Tobias M Weissenbacher
email: tobias.weissenbacher@med.uni-muenchen.de

Madeleine Zschage

Wolfgang Janni

Udo Jeschke

Thomas Dimpfl

Doris Mayr

Brigitte Rack

Christian Schindlbeck

Klaus Friese

Darius Dian

Á M Zschage

Á U Jeschke

Á B Rack

Á C Schindlbeck

Á K Friese

Á D Dian

I Frauenklinik

W Janni Frauenklinik

Multicentric and multifocal versus unifocal breast cancer: is the tumor-node-metastasis classification justified?

Keywords: Breast cancer, Overall survival, Multicentric, Multifocal, Prognostic factor, Prognosis

Introduction

For breast cancer classification, tumor-node-metastasis (TNM) staging has been the standard for more than 50 years. During this time, however, the classification has changed substantially. In 2003, the 6th Edition of the TNM classification was established [START_REF] Benson | The TNM staging system and breast cancer[END_REF][START_REF] Cserni | New TNM classification of breast tumors[END_REF][START_REF] Escobar | The 2003 revised TNM staging system for breast cancer: results of stage re-classification on survival and future comparisons among stage groups[END_REF][START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF]. The prognostic relevance of the T category is a constant factor in regard to the current and past classification [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF]. Survival of patients with breast cancer depends on two different types of prognostic factors: tumor size as a marker of tumor biology and as a time-dependent phenomenon, and biological factors (i.e., histological grade), which represent tumor aggressiveness [START_REF] Bundred | Prognostic and predictive factors in breast cancer[END_REF]. Other prognostic factors are the estrogen and progesterone receptor status as well as the number of mitotic figures per ten high power fields (MF/10HPF) [START_REF] Dabakuyo | Population-based study of breast cancer survival in Cote d'Or (France): prognostic factors and relative survival[END_REF][START_REF] Younes | Stratified multivariate analysis of prognostic markers in breast cancer a: preliminary report[END_REF]. The treatment follows worldwide prevailing suggestions, including those of the TNM system. Not only is the TNM classification changing, treatment recommendations and the treatments themselves have been modified as well.

Breast conserving treatment, once a controversial issue, is now an established alternative to modified radical mastectomy for surgically manageable breast cancer. Established prognostic factors [START_REF] Bundred | Prognostic and predictive factors in breast cancer[END_REF][START_REF] Duraker | Prognostic value of the 2002 TNM classification for breast carcinoma with regard to the number of metastatic axillary lymph nodes[END_REF][START_REF] Comas | Prognostic value of the interpectoral lymph nodes in breast cancer. A 20-year survival study[END_REF] as well as new possible factors, such as the E-cadherin transcriptional repressor Snail or the c-Jun activation domain binding protein-1 (Jab1), are multifunctional signaling proteins that are controversially discussed in the literature [START_REF] Wanami | Vascular endothelial growth factor-A stimulates snail expression in breast tumor cells: Implications for tumor progression[END_REF][START_REF] Wang | Jab1 is a target of EGFR signaling in ER-alpha negative breast cancer[END_REF][START_REF] Yildirim | Serum big endothelin-1 levels in female patients with breast cancer[END_REF]. Despite these ongoing modifications, tumor size in multicentric/multifocal tumors is only assessed by the largest tumor focus and does not reflect the cumulative tumor volume.

The aim of this study was to compare the prognosis of multicentric/multifocal tumors with unifocal tumors with apparently identical tumor size according to TNM staging.

Patients and methods

Patients

Based on a consecutive patient cohort consisting of 5,691 patients who were documented and surgically treated for primary breast cancer between 1963 and 2007, two equivalent groups of 288 early breast cancer patients with multicentric/multifocal versus unifocal tumors were selected using a matched-paired analysis (see ''Statistical analysis'' section below). Study sites were the Departments of Gynecology of the University Hospitals in Berlin-Charlottenburg and Munich-Innenstadt (1987-2007).

Work up modalities for verifying unifocality and specifying data on multifocality/centricity included clinical examinations, ultrasonics, and x-ray. In addition, a nuclear magnetic resonance imaging (NMRI), galactography, or pneumocystography were performed if necessary.

Data were contemporaneously gathered for the unifocal and multicentric/multifocal tumors. To be eligible, patients were required to be free of disease, and they must have been treated at the study sites at the time of primary diagnosis of resectable breast cancer. The tumor stage at primary diagnosis was classified according to the UICC TNM classification [START_REF] Sobin | TNM classification of malignant tumors. A comparison between the new (1987) and the old editions[END_REF]. The Bloom-Richardson system was used for histopathological grading, and match criteria were tumor size, grading, and hormone receptor status, all of which were equally distributed in both groups (P = 1.0). We consciously disclosed the lymph node statuses (LNS) as matching criteria to evaluate differences in LNS in respect to focality.

Treatment

The primary surgical treatment consisted of either breast conservation or modified radical mastectomy. Routine axillary dissections were performed on lymph nodes of levels I and II, while those of level III were excised only in cases expressing macroscopic metastatic of the lower levels. For the diagnosis of lymph node metastasis, single embedded lymph nodes were screened at up to three levels. Telecobalt radiation therapy was administered in patients treated with breast conservation. Chest-wall irradiation was given to all mastectomy patients before 1979, while patients after this date received chest-wall irradiation in cases with extensive lymphangiosis carcinomatosa surrounding the tumors. The median absorbed dose in the target volume was 50 Gy fractionated in 25 fractions in all patients who received irradiation.

According to the guidelines of the Cancer Treatment Center of Munich, patients received chemotherapy in cases of lymph node involvement. Depending on the hormone receptor status they received adjuvant endocrine therapy with tamoxifen (20 mg-30 mg/day). The guidelines for chemotherapy and cytostatic regime were changing dramatically during the observation time of the study. In our patient series, the most common type of chemotherapy consisted of six cycles of CMF, cyclophosphamide (600 mg/m 2 body-surface area), methotrexate (40 mg/m 2 ), and 5-fluoruracil (600 mg/m 2 ) every 21 days. However, this regime was not investigated as adjuvant systemic chemotherapy in a randomized trial. Tamoxifen was given to hormone receptor-positive postmenopausal patients. Premenopausal patients also received GnRH-analogs in the later years of the 44-year time period. Aromatase inhibitors were used in case of contraindications.

Patients were evaluated over a very long period (44 years). As mentioned above, during this time surgical, radiation oncology, and chemotherapy treatment options changed dramatically. However, we did not separate into age-dependent cohorts. Due to the large number of patients, we assumed a normal distribution. A matchedpair-analysis was used to determine the overall and relapsefree survival by comparing multicentric/multifocal versus unifocal breast cancer. Different prognostic factors were correlated with breast cancer-specific survival and relapsefree survival using a multivariate Cox regression analysis.

Statistical analysis

Data were entered into the database in a coded fashion. An error-correcting and plausibility check was performed before starting the statistical analysis. Contingency tables were constructed, analyzed by a chi-square test to evaluate differences of possible non-parametric data. Because of the uneven distribution of prognostic factors in our original patient group of 576 cases that met the match criteria, a matched-pair analysis was performed. A total of 288 pairs of patients, each consisting of one patient with unifocal and one with multicentric/multifocal tumor lesions, were selected according to the highest degree of equivalence in the following hierarchical and sequential order: tumor size at the time of primary diagnosis, histopathological grading, and hormone receptor status. Each parameter was required to have a P value [ 0.50 to achieve intergroup homogeneity. We deliberately matched criteria characterizing the patients at the time of primary diagnosis.

Data quality was controlled by verifying all reported immunocytochemical and histopathological results, event reports during follow-up, and by reexamining the original data files. As a starting point the day of the primary operation and as an end point occurrence of relapse or cancer-associated death were chosen. Kaplan-Meier life table curves were constructed to estimate survival and were compared using the log-rank test. Cox's regression analysis was used for multivariate analyses, and P values less than 0.05 were considered significant. The computer software ''Statistical Package for the Social Sciences 15.0'' (SPSS Inc., Chicago, USA) was used.

Follow-up

Patients attended the oncology outpatient clinic of our institution at least every 3 months during the first 2 years after primary diagnosis or after a local recurrence, every 6 months for the following 2 years, and every year thereafter. Local recurrence was defined as chest-wall recurrence in case of mastectomy and in-breast recurrence in patients with breast saving treatment. Ipsilateral lymph node metastases were defined as local recurrence.

Follow-up included a clinical examination and yearly mammography. Chest radiography, ultrasound of the liver, bone scan, blood tests, and a computed tomography of the appropriate region were performed according to tumor manifestation and/or symptoms. If patients sought care elsewhere, data were collected from the family physicians or gynecologists in charge. All deaths were verified by referring to the German Cancer Register [START_REF] Altmann | Application of a standard methodology for the development of messages and aspects of realization in the area of tumour documentation[END_REF] and classified according to the cancer-associated cause. The mean observation time was 70 months (5.8 years, range: 19-321 months).

Results

A total of 288 pairs, each consisting of one patient with unifocal disease and one patient with multifocal/-centric breast cancer, were created by matched-pair analysis to achieve statistical balance of the major prognostic factors between both groups. All Match criteria (tumor size, grading, and hormone receptor status) were equally distributed in both groups (P = 1.0). Furthermore, no significant difference was found between the two groups in terms of systemic therapy and primary operation. The primary patient characteristics of both groups are shown in Table 1. According to the histological tumor type, there was a significant difference in the incidence of invasive lobular cancer in the multifocal group in comparison to the unifocal group. All the 92 patients had an invasive lobular cancer, 79 of which had a multifocal/-centric disease, whereas only 13 had a unifocal breast cancer. These results were similar concerning invasive ductal tumors. Regarding systemic therapy, 297 patients are included in the multicentric/multifocal group and 295 are in the unifocal group which exceeds the total number of 288. Some patients received only endocrine or only cytostatic therapy, whereas some individuals underwent both types of systemic therapy. Furthermore, patients with multicentric/multifocal disease were demonstrated to have a higher rate of chemotherapy compared to patients with unifocal breast cancer. In all, 101 women (35.1%) received cytostatic therapy in the multicentric/multifocal group, whereas only 65 patients (22.6%) did so in the unifocal group. In contrast, 61 patients (21.2%) with unifocal disease received endocrine therapy as compared to only 51 patients (17.7%) in the multicentric/multifocal group. Considering these treatment variations, a subgrouping was done for the survival curves (Figs. [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF][START_REF] Bundred | Prognostic and predictive factors in breast cancer[END_REF][START_REF] Dabakuyo | Population-based study of breast cancer survival in Cote d'Or (France): prognostic factors and relative survival[END_REF][START_REF] Younes | Stratified multivariate analysis of prognostic markers in breast cancer a: preliminary report[END_REF].

Univariate analysis

At the mean follow-up time of 70 months (5.8 years, range: 19-321 months), the mean breast cancer-specific survival time of 221.6 months in patients with unifocal disease (206.7-236.4, 95% The confidence Interval [CI]) was significant higher compared to patients with multicentric/multifocal tumors (203.3 months, 184.1-222.5, 95% CI, P \ 0.0001, log-rank test) (Fig. 1). In patients with a unifocal tumor, the relapse-free survival was 205.9 months (190.7-221.2, 95% CI), as opposed to 169.6 months (150.8-188.5, 95% CI, P \ 0.0001, log-rank test) in patients with multicentric/multifocal disease. Twenty-one of 288 patients (7.3%) with unifocal disease developed a local recurrence compared to 50 of 288 (17.4%) with multifocal tumors (P \ 0.0001). Occurrence of distant metastases was observed in 61 multicentric/ multifocal diseased patients, as compared to 36 patients in the unifocal group (Figs. 2,3).

The univariate Kaplan-Meier analyses of patients with multicentric/multifocal diseases receiving chemotherapy (N = 101) was significantly different from the unifocal group (N = 65) (P = 0.001). Regarding hormonal therapy, no statistical correlation was identified (P = 0.34, Log rank). Fifty-one patients within the multicentric/ multifocal group received hormone therapy, compared to 61 in the unifocal group. Figures 4,5,6, and 7 illustrate the differences in overall survival comparing focality according to hormone therapy and chemotherapy (Figs. [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF][START_REF] Bundred | Prognostic and predictive factors in breast cancer[END_REF][START_REF] Dabakuyo | Population-based study of breast cancer survival in Cote d'Or (France): prognostic factors and relative survival[END_REF][START_REF] Younes | Stratified multivariate analysis of prognostic markers in breast cancer a: preliminary report[END_REF]. Overall survival was significantly better in patients with unifocal breast cancer compared to multicentric/multifocal disease, either with or without hormone therapy (P = 0.002). Regarding chemotherapy, overall survival was different in patients with multicentric/ multifocal disease compared to unifocal disease but the results did not reach statistical significance (P = 0.07) (Figs. 4,5,6,7).

Multivariate analysis

All of the major established prognostic factors for overall survival as well as for the occurrence of local relapses or metastases (lymph node metastases, grading, tumor size, age, hormone receptor status, chemotherapy, hormonal therapy, and radiation) were included in the multivariate analysis to evaluate the impact of focality. For the Cox multivariate regression analysis regarding breast cancerspecific survival, local or systemic relapse showed that multicentricity/multifocality is a significant independent predictor of reduced breast cancer-specific survival as well as of reduced relapse-free survival (Tables 2,3). Relapse was defined as overall recurrence, including local, regional, and distant recurrence of disease, Tumor size, grading, and lymph node status were also significant independent predictors. Hormone receptor status and age failed to demonstrate significance for the prediction of overall survival (P = 0.80 for progesterone receptor status, P = 0.43 for estrogen receptor status, and P = 0.21 for age). In addition, chemotherapy and radiation showed no statistical significance by multivariate analysis in regard to overall survival. Hormonal therapy was significant concerning overall survival (P = 0.002). Concerning disease recurrence, neither chemotherapy, hormonal therapy nor radiation showed any significance (Tables 2,3).

Discussion

The TNM classification has changed over time to provide more reliable prognostic information. Tumor size, lymph node involvement, and metastases are the key elements taken into account. Especially, LNS is known to be a relevant factor convening disease free survival (DFS) and the development of metastases. In our study, we consciously utilized this parameter as matching criteria to evaluate differences in LNS with respect to focality. Looking at the multivariate Cox Regression Analysis in our study, LNS is shown to be a highly significant risk factor for all evaluated events.

Distant Metastases

Several authors suggest using the TNM 6 classification [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF][START_REF] Duraker | Prognostic value of the 2002 TNM classification for breast carcinoma with regard to the number of metastatic axillary lymph nodes[END_REF][START_REF] Gusterson | The new TNM classification and micrometastases[END_REF][START_REF] Johnson | TNM classification Veronesi et al[END_REF][START_REF] Savran | (2006) Stages of breast cancer[END_REF][START_REF] Singletary | Revision of breast cancer staging: the 6th edition of the TNM classification[END_REF], including the definition of micrometastases. A transformation from the current rigid categorization into a more flexible system is also recommended [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF] because several additional factors should be considered during the decision-making process.

Diagnosis and treatment have also changed over the years for multicentric/multifocal breast tumors, although at a much slower pace. Breast conserving treatment is now an established alternative to radical mastectomy. As for tumors with more than one lesion, treatment suggestions are currently changing. Many researchers support the extension of conservative surgery for multicentric/multifocal tumors [START_REF] Gentilini | Conservative surgery in patients with multifocal/multicentric breast cancer[END_REF][START_REF] Gentilini | Conservative approach for breast cancer. The experience of the European Institute of Oncology[END_REF][START_REF] Spinelli | Multicentric breast tumour: an anatomical-clinical study of 100 cases[END_REF][START_REF] Wakabayashi | Unilateral multicentric breast carcinoma studied by whole mammary gland serial sectioning[END_REF]. Nevertheless, the indications should be accurately applied as some authors report a higher rate of recurrence after conservative treatment [START_REF] Fowble | The role of mastectomy in patients with stage I-II breast cancer presenting with gross multifocal or multicentric disease or diffuse microclassifications[END_REF].

In terms of TNM classification, however, the tumor size in multicentric/multifocal breast carcinomas still refers to the diameter of the largest lesions. This study examined two groups of patients with multicentric/multifocal and unifocal breast cancer. Both groups were well balanced for the established prognostic indicators-tumor size, grading, and hormone receptor status at the time of presentationand equally distributed in both groups (P = 1.0). Our longterm findings, with a median follow-up of 70 months (5.8 years), demonstrated that multicentricity/multifocality are significant independent predictors for reduced breast cancer-specific survival as well as reduced relapse-free survival.

These findings are consistent with several studies [START_REF] Andea | Correlation of tumor volume and surface area with lymph node status in patients with multifocal/multicentric breast carcinoma[END_REF][START_REF] Andea | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF][START_REF] Sobin | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF][START_REF] Sobin | Multifocal/multicentric breast carcinoma[END_REF][START_REF] Vlastos | Impact of multicentricity on clinical outcome in patients with T1-2, N0-1, M0 breast cancer[END_REF]. To our knowledge, however, no other study to date has examined the overall survival as well as the relapse-free survival. Other studies have shown, that if aggregate diameters are used, unifocal and multifocal breast carcinomas are similar with respect to the frequency of regional lymph node metastasis. The currently used algorithms, which employs the diameter of the largest nodule, result in the downplaying of multifocal breast carcinomas due to the underestimation of actual tumor size [START_REF] Andea | Correlation of tumor volume and surface area with lymph node status in patients with multifocal/multicentric breast carcinoma[END_REF][START_REF] Andea | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF]. It has been suggested that studies should indicate the number of macroscopically recognizable tumors in parentheses after the T classification [START_REF] Sobin | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF][START_REF] Sobin | Multifocal/multicentric breast carcinoma[END_REF]. Andea et al. [START_REF] Andea | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF] showed that, when a combined diameter assessment was used, the frequency of lymph node positivity was not significantly different in multifocal versus unifocal cases. Coombs and Boyages [START_REF] Coombs | Multifocal and multicentric breast cancer: does each focus matter?[END_REF] confirmed this finding. Concerning this investigation, multicentric and multifocal breast cancer is associated with increased nodal involvement compared to similar unifocal disease and the tendency of breast tumors to metastasize is a reflection of the total tumor load [START_REF] Coombs | Multifocal and multicentric breast cancer: does each focus matter?[END_REF]. When looking only at the largest diameter, an understaging of multifocal breast carcinomas results [START_REF] Andea | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF]. Evaluating patients characteristics (Table 1), 30.9% of BC patients in the multicentric/mulitfocal group were classified as pN2 and pN3 in our study. This was more than twice as much as in the unifocal group with 14.2%. This might result in facilitating treatment decisions and, subsequently, in the outcome of the multicentric/mulitfocal group. All of the previously mentioned studies reveal a clear coherence between tumor load and lymph node involvement in a subsequently poor outcome.

In contrast, Fajdic et al. [START_REF] Fajdic | Unilateral multicentric breast cancer[END_REF] did not show an association between size and increased risk for a poor prognosis of multicentric carcinomas. This study included 296 breast cancer patients, but only 29 were histopathologically proven as multicentric tumors, and significantly lower survival rates were shown in all advanced tumor stages. The finding of this study is reliable for lower tumor stages. By retrospectively analyzing 706 patients who received neoadjuvant anthracycline-based chemotherapy, clinically detecting multicentric/multifocal disease in this study also failed to predict an inferior outcome [START_REF] Oh | Locoregional control of clinically diagnosed multifocal or multicentric breast cancer after neoadjuvant chemotherapy and locoregional therapy[END_REF]. Even when cancer stages were reaching from levels I-III, the prognostic value was masked by the neoadjuvant chemotherapy, which does not collectively reflect the findings of our study. The results of this study should, therefore, be interpreted with caution.

Vlastots et al. [START_REF] Vlastos | Impact of multicentricity on clinical outcome in patients with T1-2, N0-1, M0 breast cancer[END_REF] examined 284 patients and detected multicentric breast cancer in 60 patients. There was a significant difference between groups in estrogen receptor and axillary lymph node positivity, but these did not significantly contribute to the outcome of the multivariate analysis. Therefore, multicentricity in this study did not increase the risk of poor outcome. In contrast, our study showed no significant differences regarding hormone receptor status and patient age for the prediction of overall survival. We found a reduced breast cancer-specific survival as well as reduced relapse-free survival (Figs. 1,2). These findings are in contrast to Vlastots et al. [START_REF] Vlastos | Impact of multicentricity on clinical outcome in patients with T1-2, N0-1, M0 breast cancer[END_REF].

The tendency of breast tumors to metastasize [START_REF] Coombs | Multifocal and multicentric breast cancer: does each focus matter?[END_REF] is a reflection of the total tumor load. Failure to measure the additional tumor burden provided by multiple small foci may underestimate the disease. This may deny patients the opportunity of adjuvant therapies if the contribution of the smaller foci to the incidence of node positivity and survival is ignored. This hypothesis is supported by our findings. The factors correlated to multicentricity were shown to be, in a retrospective study by Anastassiades et al. [START_REF] Anastassiades | Multicentricity in breast cancer. A study of 366 cases[END_REF] the age of the patient, the size, and the histological type of the tumor, levels of the progesterone receptors more than 50 fmol/mg of protein, and the amount of solid tissue in the breasts. In contrast, our results are not comparable due to the fact that the above elements represent our match criteria.

Many studies point out an increased risk for metastatic dissemination and, therefore, poorer outcome, a finding that is not currently reflected in the TNM classification [START_REF] Fowble | The role of mastectomy in patients with stage I-II breast cancer presenting with gross multifocal or multicentric disease or diffuse microclassifications[END_REF][START_REF] Andea | Correlation of tumor volume and surface area with lymph node status in patients with multifocal/multicentric breast carcinoma[END_REF][START_REF] Andea | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF][START_REF] Sobin | Pathologic analysis of tumor size and lymph node status in multifocal/multicentric breast carcinoma[END_REF][START_REF] Sobin | Multifocal/multicentric breast carcinoma[END_REF]. As mentioned above, the prognostic relevance of the T category is a constant factor in regard to current and past classification [START_REF] Veronesi | Rethinking TNM (2006) breast cancer TNM classification for treatment decision-making and research[END_REF] and survival of patients with breast cancer depends on several prognostic factors. Tumor size as a marker of tumor biology and as a time-dependent phenomenon, represents a very important prognostic factor, but might, according to our results, be misrepresented. On the basis of the presented data, we conclude that in breast cancer multicentric/multifocal disease is associated with a worse prognosis compared to unifocal disease with an identical tumor size. In summary, the combined tumor volumes appear to be more decisive rather than only the largest tumor diameter. The total tumor volume seems to be of decisive importance rather than the largest tumor diameter. We hypothesize that multicentric/multifocal BC is an independent BC risk factor and should be included in the risk assessment by re-evaluating the current TNM classification of the UICC.

Fig. 3 FocalityFig. 4 Fig. 2 Fig. 1

 3421 Fig. 3 Kaplan-Meier occurrence of distant metastasis analysis among patients with multicentric/Multifocal and unifocal breast cancer as the time of follow-up (in month [mon])

Fig. 5 Fig. 6 Fig. 7

 567 Fig. 5 Kaplan-overall survival with hormone therapy analysis among patients with multicentric/multifocal and unifocal breast cancer as the time of follow-up (in month [mon])

Table 1

 1

	Patient characteristics at primary diagnosis	Multicentric/multifocal (%)	Unifocal (%)	P Value
	Number of patients	288	288
	Age	54.0	57.4	.001
	Lymph node metastases			.0001
	Absent (N0)	139 (48.3)	168 (58.3)
	1-3 axillary LNM (pN1)	60 (20.8)	79 (27.4)
	4-9 axillary LNM (pN2)	47 (16.3)	26 (9.0)
	Caxillary LNM (pN3)	42 (14.6)	15 (5.2)
	Histological type			.0001
	Ductal	207 (71.9)	274 (95.1)
	Lobular	79 (27.4)	13 (4.5)
	Ductal-lobular	2 (0.7)
	Other		1 (0.3)
	Menopausal status			.001
	Premenopausal	138 (47.9)	108 (37.5)
	Postmenopausal	150 (52.1)	180 (62.5)
	Primary operation			.047
	Breast conserving	124 (43.1)	145 (50.3)
	Mastectomy	164 (56.9)	143 (49.7)
	Radiotherapy			.001
	Performed	118 (41.0)	167 (58.0)
	Not performed	170 (59.0)	121 (42.0)
	Systemic therapy			.027
	None	145 (50.3)	169 (58.7)
	Endocrine	51 (17.7)	61 (21.2)
	Cytostatic	101 (35.1)	65 (22.6)
	Local recurrence			.001
	No recurrence	238 (82.6)	267 (92.7)
	Recurrence	50 (17.4)	21 (7.3)
	Distant metastasis			.004
	Absent	227 (78.8)	252 (87.5)
	Distant metastasis	61 (21.2)	36 (12.5)

Table 2

 2 Multivariate Cox regression analysis for breast cancer-specific survival

	Variables	Significance	Wald statistics	HR
	Tumor size a	.013	6.2	1.02
	Lymph node metastases a	.0001	36.8	1.08
	Grading	.044	4.04	2.42
	Focality	.016	5.8	1.57
	Chemotherapy	0.07		0.
	Hormonal therapy	0.002		0.002
	Radiation	0.157
	Age	0.013		0.013

a At time of primary treatment HR Hazard ratio

Table 3

 3 Multivariate Cox regression analysis for overall recurrence

	Variables	Significance	Wald statistics	HR
	Tumor size a	.005	7.8	1.08
	Lymph node metastases a	.0001	37.8	1.07
	Radiation therapy	.028	4.8	.71
	Focality	.0001	12.1	1.74
	Chemotherapy	0.761
	Hormonal therapy	0.107
	Radiation	0.151
	Age	0.666
	a At time of primary treatment
	HR Hazard ratio

Breast Cancer Res Treat (2010) 122:27-34

Acknowledgment We would like to thank Dr. Steven S. Witkin (Weill Cornell Medical College, New York, USA) for his help with the manuscript.


Guillaume Moulis

Haleh Bagheri

Jacques Saint Martory

Pascale Bernard

Jean-Louis Montastruc

Jacques Saint

Very late relapse of dapsone-induced methaemoglobinemia

Keywords:

Dapsone, a sulfone active against a wide range of bacteria, is mainly used for Mycobacterium leprae. Moreover, because of its immunosuppressive effect, it is used for many auto-immune diseases [START_REF] Anonymous | ) Martindalethe complete drug reference[END_REF]. Methaemoglobinemia is an expected adverse drug reaction of dapsone, seldom symptomatic. Long-term administration of dapsone at standard doses (100 mg/day) results in methaemoglobinemia in about 15% of patients [START_REF] Coleman | The use of cimetidine to reduce dapsone-dependent methaemoglobinaemia in dermatitis herpetiformis patients[END_REF]. When symptomatic (dyspnea, cyanosis, neurological disorders), methaemoglobinemia is treated with methylene blue (1-2 mg/kg) [START_REF] Turner | The recognition, physiology, and treatment of medication-induced methemoglobinemia: a case-report[END_REF]. Rebounds have been described after dapsone withdrawal, with a maximum delay of 64 h [START_REF] Macdonald | Acute dapsone intoxication: a pediatric case-report[END_REF]. We report a case of dapsone-induced methaemoglobinemia which relapsed 10 days after drug withdrawal.

A 25-year-old woman was diagnosed with systemic lupus erythematosus in January 2009. Organs involved were skin and joints, and hydroxychloroquine alone was prescribed. In October 2009, she developed a bullous dermatitis and dapsone was added to hydroxychloroquine on 26 October 2009. About 1 month later, on 24 November, she was admitted to the hospital for severe dyspnea. Physical examination was normal except for cutaneous cyanosis. Chest computed tomography and electrocardiogram were normal. Arterial blood gas analysis showed pH: 7.63, PaO 2 : 110 mmHg, PaCO 2 : 15 mmHg, haemoglobin: 11 g/dL, SaO 2 : 99.1%; methaemoglobin: 17%. Methylene blue (2 mg/kg) was injected intravenously. Twenty minutes later, the patient was asymptomatic and arterial methaemoglobin was 4%. Dapsone was discontinued and the patient left the hospital on 26 November. Ten days later, she consulted for relapse of dyspnea. Arterial blood gas analysis showed methaemoglobin at 14%. No symptomatic treatment was done and methaemoglobin rate decreased spontaneously to 5% the next day. Hydroxychloroquine was stopped. The compliance of the patient was not doubtful, and no other cause of intoxication causing methaemoglobinemia was found.

Methaemoglobinemia in dapsone intoxication is due to the ability of dapsone hydroxylamine, a metabolite of the drug, to react with haemoglobin (Fe 2+ ) to form methaemoglobin (Fe 3+ ) [START_REF] Kramer | Mechanism of methaemoglobin formation by diphenylsulphones[END_REF]. Methaemoglobin is unable to bind dioxygen, which results in functional anaemia. The clinical symptoms depend on the level of methaemoglobinemia: asymptomatic if less than 15%; cyanosis for levels above15%; headache, dyspnea, nausea, tachycardia, weakness for levels above 20%; coma for levels above 45%; and a high mortality rate is associated with levels above 70% [START_REF] Curry | Methemoglobinemia[END_REF]. Methylene blue could act as a cofactor of the enzyme NADPH-methaemoglobin reductase, converting a methaemoglobin molecule into a haemoglobin molecule.

Regarding the relationship with hydroxychloroquine, the delay of onset of methaemoglobinemia seems too long (10 months). To our knowledge, no observation of hydroxychloroquine-induced methaemoglobinemia has been reported in the literature. We only found some cases of chloroquine-induced methaemoglobinemia, generally revealing a genetic deficiency in NADPH-cytochrome b5 reductase [START_REF] Sharma | Unusual life-threatening adverse drug effects with chloroquine in a young girl[END_REF][START_REF] Cohen | Methemoglobinemia provoked by malaria prophylaxis in Vietnam[END_REF].

The delay of the methaemoglobinemia relapse after dapsone withdrawal may be explained by the long elimination half-time (20-30 h) [START_REF] Zuidema | Clinical pharmacokinetics of dapsone[END_REF], which also depends on genetic polymorphisms of cytochrome P450, particularly CYP2E1, which produces the hydroxylamine metabolite [START_REF] Mitra | Metabolism of dapsone to its hydroxylamine by CYP2E1 in vitro and in vivo[END_REF]. Furthermore, a linear relationship has been demonstrated between hydroxylamine-dependent methaemoglobin formation and reduction of hydroxylamine to dapsone in human red blood cells in vitro [START_REF] Coleman | Reduction of dapsone hydroxylamine to dapsone during methaemoglobin formation in human erythrocytes in vitro[END_REF]. According to these authors, there may be a cycle between hepatic oxidation of dapsone to its hydroxylamine and reduction to the parent amine within the red cells, which may lead to re-oxidation by hepatic cytochrome P450 and thus contribute to the persistence of the drug in vivo.

These pharmacokinetic particularities leading to a late relapse of dapsone-induced methaemoglobinemia should be kept in mind by physicans for appropriate treatment.

G. Moulis Service de Médecine Interne, Centre Hospitalier Universitaire de Toulouse, Université Paul-Sabatier, Toulouse, France e-mail: gmoulis@hotmail.com G. Moulis : H. Bagheri : J.-L. Montastruc Service de Pharmacologie Clinique, Centre Midi-Pyrénées de Pharmacovigilance, de Pharmacoépidémiologie et d'Informations sur le Médicament, Unité de Pharmacoépidémiologie EA 3696, Centre Hospitalier Universitaire de Toulouse, Université de Toulouse, Faculté de Médecine, Toulouse, France J. Saint Martory Service de Médecine Interne, Centre Hospitalier d 'Albi, Albi, France P. Bernard Service de Pharmacie, Clinique de l'Union, Saint-Jean de l'Union, France G. Moulis : H. Bagheri (*) Service de Pharmacologie Clinique, Faculté de Médecine, 37 Allées Jules-Guesde, 31000 Toulouse, France e-mail: bagheri@cict.fr Eur J Clin Pharmacol (2010) 66:645-646 DOI 10.1007/s00228-010-0801-x


Laurent Boudin

AND Aurore Mercier
email: mercier@ann.jussieu.fr

Francesco Salvarani
email: francesco.salvarani@unipv.it

Conciliatory and contradictory dynamics in opinion formation

come

Introduction

The kinetic approach in sociophysics is a promising line of research to explain collective behaviours in a simple, but mathematically solid, way.

A kinetic model consists in a set of partial integro-differential equations that govern the time evolution of a probability density function, which fully describes the system. The independent variables of the unknown function are the time, and other physical quantities which are relevant to the problem. When dealing with rarefied gases, the typical independent variables are the position and the velocity of the gas particles. In opinion dynamics, a common independent variable is the opinion -or the agreement -with respect to a question.

The introduction of kinetic models in the sociophysical literature started some years ago [START_REF] Helbing | Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory[END_REF][START_REF] Helbing | Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models[END_REF][START_REF] Helbing | A mathematical model for the behavior of individuals in a social field[END_REF]. This methodology has recently experienced a renewal of attention [START_REF] Toscani | Kinetic models of opinion formation[END_REF][START_REF] Aletti | First-order continuous models of opinion formation[END_REF][START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF][START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF][START_REF] Comincioli | A Boltzmann-like equation for choice formation[END_REF][START_REF] Düring | Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders[END_REF][START_REF] Boudin | Kinetic model for multidimensional opinion formation[END_REF] for many reasons, see also the review article [START_REF] Boudin | Modelling opinion formation by means of kinetic equations[END_REF].

In this article we study the time evolution of the opinion, with respect to questions of binary type (e.g. a referendum), in a closed community. This problem is a classical issue of sociophysics, and many authors have considered it, see, for example, [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF][START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF][START_REF] Toscani | Kinetic models of opinion formation[END_REF][START_REF] Comincioli | A Boltzmann-like equation for choice formation[END_REF][START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF][START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF][START_REF] Boudin | Kinetic model for multidimensional opinion formation[END_REF].

In our model, based on a kinetic approach, the unknown is a probability density function that depends on two independent variables: time and opinion of the agent regarding the aforementioned binary question. Each individual of the population can modify his opinion only through the binary exchange of ideas with another member of the community.

Nevertheless, even if we only take into account this elementary phenomenon of opinion exchange, the collective behaviour of the population with respect to the binary question is far from being simple. Indeed, it is well accepted in the literature that many different behaviours concerning the dynamics of opinion formation depend on the fact that the way people think is not uniform. A realistic model should therefore include as many binary interaction rules between individuals as mental paths inside the population.

For simplicity reasons, we only consider here two ways of thinking. As we shall show, even this simplified situation leads to interesting phenomena.

The first psychological attitude is typical of individuals which tend to compromise after an opinion exchange. This behaviour is widely recognized, and it can be considered as the most common one [START_REF] Ben-Naim | Bifurcation and patterns in compromise processes[END_REF][START_REF] Ben-Naim | Unity and discord in opinion dynamics[END_REF][START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF][START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF]. Besides, it is also the main common feature of the kinetic approaches, as it is emphasized in [START_REF] Boudin | Modelling opinion formation by means of kinetic equations[END_REF].

The other mental path is completely different, and leads to psychological dynamics of contradictory type. This behaviour has been proposed by Galam in [START_REF] Galam | Contrarian deterministic effects on opinion dynamics: "the hung elections scenario[END_REF][START_REF] Galam | From 2000 Bush-Gore to 2006 Italian elections: voting at fifty-fifty and the contrarian effect[END_REF], and is based on the fact that some people are deliberately opposed to the choice of the interlocutors, whatever that choice may be.

These two psychological behaviours are translated in our paper by different kinds of collisional rules of kinetic type, which permit to obtain, in a deterministic way, the post-interaction opinions starting from the precollisional ones.

It is worth noting that the collision rules proposed here are only examples of possible behaviours. Actually, the psychology of an individual cannot be considered as a mechanical system and it may vary interaction by interaction. It is moreover clear that, in a real situation, the phenomenology is much more intricate (the effect of mass media can be decisive, for example [START_REF] Boudin | Kinetic model for multidimensional opinion formation[END_REF]). Nevertheless, as we shall see, the presence of only two fixed behaviours in the context of interpersonal communication is already enough to explain many interesting phenomena, such as the concentration towards some particular opinions or the cyclic (in time) behaviour of the distribution function, two phenomenologies that have been sociologically observed [START_REF] Galam | From 2000 Bush-Gore to 2006 Italian elections: voting at fifty-fifty and the contrarian effect[END_REF].

In our model, we suppose that the population is closed. It means that the total number of individuals is constant. This assumption does not imply a great limitation, since the characteristic times of opinion evolution are very small with respect to typical characteristic times in population dynamics. Moreover, we suppose that the probability of a binary interaction is constant. In a structured society, this hypothesis is not true. Indeed, people are normally involved in a social network, and hence some meetings are much more probable than other ones. However, the influence of a network cannot be explained as a function of opinion and time. Since we restrain ourselves to consider only those two independent variables, it seems logical to treat, as a first approximation, the probability of a binary interaction as a constant. We finally note that our description is well adapted only when the size of the population is large enough: this is the key-point of any kinetic model.

All the previous assumptions have a double effect. On the one hand, they reduce the applicability of the model, on the other hand, by simplifying the phenomena taken into account, they allow to build a model which is tractable from a mathematical point of view.

The organization of the paper is the following. We first describe our model, and put in light different collision rules for both groups of agents. Then we prove an existence result for the considered problem. Eventually, we present some relevant numerical tests and provide an analysis of the quantitative results from a sociological point of view.

The kinetic model

In this paper, we study the opinions regarding a binary question (e.g. a referendum) in a non-homogeneous population composed by individuals with different psychologies. We aim to estimate how different possible reactions with respect to the opinion exchange can influence its global behaviour.

In the following, Ω denotes the open interval (-1, 1). We describe the opinion by means of a continuous variable x ∈ Ω, where x = -1 and x = 1 are identified with the two extreme positions. Any intermediate value between those values means that the corresponding individual partially agrees with the opinion labelled with the same sign, with a degree of conviction which is proportional to |x|. If x = 0, the corresponding individual has no preference with respect to the question.

We describe the population by using a kinetic approach. The main tool of the model is the concept of distribution function, a quantity which depends on time t and opinion variable x, and on the features of the population. Its time evolution, governed by a partial differential equation, then allows to forecast the behaviour of the system. Since the population is nonhomogeneous, a possible strategy of description consists in stratifying the individuals with respect to their psychological reactions during the opinion exchange process.

We assume that the members of the population belong to two different groups, and introduce two associated distribution functions which separately describe each group: the population can be split into conciliatory people and contradictory ones (the precise definitions is given later on). This assumption is the simplest one that allows to investigate the effects of different psychologies on the global behaviour of the population. A generalization of the kinetic approach to more complex situations is obviously possible. In that case, the distribution function is of vectorial type, with as many entries as the possible psychological behaviours inside the population. Note that we do not pay attention to the spatial structure of the closed community, which is assumed interlinked. The two groups are respectively described by the density functions f = f (t, x) and g = g(t, x). Both are defined on R + × Ω.

Once individuated the population to study, if the opinions are defined on a subdomain D of Ω, the integrals D f (t, x) dx and D g(t, x) dx represent the number of conciliatory and contradictory individuals with opinion included in D at time t. Note that, in order to give a sense to the previous considerations, we must suppose that f and g satisfy f (t, •) ∈ L 1 (Ω) and g(t, •) ∈ L 1 (Ω) for all t ∈ R + .

As sketched in the introduction, we only take into account one process of opinion formation given by the interaction between agents, who exchange their point of view and influence themselves. Moreover, we suppose that the interactions between individuals are only of binary type. Multiple interactions can be seen as the result of a chain of binary exchanges.

We model this binary process by borrowing the collisional mechanism of a typical interaction in the kinetic theory of gases: whereas in rarefied gas dynamics the particles exchange momentum and energy in such a way that the principles of classical mechanics are satisfied, here the collisions between individuals allow the exchange of opinions. Since there are two categories of people within the population, we define three types of interactions. We assume that the collision mechanisms do not destroy the bounds of the interval Ω. We shall present the collision rules in Section 4.

Each post-collisional opinion can be written as a function of the precollisional opinions and depends on the psychologies of the individuals. The interactions between individuals are described by a collisional integral of Boltzmann type, which has the classical structure of a dissipative Boltzmann kernel. Each collisional integral can be viewed as composed of two parts: a gain term, which quantifies the exchanges of opinion between individuals which give, after the interaction with an other individual, the opinion x, and a loss term, which quantifies the exchanges of opinion where an individual with pre-collisional opinion x experiences an interaction with another member of the population.

It is apparent, in general, that the existence of a pre-collisional pair, which returns a given post-collisional pair after interaction, is not guaranteed, unless we suppose that the collisional rule is a diffeomorphism of Ω2 onto itself. Unfortunately, this assumption is not easy to satisfy. In order to overcome this difficulty, we use a weak form of our problem, in the variable x only, which seems a natural framework for such collision rules, as in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF].

The weak form of the collision kernels are presented below, where ϕ = ϕ(x) ∈ C 0 ( Ω) is a test function.

Conciliatory-conciliatory interactions. Let x, x * ∈ Ω the pre-collisional opinions of two conciliatory agents, and x Q , x Q * ∈ Ω the opinions after the interaction.

We denote by Q(f, f ) the associated kernel, which is defined by

(1) Q(f, f ), ϕ = β Q Ω 2 f (t, x)f (t, x * ) ϕ x Q -ϕ(x) dx dx * ,
and

Q + (f, f ) the gain part of Q(f, f ), namely Q + (f, f ), ϕ = β Q Ω 2 f (t, x)f (t, x * )ϕ x Q dx dx * .
Conciliatory-contradictory interactions. Let x, x * ∈ Ω the respective pre-collisional opinions of a conciliatory agent and a contradictory one before an interaction, and x R , x R * ∈ Ω the opinions after collision. We denote by R 1 (f, g) and R 2 (f, g) the associated kernels, which respectively contribute to the time evolution of f and of g. They are defined by

R 1 (f, g), ϕ = β R Ω 2 f (t, x)g(t, x * ) ϕ x R -ϕ(x) dx dx * , (2)
R 2 (f, g), ϕ = β R Ω 2 f (t, x)g(t, x * ) ϕ x R * -ϕ(x) dx dx * . (3)
Their gain parts are respectively R + 1 (f, g) and R + 2 (f, g), i.e.

R + 1 (f, g), ϕ = β R Ω 2 f (t, x)g(t, x * )ϕ x R dx dx * , R + 2 (f, g), ϕ = β R Ω 2 f (t, x)g(t, x * )ϕ x R * dx dx * .
Contradictory-contradictory interactions. Let x, x * ∈ Ω the pre-collisional opinions of two contradictory individuals, and x S , x S * ∈ Ω the postcollisional ones.

We eventually define the associated kernel S(g, g) by ( 4)

S(g, g), ϕ = β S Ω 2 g(t, x)g(t, x * ) ϕ x S * -ϕ(x) dx dx * .
The associated gain term is denoted by S + (g, g), and defined by

S + (g, g), ϕ = β S Ω 2 g(t, x)g(t, x * )ϕ x S * dx dx * .
Analytical form of the model. In all collisional terms, the parameters β Q , β R and β S govern the probability that the associated interaction can occur. In our model, we choose them as constants. This is the simplest possible assumption, which means that the probability of interaction of two individuals does not depend on their respective opinions. Of course, other choices, based on sociological considerations, are possible. Note that, in (1)-( 4), the post-collisional opinions

x Q , x Q * , x R , x R * , x S and x S
* only appear as an argument of the test-function ϕ. It is also clear that the collision operators only act on the opinion variable. Moreover, if we suppose that f (t, •) and g(t, •) lie in L 1 (Ω), then the gain and loss parts of the operators, and consequently, the operators themselves, also lie in L 1 (Ω) for any t.

Let T > 0. The evolution of f and g is given by the following system of integro-differential equations, in the weak sense in the variable x, where both f in and g in are L 1 (Ω) and nonnegative,

∂ t f = Q(f, f ) + R 1 (f, g), (5) ∂ t g = R 2 (f, g) + S(g, g), (6)
with the initial conditions ( 7)

f (0, x) = f in (x), g(0, x) = g in (x) a.e. x ∈ Ω.
Equations ( 5)-( 6) take sense for almost every t ∈ [0, T ], for test-functions ϕ ∈ C 0 ( Ω).

Mathematical properties

This section is devoted to state and study some mathematical properties of ( 5)- [START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF], which do not depend on the collision rules. We first obtain some a priori estimates and then deduce a result which ensures the existence of weak solutions to ( 5)-( 7).

Proposition 1. Let (f, g) be a nonnegative weak solution to (5)-( 7) with nonnegative initial data f in , g in ∈ L 1 (Ω). Their respective total masses are conserved, i.e. for almost every t ∈ [0, T ],

f (t, •) L 1 (Ω) = f in L 1 (Ω) , g(t, •) L 1 (Ω) = g in L 1 (Ω) .
Proof. We take ϕ = 1 as the test function in ( 5)-( 6).

Proposition 1 means that there is conservation of both species of the population. This property is not realistic if we consider long-time forecasts. Indeed, in such situations, we should also consider processes of birth and death, which also lead to oscillations in the total number of individuals. But usually, as in the case of elections or referendums, the interest of such models is to deduce short-term forecast by using, as an initial datum, poll results. The quantities of interest are then the macroscopic observables

Ω - f (t, x) dx, Ω + f (t, x) dx, Ω - g(t, x) dx, Ω + g(t, x) dx,
where Ω -= (-1, 0) and Ω + = (0, 1). In fact, we shall numerically study those integrals in Section 5.

Since |x| ≤ 1, from the mass conservation, we immediately deduce that all the moments of both f and g are bounded.

Corollary 1. Let (f, g) be a nonnegative weak solution to (5)-( 7) with nonnegative initial data f in , g in ∈ L 1 (Ω). Then we have, for almost every t ∈ [0, T ],

Ω x n f (t, x) dx ≤ f in L 1 (Ω) , Ω x n g(t, x) dx ≤ g in L 1 (Ω) .
In order to prove the existence of weak solutions to ( 5)-( 7), we first need the following proposition.

Proposition 2. Let µ 1 , µ 2 be nonnegative constants, σ 1 , σ 2 nonnegative functions in C 0 ([0, T ]; L 1 (Ω)), and u in , v in nonnegative initial data in L 1 (Ω). The system

(8) ∂ t u + µ 1 u = σ 1 , ∂ t v + µ 2 v = σ 2 ,
with initial conditions

(9) u(0, •) = u in , v(0, •) = v in , has a unique solution (u, v) ∈ C 0 ([0, T ]; L 1 (Ω)) 2 .
Moreover, both u and v are nonnegative.

Proof. Let us note that ( 8)-( 9) is a linear ordinary differential system. The Cauchy-Lipschitz theorem implies the existence and uniqueness of the solution to ( 8)- [START_REF] Comincioli | A Boltzmann-like equation for choice formation[END_REF]. By using Duhamel's formula, we also know that this solution satisfies, for any t and for a.e. x,

u(t, x) = u in (x) e -µ 1 t + t 0 e µ 1 (s-t) σ 1 (s, x) ds, v(t, x) = v in (x) e -µ 2 t + t 0 e µ 2 (s-t) σ 2 (s, x) ds.
The explicit form of the solution ensures the nonnegativity and time continuity of u and v with respect to t.

Thanks to the previous result, we can now prove the following existence theorem, whose proof is quite similar to the one of the main result in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF].

Theorem 1. Let f in , g in be nonnegative functions in L 1 (Ω). Then there exists (f, g) ∈ L ∞ (0, T ; L 1 (Ω)) × L ∞ (0, T ; L 1 (Ω)) which solves (5)-( 6) with initial conditions [START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF], where the equations take sense in D ′ (-T, T ).

Proof. Let us set

̺ f = Ω f in (x * ) dx * ≥ 0, ̺ g = Ω g in (x) dx ≥ 0.
We consider the sequence (f n , g n ) n∈N inductively defined by f 0 = 0, g 0 = 0, and, for n ≥ 1, as weak solutions of

∂ t f n+1 + (β Q ̺ f + β R ̺ g )f n+1 = Q + (f n , f n ) + R + 1 (f n , g n ), ( 10
)
∂ t g n+1 + (β R ̺ f + β S ̺ g )g n+1 = R + 2 (f n , g n ) + S + (g n , g n ), (11)
altogether with the initial conditions f n (0, •) = f in and g n (0, •) = g in .

With a constant test function equal to 1, it is clear that, for all n ∈ N, we have

Ω f n dx ≤ ̺ f , Ω g n dx ≤ ̺ g .
The existence of f n and g n in C 0 ([0, T ]; L 1 (Ω)) as nonnegative solutions of ( 10)-( 11) is obtained by induction thanks to Proposition 2, remember-

ing that Q + (f n , f n ), R 1 + (f n , g n ), R 2 + (f n , g n ) and S + (g n , g n ) all belong to C 0 ([0, T ]; L 1 (Ω)).
We next prove, by induction again, that (f n ) and (g n ) are non decreasing sequences. It is first clear that f 1 ≥ 0 = f 0 and g 1 ≥ 0 = g 0 . Then we suppose that f n ≥ f n-1 and g n ≥ g n-1 . We can write, in a weak sense,

∂ t (g n+1 -g n ) + (β R ̺ f + β S ̺ g )(g n+1 -g n ) = P (f n , f n-1 , g n , g n-1
), where

P = R + 2 (f n , g n ) + S + (g n , g n ) -[R + 2 (f n-1 , g n-1 ) + S + (g n-1 , g n-1 )]. It is clear that P lies in C 0 ([0, T ]; L 1 (Ω)
) and is nonnegative. Again, Proposition 2 ensures that g n+1 -g n is a nonnegative solution of the previous equation. In the same way, we also obtain that f n+1 ≥ f n .

Therefore, by monotone convergence, there exist f , g ∈ L ∞ (0, T ; L 1 (Ω)), such that (f n ) and (g n ) converge to f and g, almost everywhere and in L ∞ (0, T ; L 1 (Ω)).

We still have to prove that (f, g) satisfies the initial conditions [START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF], which is quite clear, and solves ( 5)-( 6) in a distributional sense. Let us choose a test function ϕ ∈ C 0 ( Ω), and a test function in D([-T, T ]). Equations ( 5)-( 6) can be written under a weak form, using these test functions. We investigate what happens when n → +∞ in this formulation.

First, the time derivatives and initial data do not induce any difficulty, when n → +∞. To treat the linear term with the indices n + 1, we only have to use Proposition 1 to obtain the loss terms of the collision kernels. Eventually, we have to deal with the nonlinear terms involving f n and g n . For instance, if we set M = sup{|ϕ(x)|; x ∈ Ω}, we can note that

Ω 2 |f n (t, x)g n (t, x * ) -f (t, x)g(t, x * )| ϕ(x ′ ) dx dx * ≤ ̺ g M f n (t, •) -f (t, •) L 1 (Ω) + ̺ f M g n (t, •) -g(t, •) L 1 (Ω) ,
which goes to 0 when n → +∞. Using this argument or a similar one, it is easy to recover the gain terms of the collision kernels.

We are then able to let n go to +∞ in the weak formulation of ( 5)-( 6) and obtain the required result.

The collision rules

This section is devoted to describe, in a precise way, the psychological dynamics that we take into account. For each individual, the exchange of opinions with another member of the population is represented, in the model, by a collisional rule that quantifies the modifications in the opinions originated by the exchange itself.

In what follows, (x, x * ) ∈ Ω2 are the pre-collisional opinions of the agents, whereas (x ′ , x ′ * ) ∈ Ω2 represent their opinion after the discussion, where the primes denote the types of variables Q, R and S.

As we shall see, the choice of the collision rules is crucial and heavily conditions the time evolution of the distribution functions given by Equations ( 5)- [START_REF] Boudin | The quasi-invariant limit for a kinetic model of sociological collective behavior[END_REF].

We here propose two models, based on two different psychological mechanisms for contradictory individuals, which allow to describe two types of collective behaviour observed in real situations. Both psychological rules translate the idea that the contradictory way of thinking tends to oppose the effects of the consensus rule.

An essential ingredient of both collision rules is the attraction function η, a smooth function which describes the degree of attraction of the average opinion with respect to starting opinion of the individual. In the first model, we introduce also the reaction function α, a smooth function which modulates the reaction of a contradictory individual during the exchange process.

In order to make the models unaffected by the change of label of the two extreme opinions, we shall always assume that both η and α are even. Moreover, we suppose that η : Ω → R and α : Ω → R are C 1 , and such that 0 ≤ η < 1, 0 < α ≤ 1. Consequently, the interactions do not destroy the bounds of the interval Ω.

In order to translate the idea, well accepted in the literature [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF][START_REF] Toscani | Kinetic models of opinion formation[END_REF][START_REF] Galam | Rational group decision making: A random field Ising model at T = 0[END_REF][START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF], that extreme opinions are more stable than moderate ones for conciliatory individuals, we suppose that η ′ (x) ≥ 0 when x ≥ 0. We also assume that the attraction and reaction functions are such that the Jacobians of the collisional mechanisms are always non zero.

We present two different models, which we respectively name the twist and swing models. This choice of nomenclature will be clear by observing the numerical results of Section 5. We keep the notations defined in 2 for the post-collisional opinions in each case.

4.1. Twist model. In the following paragraphs, we detail the involved collision rules.

Exchange of opinions between two conciliatory individuals. Let x,

x * ∈ Ω the opinions of the two conciliatory agents before an interaction. The interaction is described by the rule defined in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF]: the stronger opinions are less attracted towards the average than the weaker ones. The mechanism which restitutes the post-collisional opinions x Q , x Q * is given by

x Q = x + x * 2 + η(x) x -x * 2 , ( 12
)
x Q * = x * + x 2 + η(x * ) x * -x 2 . (13)
4.1.2. Exchange of opinions between a conciliatory individual and a contradictory one. Let x, x * ∈ Ω the respective opinions of a conciliatory agent and a contradictory one before an interaction. Whereas the conciliatory individual still follows the consensus rule [START_REF] Galam | Rational group decision making: A random field Ising model at T = 0[END_REF], the post-collisional contradictory opinion is computed using the value given by ( 13), and then somehow taking the opposite value, to model the contradictory effect. The collision rule that individuates the post-collisional opinions x R , x R * writes

x R = x + x * 2 + η(x) x -x * 2 , ( 14
) x R * = -α(x * ) x * + x 2 + η(x * ) x * -x 2 . ( 15
)
4.1.3. Exchange of opinions between two contradictory individuals. Let x, x * ∈ Ω the opinions of the two contradictory agents before an interaction. The opinion exchange gives a pair of post-collisional opinion variables x S , x S * , which are given by

x S = -α(x) x + x * 2 + η(x) x -x * 2 , (16)
x S * = -α(x * )

x * + x 2 + η(x * ) x * -x 2 . (17) 4.2
. Swing model. This model substantially differs from the previous one because of the psychological behaviour of contradictory individuals.

Exchange of opinions between two conciliatory individuals.

The interaction between two conciliatory individuals is still defined by ( 12)-(13).

4.2.2.

Exchange of opinions between a conciliatory individual and a contradictory one. Let x, x * ∈ Ω the respective opinions of a conciliatory agent and a contradictory one before an interaction. After interaction, the contradictory individual tends to oppose the post-collisional opinion of his peer, i.e. the post-collisional opinions x R , x R * are defined by

x R = x + x * 2 + η(x) x -x * 2 , ( 18
) x R * =                1 - (1 -x R )(1 -x * ) (1 -x) if x < x * , x * if x = x * , (1 + x R )(1 + x * ) (1 + x) -1 if x > x * . (19)
We note that the post-collisional opinion x R * is well defined because the range of the validity of the formulae prevents the denominators from vanishing. The interaction clearly does not destroy the bounds of Ω. Finally, the continuity of the mechanism with respect to (x, x * ) is also ensured when x = x * . 4.2.3. Exchange of opinions between two contradictory individuals. Let x, x * ∈ Ω the pre-collisional opinions of two contradictory agents. The interaction leading to the post-collisional opinions x S , x S * is given by

x S =      x -1 2 + η(x) x + 1 2 if x ≤ x * , x + 1 2 + η(x) x -1 2 if x > x * , (20)
x

S * =      x * + 1 2 + η(x * ) x * -1 2 if x ≤ x * , x * -1 2 + η(x * ) x * + 1 2 if x > x * . (21)
It can be seen as the opposite of the consensus rule, i.e. the post-collisional opinions are obtained using the consensus rule ( 12)-( 13) with the extreme opinions ±1.

4.3.

On the collision mechanisms. The sets of attraction and reaction functions are not empty. Indeed, a possible choice for functions η and α is η(x) = H and α(x) = A for x ∈ Ω, where the constants H and A both satisfy 0 < H, A < 1. We can easily check that, in that case, the Jacobians of all the collisions rules ( 12)-( 21) are defined and nonzero.

Numerical tests

In this section, we present some numerical results concerning both models presented above. The computations were performed by using a numerical code written in C.

We consider a regular subdivision (x 0 , • • • , x N ) of Ω, with N ≥ 1. The functions f and g are computed at the center x i+1/2 of each interval [x i , x i+1 ], 0 ≤ i ≤ N -1, and we choose N = 1000.

Since f in L 1 (Ω) and g in L 1 (Ω) may not have the same order of magnitude, we replace f and g respectively by

f := f f in L 1 (Ω)
, and g := g g in L 1 (Ω) .

Then f and g solve the same kind of equations ( 5)-( 6) as f and g, but

β Q is replaced by β Q × f in L 1 (Ω) , β S by β S × g in L 1 (Ω) , β R by β R 1 := β R × g in L 1 (Ω) for R 1 and β R by β R 2 := β R × f in L 1 (Ω) for R 2 .
It is then quite clear that both f (t, •) L 1 (Ω) and g(t, •) L 1 (Ω) remain constant and equal to 1, for almost every t.

To numerically perform the collisions, we use a slightly modified Bird method [START_REF] Bird | Molecular gas dynamics and the direct simulation of gas flows[END_REF], as in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF]. Note that our scheme of course prevents the opinions from going out of Ω, and conserves the population of each group.

As we already explained, we are mainly interested in the computation of the quantities

I -(f + g) = 0 -1 (f (t, x) + g(t, x)) dx, I + (f + g) = 1 0 (f (t, x) + g(t, x)) dx.
When normalized to f in + g in L 1 (Ω) , they can be seen as the fraction of agents who respectively favour negative and positive opinions.

In each numerical tests, the collision frequencies are set to β S = 1, β R = 2, β Q = 5. That means that we always consider that the interactions involving contradictory people are less frequent that the ones involving conciliatory individuals. Besides, the initial data are chosen such that f in L 1 (Ω) = 1 and g in L 1 (Ω) = 0.1, so that conciliatory individuals are majority, but contradictory people are a significant part of the whole population. We also tried smaller values of g in L 1 (Ω) . The same kind of behaviours shown below are recovered, but the time scales get smaller and the interesting transient effects cannot really be pointed out.

Eventually, we choose, for x ∈ Ω, η(x) = (1+x 2 )/4, and α(x) = (1+x 2 )/2 when required. It is quite difficult to find a precise sociological meaning for those choices. We refer to [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF] for the discussion about the choice of η. Nevertheless, both functions satisfy the required assumptions, and we emphasize that we have numerically checked that all the Jacobians were nonzero.

5.1. Twist model.

5.1.1.

Uniform contradictory group within an uncentred conciliatory population. In this first test, we show the behaviour of the model with an Heaviside-step-like initial datum for conciliatory individuals, and a constant initial datum for contradictory people:

(22) f in (x) = 2 if x < -0.5, 0 if x > -0.5, g in (x) = 0.05.
In Figure 1, we plot both graphs of distribution functions at an arbitrary time t = 20, and we note that f and g are already close to their equilibrium state. As expected, the initial conditions for both groups do not have any influence after a transient time. The distribution functions f and g asymptotically get a Dirac-mass shape. Figure 2 ensures that the Dirac mass centres for both f and g go to 0, since I + (f + g) goes to f in + g in L 1 (Ω) /2 = 0.55. Of course, convergence to the equilibrium for g is slower than the one for f . The twist model then clearly results in a centred population with no precise opinion, since both Dirac masses are asymptotically centred at 0. It is interesting to note that, in the contradictory-free diffusionless case described in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF], when g does not appear, f also converges to a Dirac mass, but its centre is given by the average opinion of the initial conciliatory population. 5.1.2. Disjoint-supported initial data. Let us now investigate the case when the support sets of the initial data are disjoint. Indeed, in that situation, the support sets of f and g may still remain disjoint because of the collision rules involving contradictory people. More precisely, we consider the following initial data:

(23) f in (x) = 2 if x < -0.5, 0 if x > -0.5, and
g in (x) = 0 if x < 0.5, 0.2 if x > 0.5.
Again, both functions f and g fastly converge towards Dirac masses centred at 0, and the conclusion we obtained in 5.1.1 seems to hold. However, for small times, such as t = 20, each conciliatory individual has a positive opinion and each contradictory individual a negative one (see Figure 3), a situation which is significantly different with respect to the initial data. After a transient period, where a double reversal of the majority takes place, the system reach an equilibrium configuration. In fact, this effect could be found in 5.1.1 too, see Figure 2.

As a conclusion, the twist model forecasts situations which always lead to a fifty/fifty result. The phenomenon of "hung elections" [START_REF] Galam | Contrarian deterministic effects on opinion dynamics: "the hung elections scenario[END_REF] is the natural issue of this model. But one of the main interest of our model is its behaviour during a transient time, with unexpected results due to contradictory individuals. 5.2. Swing model.

5.2.1.

Uniform contradictory group within an uncentred conciliatory population. We start with the first same numerical test as in 5.1.1, i.e. initial data given by ( 22). As we can see in Figure 5, the distribution function f is a Dirac mass again, but, this time, its centre seems to randomly swing. Thus, at t = 30, each conciliatory individual has a positive opinion, and at t = 90, each one of them has a negative opinion. This behaviour is of course linked to the one of g, which seems to converge towards the sum of two Dirac masses, respectively centred at -1 and 1. But in fact, the mass of each Dirac function does not remain constant with respect to t.

Those behaviours are very far from the results given by the twist model. Figure 6 shows that this time, we may not get a fifty/fifty situation, and no tendency points out: when time grows, I + (f + g) can randomly go below or above f in + g in L 1 (Ω) /2 = 0.55, and there is no asymptotic equilibrium. In Figure 7, we can see that f first concentrates near 0.6, and then have a swinging behaviour, with variable opinion clusters with respect to time. The graph of g is only studied near ±1: indeed, g = 0 outside the domains which are shown on Figure 8. The contradictory distribution function g seems to converge towards a sum of two Dirac masses centred at ±1, but their respective masses do not remain constant.

Eventually, in Figure 9, we note that there are only two changes of majority in the opinion, and they both happen at small times (t < 200). Beyond that time, there is no majority change anymore: even if the majority rate itself is modified, it stays below 0.55. That seems to be significantly different from the case when contradictory people are initially uniformly distributed, where majority changes can happen at any time. Nevertheless, those majority changes are not very frequent anyway, so when we perform a computation with higher final times, in fact, we can recover this possibility of majority change. It ascertains the idea that there is no asymptotic equilibrium at all, and that there exist psychological behaviours which can lead to unpredictable election resultas, where the majority can quickly change.

Figure 1 .

 1 Figure 1. Twist model: graphs of (a) f and (b) g at t = 20 with initial data (22).

Figure 2 .

 2 Figure 2. Twist model: graph of I + (f + g) w.r.t. t with initial data (22).

Figure 3 .

 3 Figure 3. Twist model: graphs of (a) f and (b) g at t = 20 with initial data (23).

Figure 4 .

 4 Figure 4. Twist model: graph of I + (f + g) w.r.t. t with initial data (23).

Figure 5 .

 5 Figure 5. Swing model: graphs of (a) f and (b) g, at various times, with initial data (22).

Figure 6 .

 6 Figure 6. Swing model: graph of I + (f + g) w.r.t. t with initial data (22).

5. 2 . 2 .

 22 Disjoint-supported initial data. Let us now study the behaviour of the swing model with the set of initial data (23).

Figure 7 .

 7 Figure 7. Swing model: graph of f , at various times, with initial datum (23).

Figure 8 .

 8 Figure 8. Swing model: graph of g near (a) -1 and (b) +1, at various times, with initial datum (23).

Figure 9 .

 9 Figure 9. Swing model: graph of I + (f + g) w.r.t. t with initial data (23).

Acknowledgements. The authors wish to thank Serge Galam for the discussions they shared on the topic of this article. The second author is grateful to the University of Pavia, where part of this paper has been written, for its hospitality.