer Ω -= R × [0, +∞[ separated by an horizontal interface Γ (see Fig. 1). We first describe the equations in the two layers ( 1.1 and 1.2) and the transmission conditions on the interface Γ ( 1.3), then we present the Green problem from which we compute the analytical solution ( 1.4).

The equation of acoustics

In the acoustic layer we consider the second-order formulation of the wave equation with a point source in space, a regular source function f in time and zero initial conditions:

                 P + -V + 2 ∆P + = δ x δ y-h f (t), in Ω + ×]0, T ], Ü + = - 1 ρ + ∇P + , in Ω + ×]0, T ], P + (x, y, 0) = 0, Ṗ + (x, y, 0) = 0, in Ω + U + (x, y, 0) = 0, U+ (x, y, 0) = 0, in Ω + (1)
where P + is the pressure;

U + is the displacement field;

V + is the celerity of the wave;

ρ + is the density of the fluid.

INRIA

Biot's Model

In the second layer we consider the second order formulation of the poroelastic equations [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low-frequency range[END_REF][START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. II. higher frequency range[END_REF][START_REF] Biot | Mechanics of deformation and acoustic propagation in porous media[END_REF]]

                                   ρ -Ü - s + ρ - f Ẅ --∇ ó Σ -= 0, in Ω -×]0, T ], ρ - f Ü - s + ρ - w Ẅ -+ 1 K - Ẇ -+ ∇P -= 0, in Ω -×]0, T ], Σ -= λ -∇ ó U - s I 2 + 2µ -ε(U - s ) -β -P -I 2 , in Ω -×]0, T ], 1 m -P -+ β -∇ ó U - s + ∇ ó W -= 0, in Ω -×]0, T ], U - s (x, 0) = 0, W -(x, 0) = 0, in Ω -, U - s (x, 0) = 0, Ẇ -(x, 0) = 0, in Ω -, (2)
with

(∇ ó Σ -) i = 2 j=1 ∂Σ - ij ∂x j ∀ i = 1, 2. As usual I 2 is the identity matrix of M 2 (IR)
and ε(U - s ) is the solid strain tensor defined by:

ε ij (U ) = 1 2
∂U i ∂x j + ∂U j ∂x i .
In (2), the unknowns are:

U - s the displacement field of solid particles;

W -= φ -(U - f -U - s )
, the relative displacement, U - f being the displacement field of fluid particle and φ -the porosity; P -, the fluid pressure; Σ -, the solid stress tensor.

The parameters describing the physical properties of the medium are given by:

ρ -= φ -ρ - f + (1 -φ -)ρ -
s is the overall density of the saturated medium, with ρ - s the density of the solid and ρ - f the density of the fluid;

ρ - w = a -ρ - f /φ -,
where a -is the tortuosity of the solid matrix;

K -= κ -/η -
, where κ -is the permeability of the solid matrix and η is the viscosity of the fluid;

m -and β -are positive physical coefficients:

β -= 1 -K - b /K - s and m -= φ -/K - f + (β --φ -)/K - s -1
, where K - s is the bulk modulus of the solid, K - f is the bulk modulus of the fluid and K - b is the frame bulk modulus;

µ -is the frame shear modulus, and λ -= K - b -2µ -/3 is the Lamé constant.

Transmission conditions

Let n be the unitary normal vector of Γ outwardly directed to Ω -. The transmission conditions on the interface between the acoustic and porous medium are [START_REF] Carcione | Wave Fields in Real Media : Wave propagation in Anisotropic, Anelastic and Porous Media[END_REF]:

     W -ó n = (U + -U - s ) ó n, P -= P + , Σ -n = -P + n. (3)

The Green problem

We won't compute directly the solution to (1-2-3) but the solution to the following Green problem:

p+ -V + 2 ∆p + = δ x δ y-h δ t , in Ω + ×]0, T ], ( 4a
) ü+ = - 1 ρ + ∇p + , in Ω + ×]0, T ], (4b)
ρ -ü- s + ρ - f ẅ--∇ ó σ -= 0, in Ω -×]0, T ], (5a) ρ
- f ü- s + ρ - w ẅ-+ 1 K - ẇ-+ ∇p -= 0, in Ω -×]0, T ], (5b)
σ -= λ -∇ ó u - s I 2 + 2µ -ε(u - s ) -β -p -I 2 , in Ω -×]0, T ], (5c)
1 m -p -+ β -∇ ó u - s + ∇ ó w -= 0, in Ω -×]0, T ], (5d)
w -ó n = (u + -u - s ) ó n, on Γ×]0, T ] (6a)
p -= p + , on Γ×]0, T ] (6b) σ -n = -p + n. on Γ×]0, T ] (6c)
The solution to (1-2-3) is then computed from the solution to the Green Problem thanks to a convolution by the source function. For instance we have:

P + (x, y, t) = p + (x, y, .) * f (.) = t 0 p + (x, y, τ )f (t -τ ) dτ INRIA
(we have similar relations for the other unknowns). We also suppose that the poroelastic medium is non dissipative, i.e the viscosity η -= 0. Using the equations (5c, 5d) we can eliminate σ -and p -in (5) and we obtain the equivalent system:

   ρ -ü- s + ρ - f ẅ--α -∇(∇ ó u - s ) + µ -∇ × (∇ × u - s ) -m -β -∇(∇ ó w -) = 0, ρ - f ü- s + ρ - w ẅ--m -β -∇(∇ ó u - s ) -m -∇(∇ ó w -) = 0, (7)
with

α -= λ -+ 2µ -+ m -β -2 .
And using the equation (4b) the transmission conditions (6) are rewritten as:

                     ü- s y + ẅ- y = - 1 ρ + ∂ y p + , -m -β -∇ ó u - s -m -∇ ó w -= p + , ∂ y u - sx + ∂ x u - sy = 0, (λ -+ m -β -2 )∇ ó u - s + 2µ -∂ y u - sy + m -β -∇ ó w = -p + . (8)
We split the displacement fields u - s and u - f into irrotational and isovolumic fields (P-wave and S-wave):

u - s = ∇Θ - u + ∇ × Ψ - u ; w -= ∇Θ - w + ∇ × Ψ - w . (9)
We can then rewrite system [START_REF] Diaz | Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées[END_REF] in the following form:

             A -Θ--B -∆Θ -= 0, y < 0 Ψ- u -V - S 2 ∆Ψ - u = 0, y < 0 Ψ- w = - ρ - f ρ - w Ψ- u , y < 0 ( 10
)
where Θ -= (Θ - u , Θ - w ) t , A -and B -are 2 × 2 symmetric matrices:

A -= ρ -ρ - f ρ - f ρ - w ; B -= λ -+ 2µ -+ m -(β -) 2 m -β - m -β - m - , and
V - S = µρ - w ρ -ρ - w -ρ - f 2
is the S-wave velocity.

We multiply the first equation of system [START_REF] Ezziani | Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques[END_REF] by the inverse of A. The matrix A -1 B is diagonalizable:

A -1 B = PDP -1
, where P is the change-of-coordinates matrix,

D = diag(V - P f 2 , V - P s
2 ) is the diagonal matrix similar to A -1 B, V - P f and V - P s are respectively the fast P-wave velocity and the slow P-wave velocity (V P s < V P f ).

RR n 6509

Using the change of variables Φ = (Φ P f , Φ P s ) t = P -1 Θ, we obtain the uncoupled system on fast P-waves, slow P-waves and S-waves:

             Φ -D∆Φ = 0, y < 0 Ψu -V - S 2 ∆Ψ u = 0, y < 0 Ψ w = - ρ - f ρ - w Ψ u , y < 0 (11)
Finally, we obtain the Green problem equivalent to [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] Carcione | Wave Fields in Real Media : Wave propagation in Anisotropic, Anelastic and Porous Media[END_REF][START_REF] De Hoop | The surface line source problem[END_REF]:

         p+ -V + 2 ∆p + = δ x δ y-h δ t , y > 0 Φ- i -V - i 2 ∆Φ - i = 0, i ∈ {P f, P s, S} y < 0 B(p + , Φ - P f , Φ - P s , Φ - S ) = 0, y = 0 ( 12
)
where we have set Φ S = Ψ u in order to have similar notations for the P f , P s and S waves.

The operator B represents the transmission conditions on Γ:

B     p + Φ - P f Φ - P s Φ - S     =              1 ρ + ∂ y (P 11 + P 21 ) ∂ 3 ytt (P 12 + P 22 ) ∂ 3 ytt ( ρ - f ρ - w -1) ∂ 3 xtt 1 m -(β -P 11 + P 21 ) V 2 P f ∂ 2 tt m -(β -P 12 + P 22 ) V 2 P s ∂ 2 tt 0 0 2P 11 ∂ 2 xy 2P 12 ∂ 2 xy ∂ 2 yy -∂ 2 xx 1 B 42 B 43 -2µ -∂ 2 xy                         p + Φ - P f Φ - P s Φ - S           
where P i,j , i, j = 1, 2 are the components of the change-of-coordinates matrix P, B 41 and B 43 are given by:

B 42 = (λ -+ m -β -2 )P 11 + m -β -P 21 V - P f 2 ∂ 2 tt + 2µ -P 11 ∂ 2 yy , B 43 = (λ -+ m -β -2 )P 12 + m -β -P 22 V - P s 2 ∂ 2 tt + 2µ -P 12 ∂ 2 yy .
To obtain this operator we have used the transmission conditions ( 8), the change of variables ( 9) and the uncoupled system [START_REF] Feng | High-frequency acoustic properties of a fluid/porous solid interface. ii. the 2d reflection green's function[END_REF].

Moreover, we can determine the solid displacement u - s by using the change of variables (9) and the fluid displacement u + by using (4b).

Expression of the analytical solution

To state our results, we need the following notations and definitions: INRIA 1. Definition of the complex square root. For q ∈ C\IR -, we use the following definition of the square root g(q) = q 1/2 : g(q) 2 = q and ℜe[g(q)] > 0.

The branch cut of g(q) in the complex plane will thus be the half-line defined by {q ∈ IR -} (see Fig. 2). In the following, we use the abuse of notation g(q) = i √ -q for q ∈ IR -.

ℑm(q)

ℜe(q) -π +π 2. Definition of the functions κ + and κ - i . For i ∈ {P f, P s, S} and q ∈ C, we define the functions

κ + := κ + (q) = 1 V + 2 + q 2 1/2 and κ - i := κ - i (q) = 1 V - i 2 + q 2 1/2 .
3. Definition of the reflection and transmission coefficients. For a given q ∈ C, we denote by R(q), T P f (q), T P s (q) and T S (q) the solution to the linear system

A(q)          R(q) T P f (q) T P s (q) T S (q)          = - 1 2κ + V + 2           κ + (q) ρ + 1 0 1           , ( 13
)
where the matrix A(q) is defined for q ∈ C by:

A(q) =             - κ + (q) ρ + κ - P f (q)(P 11 + P 21 ) κ - P s (q)(P 12 + P 22 ) i q 1 - ρ - f ρ - w 1 m - V - P f 2 β -P 11 + P 21 m - V - P s 2 β -P 12 + P 22 0 0 -2i q κ - P f (q) P 11 -2i q κ - P s (q) P 12 κ - S 2 (q) + q 2 1 A 4,2 (q) A 4,3 (q) 2i qµ -κ - S (q)             , RR n 6509 with A 4,2 (q) = λ -+ m -β -2 P 11 + m -β -P 21 V - P f 2 + 2µ -κ - P f (q) 2 P 11 , A 4,3 (q) = λ -+ m -β -2 P 12 + m -β -P 22 V - P s 2 + 2µ -κ - P s (q) 2 P 12 ,
We also denote by V max the greatest velocity in the two media:

V max = max(V + , V P f , V P s , V S ).
We can now present the expression of the solution to the Green Problem:

Theorem 2.1. The pressure and the displacement in the top medium are given by

p + (x, y, t) = p + inc (x, y, t)+p + ref (x, y, t) and u + (x, y, t) = t 0 ν + inc (x, y, τ ) dτ + t 0 ν + ref (x, y, τ ) dτ,
and the displacement in the bottom medium is given by

u - s = t 0 ν -(x, y, τ )dτ with ν -= ν - P f + ν - P s + ν - S ,
where p + inc and ν + inc are respectively the pressure and the velocity of the incident wave and satisfy:

                   p + inc (x, y, t) = 1 2πV + 2 t 2 -t 2 0 , ν inc,x (x, y, t) = tx 2πV + 2 ρ + r t 2 -t 2 0 , ν inc,y (x, y, t) = t(y -h) 2πV + 2 ρ + r t 2 -t 2 0 , if t > t 0
p inc (x, y, t) = 0 and ν inc (x, y, t) = 0, else.

We set here r = (x 2 +(y -h) 2 ) 1/2 and t 0 = r/V + denotes the arrival time of the incident wave.

INRIA p + ref and ν + ref are respectively the pressure and the velocity of the reflected wave and satisfy:

                       p + ref (x, y, t) = - ℑm [κ + (υ(t))R(υ(t))] π t 2 0 -t 2 , ν + ref,x (x, y, t) = - ℑm [i υ(t)κ + (υ(t))R(υ(t))] πρ + t 2 0 -t 2 , ν + ref,y (x, y, t) = - ℑm κ + 2 (υ(t))R(υ(t)) πρ + t 2 0 -t 2 , if t h < t ≤ t 0 and x r > V + V max ,                        p + ref (x, y, t) = ℜe [κ + (γ(t))R(γ(t))] π t 2 0 -t 2 , ν + ref,x (x, y, t) = ℜe [i γ(t)κ + (γ(t))R(γ(t))] πρ + t 2 0 -t 2 , ν + ref,y (x, y, t) = ℜe κ + 2 (γ(t))R(γ(t)) πρ + t 2 -t 2 0 , if t > t 0 , p ref (x, y, t) = 0 and ν ref (x, y, t) = 0, else.
We set here r = (x 2 + (y + h) 2 ) 1/2 , t 0 = r/V + denotes the arrival time of the reflected volume wave,

t h = (y + h) 1 V + 2 - 1 V 2 max + |x| V max ,
denotes the arrival time of the reflected head wave and the complex functions υ := υ(t) and γ := γ(t) are defined by

                 υ(t) = -i   y + h r 1 V + 2 - t 2 r 2 + xt r 2   for t h < t ≤ t 0 and x < 0, υ(t) = i   y + h r 1 V + 2 - t 2 r 2 - xt r 2   for t h < t ≤ t 0 and x ≥ 0, and
γ(t) = -i x r 2 t + y + h r t 2 r 2 - 1 V + 2 for t > t 0 .
RR n 6509 ν - P f is the velocity of the transmitted P f wave and satisfies:

           ν - P f,x (x, y, t) = - P 11 π ℜe i υ(t)T P f (υ(t)) dυ dt (t) , ν - P f,y (x, y, t) = P 11 π ℜe κ - P f (υ(t))T P f (υ(t)) dυ dt (t) , if t h < t ≤ t 0 and |ℑm (υ(t 0 ))| > 1 V max          ν - P f,x (x, y, t) = - P 11 π ℜe i γ(t)T P f (γ(t)) dγ dt (t) , ν - P f,y (x, y, t) = P 11 π ℜe κ - P f (γ(t))T P f (γ(t)) dγ dt (t) , if t > t 0 ν - P f (x, y, t) = 0, else.
Here t 0 denotes the arrival time of the P f wave (its calculation is detailed in appendix) and t h is the arrival time of the P f head wave:

t h = -y 1
V - P f 2 - 1 V 2 max + h 1 V + 2 - 1 V 2 max + |x| V max .
For t h < t ≤ t 0 , the function υ := υ(t) is defined as the only root of q ∈ C → F(q, t) = -y 1

V - P f 2 + q 2 1/2 + h 1 V + 2 + q 2 1/2 + iqx -t,
such that ℑm x dυ(t) dt ≤ 0. For t > t 0 , the function γ := γ(t) is defined as the only root of q ∈ C → F(q, t) whose real part is positive. ν - P s is the velocity of the transmitted P s wave and satisfies:

           ν - P s,x (x, y, t) = - P 12 π ℜe i υ(t)T P s (υ(t)) dυ dt (t) , ν - P s,y (x, y, t) = P 12 π ℜe κ - P s (υ(t))T P s (υ(t)) dυ dt (t) , t h < t ≤ t 0 and |ℑm (υ(t 0 ))| > 1 V max ,          ν - P s,x (x, y, t) = - P 12 π ℜe i γ(t)T P s (γ(t)) dγ dt (t) , ν - P s,y (x, y, t) = P 12 π ℜe κ - P s (γ(t))T P s (γ(t)) dγ dt (t) , if t > t 0 , ν - P s (x, y, t) = 0, else.
Here t 0 denotes the arrival time of the P s wave and t h is the arrival time of the P s head wave:

t h = -y 1
V - P s 2 - 1 V 2 max + h 1 V + 2 - 1 V 2 max + |x| V max .
INRIA For t h < t ≤ t 0 , the function υ := υ(t) is defined as the only root of

q ∈ C → F(q, t) = -y 1
V - P s 2 + q 2 1/2 + h 1 V + 2 + q 2 1/2 + iqx -t,
such that ℑm x dυ(t) dt ≤ 0. For t > t 0 , the function γ := γ(t) is defined as the only root of q ∈ C → F(q, t) whose real part is positive.

ν -
S is the velocity of the transmitted S wave and satisfies:

           ν - S,x (x, y, t) = 1 π ℜe κ - S (υ(t))T S (υ(t)) dυ dt (t) , ν - S,y (x, y, t) = 1 π ℜe i υ(t)T S (υ(t)) dυ dt (t) , if t h < t ≤ t 0 and |ℑm (υ(t 0 ))| > 1 V max ,            ν - S,x (x, y, t) = 1 π ℜe κ - S (γ(t))T S (γ(t)) dγ dt (t) , ν - S,y (x, y, t) = 1 π ℜe i γ(t)T S (γ(t)) dγ dt (t) , if t > t 0 , ν - S,x (x, y, t) = 0, else.
Here t 0 denotes the arrival time of the S wave and t h is the arrival time of the S head wave:

t h = -y 1 V - S 2 - 1 V 2 max + h 1 V + 2 - 1 V 2 max + |x| V max .
For t h < t ≤ t 0 , the function υ := υ(t) is defined as the only root of

q ∈ C → F(q, t) = -y 1 V - S 2 + q 2 1/2 + h 1 V + 2 + q 2 1/2 + iqx -t,
such that ℑm x dυ(t) dt ≤ 0. For t > t 0 , the function γ := γ(t) is defined as the only root of q ∈ C → F(q, t) whose real part is positive.

Remark 2.1. For the practical computations of the displacement, we won't have to explicitly compute the primitive of the velocities ν, which would be rather tedious, since

t 0 ν(τ ) dτ * f = ν * t 0 f (τ ) dτ .
Therefore, we'll only have to compute the primitive of the source function f .

RR n 6509 3 Proof of the theorem

To prove the theorem, we use the Cagniard-de Hoop method (see [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] De Hoop | The surface line source problem[END_REF][START_REF] Van Der Hijden | Propagation of transient elastic waves in stratified anisotropic media[END_REF][START_REF] Pao | The generalized ray theory and transient response of layered elastic solids[END_REF][START_REF]Etude dans le domaine temporel de la propagation d'ondes elastiques en milieux stratifiés ; modèlisation de la reponse du thorax a un impacts[END_REF]), which consists of two steps:

1. We apply a Laplace transform in time, ũ(x, y, s) = +∞ 0 u(x, y, t) e -st dt, and a Fourier transform in the x variable, û(k x , y, s) = +∞ -∞ ũ(x, y, s) e i kxx dx to [START_REF]Etude dans le domaine temporel de la propagation d'ondes elastiques en milieux stratifiés ; modèlisation de la reponse du thorax a un impacts[END_REF] in order to obtain an ordinary differential system whose solution Ĝ(k x , y, s) can be explicitly computed ( 3.1); 2. we apply an inverse Fourier transform in the x variable to G:

G(x, y, s) = 1 2π +∞ -∞ Ĝ(k x , y, s) e -i kxx dk x .
and, using tools of complex analysis, we turn this inverse Fourier transform into the Laplace transform of some function H( §, †, ⊔):

G(x, y, s) = 1 2π
+∞ 0 H(x, y, t) e -st dt.

Then, using the injectivity of the Laplace transform, we identify G(x, y, t) to H(x, y, t) ( 3.2).

The solution in the Laplace-Fourier plane

Let us first apply a Laplace transform in time and a Fourier transform in the x variable to [START_REF]Etude dans le domaine temporel de la propagation d'ondes elastiques en milieux stratifiés ; modèlisation de la reponse du thorax a un impacts[END_REF] to obtain

                   s 2 V + 2 + k 2 x p+ - ∂ 2 p+ ∂y 2 = δ(y -h) V + 2 , y > 0, s 2 V - i 2 + k 2 x Φ- i - ∂ 2 Φ- i ∂y 2 = 0, i ∈ {P f, P s, S} y < 0, B(p + , Φ- P f , Φ- P f , Φ- S ) = 0 y = 0, ( 14
)
where B is the Laplace-Fourier transform of the operator B.

From the two first equations of ( 14), we deduce that the solution (p + , ( Φi ) i∈{P f,P s,S} ) is

INRIA such that                                    p+ = p+ inc + p+ ref , p+ inc (k x , y, s) = e -s|y-h| 1 V + 2 + k 2 x s 2 1/2 sV + 2 1 V + 2 + k 2 x s 2 1/2 , p+ ref (k x , y, s) = R(k x , s)e -sy 1 V + 2 + k 2 x s 2 1/2 , Φ- i (k x , y, s) = T i (k x , s)e sy 1 V - i 2 + k 2 x s 2 1/2 , i ∈ {P f, P s, S}. (15)
where the coefficients R and T i are computed by using the last equation of ( 14):

B(p + ref , Φ- P f , Φ- P s , Φ- S ) = -B(p + inc , 0, 0, 0).
Using the expressions of p+ inc , p+ ref and Φi , we obtain that R(k x , s), T P f (k x , s), T P s (k x , s), T S (k x , s) are solution to

A k x s          sR(k x , s) s 3 T P f (k x , s) s 3 T P s (k x , s) s 3 T S (k x , s)          = - e -shκ + ( kx s ) 2V + 2 κ + kx s           κ + kx s ρ + 1 0 1           . ( 16
)
From the definition of the functions R(q), T P f (q), T P f (q) and T P s (q) we deduce that

         R kx s T P f kx s T P s kx s T S kx s          =          sR(k x , s) s 3 T P f (k x , s) s 3 T P s (k x , s) s 3 T S (k x , s)          e shκ + ( kx s ) . (17)
Finally, we rewrite p+ inc , p+ ref and Φi under the form

                           p+ = p+ inc + p+ ref , p+ inc (k x , y, s) = 1 sV + 2 κ + kx s e -s|y-h|κ + ( kx s ) , p+ ref (k x , y, s) = 1 s R k x s e -s(y+h)κ + ( kx s ) , Φ- i (k x , y, s) = 1 s 3 T i k x s e -s(yκ - i ( kx s )-hκ + ( kx s )) , i ∈ {P f, P s, S}. (18)
RR n 6509

From (4b) and ( 9), we deduce the expression of the displacement field:

                                   û+ = û+ inc + û+ ref , û+ inc,x = i k x ρ + s 2 p+ inc , û+ inc,y = sign(h -y) κ + kx s ρ + s p+ inc , û+ ref,x = i k x ρ + s 2 p+ ref , û+ ref,y = κ + kx s ρ + s p+ ref , û- sx = -i k x P 11 ΦP f -i k x P 12 Φ- P s + sκ - S k x s Φ - S û- sy = sκ - P f k x s P 11 Φ- P f + sκ - P s k x s P 12 Φ- P s + i k x Φ - S ( 19
)
In the following we only detail the computation of ûsx,P s = -i k x P 12 Φ-P s , since the computation of the other terms is very similar.

The Laplace transform of the solution

Let us now apply an inverse Fourier transform in the x variable to ûsx,P s and set

k x = qs to obtain ũ- sx,P s = - +∞ -∞ i qP 12 2sπ T P s (q)e -s(-yκ - P s (q)+hκ + (q))+i qx) dq = - P 12 2sπ +∞ -∞ Ξ(q) dq, (20)
with Ξ(q) = i q T P s (q)e -s(-yκ - P s (q)+hκ + (q))+i qx) .

The key point of the Cagniard-de Hoop method is to turn this Fourier integral into a Laplace integral by finding a path Γ in the complex plane such that -yκ - P s (q) + hκ + (q) + i qx = t ∈ IR + , ∀q ∈ Γ. This amounts to compute the roots q of the function

F(q, t) = -y 1
V - P s 2 + q 2 1/2 + h 1 V + 2 + q 2 1/2 + iqx -t,
for t ∈ IR + . We recall in appendix A some properties of the function F; from properties A.1, A.3 and A.2 we can define the function γ(t) ∈ C for t > t 0 as the only root of F(., t) whose real part is positive, where t 0 is the arrival time of the P s volume wave (we recall in appendix B the practical computation of t 0 ). Moreover, for a given R ∈ IR + , we define

Γ + R = {p = γ(t) | t 0 < t < R}, Γ - R = {p = -γ(t) | t 0 < t < R} and Γ ± = lim R→∞ Γ ± R .
We represent the paths Γ ± R in Fig. 3 in the case x < 0 (for x ≥ 0 the path would be similar but in the half plane ℑm(q) ≤ 0). The shape of Γ ± is given by the following property: Property 3.1. When R tends to infinity, Γ ± R admits an asymptote of equation

p = ±(y -h) -i x (y -h) 2 + x 2 . INRIA Γ - R Γ + R γ(t 0 ) -|γ -(R)| |γ + (R)| D R ℑm(q) ℜe(q) C - R C + R i V max Figure 3: Integration path if |γ(t 0 )| < 1/V max .
We now have to decide whether this path intersects the branch cuts of the functions κ + and κ - i . From the definition of the complex square root, we deduce that the branch cuts of κ + are the two half lines {q ∈ C | ℜe(q) = 0 and |ℑm(q)| ≥ 1 V + } and the branch cuts of κ - i are the two half lines

{q ∈ C | ℜe(q) = 0 and |ℑm(q)| ≥ 1 V - i }.
and, since only γ(t 0 ) is purely imaginary, the path cross the branch cuts if and only if

|γ(t 0 )| ≥ 1 V max .
We have then have to consider two possibilities: 3). By Cauchy's theorem we have

if |γ(t 0 )| < 1/V max : then Γ R does not
D R Ξ(q) dq + C + R Ξ(q) dq + Γ - R Ξ(q) dq - Γ + R Ξ(q) dq + C - R Ξ(q) dq = 0,
and, using Jordan's lemma, we obtain:

lim R→∞ C ± R Ξ(q) dq = 0. so that +∞ -∞ Ξ(q) dq = Γ + Ξ(q) dq - Γ - Ξ(q) dq.

RR n 6509

Next, using the change of variable q = γ(t) on Γ + and q = -γ(t) on Γ -, we end up with

+∞ -∞ Ξ(q) dq = +∞ t 0 i γ(t)T P s (γ(t)) dγ(t) dt e -st dt - +∞ t 0 i γ(t)T P s (-γ(t)) dγ(t) dt e -st dt.
From the definition of T P s , we check, after some calculation, that and can therefore be integrated.

if |γ(t 0 )| ≤ 1/V max ; in this case, the path does intersect the branch cut and we have to consider an additional path Υ to bypass the half-lines defined by

{q ∈ C | ℜe(q) = 0 and |ℑm(q)| ≥ 1 V max }.
This path must obviously satisfy the condition -yκ - P s (q) + hκ + (q) + i qx = t ∈ IR + , ∀q ∈ Υ.

If x < 0, ℑm(γ(t 0 )) ≥ 0 (property A.2), so that γ(t 0 ) lies on the branch cut q ∈ C | ℜe(q) = 0 and ℑm(q) > 1 V max .

INRIA Therefore, using properties A.4 and A.5, we define υ(t), for t h ≤ t ≤ t 0 , as the only root of F(., t) such that

ℑm dυ(t) dt = ℑm      -y υ(t) 1 V - P s 2 + υ 2 (t) 1/2 + h υ(t) 1 V + 2 + υ 2 (t) 1/2 + i x      > 0,
where

t h = -y 1
V - P s 2 - 1 V 2 max + h 1 V + 2 - 1 V 2 max + |x| V max .
If x > 0, we define υ(t), for t h ≤ t ≤ t 0 , as the only root of F(., t) such that

ℑm dυ(t) dt = ℑm      -y υ(t) 1 V - P s 2 + υ 2 (t) 1/2 + h υ(t) 1 V + 2 + υ 2 (t) 1/2 + i x      < 0.
We are now able to define the paths Υ ± R and Υ ± by:

Υ ± R = υ(t) ± 1 R | t h < t < t 0 and Υ ± = lim R→+∞ Υ ± R
We represent the paths Γ ± R and Υ ± R in Fig. 4 in the case x < 0 (for x ≥ 0 the path would be similar but in the half plane ℑm(q) ≤ 0.). We define the segment

D R = {p ∈ IR, | |p| ≤ |γ(R)|}
and we close the path by the two arcs of circle C - R and C + R of radius γ(R) linking Γ ± R and D R and the half circle c R linking Υ + R to Υ - R (see Fig. 4). By Cauchy's theorem:

D R Ξ(q) dq + C + R Ξ(q) dq - Γ + R Ξ(q) dq + Γ - R Ξ(q) dq - Υ + R Ξ(q) dq + Υ - R Ξ(q) dq + C - R Ξ(q) dq + c R Ξ(q) dq = 0.
Using once again Jordan's Lemma, we prove that the integrals over C - R , C + R and c R vanish when R tends to infinity and that

+∞ -∞ Ξ(q) dq = Γ + Ξ(q) dq - Γ - Ξ(q) dq + Υ + Ξ(q) dq - Υ - Ξ(q) dq.
The calculation of the integral over Γ ± is done as in the first case and we only focus on the calculation over Υ ± . Let us now use the change of variable q = υ - P s (t) -1/R on Υ - R and q = υ - P s (t) + 1/R on Υ + R to obtain:

Υ - R Ξ(q) dq = t 0 t h i υ(t) - 1 R T P s υ(t) - 1 R dυ(t) dt e -st dt Υ + R Ξ(q) dq = t 0 t h i υ(t) + 1 R T P s υ(t) + 1 R dυ(t) dt e -st dt. γ(t 0 ) Γ - R Γ + R c R Υ - R Υ + R -|γ(R)| |γ(R)| D R ℜe(q) C - R C + R i V max i V - P s ℑm(q) Figure 4: Integration path if |γ(t 0 )| > 1/V max . Because of the branch cut, it is clear that lim R→+∞ T P s (υ(t) + 1/R) = lim R→+∞ T P s (υ(t) -1/R),
however, as υ(t) is imaginary, υ(t) -1/R = -υ(t) + 1/R and T P s (υ(t) -1/R) = T P s (υ(t) + 1/R), so that

Υ + Ξ(q) dq - Υ - Ξ(q) dq = - t 0 t h 2υ(t)

Numerical illustration

To illustrate the use our results, we have compared our analytical solution to a numerical one obtained by C. Morency and J. Tromp [START_REF] Morency | Spectral-element simulations ofwave propagation in porous media[END_REF]. We consider an acoustic layer with a density ρ + = 1020 kg/m 3 and a celerity V + = 1500 m/s on top of a poroelastic layer whose characteristic coefficients are: The source is located in the acoustic layer, at 500 m from the interface. It is a point source in space and a fifth derivative of a Gaussian of dominant frequency f 0 = 15 Hz:

f (t) = 4.10 10 π 2 f 2 0 9 t - 1 f 0 + 4 π 2 f 2 0 t - 1 f 0 3 -4 π 4 f 4 0 t - 1 f 0 5 e -π 2 f 2 0 t-1 f 0 2 .
We compute the solution at two receivers, the first one is in the acoustic layer, at 533 m from the interface; the second one is in the poroelastic layer, at 533 m from the interface; both are located on a vertical line at 400 m from the source (see Fig. 5). We represent the y component of the displacement from t = 0 to t = 1 s. on Fig 6 . The left picture represents the solution at receiver 1 while the right picture represents the solution at receiver 2. On both pictures the blue solid curve is the analytical solution and the red dashed curve is the numerical solution.

Both pictures show a good agreement between the two solutions, which validates the numerical code.

INRIA 5 Conclusion

We provided the complete solution (reflected and transmitted wave) of the propagation of wave in a stratified 2D medium composed of an acoustic and a poroelastic layer and we used it to validate a numerical code. In a forthcoming paper we will use this solution as a basis to derive the solution in a three dimensional medium. We will also extend the method to the propagation of waves in heterogeneous poroelastic medium in two and three dimensions.

Figure 1 :

 1 Figure 1: Configuration of the study

Figure 2 :

 2 Figure 2: Definition of the function x → (x) 1/2

  intersect the branch cuts of the functions κ - i or κ + . We define the segment D R = {p ∈ IR, | |p| ≤ |γ(R)|} and we close the path by the two arcs of circle C - R and C + R of radius γ(R) linking Γ ± R and D R (see Fig.

  the solid density ρ - s = 2500 kg/m 3 ; the fluid density ρ - f = 1020 kg/m 3 ; the porosity φ -= 0.4; the tortuosity a -= 2; the solid bulk modulus K - s = 16.0554 GPa; the fluid bulk modulus K - f = 2.295 GPa; the frame bulk modulus K - b = 10 GPa;the frame shear modulus µ -= 9.63342 GPa; so that the celerity of the waves in the poroelastic medium are:for the fast P wave, V - P f = 3677 m/s for the slow P wave, V - P s = 1060 m/s for the ψ wave, V - S = 2378 m/s.

Figure 6 :

 6 Figure 5: Configuration of the experiment

1 The model problem 3 1 . 1 1 . 3 6 2of the analytical solution 8 3A

 111368 The equation of acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Biot's Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Transmission conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The Green problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expression Proof of the theorem 14 3.1 The solution in the Laplace-Fourier plane . . . . . . . . . . . . . . . . . . . . 14 3.2 The Laplace transform of the solution . . . . . . . . . . . . . . . . . . . . . . 16 Properties of the function F(q, t) 23 B Definition of the arrival time of the transmitted waves.

RR n 6509

Acknowledgments

We thanks Christina Morency who provided us the numerical solutions we have used to validate our analytical solution.

A Properties of the function F (q, t)

We recall in this section some properties of the function (see [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] Van Der Hijden | Propagation of transient elastic waves in stratified anisotropic media[END_REF][START_REF] Diaz | Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées[END_REF]):

Property A.1. For each t ∈ IR + , F(., t) admits at most two roots.

Property A.2. There is one and only one t 0 ∈ IR + such that F(., t 0 ) admits a double root q 0 . This root is purely imaginary and t 0 corresponds to the physical arrival time of the wave. Moreover ℑm(q 0 ) ≤ 0 if x ≥ 0 and ℑm(q 0 ) > 0 if x < 0.

Property A.3. For t > t 0 , F(., t) admits exactly two roots, which have the same imaginary part, positive if x < 0 and negative else, and an opposite non-zero real part.

there exists a time t h ≤ t 0 , such that for t h ≤ t ≤ t 0 , the function F(., t) admits exactly two imaginary roots q 1 (t) and q 2 (t). The time t h is such that

and corresponds to the physical arrival time of the head wave. It can be computed by using the relation

Property A.5. The roots q 1 (t) and q 2 (t) satisfy ℑm(q 1 (t)) ∈ [1/V max , ℑm(γ(t 0 ))] and ℑm(∂ t q 1 (t)) > 0 ℑm(q 2 (t)) ∈ [ℑm(γ(t 0 )), -1/V max ] and ℑm(∂ t q 2 (t)) < 0

RR n 6509 B Definition of the arrival time of the transmitted waves.

We detail in this section the computation of the arrival time of the transmitted P s wave at point (x, y). We first have to determine the fastest path from the source to the point (x, y): we search a point ξ 0 on the interface between the two media which minimizes the function

(see Fig. 7). This leads us to find ξ 0 such that

From a numerical point of view, the solution to this equation is done by computing the roots of the following fourth degree polynomial

ξ 0 is thus the only real root of this polynomial located between 0 and x which is also solution to (22). Once ξ 0 is computed, we can define

Source