Introduction

Blastic plasmacytoid dendritic cell neoplasm (BPDCN) is an aggressive neoplasm derived from plasmacytoid dendritic cells. [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF] In 2008, BPDCN was classified by the World Health Organization (WHO) as a distinct entity in the group of "acute myeloid leukemia (AML) and related precursor neoplasms". [START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF] Although elderly subjects are principally affected, BPDCN can also arise in young adults and children. [START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Approximately 90% of patients exhibit cutaneous lesions at diagnosis, which upon microscopic analysis appear as a dermal infiltrate of immature blastic cells with features of plasmacytoid dendritic cells. [START_REF] Julia | Blastic plasmacytoid dentritic cell neoplasm: clinical features in 90 patients[END_REF][START_REF] Petrella | TCL1 and CLA expression in agranular CD4/CD56 hematodermic neoplasms (blastic NK-cell lymphomas) and leukemia cutis[END_REF] Malignant cells isolated from skin, lymph nodes, bone marrow, spleen and/or other tissues usually express the following markers: interleukin-3 receptor alpha (IL-3Ra or CD123), BDCA2 (CD303), BDCA4 (CD304), TCL1 and ILT7. [START_REF] Angelot-Delettre | Intracytoplasmic detection of TCL1--but not ILT7-by flow cytometry is useful for blastic plasmacytoid dendritic cell leukemia diagnosis[END_REF][START_REF] Garnache-Ottou | Extended diagnostic criteria for plasmacytoid dendritic cell leukaemia[END_REF][START_REF] Marafioti | Novel markers of normal and neoplastic human plasmacytoid dendritic cells[END_REF] Currently, there is no consensus regarding the optimal treatment modality for BPDCN. Several treatments, including multi-agent chemotherapy regimens, symptomatic approaches (e.g. local radiation [START_REF] Petrella | Blastic NK-cell lymphomas (agranular CD4+CD56+ hematodermic neoplasms): a review[END_REF] ), and intensive chemotherapy with allogeneic hematopoietic cell transplantation, [START_REF] Dalle | Blastic plasmacytoid dendritic cell neoplasm: is transplantation the treatment of choice?[END_REF][START_REF] Dietrich | Blastic plasmacytoid dendritic cell neoplasia (BPDC) in elderly patients: results of a treatment algorithm employing allogeneic stem cell transplantation with reduced conditioning intensity[END_REF][START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] are generally used to treat patients. Although chemotherapy regimens used to treat patients with acute leukemia or lymphoma are often effective at inducing an initial response, the duration of response is typically brief and recurrent disease is generally resistant to chemotherapy. BPDCN patients generally succumb to cytopenias due to tumor infiltration of the bone marrow; the median overall survival has been reported to range from 9 to 32 months irrespectively of the initial presentation of the disease. [START_REF] Gilis | Sequential combination of high dose methotrexate and L-asparaginase followed by allogeneic transplant: a first-line strategy for CD4+/CD56+ hematodermic neoplasm[END_REF][START_REF] Pagano | Blastic plasmacytoid dendritic cell neoplasm with leukemic presentation: an Italian multicenter study[END_REF][START_REF] Piccaluga | Blastic plasmacytoid dendritic cell neoplasm: is it time to redefine the standard of care?[END_REF] While longer overall survival has been reported with allogeneic hematopoietic cell transplantation, especially in younger patients, [START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF][START_REF] Dalle | Blastic plasmacytoid dendritic cell neoplasm: is transplantation the treatment of choice?[END_REF][START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF][START_REF] Piccaluga | Blastic plasmacytoid dendritic cell neoplasm: is it time to redefine the standard of care?[END_REF][START_REF] Ramanathan | A combination treatment approach and cord blood stem cell transplant for blastic plasmacytoid dendritic cell neoplasm[END_REF] many relapses have been observed after such transplants. [START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] The a-subunit of the human IL-3 receptor is a type I transmembrane glycoprotein belonging to the cytokine receptor superfamily. [START_REF] Blalock | Signal transduction, cell cycle regulatory, and anti-apoptotic pathways regulated by IL-3 in hematopoietic cells: possible sites for intervention with anti-neoplastic drugs[END_REF][START_REF] Miyajima | Cytokine receptors and signal transduction[END_REF] Interleukin-3 (IL-3) the IL-3 receptor is a heterodimer associating an a chain (CD123) and a β chain (CD131). This chain is shared by IL-3, IL-5, and granulocytemacrophage colony-stimulating factor receptors. SL-401, a novel biologic targeted therapy directed against the IL-3R, is comprised of human recombinant IL-3 joined by an acid-©2015 Ferrata Storti Foundation. This is an open-access paper. doi: 10.3324/haematol.2014.111740 The online version of this article has a Supplementary Appendix. Manuscript received on June 5, 2014. Manuscript accepted November 4, 2014. Correspondence: francine.garnache@efs.sante.fr Blastic plasmacytoid dendritic cell neoplasm is an aggressive malignancy derived from plasmacytoid dendritic cells. There is currently no accepted standard of care for treating this neoplasm, and therapeutic strategies have never been prospectively evaluated. Since blastic plasmacytoid dendritic cell neoplasm cells express high levels of interleukin-3 receptor a chain (IL3-Ra or CD123), antitumor effects of the interleukin-3 receptor-targeted drug SL-401 against blastic plasmacytoid dendritic cell neoplasm were evaluated in vitro and in vivo. The cytotoxicity of SL-401 was assessed in patient-derived blastic plasmacytoid dendritic cell neoplasm cell lines (CAL-1 and GEN2.2) and in primary blastic plasmacytoid dendritic cell neoplasm cells isolated from 12 patients using flow cytometry and an in vitro cytotoxicity assay. The cytotoxic effects of SL-401 were compared to those of several relevant cytotoxic agents. SL-401 exhibited a robust cytotoxicity against blastic plasmacytoid dendritic cell neoplasm cells in a dosedependent manner. Additionally, the cytotoxic effects of SL-401 were observed at substantially lower concentrations than those achieved in clinical trials to date. Survival of mice inoculated with a blastic plasmacytoid dendritic cell neoplasm cell line and treated with a single cycle of SL-401 was significantly longer than that of untreated controls (median survival, 58 versus 17 days, P<0.001). These findings indicate that blastic plasmacytoid dendritic cell neoplasm cells are highly sensitive to SL-401, and support further evaluation of SL-401 in patients suffering from blastic plasmacytoid dendritic cell neoplasm.

In vivo and in vitro sensitivity of blastic plasmacytoid dendritic cell neoplasm to SL-401, an interleukin-3 receptor targeted biologic agent

Fanny Angelot-Delettre, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Anne Roggy, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Arthur E. Frankel, [START_REF] Julia | Blastic plasmacytoid dentritic cell neoplasm: clinical features in 90 patients[END_REF] Baptiste Lamarthee, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Estelle Seilles, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Sabeha Biichle, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] Bernard Royer, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF][START_REF] Petrella | TCL1 and CLA expression in agranular CD4/CD56 hematodermic neoplasms (blastic NK-cell lymphomas) and leukemia cutis[END_REF] Eric Deconinck, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF][START_REF] Petrella | TCL1 and CLA expression in agranular CD4/CD56 hematodermic neoplasms (blastic NK-cell lymphomas) and leukemia cutis[END_REF] Eric K. Rowinsky, [START_REF] Angelot-Delettre | Intracytoplasmic detection of TCL1--but not ILT7-by flow cytometry is useful for blastic plasmacytoid dendritic cell leukemia diagnosis[END_REF] Christopher Brooks, [START_REF] Angelot-Delettre | Intracytoplasmic detection of TCL1--but not ILT7-by flow cytometry is useful for blastic plasmacytoid dendritic cell leukemia diagnosis[END_REF] Valerie Bardet, [START_REF] Garnache-Ottou | Extended diagnostic criteria for plasmacytoid dendritic cell leukaemia[END_REF] Blandine Benet, [START_REF] Marafioti | Novel markers of normal and neoplastic human plasmacytoid dendritic cells[END_REF] Hind Bennani, [START_REF] Petrella | Blastic NK-cell lymphomas (agranular CD4+CD56+ hematodermic neoplasms): a review[END_REF] Zehaira Benseddik, [START_REF] Dalle | Blastic plasmacytoid dendritic cell neoplasm: is transplantation the treatment of choice?[END_REF] Agathe Debliquis, [START_REF] Dietrich | Blastic plasmacytoid dendritic cell neoplasia (BPDC) in elderly patients: results of a treatment algorithm employing allogeneic stem cell transplantation with reduced conditioning intensity[END_REF] Daniel Lusina, [START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] Mikael Roussel, [START_REF] Gilis | Sequential combination of high dose methotrexate and L-asparaginase followed by allogeneic transplant: a first-line strategy for CD4+/CD56+ hematodermic neoplasm[END_REF] Françoise Solly, [START_REF] Pagano | Blastic plasmacytoid dendritic cell neoplasm with leukemic presentation: an Italian multicenter study[END_REF] Michel Ticchioni, [START_REF] Piccaluga | Blastic plasmacytoid dendritic cell neoplasm: is it time to redefine the standard of care?[END_REF] Philippe Saas, [START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF][START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF][START_REF] Ramanathan | A combination treatment approach and cord blood stem cell transplant for blastic plasmacytoid dendritic cell neoplasm[END_REF] and Francine Garnache-Ottou 1-4

ABSTRACT

labile group of amino acids to a diphtheria toxin (DT) payload that has been truncated at its receptor binding region. [START_REF] Frankel | Characterization of diphtheria fusion proteins targeted to the human interleukin-3 receptor[END_REF] Since IL-3, the natural ligand for IL-3R, binds with very high specificity and avidity, [START_REF] Rapoport | Mutational analysis of the alpha subunit of the human interleukin-3 receptor[END_REF] SL-401 is able to transport DT efficiently and preferentially to cells that overexpress IL-3R, leading to internalization followed by receptor-mediated endocytosis and localization of SL-401 to early endosomes. After cleavage of the SL-401 DT constituent in the acidic medium of endosomes, DT translocates into the cytosol and binds to ADP-ribosylated elongation factor 2, leading to blockade of protein synthesis and cell death. [START_REF] Kreitman | Recombinant immunotoxins containing truncated bacterial toxins for the treatment of hematologic malignancies[END_REF] Given the ubiquitous and high expression of IL-3R by BPDCN and the lack of therapies available to treat BPDCN, SL-401 is a potential therapeutic for BPDCN. The present study evaluated the cytotoxicity of SL-401 against patient-derived BPDCN cell lines (CAL-1 and GEN2.2) and primary BPDCN cells isolated directly from 12 patients. The investigations were performed in vitro, as well as in vivo in a murine model of BPDCN. The aim of the study was to provide further support for the use of SL-401 in patients suffering from BPDCN.

Methods

Patients' cells and cell lines

Peripheral blood or bone marrow cells were obtained for diagnostic purposes from 12 BPDCN patients (Table 1) from our national network that collects data and cells from cases diagnosed in France since 2004 (authorization number #DC-2008-713). BPDCN was diagnosed from the results of histopathology and immunostaining of cutaneous lesions, blood or bone marrow. [START_REF] Swerdlow | World Health Organisation Classification of Tumors[END_REF][START_REF] Garnache-Ottou | Extended diagnostic criteria for plasmacytoid dendritic cell leukaemia[END_REF] Two established cell lines derived from BPDCN patients were used (GEN 2.2, patent #0215927, Dr. Plumas, EFS Rhone-Alpes, Grenoble, France and CAL-1, Dr. Maeda, Nagasaki University, Japan) as well as TF/H-Ras (Prof. Frankel) and CD123 neg (MFI<800) Daudi cell lines (ACC78, DSMZ Braunschweig, Germany) as positive and negative controls, respectively. Other lymphoid and myeloid leukemic cells used to compare sensitivity to SL-401 are described in the Online Supplementary Appendix.

Drug and culture

The SL-401 drug (Stemline Therapeutics, New York, NY, USA) was stored at -80°C and tested at eight concentrations ranging from 365 pM to 0.08 fM (21 ng/mL to 0.4 ng/mL) in order to cover

+ ILT7 -CD33 + CD117 + (others My -) CD22 + (others B -) CD7 + (others T -) #7 33/M CD4 + CD56 + CD123 ++ CD303 -CD304 + 44,X,Y,13[3]/44,sl,i(7)(q10)[15]/45,XY,-15[3] yes BPDCN BM, skin, blood TCL1 + ILT7 -My -B -T - #8 70/F CD4 + CD56 + CD123 ++ CD303 + CD304 + 46,XX i(7q) yes BPDCN LN, BM, skin, TCL1 + ILT7 -My -B -T - blood, spleen #9 72/M CD4 + CD56 + CD123 ++ CD303 -CD304 + NA yes NA LN, BM, skin, blood TCL1 + ILT7 -My -B -CD2 + CD7 + (others T -) #10 63/M CD4 + CD56 + CD123 ++ CD303 + CD304 + 46XY yes BPDCN BM, skin, blood TCL1 + ILT7 -My -B -CD2 + CD7 + (others T -) #11 58/M CD4 + CD56 + CD123 ++ CD303 + 46XY yes BPDCN LN, BM, skin, CD304 + TCL1 + ILT7 + CD33 + (others My -) B -T- spleen #12
80/M CD4 + CD56 + CD123 ++ CD303 -CD304 + TCL1 + 43-44,XY,der(2) t(2;?)(q23;?),der(4)t(4;?) yes BPDCN LN, BM, skin, blood, ILT7 -My -B -CD7 + (others T -) (q35;?),5,der(7)t(7;?)(q31;?),-9,-10,-13,-15, spleen -16,del(17)(p11),+1-4mar [17];82 83,idemx2 [12];46,XY

Age (years)/gender; Results of phenotypic analysis performed on blood or bone marrow samples (flow cytometry); karyotype; presence of cutaneous lesions; histopathological diagnosis. My: myeloid markers (including myeloperoxidase, CD13, CD33, CD117, CD15, CD65, CD14, CD64); T : T lymphoid markers (including membrane CD3, intracytoplasmic CD3, CD7, CD5, CD2, CD8); B: B lymphoid markers (including intracytoplasmic CD79a

, intracytoplasmic CD22, intracytoplasmic Ig m chains, CD19, CD20, CD22, surface immunoglobulin); +: positive expression; ++: high expression; -: absence of expression; NA: not available; LN: lymph nodes; BM: bone marrow.

the concentrations obtained in vivo in patients enrolled in clinical trials. [START_REF] Frankel | Phase I clinical study of diphtheria toxininterleukin 3 fusion protein in patients with acute myeloid leukemia and myelodysplasia[END_REF][START_REF] Frankel | Activity of SL-401, a targeted therapy directed to the interleukin-3 receptor, in patients with blastic plasmacytoid dendritic cell neoplasm patients[END_REF] The effects of chemotherapy agents used in acute leukemia were also evaluated against BPDCN cells (Online Supplementary Appendix). BPDCN cells were incubated at 3x10 5 cells/mL in RPMI 1640 glutamax medium (Invitrogen, Cergy Pontoise, France) supplemented with 10% fetal calf serum (Invitrogen), 1% penicillin/streptomycin (PAA Laboratoires, Velizy Villacoublay, France) with or without SL-401 or the relevant drugs under 5% CO 2 for 18 h at 37°C.

Cytotoxicity evaluation by flow cytometry

Flow cytometry was performed using a CANTO II cytometer (BD Biosciences, San Jose, CA, USA) and DIVA 6.2 software (BD Biosciences). The cytotoxic effects of SL-401 and the various drugs were evaluated using annexin-V and 7-amino actinomycin D (AV/7AAD) and a panel of different monoclonal antibodies to gate the blastic population described in the Online Supplementary Appendix. In the mouse model, anti-mouse and anti-human CD45 plus anti-human CD123, CD4, CD56, CD304 were used to identify BPDCN human cells (Online Supplementary Appendix). A defined number of calibrated 3-mm latex beads (Flowcount beads, Beckman Coulter) was added to each sample to obtain the absolute number of circulating BPDCN cells in mice, as previously described. [START_REF] Perruche | A singleplatform approach using flow cytometry and microbeads to evaluate immune reconstitution in mice after bone marrow transplantation[END_REF]

Cytotoxicity evaluation by the MTT assay

The percentage of viable cells obtained after incubation with or without SL-401 was assessed using the MTT assay (3-(4,5dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide, Sigma Aldrich, Saint Quentin Fallavier, France). Details of the culture and following analysis are provided in the Online Supplementary Appendix.

Mice

NOD-SCID IL2Rγc-deficient (NSG) mice were irradiated (2 Gy) and inoculated intravenously 24 h later with 1x10 6 GEN2.2 cells and treated intraperitoneally 8 days later with five daily injections of SL-401 (2 mg/mouse/injection, total experimental dose 100 mg/kg) or with phosphate-buffered saline (PBS) only. Mice were monitored weekly by blood cell counts and flow cytometry (Online Supplementary Appendix).

Statistical analysis

Statistical analyses were performed using Statel software 2.6 (Adscience, Paris, France) (Online Supplementary Appendix).

Results

SL-401 is cytotoxic against blastic plasmacytoid dendritic cell neoplasm cell lines and primary cells

The viability of the two CD123 pos BPDCN cell lines, GEN2.2 and CAL-1, decreased from 62 ± 6% (range, 44-96) to 5 ± 2% (range, 0-17) after treatment with SL-401 at the experimental dose of 365 pM (21 ng/mL) for 18 h (n=9, Figure 1A). Moreover the viability of these two BPDCN cell lines decreased in a dose-dependent manner (Figure 1B). Similarly, after treatment with SL-401 at the experimental dose of 365 pM (21 ng/mL), the viability of 12 samples of freshly isolated BPDCN primary cells decreased significantly from 50 ± 4% (range, 31-71) to 10 ± 1% (range, 3-17). As expected, the CD123 neg (Daudi) cells were not sensitive to treatment with SL-401 concentrations as high as 365 pM for 18 h [viability was unchanged from 65% (range, 88-41) to 61% (range, 83-41) in cells untreated or treated with SL-401, respectively, n=3; Figure 1A,B].

SL-401 is cytotoxic against blastic plasmacytoid dendritic cell neoplasm primary cells in a concentration-dependent manner

The viability of primary malignant cells obtained from 12 BPDCN patients treated with SL-401 for 18 or 48 h decreased in a concentration-dependent manner, as assessed by flow cytometry and MTT (Figure 1C,D). As expected, CD123 neg Daudi cells were resistant to SL-401 treatment (Figure 1C,D). For patient #10, primary BPDCN cells were obtained at both the time of diagnosis and disease relapse following treatment with chemotherapy [including CHOP (cyclophosphamide, hydroxydaunorubicin, vincristine, and prednisone), methotrexate, and Lasparaginase]. Interestingly, BPDCN cells obtained at disease recurrence were slightly more sensitive to SL-401 than cells obtained at diagnosis (IC 50 , 6 fM versus 79 fM, respectively; P=0.049; Figure 1E). Thus, relapsing cells were still sensitive to SL-401 in a concentration-dependent manner, indicating that BPDCN cells retain their sensitivity to SL-401 following treatment with various cytotoxic agents, thereby suggesting a lack of cross-resistance.

SL-401 is more efficient than other tested chemotherapeutic drugs -except idarubicinat killing blastic plasmacytoid dendritic cell neoplasm cells

In order to appreciate the cytotoxic effect of SL-401 better, we also assessed the effects of other chemotherapeutic drugs. Primary malignant cells obtained from three BPDCN patients (#7, #9 and #11) were significantly more sensitive to SL-401 than to a wide variety of cytotoxic agents commonly used for the treatment of hematologic malignancies, including cytosine arabinoside, cyclophosphamide, vincristine, dexamethasone, methotrexate, Erwinia L-asparaginase, and asparaginase (Figure 2; P<0.05 for all agents). Only idarubicin was found to be more efficient than SL-401 (viability <1%, n=5; Figure 2).

Blastic plasmacytoid dendritic cell neoplasm primary cells are more sensitive to SL-401 than acute myeloid or lymphoblastic leukemia primary cells in vitro

We next assessed the viability after SL-401 treatment of primary malignant cells isolated from three previously untreated patients suffering from acute lymphoblastic leukemia and six untreated patients suffering from AML. We compared these results to those achieved following SL-401 treatment of primary BPDCN cells. The viability of the leukemic cells decreased slightly after treatment with SL-401 (365 pM for 18 h). We observed an average decrease of 13% in viability for acute lymphoblastic leukemia cells [33±20% (range, 6-73) to 26±20% (range, 4-69), n=3] and 16% for AML cells [40±8% (range, 18-75) to 36±8% (range, 11-70), n=6]. BPDCN cells were significantly more sensitive to SL-401 and had a 75% decrease in viable cells [50±5% (range, 20-71) to 11±1%, (range, 4-17) n=11; P<0.001; Figure 3].

CD123 expression correlates with SL-401 cytotoxicity in vitro

We next compared the relative expression levels of IL-3Ra (CD123) and β (CD131) chains in primary malignant

IL-3R-targeted treatment of BPDCN haematologica | 2015; 100(2)
BPDCN cells, acute leukemic cells, as well as established BPDCN cell lines to the respective sensitivities to SL-401 (Table 2). Overall, sensitivity of primary BPDCN cells was related to CD123 expression, as demonstrated by the inverse relationship between cell viability and CD123 expression (Spearman test: r = -0.58, P<0.012).

The high dependence of plasmacytoid dendritic cell lineage cells (including normal or leukemic plasmacytoid dendritic cells) to IL-3 may also contribute to the high sensitivity of BPDCN cells to deprivation of IL-3 signaling during SL-401 exposure. Importantly, no such relationship was observed for CD131 (Spearman test: r = haematologica | 2015; 100(2)

Treatment with SL-401 significantly increases the overall survival of NSG mice inoculated with blastic plasmacytoid dendritic cell neoplasm cells

Irradiated NGS mice were inoculated with the GEN2.2 cell line (1x10 6 cells per mouse) through the tail vein. Eight days after inoculation, mice were treated with a daily intra-peritoneal injection of SL-401 (2 mg/day) for 5 days or with PBS as controls (n=8 mice in 3 independent experiments). In PBS-treated control mice, the mean overall survival was 17±1 days. Treatment with five daily injections of SL-401 significantly increased the overall survival of mice compared to that of control mice (58 ± 2 days; P<0.001; Figure 4A,B). Circulating BPDCN cells were identified as human CD45 pos , CD123 pos , BDCA4 pos , CD4 pos , CD56 pos , CD3 neg , and CD34 neg cells (Figures 4C). Nearly all of the BPDCN cells expressed CD123 when mice developed the BPDCN (Figure 4C). This suggests that one course of SL-401 (2 µg/day for 5 days) is not sufficient to kill all of the BPDCN cells, rather that CD123 neg BDPCN cells emerge in response to SL-401. In PBS-treated control mice, the number of BPDCN cells progressively increased until death, whereas treatment for 5 days with SL-401 successfully reduced circulating BPDCN cells to undetectable levels for 15±3 days after treatment (Figure 4D). We monitored hemoglobin and platelet counts in mice to assess leukemic cell bone marrow involvement. In PBS-treated control mice inoculated with BPDCN, hemoglobin and platelet counts progressively decreased until death. In contrast, in treated mice, these hematologic parameters reached the levels observed in irradiated control mice that were not inocula ted Table 2. Expression of IL-3Ra and β chains on primary malignant cells from patients suffering from BPDCN, acute myeloid leukemia (AML) cells, acute lymphoid leukemia (ALL) cells and cell lines -as assessed by mean fluorescence intensity -was compared with the viability after treatment with SL-401.

Cells

Mean with BPDCN cells and not treated (Figure 4E). of cytopenia under treatment indicates an absence, or at least, a lower level of bone marrow involvement by BPDCN cells in SL-401 treated mice. Overall, SL-401 is effective in controlling BPDCN cells in vivo.

Discussion

SL-401 is a biologic agent corresponding to IL-3 genetically fused to truncated DT via a cleavable linker. This agent induces cytotoxicity by inhibiting ribosomal function, and thereby, inhibiting protein synthesis, a mechanism that is distinct from all other anticancer therapeutics. [START_REF] Frankel | Characterization of diphtheria fusion proteins targeted to the human interleukin-3 receptor[END_REF] SL-401 has been demonstrated to induce a profound cytotoxicity at picomolar and subpicomolar concentrations in AML cell lines, [START_REF] Frankel | Diphtheria toxin fused to human interleukin-3 is toxic to blasts from patients with myeloid leukemias[END_REF] as well as in a model of human AML inoculated into immunocompromised mice. [START_REF] Black | Diphtheria toxin-interleukin-3 fusion protein (DT(388)IL3) prolongs disease-free survival of leukemic immunocompromised mice[END_REF] Moreover, SL-401 is cytotoxic in vivo, in patients with advanced AML and myelosdysplastic syndrome, [START_REF] Frankel | Phase I clinical study of diphtheria toxininterleukin 3 fusion protein in patients with acute myeloid leukemia and myelodysplasia[END_REF] suggesting that SL-401 targets leukemic stem cells, as well as more mature tumor cells. In contrast, SL-401 is not cytotoxic to normal hematopoietic progenitor cells, [START_REF] Feuring-Buske | A diphtheria toxin-interleukin 3 fusion protein is cytotoxic to primitive acute myeloid leukemia progenitors but spares normal progenitors[END_REF] which has translated into a paucity of myelosuppression in clinical trials to date. [START_REF] Frankel | Phase I clinical study of diphtheria toxininterleukin 3 fusion protein in patients with acute myeloid leukemia and myelodysplasia[END_REF][START_REF] Feuring-Buske | A diphtheria toxin-interleukin 3 fusion protein is cytotoxic to primitive acute myeloid leukemia progenitors but spares normal progenitors[END_REF][START_REF] Alexander | High affinity interleukin-3 receptor expression on blasts from patients with acute myelogenous leukemia correlates with cytotoxicity of a diphtheria toxin/IL-3 fusion protein[END_REF][START_REF] Hogge | Variant diphtheria toxin-interleukin-3 fusion proteins with increased receptor affinity have enhanced cytotoxicity against acute myeloid leukemia progenitors[END_REF][START_REF] Su | Characterization of variant diphtheria toxin-interleukin-3 fusion protein, DTIL3K116W, for phase I clinical trials[END_REF] The present study was performed to evaluate and quantify the effects of SL-401 on various preclinical models of BPDCN, a malignancy that ubiquitously expresses high levels of the IL-3Ra chain, which is the target of SL-401.

The results reported here demonstrate SL-401 is highly potent against BPDCN cell lines and primary BPDCN blasts obtained from patients. Although AML cell lines and primary AML cells have demonstrated sensitivity to SL-401 with IC 50 values in the picomolar range (10 -12 M), [START_REF] Frankel | Phase I clinical study of diphtheria toxininterleukin 3 fusion protein in patients with acute myeloid leukemia and myelodysplasia[END_REF] which are lower than plasma concentrations achieved in leukemic patients undergoing treatment with SL-401, BPDCN blasts are more sensitive, with IC 50 values in the femtomolar range (10 -15 M, experimental dose 0 to 21 ng/mL). [START_REF] Angelot-Delettre | The IL-3Ra-targeted drug SL-401 selective-ly kills blastic plasmacytoid dendritic cell neoplasm cells[END_REF] In addition, SL-401 produced a robust antitumor effect in an in vivo xenograft model using human BPDCN cells. This also indicates a potential good therapeutic index, as well as systemic activity, since mice survived more than 40 days after SL-401 treatment. The high sensitivity of BPDCN to SL-401 and the potential good therapeutic index of this agent likely reflect the high specificity of the IL-3 ligand component of SL-401 for CD123, in addition to the mechanism of action and potency for its DT payload. Since the IL-3 component of SL-401 is bound via an amino acid linker to a DT for which the receptor binding site is truncated, free DT is essentially inert from a toxicity standpoint. DT can only be delivered intracellularly following the binding of SL-401 to the IL-3R via IL-3 and internalization. Since IL-3Ra expression is limited to only a few normal tissues (plasmacytoid dendritic cells and basophils) and, in contrast, the receptor is overexpressed by BPDCN cells, SL-401 can potentially confer high therapeutic indexes for patients. Additionally, SL-401 is not a substrate for p-glycoprotein and other efflux pumps, and thus, cannot be excluded from the blastic cells. Moreover, its mechanism, binding to ADPribosylated elongation factor 2, thereby uncoupling protein synthesis blockade, does not overlap with other agents currently used. [START_REF] Kreitman | Recombinant immunotoxins containing truncated bacterial toxins for the treatment of hematologic malignancies[END_REF] IL-3 has also been shown to be a critical survival factor for plasmacytoid dendritic cells. [START_REF] Grouard | The enigmatic plasmacytoid T cells develop into dendritic cells with interleukin (IL)-3 and CD40-ligand[END_REF] Thus, interference of this pathway by SL-401 may explain the high sensitivity of BPDCN compared with the sensitivity of other myeloid and lymphoid leukemic cells.

There is still no consensus on the best therapeutic approach for BPDCN, [START_REF] Jegalian | Blastic plasmacytoid dendritic cell neoplasm in children: diagnostic features and clinical implications[END_REF][START_REF] Dietrich | Blastic plasmacytoid dendritic cell neoplasia (BPDC) in elderly patients: results of a treatment algorithm employing allogeneic stem cell transplantation with reduced conditioning intensity[END_REF][START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF][START_REF] Gilis | Sequential combination of high dose methotrexate and L-asparaginase followed by allogeneic transplant: a first-line strategy for CD4+/CD56+ hematodermic neoplasm[END_REF][START_REF] Pagano | Blastic plasmacytoid dendritic cell neoplasm with leukemic presentation: an Italian multicenter study[END_REF][START_REF] Amor | Successful non-myeloablative allogeneic HLA-identical stem cell transplantation for CD4/CD56 positive acute leukemia[END_REF][START_REF] Fontaine | Haematodermic CD4+CD56+ neoplasm: complete remission after methotrexateasparaginase treatment[END_REF][START_REF] Gruson | Lasparaginase with methotrexate and dexamethasone is an effective treatment combination in blastic plasmacytoid dendritic cell neoplasm[END_REF][START_REF] Leitenberger | CD4+ CD56+ hematodermic/plasmacytoid dendritic cell tumor with response to pralatrexate[END_REF][START_REF] Narita | Plasmacytoid dendritic cell leukemia with potent antigen-presenting ability[END_REF] and, overall, BPDCN remains a chemotherapy-resistant disease and may also resist the graft-versus-leukemia effect, since 32% of patients relapse after allogeneic hematopoietic cell transplantation according to a recent study. [START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] Thus, targeted or immune-based therapies are alternative strategies to treat this aggressive leukemia. [START_REF] Agliano | Therapeutic effect of lenalidomide in a novel xenograft mouse model of human blastic NK cell lymphoma/blastic plasmacytoid dendritic cell neoplasm[END_REF][START_REF] Menezes | Exome sequencing reveals novel and recurrent mutations with clinical impact in blastic plasmacytoid dendritic cell neoplasm[END_REF][START_REF] Sapienza | Molecular profiling of blastic plasmacytoid dendritic cell neoplasm reveals a unique pattern and suggests selective sensitivity to NF-kB pathway inhibition[END_REF] Here, we propose that SL-401 is an efficient target-based therapy available for clinical trials and has already been shown to have favorable effects in patients with refractory or relapsed AML or myelodysplastic syndrome, although expression of CD123 is lower on myeloid blasts than on BPDCN cells. [START_REF] Frankel | Phase I clinical study of diphtheria toxininterleukin 3 fusion protein in patients with acute myeloid leukemia and myelodysplasia[END_REF] We recently published data from a phase I/II clinical study involving 11 BPDCN patients. These data showed that a single cycle of SL-401 induced major responses in 78% of the patients. [START_REF] Frankel | Activity of SL-401, a targeted therapy directed to the interleukin-3 receptor, in patients with blastic plasmacytoid dendritic cell neoplasm patients[END_REF] The way to use SL-401 in BPDCN patients must, howe ver, be discussed in the light of data from literature obtained in such patients. SL-401 can be used to consolidate the effects of first-line chemotherapy, reducing the number of relapses, which always occur after chemothe rapy treatment. The combination of SL-401 with chemotherapy should make it possible to reduce chemotherapy doses, and consequently, their adverse effects, which are significant in elderly patients with comorbidities (i.e., most of the patients with BPDCN). We showed here that idarubicin, at the dose we tested, induces a relevant level of cytotoxic activity in vitro whereas cytosine arabinoside does not. This confirms recent data showing that the BPDCN cell line CAL-1 is resistant to cytosine arabinoside. [START_REF] Sapienza | Molecular profiling of blastic plasmacytoid dendritic cell neoplasm reveals a unique pattern and suggests selective sensitivity to NF-kB pathway inhibition[END_REF] Intrathecal injection of SL-401 could also be of interest since there are frequently patients with central nervous system relapse [START_REF] Hertler | An immunotoxin for the treatment of T-acute lymphoblastic leukemic meningitis: studies in rhesus monkeys[END_REF] and the molecular weight of SL-401 (57 KDa) predicts no diffusion across the blood-brain barrier. For patients who undergo allogeneic hematopoietic cell transplantation, SL-401 treatment can be used before allografting to minimize the level of minimal resid-ual disease, which is the most important prognostic factor in a recent study on allografted BPDCN patients, [START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] or as a consolidation treatment after allogeneic hematopoietic cell transplantation. [START_REF] Roos-Weil | Stem cell transplantation can provide durable disease control in blastic plasmacytoid dendritic cell neoplasm: a retrospective study from the European Group for Blood and Marrow Transplantation[END_REF] In support of this latter use, we observed that, in vitro, the blastic cells from a relapsing BPDCN patient were still sensitive to SL-401-mediated death. Moreover, the IL-3Ra chain (CD123) was still expressed on the surface at relapse (CD123 MFI at diagnosis: 5509 versus 5137 at relapse, patient #10). In the mouse model after one course of SL-401, all the BPDCN cells at relapse also expressed CD123. This supports the hypothesis that patients suffering from BPDCN can be treated with SL-401 as first-line or second-line therapy and, maybe, even with several courses of SL-401. Overall, immune-based therapy using SL-401 appears to be an appropriate way to treat BPDCN patients. New approaches based on immunomodulators [START_REF] Sapienza | Molecular profiling of blastic plasmacytoid dendritic cell neoplasm reveals a unique pattern and suggests selective sensitivity to NF-kB pathway inhibition[END_REF] or demethylating agents [START_REF] Menezes | Exome sequencing reveals novel and recurrent mutations with clinical impact in blastic plasmacytoid dendritic cell neoplasm[END_REF] must be further evaluated and compared -or associated -with SL-401.

In conclusion, we demonstrate that clinical grade SL-401, which specifically targets IL-3R, efficiently kills primary BPDCN cells in culture and significantly improves the overall survival of mice inoculated with BPDCN receiving a single cycle of SL-401. This provides a strong rationale for the use of SL-401 in the treatment of patients suffering from BPDCN. As BPDCN is a rare subtype of leukemia, an international clinical trial using SL-401 should now be conducted to validate these results prospectively.

Figure 1 .

 1 Figure 1. Sensitivity of BPDCN cells to SL-401-mediated death. (A) A representative experiment is shown. Upper panels: dot plots showing the staining of annexin V (AV FITC ) and 7-AAD, on the x-and y-axes, respectively, as assessed by flow cytometry (FC), on IL-3R non-expressing Daudi cells (negative control) and CAL-1 cells (established BPDCN patient-derived cell line) after treatment with SL-401 (365 pM) for 18 h; lower panels: primary blasts from BPDCN patient #1 after treatment with SL-401 (365 pM, right hand side panel) for 18 h or no drug treatment (no SL-401, left hand side panel). The percentages indicated in each dot plot represent viable cells (AV -/7-AAD -cells). (B) The percent viability (mean ± SEM) of the BPDCN cell lines CAL-1 (n=3) and GEN 2.2 (n=6) after treatment with SL-401 at different concentrations ranging from 0.08 fM to 365 pM (0-21 ng/mL) for 18 h. The Daudi cell line was used as a negative control. (C) Each gray line represents the percentage viability of primary blasts (AV -/7-AAD -cells) isolated from different BPDCN patients (#1-12) according to different SL-401 concentrations (from 0.08 fM to 365 pM = 0-21 ng/mL) for 18 h. The black line represents the mean of BPDCN patients' samples as a function of SL-401 concentration; the Daudi cell line was used as a negative control. (D) Viability assessed using the MTT assay: the percentage viability of primary blasts from BPDCN patients #3 and #4 and of CAL-1 and GEN 2.2 BPDCN cell lines is dependent on SL-401 concentrations. The Daudi cell line was used as a negative control and is insensitive to SL-401-mediated death whatever the concentration of the drug used. Cells were treated with SL-401 for 48 h. The values represent the results of one experiment for patients' samples or three independent experiments for cell lines. (E) Percentage (mean ± SEM) of viable primary BPDCN cells (AV -/7-AAD -cells) from patient #10 at diagnosis (n = 3) and at relapse (n=6) after incubation for 18 h with different concentrations of SL-401 or without any drug, as assessed using annexin V and 7-AAD staining and flow cytometry (FC). Untreated cells were considered as 100% viable (P=0.049).

  , P<0.93). For patient #10, the MFI of CD123 was quite similar both at diagnosis and relapse (5509 versus 5137, respectively).

Figure 2 .

 2 Figure 2. Sensitivity of primary BPDCN cells to SL-401 and other relevant chemotherapeutic agents. The mean percentages of viable cells (AV -/7-AAD -cells) from three patients (# 7, 9 and 11) after treatment with with SL-401 (365 pM), Erwinia L-asparaginase (L-ASP, 10 IU/mL), asparaginase (ASP, 10 IU/mL), methotrexate (MTX, 9.9 mM), cyclophosphamide (CYC, 100 IU/mL), cytosine arabinoside (CYT, 0.329 mM), dexamethasone (DEX, 0.637 mM), vincristine (VIN, 0.0242 mM) and idarubicin (IDA, 0.158 mM). Untreated cells were considered as 100% viable. Histograms represent the mean ± SEM of three independent experiments (*P<0.05 between SL-401 and all other drugs pairwise except IDA).

Figure 3 .

 3 Figure 3. Comparison of sensitivity of primary blasts from patients with BPDCN, acute lymphoid leukemia (ALL), and acute myeloid leukemia (AML) to SL-401-mediated death. Leukemia blasts sampled from three patients with ALL (2 B-ALL and 1 T-ALL) and six patients with AML (1 case of AML with multilineage dysplasia, 2 cases of M1 AML, 1 case of M0 AML, and 2 cases of M2 AML) and blasts from 11 BPDCN patients (#1-10 and 12) were cultured with or without SL-401 (365 pM) for 18 h. The viability was assessed by flow cytometry (AV -/7-AAD -cells). The bars represent the mean ± SEM of the percentage viable blastic cells, with untreated cells considered 100% viable. **P<0.001.

  Expression of IL-3Ra and β chains (CD123 and CD131) -measured by flow cytometry-in BPDCN cells from patients (# 1-4, 7, 9 and 10), or in different AML, ALL and cell lines (results represent the mean of 2 determinations). Percentage of viability after culture with SL-401 (experi-mental dose 365 pM= 21 ng/mL) in one experiment (▪), mean±SEM of two ( * ), three ( °) or six (#) indepen dent evaluations and IC50 are indicated. AML MLD: AML with multi lineage dysplasia; NA: not available.

F

  . Angelot-Delettre et al. 228 haematologica | 2015; 100(2)

Figure 4 .

 4 Figure 4. In vivo efficacy of SL-401 in a NSG mouse model inoculated with BPDCN cells. NSG mice were irradiated with 2 Gy and then inoculated intravenously with 1x10 6 GEN2.2 BPDCN cells on day 0. (A) Overall survival (OS) of BPDCN inoculated-mice treated with SL-401 (solid line; n=4) or with PBS (dotted line; n = 3). Treatment with SL-401 (2 mg/mouse intraperitoneally, experimental dose 100 mg/kg) performed daily for 5 days, was begun on day 7 (pink bar). OS from one representative experiments out of three is shown (P=0.3). (B) Mean OS of BPDCN inoculated-mice treated with SL-401 (n=8) or PBS (n=8) from three independent experiments (**P<0.001). (C) one example of the immunostaining of circulating peripheral blood mononuclear cells performed at day 53 prior to sacrifice. Murine (blue) and human GEN 2.2 (red) cells are distinguishable due to specific CD45 antibody expression. Human GEN2.2 BPDCN cells express CD123, BDCA4, CD4, CD56, but not CD34 or CD3. (D) BPDCN cell count values in the blood of a mouse following treatment with SL-401 (dotted line, black circles) or with PBS (blue solid line, blue square). (E) Means of hemoglobin, as well as white blood cell and platelet counts in the blood of mice following treatment with SL-401 (n=4) or saline control (n=3). The gray line represents blood parameter values in irradiated mice that were not treated with SL-401. †: means that the mice died.

  INSERM UMR1098, F25020 Besançon Cedex, France ; 2 Université de Bourgogne Franche-Comté, SFR FED4234, F25000 Besançon Cedex, France ; 3 EFS Bourgogne Franche-Comté, F25020 Besançon Cedex, France ;[START_REF] Sakashita | Usefulness of allogeneic hematopoietic stem cell transplantation in first complete remission for pediatric blastic plasmacytoid dendritic cell neoplasm with skin involvement: a case report and review of literature[END_REF] LabEX LipSTIC, ANR-11-LABX-0021, F25020 Besançon Cedex, France; 5 Southwestern Medical Center, Dallas, TX, USA; 6 CHU Besançon, Hematology, France;[START_REF] Angelot-Delettre | Intracytoplasmic detection of TCL1--but not ILT7-by flow cytometry is useful for blastic plasmacytoid dendritic cell leukemia diagnosis[END_REF] Stemline Therapeutics, Inc, 750 Lexington Avenue, 11 th Floor, New York, USA;[START_REF] Garnache-Ottou | Extended diagnostic criteria for plasmacytoid dendritic cell leukaemia[END_REF] APHP, Hopital Cochin, Paris, France; 9 CHR Metz Thionville, France; 10 Institut Curie, Hopital René Huguenin, Saint Cloud, France; 11 CH Chartres, Le Coudray, France; 12 CH Mulhouse, France; 13 APHP Hopital Avicenne, Paris, France; 14 CHU Rennes, Rennes, France; 15 CHU St Etienne, Saint Etienne, France; 16 Université de Nice-Sophia Antipolis, Nice, France; 17 CHU Besançon, CIC1431, FHU INCREASE, Besançon, France

[START_REF] Chaperot | Identification of a leukemic counterpart of the plasmacytoid dendritic cells[END_REF]

Table 1 .

 1 Characteristics of the 12 patients suffering from blastic plasmacytoid dendritic cell neoplasm (BPDCN) who provided primary BPDCN cells.

	Patient Age/	Immunophenotype	Karyotype	Cutaneous	Pathology	Sites of disease
	number gender	(flow cytometry)		lesions	diagnosis	involvement
	#1	75/M	CD4 + CD56 + CD123 ++ CD303 + CD304 +	45,XY,-5 [3], 46XY [10]	no	NA	LN, BM blood,
			My -B -CD7 + (others T -)				spleen
	#2	59/M	CD4 + CD56 + CD123 ++ CD303 -CD304 +	46,XY,del(6q)(q16q25),der(7)t(7;12)	yes	BPDCN	LN, BM skin
			CD33 + (others My -) B -CD7 + (others T -)	(p11;q1?3),?del(7)(p21),-8x2,-9x2,-12x2,
				+mar [4], 46XY [11]
	#3	81/M	CD4 + CD56 + CD123 ++ CD303 + CD304 +	46XY	yes	BPDCN	LN, BM, skin, blood
			CD33 + (others My -) CD22 + (others B -)CD2 +
			(others T -)
	#4	69/M	CD4 + CD56 + CD123 ++ CD303 + CD304 +	46XY	yes	BPDCN	skin, BM
			CD33 + (others My -)CD22 + (others B -) CD7 +
			(others T -)
	#5	63/M	CD4 + CD56 + CD123 ++ CD303 + CD304 3]	yes	BPDCN	LN, BM skin blood,
			TCL1 + My -B -T -				spleen
	#6	56/M	CD4 + CD56 + CD123 ++ CD303 -CD304 +	NA	yes	NA	BM, skin, blood
			TCL1

+ 44,X,-Y,-13[3]/44,sl,i(7)(q10)

[15]

/45,XY,-15[

Acknowledgments

This work was supported by grants from the University of Franche-Comté (BQR25JC), La Ligue Contre le Cancer (116AD.2010), the Agence Nationale de la Recherche (Labex LipSTIC, ANR-11-LABX-0021) and the Conseil Régional de Franche-Comté ("Soutien au LabEX LipSTIC" to PS). We would like to thank Sophie Perrin and the Pharmacy Department (CHRU Besançon) for their support in providing the chemotherapeutic drugs; Laboratory of Cytology (EFS BFC, Dr Françoise Schillinger); Dr Francis Bonnefoy and all the biologists and physicians who participate in the French BPDCN network.

Authorship and Disclosures Information on authorship, contributions, and financial & other disclosures was provided by the authors and is available with the online version of this article at www.haematologica.org.

IL-3R-targeted treatment of BPDCN haematologica | 2015; 100 (2)


Anne-Laure Basdevant
email: anne-laure.basdevant@ens.fr

On the equivalence of some eternal additive coalescents

Keywords: Additive coalescent, fragmentation process A, M, S, Classification, 60

research documents, whether they are published or not. The documents may come

Introduction

The paper deals with additive coalescent processes, a class of Markov processes which have been introduced first by Evans and Pitman [START_REF] Evans | Construction of Markovian coalescents[END_REF]. In the simple situation of a system initially composed of a finite number k of clusters with masses m 1 , m 2 , . . . , m k , the dynamics are such that each pair of clusters (m i , m j ) merges into a unique cluster with mass m i + m j at rate m i + m j , independently of the other pairs. In the sequel, we always assume that we start with a total mass equal to 1 (i.e. m 1 + . . . + m k = 1). This induces no loss of generality since we can then deduce the law of any additive coalescent process through a time renormalization. Hence, an additive coalescent lives on the compact set

S ↓ = {x = (x i ) i≥1 , x 1 ≥ x 2 ≥ . . . ≥ 0, i≥1 x i ≤ 1},
endowed with the topology of uniform convergence.

Evans and Pitman [START_REF] Evans | Construction of Markovian coalescents[END_REF] proved that we can define an additive coalescent on the whole real line for a system starting at time t = -∞ with an infinite number of infinitesimally small clusters. Such a process will be called an eternal coalescent process. More precisely, if we denote by (C n (t), t ≥ 0) the additive coalescent starting from the configuration (1/n, 1/n, . . . , 1/n), they proved that the sequence of processes (C n (t + 1 2 ln n), t ≥ -1 2 ln n) converges in distribution on the space of càdlàg paths with values in the set S ↓ toward some process (C ∞ (t), t ∈ R), which is called the standard additive coalescent. We stress that this process is defined for all time t ∈ R. A remarkable property of the standard additive coalescent is that, up to timereversal, its becomes a fragmentation process. Namely, the process (F (t), t ≥ 0) defined by F (t) = C ∞ (-ln t) is a self-similar fragmentation process with index of self similarity α = 1/2, with no erosion and with dislocation measure ν given by ν(x 1 ∈ dy) = (2πy 3 (1 -y) 3 ) -1/2 dy for y ∈]1/2, 1[, ν(x 3 > 0) = 0.

We refer to Bertoin [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF] for the definition of erosion, dislocation measure, and index of self similarity of a fragmentation process and a proof. Just recall that in a fragmentation process, distinct fragments evolve independently of each others.

Aldous and Pitman [START_REF] Aldous | The standard additive coalescent[END_REF] constructed this fragmentation process (F (t), t ≥ 0) by cutting the skeleton of the continuum Brownian random tree according to a Poisson point process. In another paper [START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF], they gave a generalization of this result: consider for each n ∈ N a decreasing sequence r n,1 ≥ . . . ≥ r n,n ≥ 0 with sum 1, set σ 2 n = n i=1 r 2 n,i and suppose that lim n→∞ σ n = 0 and lim n→∞ r n,i σ n = θ i for all i ∈ N.

Assume further that

i θ 2 i < 1 or i θ i = ∞. Then, it is proved in [2] that if M n = (M n (t), t ≥ 0)
denotes the additive coalescent process starting with n clusters with mass r n,1 ≥ . . . ≥ r n,n , then (M (n) (t -ln σ n ), t ≥ ln σ n ) has a limit distribution as n → ∞, which can be obtained by cutting a specific inhomogeneous random tree with a point Poisson process. Furthermore, any extreme eternal additive coalescent can be obtained this way up to a deterministic time translation.

Bertoin [START_REF] Bertoin | Eternal additive coalescents and certain bridges with exchangeable increments[END_REF] gave another construction of the limit of the process (M (n) (t -ln σ n ), t ≥ ln σ n ) in the following way. Let b θ be the bridge with exchangeable increments defined for s ∈ [0, 1] by

b θ (s) = σb s + ∞ i=1 θ i (1l {s≥V i } -s), where (b s , s ∈ [0, 1]) is a standard Brownian bridge, (V i ) i≥1 is an i.i.d. sequence of uniform random variable on [0,1] independent of b and σ = 1 -i θ 2 i . Let ε θ = (ε θ (s), s ∈ [0, 1]) be the excursion obtained from b θ by Vervaat's transform, i.e. ε θ (s) = b θ (s + m mod 1) -b θ (m),
where m is the point of [0,1] where b θ reaches its minimum. For all t ≥ 0, consider

ε (t) θ (s) = ts -ε θ (s), S (t)
θ (s) = sup 0≤u≤s ε (t) θ (u),
and define F θ (t) as the sequence of the lengths of the constancy intervals of the process (S (t) θ (s), 0 ≤ s ≤ 1). Then the limit of the process (M (n) (t -ln σ n ), t ≥ ln σ n ) has the law of (F θ (e -t ), t ∈ R). Miermont [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF] studied the same process in the special case where ε θ is the normalized excursion above the minimum of a spectrally negative Lévy process. More precisely let (X t , t ≥ 0) be a Lévy process with no positive jump, with unbounded variation and with positive and finite mean. Let X(t) = sup 0≤s≤t X t and denote by ε X = (ε X (s), s ∈ [0, 1]) the normalized excursion with duration 1 of the reflected process X -X. We now define in the same way as for b θ , the processes ε

(t) X (s), S (t)
X (s) and F X (t). Then, the process (F X (e -t ), t ∈ R) is a mixture of some eternal additive coalescents (see [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF] for more details). Furthermore, (F X (t), t ≥ 0) is a fragmentation process in the sense that distinct fragments evolve independently of each other (however, it is not necessarily homogeneous in time). It is quite remarkable that the Lévy property of X ensures the branching property of F X . We stress that there exist other eternal additive coalescents for which this property fails. Notice that when the Lévy process X is the standard Brownian motion B, the process (F B (e -t ), t ∈ R) is then the standard additive coalescent and (F B (t), t ≥ 0) is a self-similar and time-homogeneous fragmentation process.

In this paper, we study the relationship between the laws P (X) of (F X (t), t ≥ 0) and P (B) of (F B (t), t ≥ 0). We prove that, for certain Lévy processes (X t , t ≥ 0), the law P (X) is absolutely continuous with respect to P (B) and we compute explicitly the density. Our main result is the following: Theorem 1.1. Let (Γ(t), t ≥ 0) be a subordinator with no drift. Assume that E(Γ 1 ) < ∞ and take any c ≥ E(Γ 1 ). We define X t = B t -Γ t + ct, where B denotes a Brownian motion independent of Γ. Let (p t (u), u ∈ R) and (q t (u), u ∈ R) stand for the respective density of B t and X t . In particular

p t (u) = 1 √ 2πt exp(-u 2 2t
). Let S 1 be the space of positive sequences with sum 1. We consider the function h : R + × S 1 defined by

h(t, x) = e tc p 1 (0) q 1 (0) ∞ i=1 q x i (-tx i ) p x i (-tx i ) with x = (x i ) i≥1 .
Then, for all t ≥ 0, the function h(t, •) is bounded on S 1 and has the following properties:

• h(t, F (t)) is a P (B) -martingale,
• for every t ≥ 0, the law of the process (F X (s), 0 ≤ s ≤ t) is absolutely continuous with respect to that of (F B (s), 0 ≤ s ≤ t) with density h(t, F B (t)).

Let us notice that h(t, •) is a multiplicative function, i.e. it can be written as the product of functions, each of them depending only on the size of a single fragment. In the sequel we will use the notation h(t, x) = e tcx p 1 (0) q 1 (0)

x q x (-tx) p x (-tx) for x ∈]0, 1] and t ≥ 0, so we have h(t, x) = i h(t, x i ). This multiplicative form of h(t, •) implies that the process F X has the branching property (i.e. distinct fragments evolve independently of each other) since F B has it. Indeed, for every multiplicative bounded continuous function f : S ↓ → R + , for all t ′ > t > 0 and x ∈ S ↓ , we have, since h(t, F B (t)) is a P (B) -martingale,

E (X) f (F (t ′ )) F (t) = x = 1 h(t, x) E (B) h(t ′ , F (t ′ ))f (F (t ′ )) F (t) = x .
Using the branching property of F B and the multiplicative form of h(t, •), we get

E (X) f (F (t ′ )) F (t) = x = 1 h(t, x) i E (B) h(t ′ , F (t ′ ))f (F (t ′ )) F (t) = (x i , 0, . . .) .
And finally we deduce

E (X) f (F (t ′ )) F (t) = x = 1 h(t, x) i h(t, x i )E (X) f (F (t ′ )) F (t) = (x i , 0, . . .) = i E (X) f (F (t ′ )) F (t) = (x i , 0, . . .) .
Let M x (resp. M x i ) be the random measure on ]0,1[ defined by M x = i δ s i where the sequence (s i ) i≥1 has the law of F (t ′ ) conditioned on F (t) = x (resp. F (t) = (x i , 0, . . .)). Hence we have, for every bounded continuous function g : R → R,

E exp(-< g, M x >) = ∞ i=1 E exp(-< g, M x i >) ,
which proves that M x has the law of i M x i where the random measures (M x i ) i≥1 are independent. Hence the process F X has the branching property. Notice also that other multiplicative martingales have already been studied in the case of branching random walks [START_REF] Biggins | Martingale convergence in the branching random walk[END_REF][START_REF] Chauvin | KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees[END_REF][START_REF] Neveu | Multiplicative martingales for spatial branching processes[END_REF][START_REF] Kyprianou | Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis[END_REF]. This paper will be divided in two sections. The first section is devoted to the proof of this theorem and in the next one, we will use the fact that h(t, F B (t)) is a P (B) -martingale to describe an integro-differential equation solved by the function h.

Proof of Theorem 1.1

The assumptions and notation in Theorem 1.1 are implicitly enforced throughout this section.

Absolute continuity

In order to prove Theorem 1.1, we will first prove the absolute continuity of the law P (X) t of F X (t) with respect to the law P (B) t of F X (t) for a fixed time t > 0 and for a finite number of fragments. We begin first by a definition: Definition 2.1. Let x = (x 1 , x 2 , . . .) be a sequence of positive numbers with sum 1. We call the random variable y = (x j 1 , x j 2 , . . .) a size biased rearrangement of x if we have:

∀i ∈ N, P(j 1 = i) = x i ,
and by induction

∀i ∈ N\{i 1 , . . . , i k }, P(j k+1 = i | j 1 = i 1 , . . . , j k = i k ) = x i 1 -k l=1 x i l .
Notice that for every Lévy process X satisfying hypotheses of Theorem 1.1, we have

∞ i=1 F i (t) = 1 P (X)
t -a.s. (it is clear by the construction from an excursion of X since X has unbounded variation, cf [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF], Section 3.2). Hence the above definition can be applied to F X (t).

The following lemma gives the distribution of the first n fragments of F X (t), chosen with a size-biased pick:

Lemma 2.2. Let ( F X 1 (t), F X 2 (t), . . .) be a size biased rearrangement of F X (t). Then for all n ∈ N, for all x 1 , . . . , x n ∈ R + such that S = n i=1 x i < 1, we have P (X) t ( F X 1 ∈ dx 1 , . . . , F X n ∈ dx n ) = t n q 1 (0) q 1-S (St) n i=1 q x i (-tx i ) 1 -i k=1 x k dx 1 . . . dx n .
Proof. On the one hand, Miermont [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF] gave a description of the law of F X (t): let T (t) be a subordinator with Lévy measure z -1 q z (-tz)1l z>0 dz. Then F X (t) has the law of the sequence of the jumps of T (t) before time t conditioned on T (t)

t = 1.
One the other hand, consider a subordinator T on the time interval [0, u] conditioned by T u = y and pick a jump of T by size-biased sampling. Then, its distribution has density

zuh(z)f u (y -z) yf u (y) dz,
where h is the density of the Lévy measure of T and f u is the density of T u (see Theorem 2.1 of [START_REF] Perman | Size-biased sampling of Poisson point processes and excursions[END_REF]). Then, in the present case, we have [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF]).

u = t, y = 1, h(z) = z -1 q z (-tz), f u (z) = u z q z (u -zt) (cf. Lemma 9 of
Hence we get

P (X) t ( F X 1 ∈ dz) = tq z (-tz)q 1-z (zt) (1 -z)q 1 (0) dz.
This proves the lemma in the case n = 1. The proof for n ≥ 2 uses an induction. Assume that we have proved the case n -1 and let us prove the case n. We have

P (X) t ( F X 1 ∈ dx 1 , . . . , F X n ∈ dx n ) = P (X) t ( F X 1 ∈ dx 1 , . . . , F X n-1 ∈ dx n-1 )P (X) t ( F X n ∈ dx n | F X 1 ∈ dx 1 , . . . , F X n-1 ∈ dx n-1 ).
Furthermore, Perman, Pitman and Yor [START_REF] Perman | Size-biased sampling of Poisson point processes and excursions[END_REF] have proved that the n-th size biased picked jump ∆ n of a subordinator before time u conditioned by T u = y and ∆

1 = x 1 , . . . , ∆ n-1 = x n-1
has the law of a size biased picked jump of the subordinator T before time u conditioned by

T u = y -x 1 -. . . -x n-1 .
Hence we get:

P (X) t ( F X 1 ∈ dx 1 , . . . , F X n ∈ dx n ) = t n-1 q 1 (0) q 1-S n-1 (S n-1 t) n-1 i=1 q x i (-tx i ) 1 -S i tq xn (-tx n )q 1-Sn (S n t) (1 -S n )q 1-S n-1 (S n-1 t) dx 1 . . . dx n ,
where S i = i k=1 x k . And so the lemma is proved by induction.

Since the lemma is clearly also true for P (B) (take Γ = c = 0), we get:

Corollary 2.3. Let (F (t), t ≥ 0) be a fragmentation process. Let ( F1 (t), F2 (t), . . .) be a size biased rearrangement of F (t). Then for all n ∈ N, for all x 1 , . . . ,

x n ∈ R + such that S = n i=1 x i < 1, we have P (X) t ( F1 ∈ dx 1 , . . . , Fn ∈ dx n ) P (B) t ( F1 ∈ dx 1 , . . . , Fn ∈ dx n ) = h n (t, x 1 , . . . , x n ), with h n (t, x 1 , . . . , x n ) = p 1 (0) q 1 (0) q 1-S (St) p 1-S (St) n i=1 q x i (-tx i ) p x i (-tx i ) .
To establish that the law of F X (t) is absolutely continuous with respect to the law of F B (t) with density h(t, •), it remains to check that the function h n converges as n tends to infinity to h P (B) t -a.s. and in L 1 (P (B) t ). In this direction, we first prove two lemmas: Lemma 2.4. We have qy(-ty) py(-ty) < 1 for all y > 0 sufficiently small. As a consequence, if (x i ) i≥1 is a sequence of positive numbers with lim i→∞ x i = 0, then the product n i=1 qx i (-tx i ) px i (-tx i ) converges as n tends to infinity.

Proof. Since X t = B t -Γ t + tc, notice that we have

∀s > 0, ∀u ∈ R, q s (u) = E p s (u + Γ s -cs) .
Hence if we replace p s (u) by its expression 1 √ 2πs exp(-u 2 2s ), we get

q s (u) p s (u) = exp cu - c 2 s 2 E exp - Γ 2 s 2s -Γ s ( u s -c) . (1)
i.e., for all y > 0, for all t ≥ 0,

q y (-ty) p y (-ty) = exp -y(ct + c 2 2 ) E exp - Γ 2 y 2y + Γ y (t + c) . Using the inequality (c -a)(c -b) ≥ -b-a 2 2
, we have

- Γ 2 y 2y + Γ y (t + c) ≤ y(t + c) 2 2
and we deduce q y (-ty)

p y (-ty) ≤ e t 2 y 2 .
Fix c ′ ∈]0, c[, let f be the function defined by f (y) = P(Γ y ≤ c ′ y). Since Γ t is a subordinator with no drift, we have lim y→0 f (y) = 1 (indeed, Γ y = o(y) a.s., see [START_REF] Bertoin | Lévy processes[END_REF]). On the event {Γ y ≤ c ′ y}, we have

exp -y(ct + c 2 2 ) exp - Γ 2 y 2y + Γ y (t + c) ≤ exp(-y( 1 2 (c -c ′ ) 2 + t(c -c ′ )))
≤ exp(-εy),

with ε = 1 2 (c -c ′ ) 2 .
Hence, we get the upper bound q y (-ty)

p y (-ty) ≤ e -εy f (y) + (1 -f (y))e yt 2 2 .
Since f (y) → 1 as y → 0, we deduce

e -εy f (y) + (1 -f (y))e yt 2 2 = 1 -εy + o(y).
Thus, we have qy(-ty) py(-ty) < 1 for y small enough, and so the product converges for every sequence (x i ) i≥0 which tends to 0.

We prove now a second lemma: Lemma 2.5. We have lim

s→1 - q 1-s (st) p 1-s (st) = e tc .
Proof. We use again Identity (1) established in the proof of Lemma 2.4. We get:

q 1-s (st) p 1-s (st) = exp tsc - c 2 2 (1 -s) E exp - Γ 2 1-s 2(1 -s) -Γ 1-s ( ts 1 -s -c) .
For s close enough to 1, ts 1-s -c ≥ 0, hence we get

E exp - Γ 2 1-s 2(1 -s) -Γ 1-s ( ts 1 -s -c) ≤ 1
and we deduce lim sup

s→1 - q 1-s (st) p 1-s (st) ≤ e tc .
For the lower bound, we write

E exp - Γ 2 1-s 2(1 -s) -Γ 1-s ( ts 1 -s -c) ≥ E exp - Γ 1-s 2(1 -s) -Γ 1-s ( ts 1 -s -c) 1l {Γ 1-s ≤1} ≥ E exp -Γ 1-s 1 + 2ts 2(1 -s) 1l {Γ 1-s ≤1} ≥ E exp -Γ 1-s 1 + 2ts 2(1 -s) -P(Γ 1-s ≥ 1).
Since Γ t is a subordinator with no drift, lim u→0 Γu u = 0 a.s., and we have for all K > 0, lim

u→0 + E exp -K Γ u u = 1.
Hence, we get lim inf

s→1 - q 1-s (st) p 1-s (st) ≥ e tc .
We are now able to prove the absolute continuity of P (X) t

with respect to P (B)

t . Since S n = n i=1 x i converges P (B)
t -a.s. to 1, Lemma 2.4 and 2.5 imply that H n = h n (t, F1 (t), . . . , Fn (t)) converges to H = h(t, F (t)) P (B) -a.s.

Let us now prove that H n is uniformly bounded, which implies the L 1 convergence. We have already proved that there exists ε > 0 such that:

∀x ∈]0, ε[, q x (-tx) p x (-tx) ≤ 1.
Besides, it is well known that, if

X t = B t -Γ t + ct, its density (t, u) → q t (u) is continuous on R * + × R. Hence, on [ε, 1], the function x → qx(-tx) px(-tx
) is continuous and we can find an upper bound A > 0 of this function . As there are at most 1 ε fragments of F (t) larger than ε, we deduce the upper bound:

∞ i=1 q F i (-tF i ) p F i (-tF i ) ≤ A 1 ε .
Likewise, the function S → q 1-S (St) St) is continuous on [0, 1[ and has a limit at 1, so it is bounded by some D > 0 on [0, 1]. Hence we get

p 1-S (
H n ≤ A 1 ε D p 1 (0) q 1 (0) P (B) -a.s.
So H n converges to H P (B) -a.s. and in L 1 (P (B) ). Furthermore, by construction, H n is a P (B)martingale, hence we get for all n ∈ N,

E (B) (H | F1 , . . . , Fn ) = H n ,
and so, for every bounded continuous function f : S 1 → R, we have

E (X) f (F (t)) = E (B) f (F (t))h(t, F (t)) .
Hence, we have proved that, for a fixed time t ≥ 0, the law of F X (t) is absolutely continuous with respect to that of F B (t) with density h(t, F B (t)). Furthermore, Miermont [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF] has proved that the processes (F X (e -t ), t ∈ R) and (F B (e -t ), t ∈ R) are both eternal additive coalescents (with different entrance laws). Hence, they have the same semi-group of transition and we get the absolute continuity of the law of the process (F X (s), 0 ≤ s ≤ t) with respect to that of (F B (s), 0 ≤ s ≤ t) with density h(t, F B (t)).

Sufficient condition for equivalence

We can now wonder whether the measure P (X) is equivalent to the measure P (B) , that is whether h(t, F (t)) is strictly positive P (B) -a.s. A sufficient condition is given by the following proposition. Proposition 2.6. Let φ be the Laplace exponent of the subordinator Γ, i.e. ∀s ≥ 0, ∀q ≥ 0, E(exp(-qΓ s )) = exp(-sφ(q)).

Assume that there exists δ > 0 such that

lim x→∞ φ(x)x δ-1 = 0, (2)
then the function h(t, F (t)) defined in Theorem 1.1 is strictly positive P (B) -a.s.

We stress that the condition 2 is very weak. For instance, let π be the Lévy measure of the subordinator and I(x) = x 0 π(t)dt where π(t) denotes π(]t, ∞[). It is well known that φ(x) behaves like xI(1/x) as x tends to infinity (see [START_REF] Bertoin | Lévy processes[END_REF] Section III). Thus, the condition 2 is equivalent to I(x) = o(x δ ) as x tends to 0 (recall that we always have

I(x) = o(1)). Proof. Let t > 0. We must check that ∞ i=1 qx i (-tx i ) px i (-tx i ) is P (B)
t -almost surely strictly positive. Using (1), we have:

q y (-ty) p y (-ty) = exp -y(ct + c 2 2 ) E exp - Γ 2 y 2y + Γ y (t + c) . Since we have ∞ i=1 x i = 1 P (B)
t -a.s., we get

∞ i=1 q x i (-tx i ) p x i (-tx i ) ≥ exp -ct + c 2 2 ∞ i=1 E exp - Γ 2 x i 2x i + cΓ x i .
Hence we have to find a lower bound for E exp -

Γ 2 y 2y + cΓ y . Since c ≥ E(Γ 1 ), we have E exp - Γ 2 y 2y + cΓ y ≥ E exp Γ y y (E(Γ y ) - Γ y 2
) .

Set A = E(Γ 1 ) and let us fix K > 0. Notice that the event E(Γ y ) -Γy 2 ≥ -Ky is equivalent to the event Γ y ≤ (2A + K)y and by Markov inequality, we have

P(Γ y ≥ (2A + K)y) ≤ A 2A + K .
Hence we get

E exp - Γ 2 y 2y + cΓ y ≥ E exp Γ y y (E(Γ y ) - Γ y 2 )1l {Γy≤(2A+K)y} ≥ E exp(-KΓ y )1l {Γy≤(2A+K)y} ≥ E (exp(-KΓ y )) -E exp(-KΓ y )1l {Γy>(2A+K)y} ≥ exp(-φ(K)y) - A 2A + K .
This inequality holds for all K > 0. Hence, with ε > 0 and K = y -1 2 -ε , we get

E exp - Γ 2 y 2y + cΓ y ≥ exp -φ(y -1 2 -ε )y -Ay 1 2 +ε .
Furthermore, the product ∞ i=1 E exp -

Γ 2 x i 2x i + cΓ x i is strictly positive if the series ∞ i=1 1 -E exp - Γ 2 x i 2x i + cΓ x i converges. Hence, a sufficient condition is ∃ ε > 0 such that ∞ i=1 1 -exp -φ(x -1 2 -ε i )x i + x 1 2 +ε i < ∞ P (B) t -a.s.
Recall that the distribution of the Brownian fragmentation at time t is equal to the distribution of the jumps of a stable subordinator T with index 1/2 before time t conditioned on T t = 1 (see [START_REF] Aldous | The standard additive coalescent[END_REF]). Hence, it is well known that we have for all ε > 0

∞ i=1 x 1 2 +ε i < ∞ P (B)
t -a.s. (see Formula ( 9) of [START_REF] Aldous | The standard additive coalescent[END_REF]). Thus, we have equivalence between P (B) t and P (X) t as soon as there exist two strictly positive numbers ε, ε ′ such that, for x small enough

φ(x -1 2 -ε )x ≤ x
In Theorem 1.1, we have supposed that X t can be written as B t + Γ t -ct, with c ≥ E(Γ 1 ) and Γ t subordinator. We can wonder whether the theorem applies for a larger class of Lévy processes. Notice first that the process X must fulfill the conditions of Miermont's paper [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF] recalled in the introduction, i.e. X has no positive jumps, unbounded variation and finite and positive mean. Hence, a possible extension of the Theorem would be for example for X t = σ 2 B t + Γ t -ct, with σ > 0, σ = 1. In fact, it is clear that Theorem 1.1 fails in this case. Let just consider for example X t = 2B t . Using Proposition 3 of [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps[END_REF], we get that

(F X (2t), t ≥ 0) law = (F B (t), t ≥ 0).
But, it is well known that we have lim n→∞ n 2 F ↓ n (t) = t 2/π P (B) -a.s. (see [START_REF] Bertoin | On small masses in self-similar fragmentations[END_REF])

Hence, the laws P (B) t

and

P (B)
2t are mutually singular.

An integro-differential equation

Since h(t, F (t)) is the density of P (X) with respect to P (B) on the sigma-field F t = σ(F (s), s ≤ t), it is a P (B) -martingale. Hence, in this section, we will compute the infinitesimal generator of a fragmentation to deduce a remarkable integro-differential equation.

The infinitesimal generator of a fragmentation process

In this section, we recall a result obtained by Bertoin and Rouault in an unpublished paper [START_REF] Bertoin | Note sur les fragmentations[END_REF]. We denote by D the space of functions f : [0, 1] →]0, 1] of class C 1 and with f (0) = 1. For f ∈ D and x ∈ S ↓ , we set

f (x) = ∞ i=1 f (x i ).
For α ∈ R + and ν measure on S ↓ such that S ↓ (1 -x 1 )ν(dx) < ∞, we define the operator

G α f (x) = f (x) ∞ i=1 x α i ν(dy) f (x i y) f (x i ) -1 for f ∈ D and x ∈ S ↓ .
Proposition 3.1. Let (X(t), t ≥ 0) be a self-similar fragmentation with index of self-similarity α > 0, dislocation measure ν and no erosion. Then, for every function f ∈ D, the process

f (X(t)) - t 0 G α f (X(s))ds is a martingale.
Proof. We will first prove the following lemma

Lemma 3.2. For f ∈ D, y ∈ S ↓ , r ∈ [0, 1], we have f (ry) f (r) -1 ≤ 2C f e C f r(1 -y 1 ), with C f = f ′ f 2 ∞ .
Notice that, since f is C 1 on [0, 1] and strictly positive, C f is always finite.

Proof. First, we write

| ln f (ry 1 ) -ln f (r)| ≤ f ′ f ∞ (1 -y 1 )r ≤ C f (1 -y 1 )r.
We deduce then

f (ry) f (r) -1 ≤ f (ry 1 ) f (r) -1 ≤ e C f (1-y 1 )r -1 ≤ C f e C f (1 -y 1 )r.
Besides we have

ln 1 f (x 1 ) ≤ 1 f (x i ) -1 ≤ C f x i , which implies f (x) ≥ f (x 1 ) exp(-C f ∞ i=2 x i ).
Hence we get f (ry)

f (r) ≥ f (ry 1 ) f (r) exp(-C f (1 -y 1 )r) ≥ exp(-2C f (1 -y 1 )r),
and we deduce

1 - f (ry) f (r) ≤ 2C f e C f (1 -y 1 )r.
We can now prove Proposition 3.1. We denote by T the set of times where some dislocation occurs (which is a countable set). Hence we can write

f (X(t)) -f (X(0)) = s∈[0,t]∩T f (X(s)) -f (X(s-)) , as soon as s∈[0,t]∩T f (X(s)) -f (X(s-)) < ∞
For s ∈ T , if the i-th fragment X i (s-) is involved in the dislocation, we set k s = i and we denote by ∆ s the element of S ↓ according to X(s-) has been broken. Hence, we have

s∈[0,t]∩T f (X(s)) -f (X(s-)) = s∈T ∩[0,t] f (X(s-)) ∞ i=1 1l ks=i f (X i (s-)∆ s ) f (X i (s-)) -1 .
Hence, since a fragment of mass r has a rate of dislocation ν r (dx) = r α ν(dx), the predictable compensator is

t 0 ds f (X(s-)) S ↓ ν(dy) ∞ i=1 X α i (s-) f (X i (s-)y) f (X i (s-)) -1 ≤ 2C f e C f t 0 ∞ i=1 X i (s-) S ↓
(1 -y 1 )ν(dy)ds.

≤ 2C f e C f t S ↓ (1 -y 1 )ν(dy) Hence s∈[0,t]∩T f (X(s)) -f (X(s-)) < ∞ a.s.,
and thus we have

f (X(t)) -f (X(0)) = s∈[0,t]∩T f (X(s)) -f (X(s-)) , i.e. f (X(t)) -f (X(0)) = s∈T ∩[0,t] f (X(s-)) ∞ i=1 1l ks=i f (X i (s-)∆ s ) f (X i (s-)) -1 , whose predictable compensator is t 0 ds f (X(s-)) S ↓ ν(dy) ∞ i=1 X α i (s-) f (X i (s-)y) f (X i (s-)) -1 = t 0 G α f (X(s))ds.

Application to h(t, F (t))

Let F (t) be a fragmentation process and q t (x) be the density of a Lévy process fulfilling the hypotheses of Theorem 1.1. We have proved in the first section that the function

H t = h(t, F (t)) = e tc p 1 (0) q 1 (0) ∞ i=1 q F i (t) (-tF i (t)) p F i (t) (-tF i (t))
is a P (B) -martingale (since it is equal to dP (X) dP (B) |F t ). We set g(t, x) = e tcx q x (-tx) p x (-tx) for x ∈]0, 1], t ≥ 0 and g(t, 0) = 1.

Set now g(t, x) = ∞ i=1 g(t, x i (t)) for x ∈ S ↓ , t ≥ 0. So we have, as i F i (t) = 1 P (B) -a.s.,

H t = p 1 (0) q 1 (0) g(t, F (t)) for all t ≥ 0.
It is well known that if q t (u) is the density of a Lévy process

X t = B t -Γ t + ct, the function (t, u) → q t (u) is C ∞ on R * + × R. Hence (t, x) → g(t, x) is also C ∞ on R + ×]0, 1
] and in particular, for all x ∈ [0, 1], the function t → g(t, x) is C 1 and so ∂ t g(t, x) is well defined. The next proposition gives a integro-differential equation solved by the function g when g has some properties of regularity at points (t, 0), t ∈ R + . Proposition 3.3.

1. Assume that for all t ≥ 0, ∂ x g(t, 0) exists and the function (t, x) → ∂ x g(t, x) is continuous at (t, 0). Then g solves the equation:

     ∂ t g(t, x) + √ x 1 0 dy 8πy 3 (1 -y) 3 g(t, xy)g(t, x(1 -y)) -g(t, x) = 0 g(0, x) = qx(0) px(0) .
2. If the Lévy measure of the subordinator Γ is finite, then the above conditions on g hold.

Proof. Let us first notice that the hypotheses of the proposition imply that the integral

1 0 dy 8πy 3 (1 -y) 3 g(t, xy)g(t, x(1 -y)) -g(t, x)
is well defined and is continuous in x and in t. Indeed, this integral is equal to And for all y ∈]0,

1/2[, x ∈]0, 1], t ∈ R + , there exist c, c ′ ∈ [0, x] such that g(t, xy)g(t, x(1 -y)) -g(t, x) y = x(g(t, x)∂ x g(t, c) -g(t, xy)∂ x g(t, c ′ )).
Thanks to the hypothesis that the function (t, 1 2 ] and so by application of the theorem of dominated convergence, the integral is continuous in t on R + and in x on [0,1].

x) → ∂ x g(t, x) is continuous on R + × [0, 1], |x(g(t, x)∂ x g(t, c) -g(t, xy)∂ x g(t, c ′ ))| is uniformly bounded on [0, T ] × [0, 1] × [0,
We begin by proving the first point of the proposition. Recall that, according to Proposition 3.1, the generator of the Brownian fragmentation is

G 1 2 f (x) = f (x) ∞ i=1 √ x i ν(dy) f (x i y) f (x i ) -1 , with ν(y 1 ∈ du) = (2πu 3 (1 -u) 3
) -1/2 du for u ∈]1/2, 1[, ν(y 1 + y 2 = 1) = 0 (cf. [START_REF] Bertoin | Self-similar fragmentations[END_REF]).

Hence,

M t = g(t, F (t)) -g(0, F (0)) - t 0 G 1 2
g(s, F (s)) + ∂ t g(s, F (s))ds is a P (B) -martingale. Since g(t, F (t)) is already a P (B) -martingale, we get G 1 2 g(s, F (s)) + ∂ t g(s, F (s)) = 0 P (B) -a.s. for almost every s > 0, i.e. for almost every s > 0

g(s, F (s)) ∞ i=1 F 1/2 i (s) S ↓ ν(dy) g(s, F i (s)y) g(s, F i (s)) -1 + ∂ t g(s, F i (s)) g(s, F i (s)) = 0 P (B) -a.s. With F (s) = (x 1 , x 2 , . . .), we get ∞ i=1 x 1/2 i S ↓ ν(dy) g(s, x i y) g(s, x i ) -1 + ∂ t g(s, x i ) g(s, x i ) = 0 P (B) s -a.s.
Notice also that this series is absolutely convergent. Indeed, thanks to Lemma 3.2, we have

x 1/2 i S ↓ ν(dy) g(s, x i y) g(s, x i ) -1 ≤ C g,s x i S ↓ (1 -y 1 )ν(dy),
where C g,s is a positive constant (which depends on g and s), and, besides we have

g(t, x) = exp -x c 2 2 E exp - Γ 2 x 2x + Γ x (t + c) .
Thus, by application of the theorem of dominated convergence, it is easy to prove that the function

t → E exp -Γ 2 x 2x + Γ x (t + c) is derivable with derivative ∂ t E exp - Γ 2 x 2x + Γ x (t + c) = E Γ x exp - Γ 2 x 2x + Γ x (t + c) .
Notice also that this quantity is continuous in x on [0,1]. Hence we have

∀x i ∈]0, 1[, ∀s > 0, ∂ t g(s, x i ) g(s, x i ) > 0.
Thus we deduce

∞ i=1 ∂ t g(s, x i ) g(s, x i ) < ∞ P (B) s -a.s. Let define k(t, x) = ∂ t g(t, x) + √ x 1 0 dy 8πy 3 (1 -y) 3 g(t, xy)g(t, x(1 -y)) -g(t, x) .
Hence we have

∞ i=1 k(s, x i ) = 0 P (B) s -a.s. for almost every s > 0, (3) and
k : [0, 1] → R be a continuous function, such that ∞ i=1 k(x i ) = 0 P (B) t -a.s. and ∞ i=1 |k(x i )| < ∞ P (B) t -a.s. Then k ≡ 0 on [0,1].
Proof. Let F (t) = (F 1 (t), F 2 (t) . . .) be a Brownian fragmentation at time t where the sequence (F i (t)) i≥1 is ordered by a size-biased pick. We denote by S the set of positive sequence with sum less than 1. Since F (t) has the law of the size biased reordering of the jumps of a stable subordinator T (with index 1/2) before time t, conditioned by T t = 1 (see [START_REF] Aldous | The standard additive coalescent[END_REF]), it is obvious that we have ∀x ∈]0, 1 -S[, P

B) t (F 1 ∈ dx | (F i ) i≥3 ) > 0, (
where S = i≥3 F i . Let Q t be the measure on S defined by

∀A ⊂ S, Q t (A) = P (B) t ((F i ) i≥3 ∈ A)
and λ the Lebesgue measure on [0, 1]. Hence we have, for all y ∈ S -Q t -a.s.

∀x ∈]0, S[, k(x) + k(1 -S -x) + ∞ i=1 k(y i ) = 0 λ-a.s.,
where S = i y i . We choose now y ∈ S such that this equality holds for almost every x ∈]0, S[. Thus, we get that there exists a constant C = C(y) such that k(x) + k(1 -S -x) = C, for all x ∈]0, S[ λ-a.s.

Since k is continuous, this equality holds in fact for all x ∈ [0, S]. Furthermore, we have also We prove now the point 2 of Proposition 3.3.

Proof. Assume that the Lévy measure of Γ is finite. It is obvious that g has the same regularity that the function qx(-tx) px(-tx) . Recall now that we have

q x (-tx) p x (-tx) = exp -x(ct + c 2 2 ) E exp - Γ 2 x 2x + Γ x (t + c) .
Hence a sufficient condition for g to fulfill the hypotheses of Proposition 3.3 is

• u t (x) = E exp -Γ 2 x 2x + Γ x (t + c
) is derivable at 0.

• w(t, x) = u ′ t (x) is continuous at (t, 0) for t ∈ R + .

We write u t (x) = a t (x, x) with Recall that the generator of a subordinator with no drift and Lévy measure π is given for every bounded function f C 1 with bounded derivative by ∀y ∈ R + , Lf (y) = ∞ 0 (f (y + s) -f (y))π(ds), (c.f. Section 31 of [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]).

Hence, we get for all z 0 > 0, We must prove that (t, x) → u ′ t (x) is continuous at (t, 0) for t ≥ 0. For every Lévy measure π, the first term has limit 0 as (t ′ , x) tends to (t, 0) (by dominated convergence). For the second term, notice that we have for all x ∈]0, 1], exp -

∂
(Γ x + y) 2 2x + (Γ x + y)(t + c) -exp - Γ 2 x 2x + Γ x (t + c) ≤ 2 exp (t + c) 2 x 2 ,
and for all y > 0, exp -(Γx+y) 2 2x + (Γ x + y)(t + c) -exp -Γ 2 x 2x + Γ x (t + c) converges almost surely to -1 as (t ′ , x) tends to (t, 0). Hence, if π(R + ) < ∞, we deduce that the lim (t ′ ,x)→(t,0) u ′ t (x) exists (and is equal to -π(R + )).

3 ( 1 -

 31 y) 3 g(t, xy)g(t, x(1 -y)) -g(t, x) .

  ∀s ∈]0, 1[, Q t (S ∈ ds) > 0.Hence, this implies the existence for almost every s ∈]0, 1[ of a constant C s such thatk(x) + k(1 -s -x) = C s for all x ∈]0, s[.Thanks to the continuity of k, we can deduce that this property holds in fact for all s ∈ [0, 1]. Hence we have∀x, y ∈ [0, 1] 2 , such that x + y ≤ 1, k(x + y) = k(x) + k(y).So k is a linear function and since ∞ i=1 x i = 1 P (B) t -a.s., we get k ≡ 0 on [0,1].

a 2 y 2z 2 exp - Γ 2 y

 22 t (y, z) = E exp -Γ 2 y 2z + Γ y (t + c) .Since the function (y, z)→ y 2 2z 2 exp -y 2 2z + y(t + c) is bounded on R + × [0, 1], we get ∂ z a t (y, z) = E Γ 2z + Γ y (t + c) for z ∈]0, 1].

  y a t (y, z 0 ) = E(La t (Γ y , z 0 ))

	= E	0	∞	exp -	(Γ y + s) 2 2z 0	+ (Γ y + s)(t + c) -exp -	Γ 2 y 2z 0	+ Γ y (t + c)	π(ds) .
	And we deduce
	u ′ t (x) = E Γ 2 x 2x 2 exp -Γ 2 x 2x + Γ x (t + c) + E ∞ 0 exp -(Γ x + y) 2 2x + (Γ x + y)(t + c) -exp -	Γ 2 x 2x	+ Γ x (t + c)	π(dy) ,

+ε ′ .One can easily check that this condition is equivalent to (2).