Dr Philippe Robin
email: philippe.robin@chu-brest.fr

Nick Van Es

Pierre-Yves Le Roux

Matthew Rondina

Ramón Lecumberri

Mariëlle Beckers

Grégoire Le Gal

Pierre-Yves Salaun

Gre Ǵoire Le Gal

Performance of 18F-fluorodesoxyglucose positron-emission tomography/computed tomography for cancer screening in patients with unprovoked venous thromboembolism: Results from an individual patient data meta-analysis

Introduction

Venous thromboembolism (VTE) may be the first manifestation of cancer. We aimed at evaluating the performance of 18F-Fluorodesoxyglucose Positron-Emission Tomography/Computed Tomography (FDG PET/CT) for occult cancer screening in patients with unprovoked VTE.

Methods

This was a pre-specified analysis of a systematic review and individual patient data meta-analysis including prospective studies assessing cancer screening in patients with unprovoked VTE. Sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV) of FDG PET/CT were calculated based on cancer diagnosis during a 1-year follow-up period.

Results

Four studies were identified as using FDG PET/CT as part of their extensive screening strategy. Out of the 332 patients who underwent FDG PET/CT, the scan was interpreted as positive in 67 (20.2%), as equivocal in 27 (8.1%), and as negative in 238 (71.7%). Seventeen (5.1%) patients were diagnosed with cancer at inclusion or during the 12-month follow up period. All cancers were diagnosed at initial screening. Pooled sensitivity, specificity, NPV, and PPV were 87.3% (95% CI, 55.3 to 97.4), 70.2% (95% CI, 48.2 to 85.6), 98.9% (95% CI, 94.3 to 99.7), and 17.9% (95% CI, 8.5 to 33.6), respectively.

Conclusion

FDG PET/CT appears to have satisfactory accuracy indices for cancer diagnosis in patients with unprovoked VTE. In particular, it exhibits a very high negative predictive value and could be used to rule out the presence of an underlying occult malignancy in this setting. -FDG PET/CT appears to have satisfactory accuracy indices in this setting.

-Occult cancer screening using FDG PET/CT seems to have a high Negative Predictive Value.

-A negative FDG PET/CT should be reassuring for patients and clinicians in this setting.

INTRODUCTION

Because acute venous thromboembolism (VTE) may be the first manifestation of an occult malignancy, many efforts have been made to determine whether VTE patients benefit from cancer screening, with the hope of being able to detect cancer earlier and consequently improve prognosis [START_REF] Van Doormaal | Is extensive screening for cancer in idiopathic venous thromboembolism warranted? Journal of thrombosis and haemostasis[END_REF][START_REF] Prandoni | Extensive Computed Tomography versus Limited Screening for Detection of Occult Cancer in Unprovoked Venous Thromboembolism: A Multicenter, Controlled, Randomized Clinical Trial[END_REF][START_REF] Piccioli | Extensive screening for occult malignant disease in idiopathic venous thromboembolism: a prospective randomized clinical trial[END_REF][START_REF] Rieu | Systematic screening for occult cancer in elderly patients with venous thromboembolism: a prospective study[END_REF][START_REF] Carrier | Systematic review: the Trousseau syndrome revisited: should we screen extensively for cancer in patients with venous thromboembolism?[END_REF][START_REF] Carrier | Screening for Occult Cancer in Unprovoked Venous Thromboembolism[END_REF]. As all cancer types at various locations may be diagnosed in VTE patients, an extensive screening may be required.

Extensive screening, however, remains controversial for several reasons. A recent individual participant data meta-analysis (IPDMA) showed that extensive screening is able to double the proportion of cancers diagnosed at the time of an acute VTE as compared with limited screening, but that there was no significant difference after 1 year of follow-up [START_REF] Van Es | Screening for Occult Cancer in Patients With Unprovoked Venous Thromboembolism: A Systematic Review and Metaanalysis of Individual Patient Data[END_REF]. Several studies have used a combination of multiple invasive and expensive tests (e.g. abdominal and pelvic imaging, upper and lower gastrointestinal endoscopies, sputum cytology, tumor markers, mammography and Pap smear in women, and prostate-specific antigen (PSA) and transabdominal prostate ultrasound in men), making such screening strategies difficult to implement [START_REF] Piccioli | Extensive screening for occult malignant disease in idiopathic venous thromboembolism: a prospective randomized clinical trial[END_REF]. A few years ago, a systematic review had identified computed tomography (CT) as being the most promising test to improve the rate of cancer detection in VTE patients [START_REF] Carrier | Systematic review: the Trousseau syndrome revisited: should we screen extensively for cancer in patients with venous thromboembolism?[END_REF]. However, a large randomized controlled trial showed no benefit of adding a comprehensive CT of the abdomen and pelvis to a limited screening strategy, with no reduction in the rate of cancers that were missed by screening (6). 18 F-Fluorodeoxyglucose (FDG) positron emission tomography combined with computed tomography (PET/CT) is a non-invasive whole-body imaging test that is widely used for diagnosing, staging, and therapeutic monitoring of many cancers [START_REF] Bomanji | Clinical role of positron emission tomography in oncology[END_REF][START_REF] Delgado-Bolton | Meta-analysis of the performance of 18F-FDG PET in primary tumor detection in unknown primary tumors[END_REF][START_REF] Salaun | Good clinical practice recommendations for the use of PET/CT in oncology[END_REF]. FDG PET/CT could thus be an ideal test for cancer screening in patients with unprovoked VTE [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF][START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF][START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF][START_REF] Alfonso | Screening for occult malignancy with FDG-PET/CT in patients with unprovoked venous thromboembolism[END_REF]. A randomized trial failed to show a higher rate of cancer diagnosis at initial screening when adding an FDG PET/CT to limited screening.

On the other hand, the risk of subsequent cancer diagnosis during follow-up was significantly lower in patients in the FDG PET/CT group as compared with patients in the limited screening group [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF]. The trial had limited statistical power, due to the lower than expected cancer incidence rate. Other cohorts of more limited sample size also used FDG PET/CT for screening in VTE patients [START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF][START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF][START_REF] Alfonso | Screening for occult malignancy with FDG-PET/CT in patients with unprovoked venous thromboembolism[END_REF].

To gain further knowledge on the performance of FDG PET/CT for cancer screening in patients with unprovoked VTE, we sought to analyze all available data on the use of FDG PET/CT as an extensive screening tool in patients included in the aforementioned IPDMA.

METHODS

Study population

This was a pre-specified analysis of a recently published individual patient data meta-analysis (IPDMA) that included prospective studies assessing cancer screening in patients with unprovoked VTE (PROSPERO: CRD42016033371). Methods, including literature search, study selection process, and data extraction, have been previously described [START_REF] Van Es | Screening for cancer in patients with unprovoked venous thromboembolism: protocol for a systematic review and individual patient data meta-analysis[END_REF]. For this analysis, we only included studies that assessed a whole-body FDG PET/CT for the detection of occult cancer in patients with unprovoked VTE.

Three cohort studies and one randomized controlled trial were identified to address the question [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF][START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF][START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF][START_REF] Alfonso | Screening for occult malignancy with FDG-PET/CT in patients with unprovoked venous thromboembolism[END_REF]. In these studies, adult patients with unprovoked and objectively confirmed deep vein thrombosis (DVT) or pulmonary embolism (PE), in the absence of major risk factors such as recent surgery, immobilization, known cancer, or ongoing pregnancy, were prospectively included. All studies were approved by the institutional review boards of participating centers.

Types of interventions

One study enrolled patients before any screening procedure was performed [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF]. Patients were randomly assigned to a limited screening strategy (i.e medical history taking, physical examination, basic blood tests, chest radiography, and age-and sex-specific tests such as mammography or prostate-specific antigen testing), or an extensive screening strategy consisting of the limited strategy plus an FDG PET/CT [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF]. The other three studies enrolled patients after negative results on limited screening [START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF][START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF][START_REF] Alfonso | Screening for occult malignancy with FDG-PET/CT in patients with unprovoked venous thromboembolism[END_REF]. Patients were followed up for 2 years in the four studies for cancer occurrence. FDG-PET/CT were performed using Gemini GXLi, Philips; Discovery ST, General Electric; Biograph 6 LSO Pico 3D HI-REZ, Siemens Medical; Gemini GXL, Philips and Discovery 690, General Electric [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF]; Biograph Duo LSO, Siemens Medical ( 14); Biograph 4 tomograph , Siemens Medical [START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF]; Allegro, Philips [START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF].

Patients fasted for at least 6 hours before PET acquisitions, and blood glucose had to be less than 7 mmol/l (11), 7.8 mmol/l ( 14), 10 mmol/l (13), and 11.1 mmol/l (12) before injection of 370-555 MBq of 18 F-FDG. Intravenous injection was followed by a period of approximately 60 minutes when the patients remained in a quiet room. Computed tomography was performed from mid-forehead to the feet in normal shallow respiration using a low-dose setting. Intravenous iodinated contrast was not administered. Data obtained from the CT-scan [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF][START_REF] Rondina | A pilot study utilizing whole body 18 F-FDG-PET/CT as a comprehensive screening strategy for occult malignancy in patients with unprovoked venous thromboembolism[END_REF][START_REF] Alfonso | Screening for occult malignancy with FDG-PET/CT in patients with unprovoked venous thromboembolism[END_REF] or from single acquisition mode using a 137 Cs point source [START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF] were used for attenuation correction of PET data and for fusion with attenuation-corrected PET images.

FDG-PET/CT interpretation

FDG PET/CT was interpreted as positive for possible malignancy based on the following criteria: foci of non-physiological FDG uptake, not attributable to the acute venous thromboembolism (i.e. areas of vascular thrombosis or lung infarcts were excluded), with no predetermined cut-off maximum standard uptake values. In the MVTEP trial and in Beckers' study, FDG PET/CT could also be classified as equivocal when doubt existed [START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF][START_REF] Beckers | The potential role of positron emission tomography in the detection of occult cancer in 25 patients with venous thromboembolism[END_REF].

The sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV), and negative and positive likelihood ratio of FDG PET/CT were calculated. To compute these accuracy indices, patients were considered as having cancer if they were diagnosed with histologically confirmed cancer following the initial screening strategy results, or based on any diagnostic test performed during the first 12 months of follow-up.

For the purpose of this analysis, we considered patients with equivocal FDG PET/CT results as positive for possible malignancy. Indeed, in the setting of cancer screening, it appears important not to miss a cancer in patients with dubious images, who should be further explored. In a second analysis, we considered patients with equivocal FDG PET/CT results as negative for possible malignancy. FDG PET/CT was considered true-positive if a cancer was diagnosed during follow-up in a patient with FDG PET/CT interpreted as positive or equivocal for possible malignancy. FDG PET/CT was considered false positive in patients with an FDG PET/CT interpreted as positive or equivocal for possible malignancy but who were not diagnosed with malignant disease during follow-up. FDG PET/CT was considered true-negative in patients with an FDG PET/CT interpreted as negative for malignancy who were not diagnosed with malignancy during follow-up. FDG PET/CT was considered false-negative in patients with an FDG PET/CT interpreted as negative for possible malignancy who were diagnosed with malignant disease during follow-up.

Due to the low number of studies included as well as the absence of events in one study, we first considered the four studies as one cohort and present the estimates as proportions. In a second analysis, pooled estimates were obtained in random-effects meta-analysis using the bivariate model of Reitsma and colleagues [START_REF] Reitsma | Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews[END_REF] in which a continuity correction of 0.5 was used for studies with zero events. Statistical analysis was performed using R, version 3.5.1 (www.R-project.org), in particular using the mada package version 0.5.8.

10

RESULTS

Included studies

Four studies from France, The Netherlands, Spain, and United States were identified in the systematic review as using FDG PET/CT as part of their extensive screening strategy. A summary of each individual study (e.g. inclusion period, inclusion criteria, and number of included patients) is available in Table 1.

Patient Population

Altogether, 364 patients were included in the four identified studies. Of the 364 patients, 32 (8.8%) were excluded from the dataset, because they withdrew consent and refused the use of their data (n=3), did not present for their FDG PET/CT or refused to undergo the examination (n=23), were enrolled more than 90 days after their VTE diagnosis (n=4), or had their index VTE not confirmed (n=2). Characteristics of the remaining 332 patients are summarized in Table 2. Mean age was 63 years (77% were older than 50 years), 56% of the patients were men, and 66% had PE with or without DVT.

Occult cancer diagnosis

Out of the 332 analyzed patients, 67 FDG PET/CT were interpreted as positive (20.2%), 27 were interpreted as equivocal (8.1%), and 238 were interpreted as negative (71.7%). Seventeen (5.1%) patients were diagnosed with cancer at inclusion or during the 12-month follow up period. All occult cancers were detected at initial screening. Location of the 17 diagnosed cancers was as follows: prostate (n=3), pancreas (n=3), lung (n=3), colorectal (n=2), thyroid (n=2), head and neck (n=1), uterus/ovary (n=1), stomach (n=1), and testicle (n=1). After a negative FDG PET/CT, no cancer was diagnosed during the first year of follow-up. The 12-month period prevalence of occult cancer after VTE diagnosis was 5.1% (95% CI, 3.2% to 8.1%) in all analysed patients, and 6.6% (95% CI, 4.1% to 10.4%) in patients ≥ 50 years.

Accuracy of FDG PET/CT (positive + equivocal vs. negative FDG PET/CT)

The FDG PET/CT was deemed positive or equivocal for possible malignancy in 94 of 332 patients (28.3%). Of these 94 patients, 17 were diagnosed with cancer at 1 year, corresponding to a PPV of 18.1% (pooled PPV, 17.9%; 95% CI, 8.5 to 33.6). None of the 238 patients (71.7%) with a negative FDG PET/CT were diagnosed with cancer during the following year corresponding to an NPV of 100% (pooled NPV, 98.9%; 95% CI, 94.3 to 99.7). All cancers were diagnosed by PET/CT (either positive or equivocal) corresponding to a sensitivity of 100% (pooled sensitivity, 87.3%; 95% CI, 55.3 to 97.4). Specificity was 75.6% (pooled specificity, 70.2%; 95% CI, 48.2 to 85.6). The positive likelihood ratio was 4.1 and the negative likelihood ratio could not be calculated since no patients with a negative FDG PET/CT result were diagnosed with cancer.

In patients aged 50 or older (257 patients, 77%), the FDG PET/CT was deemed positive or equivocal for possible malignancy in 79 (30.7%). Of these 79 patients, 17 were diagnosed with cancer at 1 year, corresponding to a PPV of 21.5% (pooled PPV, 20.6%; 95% CI, 10.0 to 37.8). The diagnostic accuracy indices of FDG PET/CT (positive and equivocal vs. negative FDG PET/CT) in all patients, and in patients aged 50 or older are shown in Table 3.

Accuracy of FDG PET/CT (positive vs. negative + equivocal FDG PET/CT)

The FDG PET/CT was deemed positive for possible malignancy in 67/332 patients (20.2 %), of them 15 were diagnosed with cancer at one year, corresponding to a PPV of 22.4% (pooled PPV, 26.3%; 95% CI, 7.0 to 62.9). Two of the 265 patients (0.8%) with a negative or equivocal FDG PET/CT were diagnosed with cancer during the following year corresponding to a NPV of 99.2% (pooled NPV, 98.3%; 95% CI, 95.4 to 99.4). Sensitivity for the 1-year follow-up period was 88.2% (pooled sensitivity, 80.5%; 95% CI, 55.9 to 93.1) and specificity 83.5% (pooled specificity, 84.6%; 95% CI, 45.9 to 97.3). The positive likelihood ratio was 5.3 and the negative likelihood ratio was 0.14.

In patients aged 50 or older (257 patients, 77%), the FDG PET/CT was deemed positive for possible malignancy in 57 (22.4%). Of these 57 patients, 15 were diagnosed with cancer at 1 year, corresponding to a PPV of 26.3% (pooled PPV, 29.8%; 95% CI, 8.0 to 67.4). The diagnostic accuracy indices of FDG PET/CT (positive vs. negative and equivocal FDG PET/CT) in all patients, and in patients aged 50 or older are shown in Table 4.

13

DISCUSSION

In this analysis of individual patient data from four prospective management studies, we found FDG PET/CT to have satisfactory accuracy for cancer diagnosis in patients with unprovoked VTE after a 1-year follow-up period: pooled sensitivity, specificity, NPV and PPV were 87.3% (95% CI, 55.3 to 97.4), 70.2% (95% CI, 48.2 to 85.6), 98.9% (95% CI, 94.3 to 99.7), and 17.9% (95% CI, 8.5 to 33.6), respectively.

Identifying occult cancers at the time of VTE diagnosis is an important issue for patients, who want to ensure that they have no underlying cancer that might have triggered the VTE. Our analysis shows that FDG PET/CT has a very high NPV, with no patients being diagnosed with cancer during follow up after a negative screening strategy. On the other hand, one could argue that the observed PPV of 18% is relatively modest, meaning that approximately 80% of patients with a positive or equivocal FDG PET/CT will likely undergo additional testing while they do not have cancer.

However, we believe that an eighteen percent risk of cancer is substantial and justifies further investigations. Moreover, in a post-hoc analysis of the MVTEP trial, we did not find any significant increase in unnecessary invasive diagnostic procedure following the extensive screening compared with the more limited screening [START_REF] Robin | Additional testing following screening strategies for occult malignancy diagnosis in patients with unprovoked venous thromboembolism[END_REF].

What are the clinical implications of our findings? There is equipoise around the use of screening for cancer among patients with unprovoked VTE. Following the publication of two recent clinical trials and of an IPDMA [START_REF] Carrier | Screening for Occult Cancer in Unprovoked Venous Thromboembolism[END_REF][START_REF] Van Es | Screening for Occult Cancer in Patients With Unprovoked Venous Thromboembolism: A Systematic Review and Metaanalysis of Individual Patient Data[END_REF][START_REF] Robin | Limited screening with versus without (18)F-fluorodeoxyglucose PET/CT for occult malignancy in unprovoked venous thromboembolism: an open-label randomised controlled trial[END_REF] on occult cancer screening, the most recent guidance from the Scientific and Standardization Committee (SSC) of the International Society on Thrombosis and Haemostasis does not currently recommend to go beyond a limited screening comprising medical history and physical examination, basic laboratory tests, chest X-ray, and age and gender-specific recommended cancer screening in all patients with unprovoked VTE. However, the SSC acknowledges that further clinical trials are required to assess the risks and benefits of an extensive occult cancer screening program in high-risk patients with unprovoked VTE [START_REF] Delluc | Occult cancer screening in patients with venous thromboembolism: guidance from the SSC of the ISTH[END_REF]. An IPDMA tested several predictors of occult malignancy and reported that the 1-year period prevalence of occult cancer was 7-fold higher in patients aged 50 years or older (6.8%; 95% CI, 5.6 to 8.3) as compared with those younger than 50 years (1.0%; 95% CI, 0.5 to 2.3). Given the high prevalence of occult cancer in patients with unprovoked VTE older than 50 years old (~7%), this group may represent the optimal population for cancer screening [START_REF] Van Es | Screening for Occult Cancer in Patients With Unprovoked Venous Thromboembolism: A Systematic Review and Metaanalysis of Individual Patient Data[END_REF]. Our results indicate that FDG PET/CT is a promising option in this setting. Therefore, we are currently planning a multinational randomized controlled trial of an extensive screening strategy including an FDG PET/CT versus standard of care in 1276 patients older than 50 years with an unprovoked VTE. The primary outcome will be the rate of cancer diagnosis during follow-up among patients with an initial negative screening. We aim to show that the risk of subsequent cancer is significantly lower among those patients who tested negative on FDG PET/CT at the time of their acute thromboembolic event (NCT04304651).

Our study has strengths and limitations. This work was conducted by a multidisciplinary team of investigators with expertise in nuclear medicine and thrombosis from different institutions. The study included all currently available data on extensive screening based on FDG PET/CT for occult cancer in patients with unprovoked VTE. All cases of cancer were adjudicated. Among the limitations is that defining standardized criteria (e.g. standardized uptake value (SUV) cut-off) for FDG PET/CT interpretation in the specific setting of occult cancer screening may not be possible, given that patients with unprovoked VTE may develop any cancer type, location or stage. However, all studies used similar interpretation criteria based on assessment of foci of non-physiological FDG uptake. Secondly, our total sample size was relatively small despite the inclusion of all available data. The sensitivity and NPV obtained in the pooled analysis are likely conservative since a 0.5 continuity correction was used for studies with no events. Finally, PET/CT is a fast-evolving technology and data included in our analysis were obtained on devices that are now outdated. Access to imaging and cost are other challenges in an era of scarce health care resources and admittedly, further medico-economic evaluations are necessary to balance screening cost with cost savings potentially resulting from earlier cancer detection.

CONCLUSION

In summary, FDG PET/CT appears to have satisfactory accuracy for cancer diagnosis in patients with unprovoked VTE. In particular, it exhibits a very high negative predicive value and could be used to rule out the presence of an underlying occult malignancy in this setting. Two ongoing randomized controlled trials will aim to test this hypothesis (NCT04304651, NCT03937583).

  ABBREVIATIONS-CT: Computed tomography -DVT: Deep vein thrombosis -FDG PET/CT: 18 F-Fluorodeoxyglucose positron-emission tomography/computed tomography -IPDMA: Individual participant data meta-analysis -PE: Pulmonary embolism -VTE: Venous thromboembolism

Table 1 :

 1 Summary of studies included in this analysis

	20

TABLE 2 :

 2 General characteristics of the population

	Accuracy indices	After a 1-year follow-up	After a 1-year follow-up
		period (all patients)	period (patients ≥ 50
			years)
	Sensitivity, % (95% CI)	17/17 (100%; pooled	17/17 (100%; pooled
		sensitivity, 87.3%; 95%	sensitivity, 87.0%; 95%
		CI 55.3 to 97.4)	CI 53.8 to 97.5)
	Specificity, % (95% CI)	238/315 (75.6%; pooled	178/240 (74.2%; pooled
		specificity, 70.2%; 95%	specificity, 66.3%; 95%
		CI 48.2 to 85.6)	CI 40.4 to 85.1)
	Positive predictive value, % (95%	17/94 (18.1%; pooled	17/79 (21.5%; pooled
	CI)	PPV, 17.9%; 95% CI 8.5	PPV, 20.6; 95% CI 10.0
		to 33.6)	to 37.8)
	Negative predictive value, % (95%	238/238 (100%; pooled	178/178 (100%; pooled
	CI)	NPV, 98.9%; 95% CI	NPV, 98.2; 95% CI 90.9
		94.3 to 99.9)	to 99.7)
	Positive likelihood ratio (95% CI)	4.1	3.9
	Negative likelihood ratio (95% CI)	-	-

TABLE 3 :

 3 Accuracy of FDG PET/CT for the diagnosis of cancer after a 1-year follow-up period (positive or equivocal vs. negative FDG PET/CT) in all patients, and in patients aged 50 or older.

ACKNOWLEDGEMENTS

Conflict of interest: Dr. Le Gal holds an Early Researcher Award from the Province of Ontario; a Heart and Stroke Foundation of Canada (HSFC) Ontario Mid-Career Investigator Award; and the Chair on Diagnosis of Venous Thromboembolism from the Department of Medicine at the University of Ottawa. Dr. Salaun reports a grant from the French Health Ministry (PHRC Cancer 2008) during the conduct of the MVTEP study. Authors not named here have disclosed no conflicts of interest.

Author's contribution:

Drs Robin, Salaun, van Es had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.


Daniel Barlet

On Principal Value and Standard Extension of Distributions

Keywords:

.

2. The existence of a local Bernstein identity for f , due to J.E. Bjork in the analytic case (see [START_REF] Bernstein | Prolongement analytique des fonctions généralisées avec paramètres (en russe)[END_REF] and [START_REF] Björk | Ring of differential operators North Holland[END_REF]).

As an application we first deduce of this result the absence of torsion for some Dmodules generated by the distributions constructed in the first part. In fact we prove more : any distribution in such a D-module has the Standard Extension Property (compare with the result of [START_REF] Barlet | Le réseau L 2 d'un système holonome régulier[END_REF]). We recall the definition of the Standard Extension Property in the begining of the section 2.

Then we apply this Theorem to the determination of generators of the conjugate D-modules of the D-modules N λ associated to z(σ) λ , the power λ, where λ is any complex number, of the (multivalued) root of the universal equation of degree k, z k + k j=1 (-1) h σ h z k-h = 0 whose structure is studied in [START_REF] Barlet | On partial differential operators which annihilate the roots of the universal equation of degree k, math[END_REF].

2 Existence of the Principal Value

The Standard Extension Property

First, we recall the definition of this property (see [START_REF]Barlet Fonctions de type trace[END_REF] in Appendix and [START_REF] Barlet | Le réseau L 2 d'un système holonome régulier[END_REF] in Paragraph 3.1 or [START_REF] Björk | Analytical D-Modules and Applications, Mathematics and Its Applications[END_REF])

Definition 2.1.1 Let M be a complex manifold of pure dimension n + 1 and let T be a distribution on M . We shall say that T has the Standard Extension Property if the following condition are fulfilled:

1. Outside a hypersurface H in M the distribution T is a C ∞ function. It is easy to see that if the condition 2 of the previous definition is satisfied for some choice of local equation of H, it is satisfied for any other choice. Note also that this property is clearly stable by multiplication by a C ∞ function but is not stable, in general, by the action of D M . For instance the locally integrable function 1/z defines a distribution on C which has the standard extension property, but ∂ z (1/z) = iπδ 0 is a non zero torsion element in Db C .

The following result is proved in [START_REF] Barlet | Le réseau L 2 d'un système holonome régulier[END_REF] Theorem 2.1.2 Let M be a regular holonomic D M -module and let H be a hypersurface in M such that M M \H is O M \H -coherent. Let i : M → Db M be a D M -linear morphism such that i belongs to L 2 (c(M)), the L 2 lattice (see [START_REF] Barlet | Le réseau L 2 d'un système holonome régulier[END_REF]) of the conjugate module of M (see [START_REF] Kashiwara | Regular Holonomic D-modules and Distributions on a Complex Manifolds[END_REF]). Then any distribution in i(M) has the Standard Extension Property (relatively to H).

The main difficulty to use this theorem is the identification of the L 2 -lattice. When the D M -module has no O M -torsion, the L 2 -lattice is given by a local square integrability condition along H which is often rather easy to verify. But we must know "a priori" that the D M -module we are considering has no O M -torsion to use this simple characterization of the L 2 -lattice. We shall see in section 5 that the main point is to prove that there is no torsion in some of the regular holonomic D M -modules involved in our computations.

Principal Value

Let us begin by giving the precise Asymptotic Expansion Theorem for fiber-integrals proved in [START_REF] Barlet | On symmetric partial differential operators[END_REF] which we shall need below. We keep the situation and the notations introduced in the beginning of section 1.

Theorem 2.2.1 Let ϕ ∈ C ∞ c (M ) be a differential form of type (n, n) on M . Define the function Θ(s) := f =s ϕ for each s ∈ C. This function, which is C ∞ on C \ {0}, admits when s → 0 a asymptotic expansion of the form Θ(s) r,j m,m a r,j m,m |s| 2r (Log |s|) j s m sm (1)
where m, m are non negative integers, r describes a finite set R ⊂ [0, 1[∩Q and where j is an integer in [0, n + 1]. The finite set R is independent of the choice of ϕ. Moreover, this asymptotic expansion is term-wise differentiable at any order.

Remarks.

1. The continuity of the function Θ at the point s = 0 implies that for r = 0 and j ≥ 1 we have m+m ≥ 1. In fact it is proved in Proposition 6 of loc. cit. that for j ≥ 1 we have a 0,j m,0 = 0 (resp. a 0,j 0,m = 0) which implies that for r = 0 and j ≥ 1 we have m ≥ 1 and m ≥ 1 when a 0,j m,m = 0. This shows that no new type of term appears in such an expansion when we apply s∂ s or s∂ s. And, of course, neither a constant term nor a term like s m (Log|s|) j or sm (Log|s|) j with j ≥ 1 may appear in these expansions. So the functions s∂ s Θ, s∂ sΘ and ss∂ s ∂ sΘ are bounded by O(|s| γ ) for some γ > 0 when s goes to 0.

2. It is also proved, in loc. cit. that the linear map ϕ → a r,j m,m (ϕ) is a (1, 1)current on M . This current is supported by the singular set of the hypersurface {f = 0} when (r, j) = (0, 0), that is to say for terms which do not correspond to an usual term of a Taylor expansion at the origin of a C ∞ function.

It will be important in the sequel to use the following corollary of this theorem.

Corollary 2.2.2 In the situation of the previous theorem, for any differential C ∞ form ψ and ξ respectively of type (n, n + 1) and (n + 1, n + 1) with f -proper supports in M the fiber-integrals

η(s) := f =s f ψ d f = s f =s ψ d f and ζ(s) := f =s f f ξ df ∧ d f = ss f =s ξ df ∧ d f
admit asymptotic expansions of the same type as above when s goes to 0 (but see the previous remark 1).

Remark. For any compact set K in {f = 0} there exists an integer N such that we have the inclusion of sheaves in an open neighborhood of K in M

f N Ω n+1 M ⊂ df ∧ Ω n M and f N f N Ω n+1 M ∧ Ω n+1 M ⊂ Ω n M ∧ Ω n M ∧ df ∧ df.
So the only new information in the previous corollary is that we may keep m and m non negative in the expansions.

Proof. Remark first that the previous theorem is in fact local around the hypersurface {f = 0} in M and so the asymptotic expansion is valid for any C ∞ differential form of type (n, n) with f-proper support in M . Then using the fact that for such a form ϕ we have the same type of asymptotic expansion for Proof. This an easy consequence of the fact that ∂ z and ∂ z ∂ z are surjective on C ∞ (D) and that we may solve the corresponding equations with C ∞ dependence of a parameter on a relatively compact open disc D ⊂⊂ D using a fundamental solution of the corresponding operator ( see for instance [START_REF] Barlet | Cycles analytiques complexes I: théorèmes de préparation des cycles[END_REF] chapter IV Proposition 5.2.4 for details).

s∂ s Θ(s) = f =s f d ϕ df , s∂ sΘ(s) = f =s f d ϕ d f
Remark. Note that d φ1 = ψ gives also the d -case.

Theorem 2.2.4 In the situation f : M → C introduced in section 1, let α be a complex number such that its real part (α) is non negative. Then, for any positive integer N and for any C ∞ differential form ξ of type (n + 1, n + 1) with compact support in M the limit when ε > 0 goes to 0 of

T ε α,N , ξ := |f |≥ε |f | 2α f -N ξ
exists and defines a distribution (that is to say a (0, 0) current) T α,N on M .

The main argument to prove this result uses the following consequence of the previous corollary of the Asymptotic Expansion Theorem (see also [START_REF] Herrera | Residues and principal values on complex spaces[END_REF] for a proof of the proposition below using a direct computation in a desingularization of the hypersurface {f = 0}). Proof. Note first that the result is local and is obvious near a point where f does not vanish. Around a point where f vanishes, Milnor's Fibration Theorem allows to use Fubini's Theorem to compute this integral for ε small enough as follows:

|f |=ε f -N ψ = 2π 0 dθ f =s f -N ψ dθ
where ψ/dθ on {f = s = εe iθ } is equal to f ψ/id f because we have dθ = id f / f and taking in account the type of ψ and the fact that {f = s} is a complex n-dimensional sub-manifold in M for s = 0. This gives with s := εe iθ

|f |=ε f -N ψ = 2π 0 dθ f =s f -N f ψ id f . When the integral f =s f ψ/d f is O(ε N +1
) it is clear that the limit when ε goes to 0 exists and vanishes. So it is enough to show the result when we replace the fiber-integral

-i s -N s∂ sΘ = f =s f -N f ψ id f
by its asymptotic expansion at a high enough order, and then, by linearity, to prove the assertion when we replace the fiber-integral by |s| 2r (Log|s|) j s m-N sm . In this case the exponent of e iθ is given, for s = εe iθ , by :

-N + m -m and the corresponding exponent of ε is given by 2r + m + m -N.

In order to find a non zero integral between 0 and 2π we need that m = m + N and then the exponent of ε is then equal to 2r + 2m . So the only terms where the limit is not clearly equal to 0 appear when r = m = 0. But in this case the limit is again zero thanks to Remark 1 following Theorem 2.2.1.

Proof of Theorem 2.2.4. The assertion is local near each point of {f = 0} so, using Lemma 2.2.3 (in fact the remark following it) we may assume that ξ = d ψ where ψ is a C ∞ differential form of type (n, n + 1) with f-proper support. Then Stokes' Formula gives

(α -N ) |f |≥ε |f | 2α f -N df ∧ ψ f + |f |≥ε |f | 2α f -N ξ = |f |=ε |f | 2α f -N ψ.
We know that the limit of the right hand-side is 0 when ε goes to 0 thanks to Proposition 2.2.5. Now write the first integral in the left hand-side as

|s|≥ε |s| 2α s -N ds ∧ ds ss f =s f ψ d f
and using polar coordinates s = ρe iθ so ds ∧ ds = -2iρdρdθ and the asymptotic expansion of the fiber-integral of f ψ/d f we may replace this fiber-integral by its asymptotic expansion at the origin at a sufficient large order. Then, using linearity, it is enough to consider the integrability at 0 of each non zero term which only depends on the real part of α. So we have only to consider the terms for which

-N + m -m = 0.
The corresponding real part of the power of ρ is given by

-N -1 + 2r + m + m + 2 (α) = -1 + 2r + 2m + 2 (α).
This is at least equal to -1 for (α) ≥ 0 because m + r ≥ 0. Then either we have r + m + (α) > 0 and this implies the integrability at 0, or r = m = (α) = 0. And in this case, thanks to Remark 1 following Theorem 2.2.1, the term to integrate is O(|s| γ-1 ) for some γ > 0, giving again the local integrability at 0. So the limit of T ε α,N , ξ exists for each test form ξ on M when ε goes to 0. The fact that the so obtained linear form on test differential forms is a distribution is an easy exercise left to the reader.

Note that in the previous proof, we may conclude with the weaker hypothesis asking that (α) + r > 0 for any r in R ∪ {1} \ {0}. Remark that with this hypothesis (α) is not a pole for the meromorphic extension of F N,ξ (λ), ∀N ∈ N, defined below.

Remark R1. If we replace |f | 2α by |f | 2α (Log|f | 2 ) β
where β is any positive real number, the same result holds true with the same proof.

Bernstein identity and meromorphic extension

We first recall the fundamental theorem of Bernstein [START_REF] Bernstein | Prolongement analytique des fonctions généralisées avec paramètres (en russe)[END_REF], generalized to the local analytic case by Bjork [START_REF] Björk | Ring of differential operators North Holland[END_REF]. Theorem 3.0.1 Let f : (C n+1 , 0) → (C, 0) be a non zero holomorphic germ. Then there exists a monic polynomial b ∈ C[λ] and, near 0 in C n+1 , a holomorphic partial differential operator P with polynomial coefficients in λ such that the identity

P (z, ∂ z , λ)f λ+1 = b(λ)f λ (2)
holds locally outside the hypersurface {f = 0} on some open neighborhood of the origin.

The minimal monic polynomial b ∈ C[λ] such that and identity of this kind holds near 0 is called the Bernstein polynomial of f at the origin.

Recall also the fundamental result of Kashiwara [START_REF] Kashiwara | b-function and holonomic systems, rationality of roots of bfunctions[END_REF].

Theorem 3.0.2 For any non zero holomorphic germ f at the origin, the roots of the Bernstein polynomial of f at the origin are rational and negative. N by

F N,ξ (λ) := M |f | 2λ f -N ξ
is holomorphic and admits a meromorphic extension to the complex λ-plane with poles of order at most n + 1 at points in ∪ r {r -N} where r is a root of the Bernstein polynomial of f at the origin. Moreover, for any α ∈ C, the linear forms on C ∞ c (M ) (n+1,n+1) given by the coefficient

P k (λ = α, F N,ξ (λ)) of (λ -α) -k , k ∈ Z, k ≤ n + 1, in the Laurent expansion at λ = α of the meromorphic extension of F N,ξ is a distribution on M . For k ∈ [1, n + 1] this distribution has support in {f = 0} and, for k ∈ [2, n + 1] or for k ∈ [1, n + 1] and α ∈ Z, the support of this distribution is contained in the singular set of {f = 0}.
proof. The equation ( 2) implies, as b has rational coefficients, that we may find for any positive integer M a anti-holomorphic differential operator P M depending polynomially of λ such that we have

P M ( f λ+M ) = b(λ) . . . b(λ + M -1) f λ . (3)
Note that the roots of B M (λ) := b(λ) . . . b(λ + M -1) are negative rational numbers. So for M N we obtain the equality of continuous functions for (λ)

N |f | 2λ f -N = 1 B M (λ) P M (|f | 2λ f -N f M )
If P * M is the adjoint of P M this implies for (λ) N and M large enough, that for

any test differential form ξ ∈ C ∞ c (M ) (n+1,n+1) we obtain M |f | 2λ f -N ξ = 1 B M (λ) M |f | 2λ f -N f M P * M (ξ). ( 4
)
This gives the meromorphic extension to the complex λ-plane of the holomorphic

distribution ξ → M |f | 2λ f -N ξ defined for (λ) N , because P * M (ξ) is C ∞ c
in M and depends polynomially on λ and because the right hand-side of the above formula is holomorphic on the open set (λ) > -m for any given positive integer m as soon as M is large enough compare to N + m. Moreover this meromorphic extension has no pole at points which are not inside the union of the sets r -N where r is a root of b. It is easy to see that near points where f does not vanish, this meromorphic extension has no pole and that near points where f = 0 but where df does not vanish the poles of this meromorphic extension are at most simple poles at negative integers. This complete the proof.

Remark R2. Let q be a positive integer. The q-th derivative in λ of the holomorphic function F N,ξ (λ) is given, for (λ) large enough by the absolutely converging integral

M |f | 2λ (Log|f | 2 ) q f -N ξ
and the meromorphic extension of these functions allows to define, for each integer q, a meromorphic distributions on M which has no pole for (λ) ≥ 0. Now analog arguments as above give easily a generalization of the previous theorem to these cases.

Equality of the Principal Value with the value of the Meromorphic Extension

The equality theorem

We keep the notations of the introduction. The aim of this paragraph is to prove the following result:

Theorem 4.1.2 Assume that (α) ≥ 0. Then for any positive integer N we have for any test differential form ξ:

S α,N , ξ = T α,N , ξ = lim ε→0 |f |≥ε |f | 2α f -N ξ.
Proof. We want to show that the analog of the equality (4) for λ = α holds if we perform the integration only on the subset {|f | ≥ ε} with an error which goes to zero when ε goes to 0. This would be enough to complete the proof. But, of course, the error comes from the boundary terms which are integrals on {|f | = ε} appearing in the various Stokes Formulas necessary to pass from P M to its adjoint P * M . It is easy to see that such "error" terms have the following shape: a polynomial in λ with coefficient like n) and M is an integer in [0, M ]. Now using the same arguments than in the proof of Theorem 2.2.4 we see that the only non zero term in such an integral comes from the coefficient of s m sm in the asymptotic expansion at s = 0 of the function s

|f |=ε |f | 2α f -N f M ψ where ψ is in C ∞ c (M ) (n+1,
→ f =s f ψ/d f such that m -m -N -M = 0.
And this non zero term comes with a power of ε which is at least equal to 2 (α) + m + m -N + M + 2r = 2 (α) + 2m + 2M + 2r ≥ 0 and may be some (Log ε) q factor. So such term goes to 0 when ε goes to 0 when m + r + M > 0 for (α) ≥ 0 but also in the case where (α) + m + M + r = 0, thanks again to Remark 1 following Theorem 2.2.1. This concludes the proof.

Again in the previous proof, we may conclude with the weaker hypothesis asking that (α) + r > 0 for any r in R ∪ {1} \ {0}.

Remark R3. Using Remarks R1 and R2 we obtain again with the same proof, that the previous theorem is still valid if we replace |f | 2α by |f | 2α (Log|f | 2 ) q for any positive integer q.

Non torsion of the corresponding D-modules

We shall deduce from Theorem 4.1.2 an important corollary. To formulate this result we need the following definition, where we keep the situation described in the introduction. Definition 4.2.1 Let α be a complex number with a non negative real part. Let V be an holomorphic vector field on M . We define the formal action of V on |f | 2α f -N by the formula

V, |f | 2α f -N := (α -N )V (f )|f | 2α f -N -1 .
Then this defines a "formal action" of Proof. Let V be a holomorphic vector field on M . Then let T α,N be the distribution defined by

D M on the O M -module O M |f | 2α [f -1 ].
T α,N , ξ := P 0 (λ = α, M |f | 2α f -N ξ)
and let V * be the adjoint of V ; we obtain:

V (T α,N ), ξ = T α,N , V * (ξ) = lim ε→0 |f |≥ε |f | 2α f -N V * (ξ) = lim ε→0 |f |≥ε V (|f | 2α f -N )ξ
because the boundary term on {|f | = ε} has limit 0 when ε goes to 0 using the fact that

V (|f | 2α f -N ) = (α -N )V (f )|f | 2α f -N -1 ,
the same argument than in the proof of Theorem 4.1.2 and the remark that ,n+1) . This gives, using again Theorem 4.1.2, that V (T α,N ) = V (f )T α,N +1 in Db M . As holomorphic vector fields generate the O M -algebra D M , this is enough to complete the proof.

V (f )ξ is in C ∞ c (M ) (n+1
Remark R4. The generalization of the previous corollary to the cases where we replace |f | 2α by |f | 2α (Log|f | 2 ) q for any positive integer q is again an easy exercise.

The case α < 0

We shall explain now how to define, when α is a negative real number 3 a "Finite Part " of the integral

M |f | 2α f -N ξ using the Asymptotic Expansion Theorem. For a differential form ξ ∈ C ∞ c (M ) (n+1,n+1
) , we shall write as follows the asymptotic expansion when s goes to 0 of the fiberintegral

s → f =s f f ξ/df ∧ d f r,j,m,m T r,j m,m (ξ)|s| 2r s m sm (Log|s|) j
where ξ → T (ξ) is a distribution with support in {f = 0} for each r ∈ R a finite subset in [0, 1[∩Q, for j ∈ [0, n + 1] and for m, m ∈ N (see the Corollary 2.2.2 and also also Remark 1 following Theorem 2.2.1).

Theorem 4.3.1 For α a negative real number such that -α ∈ R + N the following limit exists and defines a distribution on M Then the result is analogous.

lim ε→0 {|f |≥ε} |f | 2α f -N ξ + 2iπ α+r+m <0 m=m +N T r,j m,m (ξ) ε 2(α+r+m ) 2(α + r + m ) (-Logε
Proof. Using the asymptotic expansion for the fiber-integral f =s f f ξ/df ∧ d f recalled above, the proof that the limit in the left hand-side exists is analogous to the proof of Theorem 2.2.4.

Of course, the constant term of the Laurent development at λ = α of the meromorphic extension of the function λ → M |f | 2λ f -N ξ gives also such a distribution. But in the case α < 0 the relation between the distribution defined by 3 We leave the case where α is a complex number with a negative real part as an exercise.

ξ → P 0 λ = α, M |f | 2λ f -N ξ
and the distribution defined in the previous theorem is not so clear in general, even if the difference between these two distributions has clearly its support in {f = 0}. Of course, when α satisfies the condition α + r > 0 for each r in R ∪ {1} \ {0}, not only there is no term in the sum where α + r + m ≤ 0 (using Remark 1 following Theorem 2.2.1 for the case 0) and the equality of the two distributions follows from the remark following Theorem 4.1.2. n+1) , n + 1 := p i=1 n i + 1 and where

An easy generalization and an example

Consider now p ≥ 2 domains M i in C n i +1 for i ∈ [1, p] and p holomorphic functions f i : M i → C. Then on M := p i=1 M i it is easy to prove, for α := (α 1 , . . . , α p ) satisfying (α i ) ≥ 0, ∀i ∈ [1, p], using Fubini' s Theorem, the existence of the distribution T α,N , ξ := lim ε→0 T ε α , ξ where ε := (ε 1 , . . . , ε p ) is in (R + * ) p , N := (N 1 , . . . , N p ) is in N p , ξ is in C ∞ c (M ) (n+1,
T ε α,N , ξ := p i=1 {|f i |≥ε i } p i=1 |f i | 2α i f -N i i ξ.
It is also easy to make the meromorphic extension to λ := (λ 1 , . . . , λ p ) ∈ C p of the holomorphic distributions on M defined for p i=1 { (λ i ) N i } and then to prove the following generalization of Theorem 4.1.2 to this "product" case: Theorem 4.4.1 In the product situation described above the meromorphic extension of the distribution

ξ → M p i=1 |f i | 2λ i f -N i i ξ is holomorphic near the point α ∈ C p satisfying (α i ) ≥ 0, ∀i ∈ [1,
p], and we have the equality

P 0 λ = α, M p i=1 |f i | 2λ i f -N i i ξ = T α,N , ξ
where the left hand-side denotes the value at λ = α of the meromorphic extension.

Proof. The only point to precise in order to apply the p = 1 case and Fubini's Theorem successively to prove this generalization it the following remark:

• Let M 1 × M 2 be the product of two complex manifolds M 1 × M 2 and let ξ be a C ∞ c test differential form on M 1 × M 2 and T 2 a distribution on M 2 . Then the test differential form defined on M 1 by T 2 , ξ is a C ∞ c test differential form on M 1 . So for any distribution T 1 on M 1 the distribution T 1 T 2 is well defined on M 1 × M 2 by the rule T 1 T 2 , ξ := T 1 , T 2 , ξ .
Remark. Note that the definition of T α,N implies that we have

T α,N , ξ = lim ε→0 | p i=1 f i |≥ε p i=1 |f i | 2α i f -N i i ξ.
Then, using the argument in Corollary 4.2.2 on each M i , i ∈ [1, p] we obtain that for any P ∈ D M the distribution P T α,N has the standard extension property for the hypersurface { p i=1 f i = 0}. So the sub-D M -module of Db M generated by T α,N has no torsion. Note also that this implies that the sub-D M -module generated by T α,0 is contained in the O M -module generated by the T α,N when N is in N p .

Remark R5. Again it is easy to generalize the previous theorem to the cases where we replace

|f i | 2α i by |f i | 2α i (Log|f i | 2 ) q i for any non negative integers q i , i ∈ [1, p].
An example.

Notations. Let M := C k with coordinates z 1 , . . . , z k and let π : M → N C k be the quotient by the action of the permutation group S k on M . Let σ 1 , . . . , σ k be the elementary symmetric polynomials in z 1 , . . . , z k which give a coordinate system on N . Let also ∆ := 1≤i<j≤k (z i -z j ) 2 be the discriminant which is a polynomial in σ 1 , . . . , σ k .

2 := C k-1 with coordinates x h , h ∈ [2, k]. Moreover f is in C[x 1 ] and ∆ is in C[x 2 , . . . , x k ].
Proof. The reader will see easily that ∆

= k h=2 x 2 h 2≤i<j≤k (x i -x j ) 2 .
This easy lemma allows to apply Theorem 4.4.1 to the situation described in the previous lemma.

Corollary 4.4.3 For any α ∈ C such that (α) ≥ 0 and any non negative integers q and N 1 , N 2 the distribution on M associated to the locally integrable function Proof. For our assertion on M it is enough to prove that for any holomorphic vector field V on M the distribution V (|z

X α,N 1 ,N 2 ,q := k j=1 |z j | 2α (Log|z j | 2 ) q z -N 1 j ∆ -N 2
1 | 2α z -N 1 1 ∆ -N 2 )

Some conjugate D-modules

The aim of this section is to give some examples of explicit computations of the conjugate module (in Kashiwara sense, see [START_REF] Kashiwara | Regular Holonomic D-modules and Distributions on a Complex Manifolds[END_REF]) of some regular holonomic D-modules. We consider here the case of the D-modules associated to the multivalued functions z(σ) λ for λ ∈ C where z(σ) is the root of the universal degree k equation

z k + k h=1 (-1) h σ h z k-h
where σ := (σ 1 , . . . , σ k ) is in N := C k . The structure of these regular holonomic D N -modules has been described in [START_REF] Barlet | On partial differential operators which annihilate the roots of the universal equation of degree k, math[END_REF] and we describe here for each such D-module and for each simple factor which appears in its decomposition a distribution T on N which generates the sub-D N -module of Db N which is the conjugate of the module we consider. The reader will notice that even if in each case the corresponding distribution T is rather easily constructed from the horizontal (multivalued) basis of the corresponding vector bundle with a simple pole connection associated to the D N -module under consideration, the proof that this distribution generates the conjugate module uses in a crucial way the non trivial argument of "non torsion" which is proved in the previous section (see Corollary 4.4.3).

We begin by recalling the basic results on the conjugation functor of M. Kashiwara. The following theorem is proved in [START_REF] Kashiwara | Regular Holonomic D-modules and Distributions on a Complex Manifolds[END_REF]: Notations. We consider the quotient map π : M := C k → N := C k S k C k with respective coordinates z 1 , . . . , z k and σ 1 , . . . , σ k , where σ h is the h-th elementary symmetric function of (z 1 , . . . , z k ). The vector fields on M associated to partial derivatives in z 1 , . . . , z k are denoted by ∂ z 1 , . . . , ∂ z k and the vector fields on N associated to partial derivatives in σ 1 , . . . , σ k are denoted by ∂ 1 , . . . , ∂ k . We note I the left ideal in D N generated by the following global sections:

A p,q = ∂ p ∂ q -∂ p+1 ∂ q-1 where (p, q) ∈ [1, k -1] × [2, k] T m := ∂ 1 ∂ m-1 + ∂ m E where E := k h=1 σ h ∂ h and m ∈ [2, k]
Recall that a trace function on N is a holomorphic function F such that there exists a holomorphic function f in one variable z such that

F (σ) = k j=1 f (z j )
where σ := (σ 1 , . . . , σ k ) are the elementary symmetric functions of z 1 , . . . , z k . It is proved in [START_REF] Barlet | On symmetric partial differential operators[END_REF] that the trace functions are annihilated by the left ideal I in D N and this characterizes the trace functions.

We note U -1 the vector field on N given by U -1 := k∂ 1 + k-1 h=1 (k -h)σ h ∂ h+1 which is the image by the tangent map T π to π of the vector field V -1 := k j=1 ∂ z j on M . We note U 0 the vector field on N given by U 0 := k h=1 hσ h ∂ h which is the image by T π of the vector field V 0 := k j=1 z j ∂ z j on M . We note U 1 the vector field on N given by U 1 := k h=1 (σ 1 σ h -(h + 1)σ h+1 )∂ h which is the image by T π of the vector field V 1 := k j=1 z 2 j ∂ z j on M . The left ideal J λ in D N is, by definition, the sum I + D N (U 0 -λ) and we define N λ := D J λ .

The following results which give the structure of the D N -module N λ for each λ ∈ C are proved in [START_REF] Barlet | On partial differential operators which annihilate the roots of the universal equation of degree k, math[END_REF]:

1. For each complex number λ, the D N -module N λ is holonomic and regular. 9. The right multiplication by U 1 which sends N 1 in N 0 vanishes4 on Im( U -1 ) and induces an isomorphism of N 1 onto N 0 .

2 .

 2 For each point x in H there exists an open neighborhood U of x, a local holomorphic equation {f = 0} = H ∩U of H in U such that for any test differential form ξ in C ∞ c (U ) (n+1,n+1) we have T, ξ = lim ε→0 |f |≥ε T ξ.

Proposition 2 . 2 . 5

 225 Let ψ be a C ∞ differential form of type (n, n + 1) with f-proper support in M . Then we have for any positive integer N lim ε→0 |f |=ε f -N ψ = 0.

Corollary 3 . 0 . 3

 303 Let f : (C n+1 , 0) → (C, 0) be a non zero holomorphic germ and let M be an open neighborhood of the origin on which the Bernstein identity of f is valid 2 . For any N ∈ N and any test differential form ξ ∈ C ∞ c (M ) (n+1,n+1) the function defined for λ ∈ C such that (λ)

Definition 4 . 1 . 1

 411 In the situation above we define the distribution S α,N on M by the formulaS α,N , ξ := P 0 (λ = α, M |f | 2λ f -N ξ).

  Remark that, thanks to Theorem 4.1.2, each element in O M |f | 2α [f -1 ] defines an unique distribution on M having the standard extension property along the hypersurface {f = 0}. This gives a natural O M -linear embedding of this O M -module in Db M . Corollary 4.2.2 The action of any P ∈ D M on any element in the sub-O M -module O M |f | 2α [f -1 ] ⊂ Db M coincides with the formal action defined above. In particular, the sub-D M -module generated by Z α in Db M has no torsion, where Z α is the distribution on M associated to the locally bounded function |f | 2α on M .

  ) j which extends to M the function |f | 2α f -N on {f = 0}. If -α is in R + N, write -α = r + m , r ∈ R and m ∈ N and add to the sum inside the limit in the left hand-side above the sum 2iπ n+1 j=0 T r,j m +N,m (ξ)(-Logε) j+1 /(j + 1).

Lemma 4 . 4 . 2

 442 Consider on M := C k with coordinates z 1 , . . . , z k the holomorphic functions f := z 1 and ∆ := 1≤i<j≤k (z i -z j ) 2 . Then defining new coordinates x 1 := z 1 and x h := z h -z 1 for h ∈ [2, k] we obtain a decomposition M = M 1 × M 2 where M 1 := C with coordinate x 1 and M

  generates a sub-D M -module of the D M -module Db M which has no torsion. Moreover, any distribution in this sub-D M -module has the standard extension property along the divisor {π -1 (σ k ∆(σ)) = 0} in M and is C ∞ (in fact real analytic) outside this divisor. The same properties are true along the divisor {σ k ∆(σ) = 0}, for the sub-D Nmodule of Db N generated by the distribution π * (X α,N 1 ,N 2 ,q ).

  has the standard extension property and this is clear from the remark following Theorem 4.4.1 and Remark R5. The assertion on the corresponding D N -module is consequence of the fact that if a distribution T on M has the standard extension property along the divisor {π -1 (σ k ∆(σ)) = 0} then the distribution π * (T ) has the standard extension property along the divisor {(σ k ∆(σ) = 0} because the quotient map π is a finite étale map outside {∆(σ) = 0}. Remark. If S and T are two distributions on a complex manifold N such that D N S ⊂ Db N and D N T ⊂ Db N have no O N -torsion, it is not clear that there is no O N -torsion in D N (S + T ) ⊂ Db N . But if we know that D N S and D N T contain only distributions having the standard extension property for a given hypersurface H, then this also the case for any distribution in D N S + D N T ⊂ Db N which contains D N (S + T ).

Theorem 5 . 0 . 1 (

 501 Kashiwara conjugation functor) Let N be a complex manifold, D N the sheaf of holomorphic partial differential operators on N and D N the sheaf of anti-holomorphic partial differential operator on N . Note Db N the sheaf of distributions on N . It is a left-D N -module but also a left-D N -module and these two actions commute. For each regular holonomic D N -module N the sub-D N -module c N (N ) := Hom D N (N , Db N ) is regular holonomic (as a D N -module) and the contra-variant functor c is an antiequivalence of categories which satisfies c N • c N = Id. Moreover, M. Kashiwara also obtains the following proposition which will be useful to describe the D N -modules c(N λ ): Proposition 5.0.2 ( see [13] Prop. 5) If N is a regular holonomic D N -module on a complex manifold N and if T is in c(N ), the following conditions are equivalent: a) T is an injective sheaf homomorphism of N to Db N . b) T generates c(N ) as a D N -module.

2 .

 2 The right multiplication by U -1 induces a D N -linear mapU -1 : N λ → N λ+1which is an isomorphism for each λ = -1, 0. Moreover the right multiplication by U 1 induces an isomorphism U 1 : N λ+1 → N λ for any λ = 0, -1 and we haveU 1 • U -1 = λ(λ + 1) on N λ . 3. For λ ∈ Z the D N -module N λ is simple. 4. The kernel N -1 of the D N -linear map ϕ -1 : N -1 → O N ( σ k ) defined by ϕ -1 (1) = σ k-1 /σ k is simple.5. The sub-module N 0 generated by U 1 in N 0 is simple and the quotientN 0 N 0 is isomorphic to O N ( σ k ) 6. The torsion sub-module T in N 1 is isomorphic to H 1 [σ k =0] (O N ); it is generated by the class of ∂ k U -1 in N 1 .7. The sub-module Im( U -1 )in N 1 (which is generated by U -1 ) contains T and the quotient Im( U -1 ) T is isomorphic to O N via the map ϕ 1 : N 1 → O N defined by ϕ 1 (1) = σ 1 (and then [U -1 ] → k).

8 .

 8 The quotient N 1 := N 1 Im( U -1 ) is simple and isomorphic to the quotientD N I + D N (U 0 -1) + D N U -1 .

  and for ss∂ s ∂ sΘ(s) = Let U be an open polydisc in C n and D a disc in C. Let ψ and ξ be C ∞ forms respectively of type (n, n + 1) and (n + 1, n + 1) with support in K × D where K ⊂ U is a compact set. Fix a relatively compact open disc D in D. Then there exists ϕ 1 and ϕ 2 which are C ∞ forms of type (n, n) and with support in K × D

		f f d d ϕ
	f =s	df ∧ d	f
	we see that it is enough to show that we have the local surjectivity of d , d , d d with
	f-proper supports for the type (n + 1, n), (n, n + 1) and (n + 1, n + 1) respectively.
	Using a partition of unity, this is consequence of the following lemma and the local
	parametrization theorem for the hypersurface {f = 0} 1 .
	Lemma 2.2.3 such that
	d ϕ 1 = ψ and d d ϕ 2 = ξ
	on U × D .

In order that we may choose, near each point in {f = 0}, U and D such that Ū × ∂D does not meet {f = 0} in our next lemma.

This means precisely that the Bernstein identity for f is valid in the universal cover of M \ {f = 0}, on which f λ is defined as exp(λLog f ) for any given determination of Log f .

It is proved in[START_REF] Barlet | On partial differential operators which annihilate the roots of the universal equation of degree k, math[END_REF] Formula (19) page 20, that we have U -1 U 1 = (U 0 + 1)U 0 modulo I.

10. The right multiplication by U -1 which sends N -1 to N 0 induces an isomorphism of N -1 onto N 0 .

Important remark. The point 2 recalled above shows that to study the D Nmodules N λ it is enough to consider the following cases:

• The cases (λ) ∈ [0, 1[ and λ = 0. We call it the case G.

• The cases λ = -1, 0, 1. We call them the case λ = -1, 0, 1 respectively.

Then for any λ ∈ C we reach one of these previous cases using an isomorphism given either by U N 1 or by U N -1 for a suitable N ∈ N.

Now define the following distributions on M :

For each case we define the distribution X λ := π * (X λ ) on N .

Define also the following distributions on N :

1. For the case G : Y λ := 0.

2. For the case λ = 1:

Then we have the following results:

In all cases G, -1, 0, 1 the distribution X λ defines an element in c(N λ ) via the D Nlinear map sending 1 to X λ . In case G the distribution X λ defines a generator of the (simple) D N -module c(N λ ).

In cases λ = 1 the distribution X 1 defines an element in c(N 1 ) ⊂ c(N 1 ) and gives a generator of this (simple) D N sub-module.

In cases λ = -1, 0 the simple module c(N λ ) is a quotient of c(N λ ) and the image of X λ in c(N λ ) gives a generator of this (simple) D N -module. In all cases λ = -1, 0, 1 the map sending 1 to Y λ is in c(N λ ) and the D N -module c(N λ ) is generated by X λ and Y λ .

Proof. Consider the first point. We have to prove that the left ideal J λ annihilates the distribution X λ for each case. We have also

Now the verification that in each case the distribution X λ is annihilated by the generator of the left ideal J λ is "formal" (see Corollary 4.4.3 ) because X λ is locally on the open set {σ k ∆(σ) = 0} a linear combination with anti-holomorphic coefficients of the holomorphic trace functions

Moreover it is clear that z j (σ) λ is homogeneous of degree λ on M and the vector field U 0 is equal to π * (V 0 ) where V 0 is the Euler vector field on M .

In the case G we know that c(N λ ) is a simple D N -module. So the second point of the theorem is proved as X λ is clearly not 0 in Db N .

For λ = 1, as we know that ideal J 1 = I + D N (U 0 -1) already annihilates X 1 it is enough, thanks to the point 8 recalled above, to check that U -1 (X 1 ) = 0 in Db N to conclude because we know that N 1 is simple and X 1 is not 0 in Db N . Again the fact that it is enough to check this on the open set {∆(σ) = 0} makes the computation easy :

Proof. It is proved in Proposition 4.1.7 and Lemma 4.1.8 of [START_REF] Barlet | On partial differential operators which annihilate the roots of the universal equation of degree k, math[END_REF] that the annihilator of the class of U -1 in N 1 is generated by ∂ h , h ∈ [1, k-1] and σ k ∂ k . Note that this is also the annihilator in D N of the distribution 1/σ k in Db N thanks to the previous lemma. This shows that the D N -linear map defined by [U

where the map α is the quotient map. Now, let us prove that the distribution

As σ 1 commutes with the action of D N , the annihilator of σ 1 /σ k in D N is the same than the annihilator of 1/σ k proving our claim. But we know that X 1 is a generator of c(N 1 ). So, it is enough to prove that Y 1 is in c(N 1 ) to see that with X 1 they generate this D N -module. This is proved in our next lemma and completes the proof of this corollary. Proof. The fact that Y 1 in in c(N 1 ) is easy because σ 1 is a trace function which satisfies U 0 (σ 1 ) = σ 1 , because ∂ h , h ∈ [1, k -1] and σ k ∂ k annihilate the distribution 1/σ k and it is easy to check that the generators of I and U 0 are in this left ideal of D N generated by these vector fields. Now we have also, using the same argument, that Y 0 is in c(N 0 ). To see that Y -1 is annihilated by

is a trace function on the open set {σ k ∆(σ) = 0} and that U 0 + 1 kills also this holomorphic function on this open set. This is enough to complete the proof of the lemma using again the absence of torsion. Lemma 5.0.7 The D N -linear map defined by [START_REF] Barlet | On symmetric partial differential operators[END_REF] → X 0 ∈ Db N is a generator of c(N 0 ) and the D N -linear map defined by [1] → X -1 ∈ Db N is a generator of c(N -1 ).

Proof. As we already know that the distribution X 1 is a generator of c(N 1 ) the point 9 above implies that a distribution Z in c(N 0 ) which satisfies U 1 (Z) = X 1 must induce a generator in c(N 0 ). But X 0 is such a distribution. Thanks to point 10 above and the previous result, the distribution U -1 (X 0 ) gives a generator in c(N -1 ) and we have, using as above the absence of torsion the equality

End of proof of Theorem 5.0.3. For λ = 0 we have the exact sequence of D N -modules (see point 5 above) Complement. We give in the following lemma, for each case λ = -1, 0, 1, a distribution which generates the D N -module c(N λ ).

Lemma 5.0.8 Consider the distribution Z -1 := k j=1 zj /z j on M and define Z -1 := π * (Z -1 ). Then Z -1 is a generator of c(N -1 ).

Proof. First write Z -1 = k j=1 |z j | 2 /z 2 j in order to show that any distribution in the D M -module generated by Z -1 has the standard extension property, by applying Corollary 4.4.3. Then the same is true for any distribution in the D N -module generated by Z -1 . Then it is easy to see that Z -1 is in c(N -1 ). Then we have Ū-1 (Z -1 ) = π * ( k j=1 1/z j ) = Y -1 and

So D N Z -1 is equal to c(N -1 ) thanks to Theorem 5.0.3. This is enough to conclude, thanks to Theorem 5.0.3.