MD Dario Baratti
email: dario.baratti@istitutotumouri.mi.it

MD M Vaira

PhD S Kusamura

PhD S D'amico

MD M R Balestra

MD T Cioppa

MD E Mingrone

MD M De Simone

MD M Deraco

Hyperthermic Intraperitoneal Chemotherapy (HIPEC)

Keywords: multicystic peritoneal mesothelioma, cytoreductive surgery, peritonectomy, hyperthermic intraperitoneal chemotherapy, HIPEC

come

INTRODUCTION

Tumours originating from the mesothelial cells lining the abdominal cavity cover a wide spectrum of biological aggressiveness. Among them, adenomatoid tumour is a benign lesion that mostly involves the genital tract and very unlikely recurs after local excision. [START_REF] Husain | Guidelines for pathologic diagnosis of malignant mesothelioma: a consensus statement from the International Mesothelioma Interest Group[END_REF] At the other extreme, diffuse malignant peritoneal mesothelioma is a locally aggressive and rapidly lethal malignancy. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF] Multicystic peritoneal mesothelioma (MCPM) is an exceedingly rare lesion with uncertain malignant potential and an enigmatic natural history. [START_REF] Katsube | Cystic mesothelioma of the peritoneum: a report of five cases and review of the literature[END_REF][START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF][START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF][START_REF] Sawh | Benign cystic mesothelioma of the peritoneum: a clinicopathological study of 17 cases and immunohistochemical analysis of estrogen and progesterone receptor status[END_REF][START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF] Approximately 150 cases of MCPM have been reported in the medical literature since it was first described by [START_REF] Mennemeyer | Multicystic peritoneal mesothelioma: a case report with electron microscopy of a case mimicking intra-abdominal cystic hygroma (lymphangioma)[END_REF]. [START_REF] Mennemeyer | Multicystic peritoneal mesothelioma: a case report with electron microscopy of a case mimicking intra-abdominal cystic hygroma (lymphangioma)[END_REF] According to the available information, MCPM most commonly occurs in the abdomino-pelvic cavity of reproductive age women with no history of asbestos exposure. The disease shows an indolent clinical behaviour in most cases. However, early recurrences requiring multiple surgical interventions [START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF][START_REF] Sawh | Benign cystic mesothelioma of the peritoneum: a clinicopathological study of 17 cases and immunohistochemical analysis of estrogen and progesterone receptor status[END_REF][START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF], transformation into truly malignant disease [START_REF] González-Moreno | Malignant transformation of "benign" cystic mesothelioma of the peritoneum[END_REF], lymph-node involvement [START_REF] Engohan-Aloghe | Lymph node involvement in multicystic peritoneal mesothelioma[END_REF] and even death [START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF] have been described.

The conceptual and methodological paradigm for the management of peritoneal surface malignancies is currently evolving from end-stage conditions with only palliative options to local-regional disease amenable to intensive treatment. Accordingly, an innovative treatment strategy has emerged, involving aggressive cytoreductive surgery to remove all the visible peritoneal tumour, in combination with intra-operative local-regional chemotherapy to treat the microscopic residual disease. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF][START_REF] Feldman | Analysis of factors associated with outcome in patients with malignant peritoneal mesothelioma undergoing surgical debulking and intraperitoneal chemotherapy[END_REF][START_REF] Yan | Prognostic indicators for patients undergoing cytoreductive surgery and perioperative intraperitoneal chemotherapy for diffuse malignant peritoneal mesothelioma[END_REF][START_REF] Baratti | Lymph Node Metastases in Diffuse Malignant Peritoneal Mesothelioma[END_REF] This combined approach has become a treatment option for malignant peritoneal mesothelioma in the last decade. Median survival has dramatically improved reaching approximately five years, as compared to 9-13 months in the historical case-series treated by debulking surgery and/or palliative systemic chemotherapy. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF] In few international centres, MCPM has been included among the indications for surgical cytoreduction and intraperitoneal chemotherapy. However, only a small number of patients undergoing combined treatment has been reported in the literature. [START_REF] Sethna | Peritoneal cystic mesothelioma: a case series[END_REF][START_REF] Baratti | Multicystic and well-differentiated papillary peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF][START_REF] Passot | Peritoneal mesothelioma: treatment with cytoreductive surgery combined with hyperthermic intraperitoneal chemotherapy[END_REF][START_REF] Yano | Cytoreductive surgery and intraperitoneal chemotherapy for peritoneal mesothelioma[END_REF] Furthermore, the biology of peritoneal mesothelioma has been addressed only in recent years and the features of rare sub-variants, such as MCMP, are still largely unknown. The present paper reports the collaborative effort of two Italian centres to assess safety and effectiveness of the combined treatment. Additionally, pathological and biological features of MCMP, including lymph-nodal involvement, serum and cellular marker expression, were analyzed together with their prognostic significance.

PATIENTS AND METHODS

All the patients were treated according to clinical protocols approved by the Institutional Ethics Committee of each participating institution. The informed consent form was signed by each patients. Standardized clinical data on consecutive patients were collected from prospective institutional databases and entered into a central database.

Additional information was retrieved from medical charts. The same author reviewed all the information to ensure a uniform interpretation of data. The exact number of patients from each hospital is not stated due to confidentiality issues.

Data form

A standard data form was created to retrieve information on patient characteristics (sex, age, performance status), clinical history (exposure to asbestos, presenting symptoms, methodology for diagnosis, previous therapies), operative treatment (number of peritonectomy procedures and multivisceral resections, completeness of surgical cytoreduction, methodology for intraperitoneal chemotherapy), pathological data (histological features, special studies, lymph-node metastases), length of hospital stay, grade III/IV postoperative complications according to the National Cancer Institute Common Terminology Criteria (http://ctep.cancer.gov/forms/CTCAEv3.pdf), and follow-up (date of last control, disease status, occurrence and treatment of disease relapse).

Inclusion criteria

Pathological diagnosis of MCPM was made by specialized pathologists in each institution, based on haematoxylin/eosin slides showing typical morphological features and appropriate immunohistochemical studies. [START_REF] Husain | Guidelines for pathologic diagnosis of malignant mesothelioma: a consensus statement from the International Mesothelioma Interest Group[END_REF] Patients were managed by cytoreductive surgery followed by hyperthermic intraperitoneal chemotherapy (HIPEC). In both institutions, eligibility criteria for combined treatment included: age ≤75; performance status ≤2 according to the Eastern Cooperative Oncology Group (ECOG); [START_REF] Oken | Toxicity and Response Criteria of the Eastern Cooperative Oncology Group[END_REF] no significant co-morbidities; peritoneal disease amenable to potentially complete surgical cytoreduction and no extra-abdominal or hepatic metastases at preoperative computed tomography (CT)-scan.

Operative treatment

The extent of peritoneal involvement was rated at surgical exploration using the peritoneal cancer index (PCI). PCI is a semi-quantitative score which combines lesion size (LS-0/3) with disease distribution in 13 abdomino-pelvic regions (PCI 0/39). [START_REF] Jaquet | Current methodologies for clinical assessment of patients with peritoneal carcinomatosis[END_REF] Cytoreductive surgery was performed according to the technique described by Sugarbaker. [START_REF] Sugarbaker | Peritonectomy procedures[END_REF] Briefly, the goal of the surgical cytoreduction was to remove all the visible tumor by means of diaphragmatic, parietal and pelvic peritonectomy with greater and lesser omentectomy. Depending on disease involvement of visceral surfaces, local excision and electro-coagulation of small localizations or multi-visceral resections were performed, including cholecystectomy, splenectomy, sigmoid, right or total colectomy and hysterectomy with salpingo-oophorectomy in women.

The completeness of cytoreduction (CCR) was rated at the end of the procedure, as follows: CCR-0= no visible residual disease; CCR-1= residual disease ≤2.5mm; CCR-2= residual disease >2.5mm and ≤25mm; CCR-3= residual disease >25mm. [START_REF] Simone | Semi-closed hyperthermic-antiblastic peritoneal perfusion (HAPP) in the treatment of peritoneal carcinosis[END_REF] All HIPEC procedures were performed intra-operatively, but with many variations in exposure techniques, drugs, drug doses, duration (60-90 minutes), intraperitoneal temperatures (40-42.5°C), perfusate volume (3.4-6 L), and flow rates (600-1000 mL/min.). Nine HIPEC procedures were performed using the closed-, one the open-and four an original semi-closed-abdomen technique. [START_REF] Simone | Semi-closed hyperthermic-antiblastic peritoneal perfusion (HAPP) in the treatment of peritoneal carcinosis[END_REF] The combination of cisplatin (45 mg/L) and doxorubicin (15 mg/L) was used in ten procedures, according to the results of a formal phase I study. [START_REF] Rossi | Hyperthermic intraoperative intraperitoneal chemotherapy with cisplatin and doxorubicin in patients who undergo cytoreductive surgery for peritoneal carcinomatosis and sarcomatosis: phase I study[END_REF] Two patients were treated with cisplatin alone (a total of 100 and 120 mg, respectively), one with cisplatin (25 mL/m 2 /L) plus mitomycin-C (3.3 mL/m 2 /L), and one with mitomycin-C (35 mg/m 2 ). Dose reduction up to 30% was applied to patients treated with prior systemic chemotherapy and/or extensive cytoreductive surgery.

Patients were followed postoperatively with physical examination, thoracic/abdominal CT scan, and serum marker measurements every three to six months No patient was lost at follow-up. Postoperative disease progression was confirmed at surgical exploration or by CT-scan/ultrasound-guided biopsy.

Statistics

Overall and progression-free survival was calculated from the day of cytoreduction and HIPEC to the time of death due to any cause or postoperative disease progression, according to the Kaplan-Meier method. [START_REF] Kaplan | Nonparametric estimation from incomplete observations[END_REF] Patients with uneventful postoperative course were censored at the time of last follow-up visit. The two-tailed log-rank test was used to assess the significance of the comparison between survival distributions. P value <.05 was considered significant. The following potential prognostic factors were tested with regard to overall and progression-free survival: sex, age at diagnosis, diagnosis (incidental vs. symptomatic), interval diagnosis-HIPEC, previous surgery, previous systemic chemotherapy, ECOG performance score (0 vs.1-2), time-to-progression after the last treatment, PCI, presence of ascites, CCR, baseline CA125 and CA15.3, immunohistochemical studies, number of peritonectomy procedure (≤3 vs. >3), visceral resections, HIPEC drug schedule (cisplatin and doxorubicin vs. others), cisplatin total dose (≤100mg vs. >100mg). Continuous variables were categorized into two classes using their median value as cut-off.

RESULTS

Patient characteristics

From August 1995 to October 2009, 12 patients underwent 14 procedures of cytoreduction and HIPEC in two Italian tertiary referral centres. Some of these patients were reported previously. [START_REF] Baratti | Multicystic and well-differentiated papillary peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF][START_REF] Baratti | Multicystic peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF][START_REF] Costamagna | Trattamento del mesotelioma peritoneale mediante citoriduzione e chemioipertermia intraperitoneale (CIIP)[END_REF] Patient characteristics and clinical history before comprehensive treatment are summarized in Table 1. Eleven patients were females. No-one reported a documented history of asbestos exposure. Median age at diagnosis was 39 (range 22-63). One patient was affected by Glanzman's thrombasthenia. Abdominal pain was the most common presenting symptom and final diagnosis was established by laparotomy in most instances. Every patient complained of abdominal distension and/or pain at the time of cytoreduction and HIPEC.

Before combined treatment, nine patients underwent 14 surgical procedures for MCMP (one operation in six patients, two operations in two, and four operations in one); one of them received systemic carboplatin with gemcitabine. Two partial and one macroscopically complete excision were performed in the participating institutions before the initiation of their peritoneal malignancy treatment programs; PCI range was 10-14. CCR and PCI score were difficult to ascertain retrospectively in patients operated on elsewhere. They were all considered as having received debulking surgery.

Disease progression invariably followed each of these 14 surgical attempts, with a median time-to-progression of 11 months (range 2-31).

Operative outcomes

Operative procedures are displayed in Table 2. Median PCI was 10 (range 4-26). CCR-0 cytoreduction was accomplished in 12 operations and CCR-1 cytoreduction (residual disease ≤2.5mm) in two. Twenty-three visceral resections were made during nine operations. Hysterectomy with bilateral salpingo-oophorectomy was the most commonly performed procedure, being carried out in 7 patients. Uterus and ovaries were removed in additional 3 women during previous operations. Conversely, genitalia could be spared in one reproductive age woman, according to patient will and limited disease involvement. Six colon resections were performed in four patients, and protective ileostomy in one. No operative death occurred. One grade IV small bowel perforation away from anastomotic sites required re-intervention and ileostomy. Both ileostomies were closed later.

Patho-biological features

Pathological and biological features are shown in Table 3. Eleven patients showed typical MCPM morphology, with multiple variably sized cysts separated by fibrous/adipose septa and lined by a single layer of flattened to cuboidal mesothelial cells with no or little atypia. One patient showed a mixed multicystic and papillary well-differentiated pattern, consisting of delicate papillary structures with fibrovascular core covered by cuboidal mesothelial cells.

Intra-abdominal lymph-nodes were clinically suspicious and sampled by surgeons or pathologists in six patients.

Pathological reports documented no metastatic involvement. In the remaining patients, no lymph-node was sampled or thought to be suspicious for metastatic disease.

Immunohistochemical studies were available for eight patients. In all cases, cytocheratins (CK) 5-6 and Wilm'sTumour (WT)-1 were positive and CEA and Ber-Ep4 negative. Calretinin was positive in only five patients. Mitotic count and percentage of cell expressing MIB-1 were low in all patients. Baseline serum CA125 and CA15.3 determinations were positive in one patient, respectively.

Survival and failure

Median follow-up was 64 months (range 5-148). By the time of the present analysis, all the patients are alive with no evidence of disease, including two patients who underwent the procedure twice, due to postoperative recurrence. One of them developed a widespread intra-abdominal relapse (PCI=19) at 18 months; in the second patient, a small volume recurrence (PCI=4) was incidentally diagnosed at 68 months, during hernia repair operation. Both the patients are presently free of disease after 39 and 56 months, respectively, from re-intervention. Histological and immunohistochemical features at recurrence were not remarkably different from those seen at initial cytoreduction.

Five-and 10-year Kaplan-Maier estimated progression-free survival were 90% and 72%. In figure 1, progression-free survival curves after cytoreduction and HIPEC and after 14 debulking operations carried out in nine patients before referral for combined treatment are shown; the difference was statistically significant (P<0.0001). Seventeen months after combined treatment, a multiloculated para-adnexial lesion was detected by CT-scan and trans-vaginal ultrasound in the woman with spared uterus and ovaries. Disease recurrence was suspected, but the patient refused any treatment or invasive diagnostic procedures. [START_REF] Baratti | Multicystic peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF] Serial CT-scans demonstrated the spontaneous complete regression of the cysts and the patient is now considered free of disease.

Exploratory analysis of patient-, tumour-and treatment-related data showed no significant association with survival.

Both the patients who recurred had typical MCPM; their nodes were pathologically negative. Both patients had positive calretinin and normal levels of serum CA125 and CA15.3.

DISCUSSION

To our knowledge, this is the first multi-institutional study on multicystic peritoneal mesothelioma treated by surgical cytoreduction and HIPEC. The present analysis collected twelve patients, which represent a relatively large number of such an uncommon disease entity. Our data suggest that comprehensive treatment may result in long-term disease control and even definitive cure. Furthermore, operative complication rates were similar to those reported by specialized centres performing major gastrointestinal surgery. [START_REF] Chua | Should the treatment of peritoneal carcinomatosis by cytoreductive surgery and hyperthermic intraperitoneal chemotherapy still be regarded as a highly morbid procedure?: a systematic review of morbidity and mortality[END_REF]

Rational basis for comprehensive treatment of multicystic mesothelioma

Although the histological [START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF][START_REF] Mennemeyer | Multicystic peritoneal mesothelioma: a case report with electron microscopy of a case mimicking intra-abdominal cystic hygroma (lymphangioma)[END_REF] and radiological [START_REF] Sugarbaker | Thin-walled cysts as a pathognomonic CT finding in cystic mesothelioma[END_REF] features have been extensively described, disagreement still exists regarding the natural history and treatment of choice of MCPM. Complete surgical resection has been favoured by some authors [START_REF] Katsube | Cystic mesothelioma of the peritoneum: a report of five cases and review of the literature[END_REF][START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF][START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF] and less aggressive approaches by others, due to the risk of infertility deriving from extensive pelvic surgery in women. [START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF] Radiation, systemic or intraperitoneal chemotherapy, laser vaporization, percutaneous cyst drainage, hormone-therapy, sclero-therapy, or simple observation have been used with uncertain results. [START_REF] Katsube | Cystic mesothelioma of the peritoneum: a report of five cases and review of the literature[END_REF][START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF][START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF][START_REF] Sawh | Benign cystic mesothelioma of the peritoneum: a clinicopathological study of 17 cases and immunohistochemical analysis of estrogen and progesterone receptor status[END_REF][START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF][START_REF] Mennemeyer | Multicystic peritoneal mesothelioma: a case report with electron microscopy of a case mimicking intra-abdominal cystic hygroma (lymphangioma)[END_REF] Cytoreductive surgery and HIPEC, aiming at complete tumour eradication, is a new concept. Cytoreduction to microscopic or sub-millimetric tumour provides the optimal conditions for HIPEC to target minimal residual disease.

The intraoperative setting minimizes the risk of tumour cell entrapment that could give rise to disease recurrence. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF][START_REF] Feldman | Analysis of factors associated with outcome in patients with malignant peritoneal mesothelioma undergoing surgical debulking and intraperitoneal chemotherapy[END_REF][START_REF] Yan | Prognostic indicators for patients undergoing cytoreductive surgery and perioperative intraperitoneal chemotherapy for diffuse malignant peritoneal mesothelioma[END_REF][START_REF] Baratti | Lymph Node Metastases in Diffuse Malignant Peritoneal Mesothelioma[END_REF][START_REF] Sethna | Peritoneal cystic mesothelioma: a case series[END_REF][START_REF] Baratti | Multicystic and well-differentiated papillary peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF][START_REF] Passot | Peritoneal mesothelioma: treatment with cytoreductive surgery combined with hyperthermic intraperitoneal chemotherapy[END_REF][START_REF] Yano | Cytoreductive surgery and intraperitoneal chemotherapy for peritoneal mesothelioma[END_REF] This comprehensive strategy is successfully used to treat pseudomyxoma peritonei, a disease which shares with MCPM the low biological aggressiveness and tendency to recur after incomplete excision. [START_REF] González-Moreno | Malignant transformation of "benign" cystic mesothelioma of the peritoneum[END_REF][START_REF] Sethna | Peritoneal cystic mesothelioma: a case series[END_REF][START_REF] Baratti | Multicystic and well-differentiated papillary peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF] In the current study, drugs were chosen for HIPEC based on their suitability to intra-peritoneal administration and synergistic effect with heat. No data are available on their activity against MCPM, although cisplatin and doxorubicin are largely used in systemic and local-regional treatment of peritoneal mesothelioma. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF]

Patho-biological features

In an exhaustive clinico-pathological analysis of 62 patients undergoing combined treatment at the Washington Cancer Center, 6 cases of MCPM were classified as multicystic variant of epithelial malignant mesothelioma. The authors found that all MCPM cases met the criteria for malignancy because of a clear evidence of diffuse disease distribution throughout the abdomen, invasion into peritoneal surfaces, and lymphovascular involvement. The apparent discrepancy with previous publications reporting benign behaviour was explained by the advanced disease stage and massive tumour load seen in these patients treated at a highly specialized centre. Furthermore, cytoreductive surgery provided more extensive tumour sampling from many different anatomic sites. [START_REF] Cerruto | Prognostic significance of histomorphologic parameters in diffuse malignant peritoneal mesothelioma[END_REF] A set of biological markers related to mesothelioma origin and clinical features was investigated. Due to the retrospective nature of the present study, markers were not consistently assessed in all the patients. However, interesting information were provided. The Ki-67 antigen is a nuclear protein expressed during all phases of cellular cycle, but not in non-cycling cells (G0 phase). Immunohistochemical staining with Ki-67 of Mib-1 antibodies is an excellent marker of cellular proliferation and tumour aggressiveness. [START_REF] Chua | Differential Expression of Ki-67 and Sex Steroid Hormone Receptors Between Genders in Peritoneal Mesothelioma[END_REF] Three independent studies showed low Ki-67 expression in peritoneal mesothelioma, with a median of 0.6% to 10% positive cells. [START_REF] Baratti | Lymph Node Metastases in Diffuse Malignant Peritoneal Mesothelioma[END_REF][START_REF] Zaffaroni | Survivin is highly expressed and promotes cell survival in malignant peritoneal mesothelioma[END_REF][START_REF] Chua | Differential Expression of Ki-67 and Sex Steroid Hormone Receptors Between Genders in Peritoneal Mesothelioma[END_REF]). Analogously, mitotic rate is gerally low in malignant peritoneal mesothelioma, but higher proliferative activity correlates with poor outcome. [START_REF] Baratti | Lymph Node Metastases in Diffuse Malignant Peritoneal Mesothelioma[END_REF] Ki-67 and mitotic rate were low in all our patients, as compared with the truly malignant counterpart, suggesting that poor proliferative activity may be related to the indolent MCMP behaviour. Cytokeratin5/6, which is involved in cellular differentiation, the transcription factor WT-1 and calretinin, a vitamin-Ddependent protein involved in calcium signaling, are expressed by most mesothelioma and largely used for diagnosis. [START_REF] Husain | Guidelines for pathologic diagnosis of malignant mesothelioma: a consensus statement from the International Mesothelioma Interest Group[END_REF] CK5-6 and WT-1 were positive in all our patients. Calretinin was not expressed in 3 of 8 cases, although the biological meaning of this observation is not clear, since no prognostic significance was apparent. The poor expression of CA125 and CA15.3 in MCMP suggests that serum marker measurements is not useful in this setting.

Analysis of prognostic variables

The results of the current study regarding potential determinants of outcome showed no correlation with survival, probably because of the small number of events and imbalance in prognostic variable distribution, such as optimal surgical cytoreduction. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF] Lymphatic involvement has been demonstrated to be an independent prognostic factor for malignant peritoneal mesothelioma. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF][START_REF] Baratti | Lymph Node Metastases in Diffuse Malignant Peritoneal Mesothelioma[END_REF] Nodal metastases were described in the patient with malignant transformation of MCPM, [START_REF] González-Moreno | Malignant transformation of "benign" cystic mesothelioma of the peritoneum[END_REF] and recently in a 35-year-old woman. [START_REF] Engohan-Aloghe | Lymph node involvement in multicystic peritoneal mesothelioma[END_REF] In the present series, all nodes were either pathologically negative or not clinically suspicious and not therefore sampled.

Conclusions

In summary, both literature data and the present series suggest that MCPM is a locally aggressive and border-line malignant tumour capable of transition into an invasive and potentially lethal process, rather than a benign disease.

Definitive tumour eradication by means of peritonectomy procedures and HIPEC seems to be the optimal treatment to prevent disease recurrence or malignant transformation.

Complete cytoreduction

Intraperitoneal chemotherapy

Disease status (months) Washington, DC 14 5 4 HIPEC (cis + dx) 4 NED (6-69) 1 AWD (25)* Lyon, Fr 16 3 3 HIPEC (cis + mit) 3 NED (4-56) Basingstoke, UK 17 3 3 HIPEC (cis + dx) + EPIC(cis + dx) 3 NED (NA) Turin, It 25

FIGURE LEGEND

Progression-free survival after 12 procedures of cytoreduction with hyperthermic intraperitoneal chemotherapy (HIPEC) (black line) and 14 debulking operations performed in 9 patients before combined treatment (red line).

Survival difference was statistical significant (two-tailed P-value <0.0001).

  dx= doxorubicin; mit= mitomycin-C; HIPEC= hyperthermic intraperitoneal chemotherapy; EPIC= early postoperative intraperitoneal chemotherapy; NED= no evidence of disease; AWD= alive with disease; NA= not available; *= the patient died following this report.

Table 1 .

 1 Clinical characteristics of 12 patients with multicystic peritoneal mesothelioma

		Sex Age at	Presenting	Diagnosis	Previous treatments	Time to
			diagnosis	symptom			progression
	1	M	42	incidental	laparotomy debulking	7 mo.
	2	F	29	ileus	laparotomy TAH, BSO	11 mo.
						debulking	18 mo.
						debulking	24 mo.
						debulking	31 mo.
	3	F	35	abdominal pain	laparotomy BSO, debulking	5 mo.
	4	F	63	abdominal pain	laparotomy TAH, BSO, omentectomy, small bowel	9 mo.
						and cecum resection
						syst. CBDCA +gemcitabine (6 courses)
	5	F	56	abdominal pain	laparotomy biopsy	1 mo.
						debulking	7 mo.
	6	F	50	incidental	laparotomy TAH, debulking	23 mo.
	7	F	39	ascites	laparoscopy biopsy	6 mo.
	8	F	22	abdominal pain	laparotomy partial omentectomy, debulking	27 mo.
						debulking	28 mo.
	9	F	39	umbilical hernia	laparotomy biopsy	-
	10	F	52		laparoscopy biopsy	-
	11	F	36	bleeding	laparotomy TAH left SO	25 mo.
						omentectomy, debulking	3 mo.
	12	F	28	abdominal pain	laparotomy biopsy	7 mo.
						omentectomy, left SO	12 mo.
	TAH= total abdominal hysterectomy; BSO= bilateral salpingo-oophorectomy; CBDCA= carboplatin

Table 3 .

 3 Pathological and biological features

	Variable	Categories	n.
	Pathological features (patients n=12)
	Histology	multicystic	11
		mixed multicystic and papillary	1
		well-differentiated
	Lymph-nodes	pathologically positive	-
		pathologically negative	6
		clinically neg./not sampled	6
	Mitotic count/50HPF	median, (range)	1 (0-5)
	Immunohistochemical studies (patients n=8)
	Cytocheratin5-6	Pos	8
	Calretinin	Pos	5
	WT-1	Pos	8
	Mib-1	% of positive cells, median (range)	5 (1-10)
	CEA	Pos	-
	Ber-Ep4	Pos	-
	Circulating tumour markers
	Serum CA125 (patients n= 8)	Pos	1
	Serum CA15.3 (patients n= 5)	Pos	1
	HPF= high power field; WT-1= Wilms Tumour-1; CEA= carcino-embryonic antigen.

Table 4 .

 4 Literature data on cytoreduction and hyperthermic intraperitoneal chemotherapy (HIPEC) for multicystic

	mesothelioma
	Center	Pts
		n.

Table 2. Four-teen procedures of cytoreduction and hyperthermic intra-peritoneal chemotherapy

Acknowledgment: This study was supported in part by grants from the Italian Association for Cancer Research (AIRC) and the Italian Health Ministry. The study sponsors had no involvement in the study design, in the collection, analysis and interpretation of data, in the writing of the manuscript, and in the decision to submit the manuscript for publication.

Cytoreduction and HIPEC versus debulking surgery

Due to the rarity of MCPM, direct comparative studies of treatment efficacy are virtually unfeasible. In an effort to overcome these limitations, we compared in our patients the results of cytoreduction and HIPEC with those of traditional surgery: disease recurrence occurred after all the 14 debulking operations performed before referral for combined treatment (median time-to-recurrence 11 months) and after two of 12 cytoreductions with HIPEC (median follow-up 64 months). The highly significant survival difference (P<0.0001) strongly suggests the superiority of the comprehensive approach. Since the two treatments were compared in the same population, it may be assumed that prognostic factor distribution was well-balanced. Additionally, complete tumour removal was likely feasible at the time of previous debulking, as it was accomplished at the more recent cytoreduction. However, a bias could have occurred in this setting, since the presence of active disease was a condition for patients to undergo cytoreduction with HIPEC and only cases with disease progression after previous surgery were analyzed.

The nature of MCPM as a benign, malignant or even reactive process is still debated, because of the reported association with previous abdominal surgery, inflammation, or endometriosis. [START_REF] Yan | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Malignant Peritoneal Mesothelioma: Multi-Institutional Experience[END_REF][START_REF] Katsube | Cystic mesothelioma of the peritoneum: a report of five cases and review of the literature[END_REF][START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF][START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF][START_REF] Sawh | Benign cystic mesothelioma of the peritoneum: a clinicopathological study of 17 cases and immunohistochemical analysis of estrogen and progesterone receptor status[END_REF][START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF] Consequently, criticisms may involve the use of potentially life-threatening procedures, such as cytoreduction with HIPEC. However, it has to be acknowledged that conventional treatments have often been disappointing: Ross reported disease recurrence in 14 of 25 patients, [START_REF] Mj Ross | Multilocular peritoneal inclusion cysts (so-called cystic mesotheliomas)[END_REF] Katsube in 4 of 8 [START_REF] Katsube | Cystic mesothelioma of the peritoneum: a report of five cases and review of the literature[END_REF] and Sawh in 5 of 12, [START_REF] Sawh | Benign cystic mesothelioma of the peritoneum: a clinicopathological study of 17 cases and immunohistochemical analysis of estrogen and progesterone receptor status[END_REF] with median time-to recurrence of 26-48 months. Also in a recent literature review, relapse rate was about 50%. [START_REF] Vallerie | Peritoneal inclusion cysts: a review[END_REF] Death related to MCMP was reported by Weiss and Tavassoli in two cases: a man who refused any treatment and died 12 years after diagnosis and a 6-month-old infant who died 11 months after partial resection of invasive intrahepatic MCPM with transition into malignant mesothelioma. [START_REF] Weiss | Multicystic mesothelioma. An analysis of pathologic findings and biologic behavior in 37 cases[END_REF] Malignant transformation developed over 10 years and 6 surgical attempts in a patient reported by Gonzalez-Moreno who underwent only partial cytoreduction with HIPEC, [START_REF] González-Moreno | Malignant transformation of "benign" cystic mesothelioma of the peritoneum[END_REF] and eventually died of disease progression. [START_REF] Sugarbaker | Thin-walled cysts as a pathognomonic CT finding in cystic mesothelioma[END_REF] Literature data on cytoreduction and HIPEC for MCPM are summarized in table 4. To date, the study by Sugarbaker group is the only published case-series entirely devoted to MCPM. [START_REF] Sethna | Peritoneal cystic mesothelioma: a case series[END_REF] The number of patients treated at the Washington Cancer Center has been recently updated to seven. [START_REF] Sugarbaker | Thin-walled cysts as a pathognomonic CT finding in cystic mesothelioma[END_REF] One of the participating institution reported four cases of MCPM together with papillary well-differentiated peritoneal mesothelioma. [START_REF] Baratti | Multicystic and well-differentiated papillary peritoneal mesothelioma treated by surgical cytoreduction and hyperthermic intra-peritoneal chemotherapy (HIPEC)[END_REF] Consistently with the present study, combined treatment was universally associated to favourable prognosis, except for the patient who had transition of MCMP into aggressive mesothelioma. [START_REF] González-Moreno | Malignant transformation of "benign" cystic mesothelioma of the peritoneum[END_REF][START_REF] Sugarbaker | Thin-walled cysts as a pathognomonic CT finding in cystic mesothelioma[END_REF] Conflict of interest statement The authors have no potential conflict of interest to disclose.

Procedure


Jaume Llibre
email: jllibre@mat.uab.cat

Luis Fernando Mello
email: lfmelo@unifei.edu.br

Ernesto Perez-Chavela

NEW STACKED CENTRAL CONFIGURATIONS FOR THE PLANAR 5-BODY PROBLEM

Keywords: Planar central configurations, n-body problem, stacked central configurations

A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m1, m2, m3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m4, m5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.

Introduction

The classical planar Newtonian n-body problem in celestial mechanics consists in studying the motion of n pointlike masses in a fixed plane, interacting among themselves through no other forces than their mutual gravitational attraction according to Newton's gravitational law [START_REF] Newton | Philosophi Naturalis Principia Mathematica[END_REF].

The center of mass of the system, given by n j=1 m j r j /M , where M = m 1 + • • • + m n is the total mass and r j is the position vector of the mass m j , is considered at the origin of an inertial system. Usually this inertial system is called the inertial barycentric system.

The simplest motion of the n bodies, called a homographic solution, is a motion such that the configuration of the n bodies remains the same (with respect to the inertial barycentric system) up to a dilation and a rotation of R 2 .

The first homographic solutions for the 3-body problem were found by [START_REF] Euler | De moto rectilineo trium corporum se mutuo attahentium[END_REF], for which three bodies are collinear at any time, and by [START_REF] Lagrange | Essai sur le problème de trois corps[END_REF], where the three bodies are at any time at the vertices of an equilateral triangle.

At a given instant t = t 0 the configuration of the n bodies is central if the gravitational acceleration acting on every mass point is proportional to its position (referred to the inertial barycentric system). Central configurations and homographic solutions are linked by the Laplace theorem (see for instance [START_REF] Boccaletti | Integrable systems and non-perturbative methods[END_REF][START_REF] Wintner | The Analytical Foundations of Celestial Mechanics[END_REF]): the configuration of the n bodies in a homographic solution is central at any instant of time.

If we have a central configuration, any dilation and any rotation (centered at the center of mass) of it provides another central configuration. We say that two central configurations are similar if we can pass from one to another through a dilation and a rotation. So we can study the classes of central configurations defined by the above equivalence relation. Thus the 3-body problem has exactly 5 classes of central configurations for any value of the positive masses.

Central configurations of the n-body problem are important because: they allow the computation of homographic solutions; if the n bodies are heading for a simultaneous collision then the bodies tend to a central configuration (see [START_REF] Saari | On the role and properties of central configurations[END_REF]; there is a relation between central configurations and the bifurcations of the hypersurfaces of constant energy and angular momentum (see [START_REF] Smale | Topology and mechanics II: The planar n-body problem[END_REF]. See also the Refs. [START_REF] Moeckel | On central configurations[END_REF][START_REF] Moulton | The straight line solutions of n bodies[END_REF] The main general open problem for the planar central configurations is due to [START_REF] Wintner | The Analytical Foundations of Celestial Mechanics[END_REF] and [START_REF] Smale | Mathematical problems for the next century[END_REF]: Is the number of classes of planar central configurations finite for any choice of the (positive) masses m 1 , . . . , m n ? [START_REF] Hampton | Finiteness of relative equilibria of the fourbody problem[END_REF], proved this conjecture for the 4body problem. The conjecture remains open for n > 4. But if one mass can be negative, [START_REF] Roberts | A continuum of relative equilibria in the five-body problem[END_REF], proved that there exists a oneparameter not equivalent family of planar central configurations for the 5-body problem. Also considering the particles endowed with masses and charges, [START_REF] Alfaro | Families of continua of central configurations in charged problems[END_REF] proved the existence of a continuum of central configurations in a particular 4-body problem.

Other recent papers on central configurations are due to [START_REF] Corbera | On the central configurations of the planar 1+3 body problem[END_REF], Corbera and Llibre (2010), [START_REF] Gidea | Symmetric planar central configurations of five bodies: Euler plus two[END_REF], [START_REF] Piña | Central configurations for the planar Newtonian four-body problem[END_REF][START_REF] Hampton | Stacked central configurations: new examples in the planar five-body problem[END_REF][START_REF] Hampton | Stacked central configurations: new examples in the planar five-body problem[END_REF] provides a new family of planar central configurations, called stacked central configuration, for the 5-body problem with an interesting property: the central configuration has a subset of three bodies forming a central configuration of the 3-body problem, in fact these three bodies are in an equilateral triangle, and the remaining two bodies are in the interior of the triangle and are located symmetrically with respect to a perpendicular bisector.

Recently the first and second author of this paper gave new examples of stacked central configurations of the 5-bodies which, as the ones studied by [START_REF] Hampton | Stacked central configurations: new examples in the planar five-body problem[END_REF], have three bodies in the vertices of an equilateral triangle, but the other two are on the perpendicular bisector [START_REF] Llibre | New central configurations for the planar 5-body problem[END_REF].

In this paper we find a new class of stacked central configurations in a 5-body problem which has three bodies in the vertices of an equilateral triangle and the other two are located symmetrically with respect to a perpendicular bisector in the exterior and above of the triangle, see Figure 1.

1 1 1 m 1 m 2 m 3 m 4 m 5
Figure 1. Three bodies at the vertices of an equilateral triangle and two bodies located symmetrically with respect to a perpendicular bisector.

The paper is organized as follows, in Section 2 we prove the existence and non-existence of this new class of stacked central configurations.

In section 3 we show some numerical evidence of the existence and non-existence of another families of this kind of central configurations.

Existence and non-existence of stacked central configurations

The equations of motion of the planar Newtonian n-body problem are given by ri = -

n j = 1 j = i m j r i -r j r 3 ij , (1)
for i = 1, 2, . . . , n. Here the gravitational constant is taken equal to 1, r j ∈ R 2 is the position vector of the punctual mass m j in the inertial barycentric system, and as before r ij = |r i -r j | is the Euclidean distance between r i and r j .

For the central configurations we have rj = λr j with λ = 0 for all j = 1, . . . , n. So from equation (1) we have

λr i = - n j = 1 j = i m j r i -r j r 3 ij , (2)
for i = 1, 2, . . . , n.

For the planar central configurations instead of working with equation (2) we shall use the Dziobek equations (see Hagihara 1970, p. 241)

f ij = n k = 1 k = i, j m k (R ik -R jk ) ∆ ijk = 0, (3) for 1 ≤ i < j ≤ n, where R ij = 1/r 3 ij and ∆ ijk = (r i -r j ) ∧ (r i -r k ).
As usual ∧ denotes the cross product of two vectors.

For the 5-body problem equations ( 3) is a set of ten equations. Our class of configurations with five bodies as in Figure 1 without collisions must satisfy

r 12 = r 23 = r 13 = 1, r 14 = r 25 , r 15 = r 24 , r 34 = r 35 , r 14 > r 24 , ∆ 124 = ∆ 125 , ∆ 143 = ∆ 235 , ∆ 145 = ∆ 245 , ∆ 135 = ∆ 243 .
We are also assuming ∆ 145 = 0, that is, the bodies m 1 , m 4 and m 5 are not on the same straight line.

The equation

f 45 = 0 of (3) becomes (R 14 -R 15 )∆ 145 (m 1 -m 2 ) = 0.
Thus m 1 = m 2 . Similarly the equation f 12 = 0 of (3) goes over to (R 14 -R 24 )∆ 124 (m 4 -m 5 ) = 0. Therefore m 4 = m 5 . Substituting m 1 = m 2 and m 4 = m 5 into the other eight equations of (3) it follows that

f 13 = 0 ⇔ f 23 = 0, f 15 = 0 ⇔ f 24 = 0, f 14 = 0 ⇔ f 25 = 0, f 34 = 0 ⇔ f 35 = 0,
which can be written respectively as

L = (R14 -R34)∆134 + (R15 -R34)∆135 = 0, (4) (R12 -R15)∆142 m1 + (R12 -R34)∆143 m3 + (R15 -R45)∆145 m4 = 0, (5) (R12 -R14)∆142 m1 + (R12 -R34)∆153 m3 + (R14 -R45)∆154 m4 = 0, (6) [(R12 -R14)∆134 + (R12 -R15)∆153] m1 + (R34 -R45)∆345 m4 = 0. (7)
Denote by A = [a ij ] the matrix of the coefficients of the homogeneous linear system in the variables m 1 , m 3 and m 4 defined by equations ( 5), ( 6) and ( 7). Thus

a 11 = (R 12 -R 15 )∆ 142 , a 12 = (R 12 -R 34 )∆ 143 , a 13 = (R 15 -R 45 )∆ 145 , a 21 = (R 12 -R 14 )∆ 142 , a 22 = (R 12 -R 34 )∆ 153 , a 23 = (R 14 -R 45 )∆ 154 , a 31 = (R 12 -R 14 )∆ 134 +(R 12 -R 15 )∆ 153 , a 32 = 0, a 33 = (R 34 -R 45 )∆ 345 .
In order to have a solutions different from m 1 = m 3 = m 4 = 0 the determinant of the matrix A must be zero, where

det A = (R 12 -R 34 ) [(R 12 -R 15 )∆ 153 -(R 12 -R 14 )∆ 143 ] Γ, (8) with Γ equal to (R 34 -R 45 )∆ 142 ∆ 345 +[(R 14 -R 45 )∆ 143 + (R 15 -R 45 )∆ 153 ] ∆ 154 . (9) From equation (4) one has R 14 ∆ 134 = R 34 ∆ 134 + (R 34 -R 15 )∆ 135 .
Substituting this equation into equation ( 9) it follows that Γ = 0 and therefore, from (8), we get det A = 0. So the homogeneous linear system in the variables m 1 , m 3 and m 4 , defined by equations ( 5), ( 6) and ( 7), has nontrivial solutions under the assumption L = 0, which can be viewed as a constraint on the geometry of the configuration.

m 1 m 2 m 3 P 0 P 1 P 2 P 3 P 4 R 14 = 1 R 24 = 1 R 34 = 1 R 35 = 1 ∆ 345 = 0 R 24 = R 45 R 14 = R 45 L 301 L 312 L 323 L 334 L 34∞ L 3 L ′ 3 Figure 2.
The component of the curve {L = 0} in the upper semiplane and its components L 3 and L ′ 3 .

LEMMA 1. Consider the regions

S 1 = {R 45 > R 34 > R 14 > 1}, S 2 = {1 > R 14 > R 34 > R 45 }, S 3 = {∆ 345 > 0, ∆ 134 < 0, R 34 > R 24 } and S ′ 1 , S ′ 2 , S ′
3 the symmetrical regions with respect to the mediatrix of the segment joining r 1 and r 2 . See Figure 3. The curves

L i = {L = 0} ∩ S i , L ′ i = {L = 0} ∩ S ′ i , i = 1, 2, 3
, are well-defined and are not empty. See Figure 2.

Proof. Without loss of generality we can take a coordinate system such that r 1 = (-1/2, 0), r 2 = (1/2, 0) and r 3 = (0, √ 3/2) respectively.

Thus r 4 = (x, y) and r 5 = (-x, y), where x > 0 and y ∈ R. With these coordinates we have

L(x, y) = 1 ((x + 1/2) 2 + y 2 ) 3/2 - 1 (x 2 + (y - √ 2/2) 2 ) 3/2 M 1 + 1 ((x -1/2) 2 + y 2 ) 3/2 - 1 (x 2 + (y - √ 2/2) 2 ) 3/2 M 2 ,
where

M 1 = y 2 -x √ 3 2 - √ 3 2 and M 2 = y 2 + √ 3x 2 - √ 3
2 . It is easy to see from the above equation that the curve L(x, y) = 0 is symmetric with respect to the mediatrix of the segment joining 1 and r 2 . Using Mathematica we have plotted the graph of the curve L(x, y) = 0, see Figure 2.

Define the points

Q 1 = {∆ 345 = 0} ∩ {R 24 = R 45 } ∩ S 3 , Q 2 = {R 24 = 1} ∩ {R 24 = R 45 } ∩ S 3 , Q 3 = {R 24 = 1} ∩ {R 14 = R 45 } ∩ S 3 , Q 4 = {R 34 = 1} ∩ {R 14 = R 45 } ∩ S 3 , Q 5 = {R 34 = 1} ∩ {R 24 = R 45 } ∩ S 3 ,
according to Figure 3. The symmetrical points are denoted by Q ′ i , i = 1, 2, 3, 4, 5. In [START_REF] Hampton | Stacked central configurations: new examples in the planar five-body problem[END_REF]) Hampton studied stacked central configurations when r 4 and r 5 are on the components L 1 and L ′ 1 respectively. If r 4 ∈ {∆ 234 = 0} (see Figure 3) then r 5 ∈ {∆ 135 = 0}. Thus L = (R 14 -R 34 )∆ 134 < 0. If r 4 ∈ {R 14 = R 34 } (see Figure 3) then L = (R 15 -R 34 )∆ 135 > 0. Therefore there is a connected component of L = 0 between the lines {∆ 234 = 0} and {R 14 = R 34 } denoted by L 2 . See Figure 2. Its symmetrical component is denoted by L ′ 2 . If r 4 ∈ {∆ 134 = 0} (see Figure 3) then L = (R 24 -R 34 )∆ 135 < 0. If r 4 ∈ {R 24 = R 34 } (see Figure 3) then L = (R 14 -R 34 )∆ 134 > 0. By the same argument L > 0 on the set {∆ 345 = 0}. Therefore there a connected component of L = 0 between the lines {∆ 134 = 0} and {R 24 = R 34 }∪{∆ 345 = 0} denoted by L 3 . See Figure 2. Its symmetrical component is denoted by L ′ 3 . In the coordinate system that we have introduced previously one has

Q 1 = √ 13 -1 6 , √ 3 2 , Q 2 = 1 2 , 1 , Q 3 =    1 + √ 5 4 , 1 - √ 5 -1 4 2    , Q 4 = (x 4 , y 4 ), Q 5 = (x 5 , y 5 ),
where x 4 = 1 4 √ 3 cos(π/18) + 2 cos(π/9) + sin(π/18) , y 4 = 1 2 cos(π/18) -√ 3(sin(π/18) -1) ,

x 5 = 1 2 2 + cos(π/9) -√ 3 sin(π/9),

y 5 = 1 2 cos(π/18) + √ 3(1 + sin(π/18)) .
From elementary calculations one has the following things for L:

L(Q 1 ) > 0, L(Q 2 ) < 0, L(Q 3 ) > 0, L(Q 4 ) > 0 and L(Q 5 ) < 0.
Since Q 2 , Q 3 are on the circle R 24 = 1 and L(Q 2 ) < 0, L(Q 3 ) > 0, by continuity there exists at least one point in the intersection of the curve L 3 with the circle R 24 = 1. There is numerical evidence that this point is unique, we denote it as P 2 . Let be P ′ 2 be the symmetric point respect to the mediatrix of the segment joining the particle with mass m 1 with the particle with mass m 2 .

The main result of this paper is the following.

THEOREM 2. Assume that the points r 1 , r 2 and r 3 (with positive masses m 1 , m 2 and m 3 ) are at the vertices of an equilateral triangle, whose sides have length 1, and the points r 4 and r 5 (with masses m 4 and m 5 ) are located symmetrically with respect to a perpendicular bisector, as in Figure 1. Then exist stacked central configurations for which the particles with masses m 4 and m 5 are in a small arc of L 3 near the points P 2 and P ′ 2 respectively.

Proof. When the particles 4 and 5 are at P 2 and P ′ 2 respectively, then since R 15 = R 24 = 1 we obtain a 31 = (1 -R 14 )∆ 134 < 0. Therefore the signs of the coefficients of the matrix A are given by

A P 2 ∪P ′ 2 =   0 -+ -+ + -0 +   .
By the first row of the above matrix the masses m 3 and m 4 have the same sign and by the last row of the matrix the masses m 1 and m 4 also have the same sign. So the masses m 1 , m 3 and m 4 have the same sign. Hence when the particles 4 and 5 are at P 2 and P ′ 2 respectively, there exists masses m 1 , m 3 and m 4 which form a stacked central configuration.

Now, since a 31 continuously on the mutual distances, we assure that a 31 < 0 when particles 4 and 5 are in a small neighborhood of P 2 and P ′ 2 respectively, the same is true for the other coefficients of a ij , which preserve its sign in a small neighborhood of P 2 and P ′ 2 . Using constraint L = 0 we obtain the small arc cited in the statement of the theorem. LEMMA 3. The function

a 31 = (R 12 -R 14 )∆ 134 + (R 12 -R 15 )∆ 153 is negative on the set {L = 0} {R 24 < 1}.
Proof. For our class of central configurations one has R 14 < R 24 . Substituting this inequality into the expression of a 31 we have a 31 < (∆ 143 + ∆ 135 )(R 24 -1) < 0 if r 24 > 1. Thus the lemma follows for the components of L 3 in the exterior of the circle {R 24 = 1}. Let P 0 be the vertex of the triangle at r 3 . If the particles 4 and 5 are on the curve L = 0 and close enough to P 0 , there are no stacked central configurations. In other words we have the following property PROPOSITION 4. Assume that we have the masses m 1 , m 2 and m 3 at the vertices of an equilateral triangle, whose sides have length 1, and the masses m 4 and m 5 are at L 3 = 0 and L ′ 3 respectively, located symmetrically with respect to a perpendicular bisector. If r 4 and r 5 are sufficiently close to P 0 , then there are no positive masses m 1 = m 2 , m 4 = m 5 and m 3 such that the five bodies with these masses form stacked central configurations.

Proof. With the hypothesis of the proposition, we have ∆ 142 < 0, by the other hand the point r 4 inside circle R 24 = 1 which corresponds to the region R 24 > 1, that is R 24 = R 15 < 1, therefore a 11 = (R 12 -R 15 )∆ 142 < 0 in this region and the signs of the coefficients of the matrix A are given by

A =   --- -+ + a 31 0 +   .
The first row in the above matrix implies that m 1 = m 2 , m 4 = m 5 and m 3 cannot have the same sign.

Numerical evidence of new stacked central configurations

There is numerical evidence that L 3 intersects the hyperbola R 14 = R 45 at a unique point denoted by P 4 . Also based on numerical evidences we define L 34∞ to be the unbounded connected component of L 3 with one endpoint at P 4 . As usual, L ′ 34∞ is the unbounded connected component of L ′ 3 with one endpoint at P ′ 4 .

LEMMA 5. There are no stacked central configurations when r 4 and r 5 are on L 34∞ and L ′ 34∞ , respectively.

Proof. When r 4 and r 5 are at P 4 and P ′ 4 , respectively, it is easy verify that a 21 < 0, a 22 < 0 and a 23 = 0 and this implies that the masses m 1 and m 3 have opposite signs.

On the other hand, if r 4 and r 5 are on the open arcs L 34∞ and L ′ 34∞ , respectively, we have that a 21 < 0, a 22 < 0 and a 23 < 0 and this implies that two of the masses m 1 , m 3 and m 4 have opposite signs. This proves (5).

There is numerical evidence that L 3 intersects the hyperbola R 24 = R 45 in a unique denoted as P 1 , and intersects the circle R 34 = 1 in a unique point denoted as P 3 . We now define: L 312 , L 323 and L 334 be the open arcs L 3ij of L 3 , whose endpoints are P i and P j , for i, j = 1, 3, 4. More precisely,

L 312 = L 3 ∩ {∆ 345 > 0, R 24 > 1, R 24 > R 45 , R 14 < R 45 }, L 323 = L 3 ∩ {R 24 < 1, R 24 > R 45 , R 14 < R 45 , R 34 > 1}, L 334 = L 3 ∩ {R 34 < 1, R 24 > R 45 , R 14 < R 45 }.
The same definitions are valid for L ′ 3 , see Figure 2. In order to give numerical evidence for the existence of new stacked central configurations, we have calculated the coordinates of the points P i , i = 1, 2, 3, 4 P 1 = (0.45179079888471224, 0.9022946174921869), P 2 = (0.63217335448833913, 0.9912266160486711), P 3 = (0.93395902567645331, 1.2234055072562044), P 4 = (0.95907733939463801, 1.2451548857175381). LEMMA 6. There are no stacked central configurations when r 4 and r 5 are on the components L 334 and L ′ 334 , respectively.

Proof. For r 4 ∈ L 334 and r 5 ∈ L ′ 334 the signs of the coefficients of the matrix A are given by

A L 334 ∪L ′ 334 =   -+ + --+ -0 +   .
Using the last row of the above matrix we conclude that m 1 and m 4 have the same sign. Unfortunately we cannot obtain more information the sign of the mass m 3 from the above matrix. Nevertheless, equations ( 5) and ( 7) define two planes through the origin in the space (m 1 , m 3 , m 4 ). The normal vectors of these planes are, respectively, n 1 = (a 11 , a 12 , a 13 ) and n 3 = (a 31 , 0, a 33 ). Let T = (T 1 , T 2 , T 3 ) = n 1 ∧ n 3 be the vector parallel to the straight line defined by the intersection of these two planes. Thus there exist positive masses m 1 , m 3 and m 4 as solutions of equations ( 5) and ( 7) if and only if the components of the vector T have the same sign. It is easy verify that T 1 = a 12 a 33 > 0, and T 3 = -a 12 a 31 > 0.

In order to see that T 2 = a 13 a 31 -a 11 a 33 < 0, and since we already know the coordinates of the points P 3 and P 4 , we can verify numerically that the values of T on the curve L = 0 between the points P 3 and P 4 in the interval (-0.376045, -0.368863). As not all the above components have the same sign, Lemma 6 has been proved. LEMMA 7. When r 4 and r 5 are at P 3 and P ′ 3 respectively, there are no stacked central configurations.

When r 4 and r 5 are at P 3 and P ′ 3 , the matrix A takes the form

A =   -0.664093 0 0.694115 -1.04073 0 0.00940803 -1.0721 0 0.56513   .
Then, since columns 1 and 3 in the above matrix are linearly independent the masses m 1 and m 4 must vanish. Therefore there are no stacked central configurations in this case.

The next proposition brings numerical evidence of the existence of stacked central configurations for r 4 ∈ L 323 . PROPOSITION 8. Assume that r 1 , r 2 and r 3 (with positive masses m 1 , m 2 and m 3 ) are at the vertices of an equilateral triangle whose sides have length 1, r 4 and r 5 (with positive masses m 4 and m 5 ) are located symmetrically with respect to a perpendicular bisector such that r 4 ∈ L 323 and r 5 ∈ L ′ 323 , as in Figure 2. Then there are positive masses m 1 = m 2 , m 4 = m 5 and m 3 such that the five bodies with these masses form stacked central configurations. Using the last row of the above matrix we conclude that m 1 and m 4 have the same sign. Unfortunately we cannot obtain more information about the sign of the mass m 3 from the above matrix. Nevertheless, equations ( 5) and ( 6) define two planes through the origin in the space (m 1 , m 3 , m 4 ). The normal vectors of these planes are, respectively, n 1 = (a 11 , a 12 , a 13 ) and n 2 = (a 21 , a 22 , a 23 ). Let T = (T 1 , T 2 , T 3 ) = n 1 ∧ n 2 be the vector parallel to the straight line defined by the intersection of these two planes. Thus there exist positive masses m 1 , m 3 and m 4 as solutions of equations ( 5) and ( 6) if and only if the components of the vector T have the same sign. An easy computation shows that T 1 = a 12 a 23 -a 13 a 22 < 0, T 3 = a 11 a 22 -a 12 a 21 < 0.

In order to see that T 2 = a 13 a 21 -a 11 a 23 < 0, we have verified numerically that the values of T 2 on the curve L 0 between the points P 2 and P 3 in the interval (-0.716137, -0.44343). The Proposition 8 has been proved.

Figure 3 .

 3 Figure 3. Some straight lines and curves for the calculation of the sign of L.

Proof.

  For r 4 ∈ L 323 and r 5 ∈ L ′ 323 the signs of the coefficients of the matrix A are given by A .tex; 12/03/2011; 15:43; p.12

Stacked-fourth.tex; 12/03/2011; 15:43; p.2

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Acknowledgements

The authors thank to the anonymous referees for their deep review of the original version. Their valuable comments and suggestions helped us to improve this work. The first author is partially supported by a MEC/FEDER grant number MTM2008 03437, by a CIRIT grant number 2009 SGR 410 and by ICREA Academia. The second author developed this work under the project CNPq 473747/2006-5. This work was initiated while the second author visited Universitat Autònoma de Barcelona, supported by CNPq grant 210056/2006-1. The third author has been partially supported by CONACYT-México, grant 12378.