Imène Mansouri

Maxime Raffray

Mathilde Lassalle

Florent De Vathaire

Brice Fresneau

Chiraz Fayech

Hélène Lazareth

Nadia Haddy

Sahar Bayat

Dr Cécile Couchoud
email: cecile.couchoud@biomedecine.fr

Imene Mansouri

Marie Erbault

Karen Assmann

Nelly Le Guen

Nathalie Poutignat

An algorithm for identifying chronic kidney disease in the French national health insurance claims database

Keywords: Chronic Kidney Disease, Algorithms, Validation studies, Healthcare claims databases maladie rénale chronique, algorithmes, étude de validation, SNDS CKD: chronic kidney disease CKD-EPI: Chronic Kidney Disease Epidemiology Collaboration estimated glomerular filtration rate

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chronic kidney disease (CKD) represents a heavy global health burden associated with increased mortality and morbidity and high economic impact [START_REF] Foreman | Forecasting life expectancy, years of life lost, and all-cause and cause-specific mortality for 250 causes of death: reference and alternative scenarios for 2016-40 for 195 countries and territories[END_REF][START_REF] Nichols | Health Care Costs by Type of Expenditure across eGFR Stages among Patients with and without Diabetes, Cardiovascular Disease, and Heart Failure[END_REF]. The number of individuals with CKD reached more than 700 million in 2017 worldwide, surpassing the number with diabetes mellitus [START_REF] Bikbov | Global, regional, and national burden of chronic kidney disease, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF][START_REF] James | Global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF]. The prevalence of CKD in France is unknown, with some estimates varying between 3 and 6 million, corresponding to about 10% of the French adult population, about 92 000 patients presenting end-stage kidney disease (ESKD) [START_REF] Bikbov | Global, regional, and national burden of chronic kidney disease, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF][START_REF]Maladie renale chronique[END_REF][START_REF] Couchoud | rapport REIN Registry -2019[END_REF]. Solid data on CKD prevalence in the general population and tools for identifying CKD cases before RRT are lacking And yet, health system planning and policy-making requires careful assessment of CKD epidemiology to develop efficient and cost-effective care strategies.

Health claims databases have long been used to efficiently estimate the prevalence of diseases.

These data represent a useful source of information for policy-makers regarding the management of chronic diseases including diabetes, cancer and cardiovascular diseases [START_REF] Lipscombe | Identifying diabetes cases from administrative data: a population-based validation study[END_REF][START_REF] Bello | Use of administrative databases for health-care planning in CKD[END_REF][START_REF] Saczynski | A systematic review of validated methods for identifying heart failure using administrative data: DETECTION OF CONGESTIVE HEART FAILURE IN CLAIMS[END_REF][START_REF] Choma | An algorithm to identify incident myocardial infarction using Medicaid data[END_REF][START_REF] Ajrouche | Evaluation of algorithms to identify incident cancer cases by using French health administrative databases: Cancer incidence in the sniiram/egb databases[END_REF]. The analysis of these databases could provide insight into the global burden of CKD and allow for evaluating treatment strategies aimed at slowing its progression. Nevertheless, even though the identification of patients having renal replacement therapy (RRT) in health claims databases is fairly straightforward, identifying other stages of CKD remains challenging.

A systematic review that analyzed several algorithms for CKD based on both diagnostic and procedural codes in 25 administrative databases across 8 countries found poor algorithm performance, yielding low sensitivity and positive predictive value [START_REF] Vlasschaert | Validity of Administrative Database Coding for Kidney Disease: A Systematic Review[END_REF]. Another study that identified CKD with diagnostic and procedural codes in Dutch hospital-based database, found higher sensitivity among younger patients and those with advanced CKD [START_REF] Van Oosten | The validity of Dutch health claims data for identifying patients with chronic kidney disease: a hospital-based study in the Netherlands[END_REF]. Only a few other studies in Italy and Canada focused on developing algorithms with higher sensitivity based on Accepted manuscript prescription of specific drugs, medical procedures and hospitalizations related to CKD from healthcare claims data [START_REF] Bello | Use of administrative databases for health-care planning in CKD[END_REF][START_REF] Marino | Prevalence of chronic kidney disease in the Lazio region, Italy: a classification algorithm based on health information systems[END_REF][START_REF] Fleet | Detecting chronic kidney disease in population-based administrative databases using an algorithm of hospital encounter and physician claim codes[END_REF].

In France, the national REIN registry (Renal Epidemiology and Information Network ) includes all patients receiving RRT for ESKD. France also has a nationwide health claims database.

Unfortunately, results of biological tests (including blood creatinine levels) are not available in this database This study aimed to propose and describe an algorithm for the identification of all stage CKD using the French health claims database and assess its performance and utility using data from two different populations: confirmed ESKD cases (REIN registry) one and two years prior RRT and survivors of childhood cancer from the (French Childhood Cancer Survivor Study [FCCSS] cohort)

Methods

The French administrative healthcare database (SNDS)

The SNDS consists of two main databases: the hospital discharge summaries database (PMSI) and the national health insurance claims database (DCIR) and covers 98.8% of the French population, over 66 million persons, from birth (or immigration) to death [START_REF] Moulis | French health insurance databases: What interest for medical research?[END_REF][START_REF] Tuppin | French national health insurance information system and the permanent beneficiaries sample[END_REF][START_REF] Bezin | The national healthcare system claims databases in France, SNIIRAM and EGB: Powerful tools for pharmacoepidemiology[END_REF]. The PMSI database includes primary, related and associated diagnoses for all private or public medical, obstetric and surgical hospitalizations. These diagnoses are coded according to the International Statistical Classification of Diseases and Related Health Problems, 10 th Revision (ICD-10) [START_REF] Tuppin | French national health insurance information system and the permanent beneficiaries sample[END_REF].

The date and duration of hospitalization are included. Medical procedures performed during the hospitalization are coded according to the French Common Classification of Medical Acts (CCAM), diagnostic-related groups, as well as highly expensive drugs. The DCIR database includes data on all reimbursed ambulatory care including consultations, medical procedures Accepted manuscript coded according to the French CCAM, prescribed medications coded according to the Anatomical Therapeutic Classification, and laboratory biological tests coded according to the French Nomenclature of Biological Acts. In addition to including records of all reimbursed ambulatory care, the DCIR contains a list of long-term diseases that allow full reimbursement of costs related to these conditions, with start and end dates. Clinical and biological test results are not available in the database.

The identification of CKD cases in the French SNDS was based on querying all hospital discharge claims, ambulatory care claims, and medication-dispensing data, in private or in public structures.

CKD case definition algorithm

A group of experts in nephrology, renal epidemiology and healthcare claims databases met several times to define a list of items and combinations of items that could be related to CKD.

Inclusions of items was made by unanimous decisions. The aggregation of all these items defined the so-called "algorithm". The information on whether a patient may have CKD (identification The details of each CKD identification item for each component of the algorithm (nomenclature and codes) are in Supplementary Tables 1 for certain (the item is self-sufficient) , probable (high probability of being related to CKD, a combination of probable items is required to pass to the certain level) and possible (a combination of possible items is required to pass to the probable level) CKD cases.

Accepted manuscript

Certain items consisted of 1) hospitalization with at least one diagnosis of CKD: ICD-10 codes N00-N08 (Glomerular diseases), N11 and N13-N16 (Renal tubulo-interstitial diseases), N18 (CKD), E102 (Type 1 diabetes with diabetic CKD), E112 (Type 2 diabetes with diabetic CKD), T861 (Kidney transplant failure and rejection), Z49 (Care involving dialysis) and Z940 (Kidney transplantation); 2) at least two consultations with a nephrologist during one calendar year; 3) combinations of prescribed medications used in treating CKD including erythropoiesisstimulating agents, drugs for treating hyperkalemia and hyperphosphatemia, angiotensinconverting enzyme inhibitors, iron, antacids with sodium bicarbonate, vitamin D, calcium, high doses of diuretics and hepatitis B vaccine with the specialty of the prescriber; 4) medical acts involving RRT by dialysis or kidney transplantation and creation of arteriovenous fistula; and 5) different combinations of biological tests involved in the diagnosis and/or follow-up of CKD: creatinine clearance, complete blood electrolytes, blood urea nitrogen, parathyroid hormone blood test, serum S-25-hydroxyvitamin D, hepatitis B surface antibody dosage and urine testing for protein.

The association of at least two of the following probable items also led to a certain identification of CKD: 1) other hospitalization probably related to CKD with the following diagnostic codes: I13 (Hypertensive heart and renal disease), I151 (Hypertension secondary to other renal disorders), N171 (Acute renal failure with medullary necrosis), N280 (Ischemia and infarction of kidney) and Q61 (Cystic kidney disease); 2) other medication delivery related to CKD prescribed by a nephrologist; 3) biological tests related to CKD prescribed by a nephrologist; and 4) medication delivery for CKD prescribed by a nephrologist and with different medical procedures related to the creation or the surgical repair of arteriovenous fistula, renal biopsy and arterial Doppler imaging.

Accepted manuscript

Study population and data sources

French Childhood Cancer Survivor Study cohort (FCCSS)

The FCCSS cohort includes 7670 5-year childhood cancer survivors who received treatment from 1942 to 2000 for solid cancers or lymphomas before age 21 in several French centers. Among them, 4567 treated at Gustave Roussy Hospital alive in 2012 were eligible for a long-term follow-up (LTFU) visit. A total of 1002 (22%) attended the long-term follow-up (LTFU) between 2012 and 2018. Systematic screening offered by the LTFU clinic included clinical examination and urine and biological testing [START_REF] Haghiri | Long-term follow-up of high-risk neuroblastoma survivors treated with high-dose chemotherapy and stem cell transplantation rescue[END_REF]. Serum levels of creatinine and markers of kidney damage (proteinuria, hematuria, calcium, phosphate, glycosuria, phosphorus reabsorption rate etc.) were reported in medical records. CKD was defined according to the Kidney Disease: Improving Global Outcomes definition as functional abnormalities (tubulopathies, proteinuria…) of the kidney regardless of estimated glomerular filtration rate (eGFR) or eGFR < 60 ml/min/1.73m² [START_REF] Stevens | Evaluation and Management of Chronic Kidney Disease: Synopsis of the Kidney Disease: Improving Global Outcomes 2012 Clinical Practice Guideline[END_REF]. GFR was estimated according to the Chronic Kidney Disease Epidemiology Collaboration equation [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF]. Partial nephrectomy without functional consequence was not considered as CKD. All cases were confirmed by an expert. Information on the diagnosis of any CKD was extracted from medical records and considered the gold standard of confirmed CKD. A total of 867 childhood cancer survivors from the FCCSS cohort with at least one LTFU visit had available outpatient data in the SNDS (Supplementary Figure 1).

The FCCSS protocol was approved by the INSERM national ethics committee and the French National Agency regulating Data Protection (CNIL no. 902 287). Consent was obtained from patients, parents or guardians according to national research ethics requirements.

Accepted manuscript

The REIN registry: Confirmed ESRD adult patients with RRT Since 2012, the REIN registry has gathered data on all new ESKD patients who started RRT in metropolitan France and its overseas territories. The registry includes data on patient identification (age, sex, and postcode of the place of residence), comorbidities (e.g., cardiovascular diseases, diabetes, cancer), and characteristics at RRT start (eGFR, hemoglobin and serum albumin levels, planned or emergency dialysis, center identification etc.) [START_REF] Couchoud | The renal epidemiology and information network (REIN): a new registry for end-stage renal disease in France[END_REF]. Patients are followed, and specific events, such as placement on a waitlist for kidney transplantation, kidney transplantation and death, are recorded. To obtain information on patients' healthcare consumption before RRT, this population was linked to the SNDS by using a deterministic and iterative linkage method that was previously described [START_REF] Raffray | Linking disease registries and nationwide healthcare administrative databases: the French renal epidemiology and information network (REIN) insight[END_REF]. Two years of healthcare consumption data prior RRT were extracted for all adults (≥ 18 years old) with ESKD who were included in the REIN registry and started RRT in France in 2015. . Data for long-term diseases were not available for these patients.

Statistical analysis

After specifying the algorithm, it was locked and following analysis were performed.

Medical data from the subset of the FCCSS cohort who attended at least one LTFU visit at Gustave Roussy LTFU Clinic and with data from the French SNDS were compared at an individual level. The algorithm was applied, including different combinations of codes, with medical records as the gold standard for determining a CKD case. The final identification of patients with CKD in the SNDS was based on items considered certain and/or the combinations of at least two probable items.

To evaluate the performance of the algorithm in identifying diagnosed CKD, different indicators were calculated: sensitivity (Se), specificity (Sp), positive predictive value (PPV), negative

Accepted manuscript

predictive value (NPV), accuracy and Cohen's kappa coefficient (k-coefficient) and their 95% confidence interval (CI). Se was calculated as the proportion of cases classified as positive by both the algorithm and medical record review, or "true positives" (TPs), as compared with all CKD cases identified by the gold standard (medical record review). Sp was calculated as the proportion of cases without CKD identified by both the algorithm and the gold standard, or "true negative" (TNs), as compared with all negative cases by the gold standard. PPV was calculated as the proportion of TPs divided by all potential CKD cases identified by the algorithm and medical record review. NPV was defined as the number of TNs divided by the number of patients with a negative classification for CKD by the algorithm and the gold standard.

For the population of confirmed ESKD cases extracted from the REIN registry, the sensibility of the algorithm was calculated as the proportion of cases identified by the algorithm compared to the total number of ESRD cases recorded by the REIN registry (gold standard). The final classification of patients with CKD in the SNDS was based on items considered certain, probable or possible.

The algorithm was used with the available SNDS healthcare data separately for 1 and 2 years before RRT start. We then assessed the contribution of each component of the algorithm by using Venn diagrams [START_REF] Heberle | InteractiVenn: a web-based tool for the analysis of sets through Venn diagrams[END_REF].

All statistical analyses involved using SAS 9.4.

Results

Validation of the algorithm in the subset of FCCSS cohort with LTFU visit

In the FCCSS cohort, 1002 patients had an LTFU visit and available data on renal function at the date of the visit; 135 were excluded because of pairing failure with the health insurance database Accepted manuscript (Supplementary Figure 1). The characteristics of the validation population (n = 867) were compared to those of the excluded population. The groups did not significantly differ in type of primary cancer malignancy. However, the validation sample was slightly younger (median age 35.4 [IQR 2.8-49.7]) and more frequently had a diagnosis of primary childhood malignancy in recent years or hypertension than the excluded population (Supplementary Table 2). When the validation cohort (n=867) was compared to the 3535 excluded patients who never had a LTFU visit, females, CNS (Central Nervous System) tumor survivors and patients with comorbidities showed up more for LTFU visits.

A total of 59 childhood cancer survivors had CKD confirmed by clinicians during the LTFU visit including 4 ESKD, detailed description is shown in Supplementary Table 3. In the French administrative healthcare database (SNDS), among them, 29 (49.2%) cases were coded as certain with the algorithm due to a hospitalization diagnoses, 25 (42.4%) were coded as certain because of physician claims and 21 (35.6%) by long-term illness exemption due to severe or chronic nephropathy (Supplementary Table 4).

A total of 67 patients were considered as CKD by the algorithm (at least one certain items or at least 2 probable items). Therefore, for identifying confirmed CKD cases (all stages), in the FCCS cohort, the algorithm Se was 78% (95% CI 67.4-88.5), Sp 97.4% (95% CI 96.3-98.5), NPV 94.8% (87.5-99.3) and PPV 68.7% (57.6-79.8) (Table 1). Concerning level of agreement with the gold standard, the k-coefficient for the algorithm was 0.79 (95% CI 0.61-0.80). When restricting the analysis to survivors of nephroblastoma (Wilm's tumors) (n=127), both sensitivity and specificity remained similar, at 78.4% (95% CI 65.1-91.6) and 97.6% (95% CI 96.4-98.7) respectively. Analysis of false-negative and false-positive cases are provided in Supplementary tables 5 and 6.

Accepted manuscript

The sensitivity was significantly higher with our algorithm compared to the used of hospital claims alone due to reclassification of false negative (Table 1).

Sensitivity of the algorithm with the confirmed ESKD cases from the REIN registry

Among the 11083 patients from the REIN registry who started RRT in 2015 in France, data for 9627 (86.8%) were linked with the SNDS; 134 did not have any healthcare consumption in the SNDS database before RRT start and were considered inherently undetectable by the algorithm and thus were excluded from this validation analysis. Hence, 9493 patients were included in the analysis (Supplementary Figure 2).

The algorithm identified 8885 (93.6%) of the confirmed incident ESKD cases from the REIN registry as certain cases when considering 1 year before RRT start. Only 107 (1.1%) confirmed cases were not identified as cases.

The period considered was of importance: the algorithm identified 5526 (55.1%) confirmed cases as certain cases when considering the 2 years before RRT start (Table 2).

The proportion of certain cases identified by the Hospitalization diagnoses and Diagnosis-related groups components of the algorithm greatly increased between the 2 periods. Figure 1 shows the evolution of cases identified by the algorithm according to their status (certain, probable, possible) between 2 years and 1 year before RRT start. Most probable, possible and previously undetected cases were identified as certain cases in the year before RRT start (95%, 87% and 81%, respectively). The contribution of each component is presented in figure 2.

Accepted manuscript

Discussion

Accurately identifying patients with CKD at the national level is an ambitious challenge in CKD epidemiology but ultimately critical in healthcare policy-making and evaluation. Using healthcare databases is a promising perspective. The algorithm based on healthcare data we present in this article is a first and important step toward this goal. With validation in 2 different populations and contexts, we show good performance of this algorithm.

Previous algorithms have been developed to detect CKD patients in healthcare claims databases. Some algorithms benefit from serum creatinine results, which are of great value to detect CKD cases; an example is the Alberta Kidney Disease Network (AKDN) database, which combined administrative databases with laboratory data [START_REF] Bello | Use of administrative databases for health-care planning in CKD[END_REF]. However, serum creatinine value is lacking in many healthcare claims databases, including the French health insurance databases. Other authors used algorithms based on diagnosis at hospital discharge to identify CKD: this was the case for the 16 studies included in a systematic review published in 2010 [START_REF] Vlasschaert | Validity of Administrative Database Coding for Kidney Disease: A Systematic Review[END_REF]. In such selected populations, Sp is high but Se is poor. This approach is not conclusive in evaluating the burden of CKD in the general population because it selects only hospitalized patients, who may not be representative. The performance of 11 diagnostic codes and their combination was analyzed with 7 databases in Ontario, Canada [START_REF] Fleet | Detecting chronic kidney disease in population-based administrative databases using an algorithm of hospital encounter and physician claim codes[END_REF]. The results showed high Sp but low Se, especially in early-stage CKD. In our study, Se was lower when only hospital claims were used. Only one recent algorithm used information on medications and outpatient services combined with that from a Hospital Discharge Registry and a Ticket Exemption Registry [START_REF] Marino | Prevalence of chronic kidney disease in the Lazio region, Italy: a classification algorithm based on health information systems[END_REF]. This algorithm identified 99,457 individuals with CKD (mean age 71 years, 55.8% males). The exclusive contributions of each regional source were 35,047 (35.2%) from the Outpatient Specialist Service Information System, 27,778 (27.9%) from the Hospital Discharge Registry, 4143 (4.2%) from the Ticket Exemption Registry and Accepted manuscript 463 (0.5%) from a Drug Dispensing Registry; 5.1% of cases were found in all databases. However, because of the lack of a gold standard, this algorithm was validated in only dialysis patients.

The low performance of these algorithms to correctly identify CKD patients (TP rate) may be due to a high number of false negatives. Indeed, CKD remains a silent disease for a long time and associated with non-specific symptoms, so its diagnosis is difficult for health professionals. This situation could explain the lack of specific healthcare consumption until advanced stages of CKD and for some patients close to RRT as well by some quality issues in coding. Our algorithm showed good performance in the FCCSS cohort, with Se > 70% and Sp > 97%. False negatives were mostly patients with CKD stage 2 and renal tumor (Supplementary table 2). The high sensitivity found in the FCCSS could be explained by the inclusion of younger patients (median age 35.4 years old). Similar results were also reported in a study based on the Dutch hospital-based database [START_REF] Van Oosten | The validity of Dutch health claims data for identifying patients with chronic kidney disease: a hospital-based study in the Netherlands[END_REF]. Also, CKD in this population could be related to risk factors different from those in the general population. Nevertheless, the diagnosis and management of CKD were based on the KDIGO guidelines as in the general population Second, the LTFU guidelines for survivors undergoing unilateral renal surgery recommends an annual assessment of renal function, which may lead to a possible over-diagnosis bias of CKD in survivors of renal malignancies [START_REF]Long Term Follow-up Guidelines For Survivors of Childhood, Adolescent And Young Adult Cancers[END_REF]. Furthermore, only 22% of survivors included in the FCCSS cohort alive in 2012 had an LTFU visit and a renal function assessment; 75.4% (49 patients) of those with confirmed CKD received the diagnosis during the LTFU visit. This observation emphasizes the crucial role of this visit in the LTFU of childhood cancer survivors.

Because the REIN registry ESRD cohort consisted of only ESRD patients (i.e., no negative cases), Se could not be estimated. However, this analysis allows for showing that our algorithm is performant in more severe disease stages, close to RRT. Indeed, the algorithm correctly identified 93.6% confirmed incident cases of ESRD during the year before RRT. Of note, the 107 (1.1%) confirmed cases not Accepted manuscript identified as cases by the algorithm represent patients with late referral and without any health consumption before RRT. As RRT drew near, medical acts and hospital diagnosis were more prominent as sources of identification. During the 2 years before RRT start, medications and visits with a nephrologist were more frequent sources of identification.

Our study may suffer from the following limits. The validation was made in a selected small cohort that may not be representative of the general population. The classification of the items in certain, probable or possible is rather subjective and may be discussed. Sensitivity analyses are planned. Finally, our algorithm was developed by a group of experts and not data-driven. Although it seems to present good performance as is, this methodological challenge of CKD identification is an iterative process and will be updated regularly. For example, the pool of items classified as indicative of possible cases of CKD (supplementary table 1) was not used here to identify patients with CKD and represent an area of further research.

Nevertheless, identifying milder stages of CKD can be challenging because it requires more complex and advanced case-finding algorithms. Future research will address the ability of the algorithm to detect all CKD stages and classify individuals at early, advanced or late stage of CKD as well as the use of other populations and contexts for further validation.

In conclusion, our algorithm showed good performance among young patients and those with ESKD in the two last years prior to RRT. Because it is not based on lab results, it can be used in various contexts, especially in big medico-administrative databases. Further improvements and other validations in various populations are planned. Accepted manuscript

Accepted manuscript

Accepted manuscript

  item) was searched in different components of the SNDS: a) Long-term diseases, b) Physician claims (consultations), c) Drug delivery, d) Biological tests, e) Diagnosis-related groups, f) Hospitalization diagnoses and e) Medical acts.

Figure 1 :

 1 Figure 1: Flow of confirmed ESRD new cases identified by the algorithm between the second and first year before RRT start. Abbreviations: ESRD: end-stage renal disease; RRT: renal replacement therapyThe contribution of the algorithm to the identification of certain cases differed across its components

Figure 2 :

 2 Figure 2: Origin of the cases identified as certain in the confirmed end-stage kidney disease (ESKD) population according to the component of the algorithm (Consultation, Medication delivery, Biological tests, Medical acts, Hospitalization diagnoses, Diagnosis-related groups [DRG]) in the first (A) and second (B) year before renal replacement therapy start (N=8885)

Table 1 .

 1 Test characteristics of the CKD case definition algorithms applied in the French SNDS using confirmed CKD cases in the subset of FCCSS cohort with at least one LTFU visit as the gold standard

		N TP FP	TN	FN		Se (%)	Sp (%)	PPV (%)	NPV (%)	Acc (%)	k (%)
							(95% CI)	(95% CI)	(95% CI)	(95% CI)	(95% CI)	(95% CI)
	Based only on hospital claims	86 7	29 19	789	30 49.2 (36.4-61.9)	97.6	(96.6-98.7)	60.4	(46.6-74.3)	96. 3	(95.1-97.6)	94. 3	(92.8-95.9)	0.5 1	(0.39-0.63)
	Identification	867 46	21	787	13 78.0	(67.4-	97.4	(96.3-	68.7	(57.6-	98.4 (97.5-	96.1 (94.8-	0.79 (0.61-0.80)
	algorithm							88.5)		98.5)		79.8)		99.3)		97.4)
	Identification	740 29	17	686	8	78.4	(65.1-	97.6	(96.4-	63.0	(49.1-	98.3 (97.3-	96.6 (95.3-	0.75 (0.59-0.90)
	algorithm							91.6)		98.7)		77.0)		99.2)		97.9)
	excluding
	survivors of
	renal
	malignanciest
	umor
	Identification	127 17	4	101	5	77.3	(59.8-	96.2	(92.5-	81.0	(64.7-	95.3 (91.3-	92.9 (88.4-	0.75 (0.59-0.90)
	algorithm							94.8)		99.8)		97.8)		99.3)		97.4)
	among
	survivors of
	renal
	malignancies

FCCSS: French Childhood Cancer Survivor Study; LTFU : long-term follow-up, N; total number of subjects included in the validation sample, TP; true positive, FP; false positive, Se; sensitivity, Sp; specificity, PPV; positive predictive value, NPV; negative predictive value, Acc; accuracy, k; Cohen's kappa coefficient, 95% CI; confidence interval

Table 2

 2 Proportion of confirmed ESKD incident cases in 2015 in France identified by the algorithm in the SNDS according to the time frame considered before renal replacement therapy (RRT) start.

	Component of the algorithm and case status	Adult patients starting RRT in 2015 in France
			(confirmed ESKD cases)
			N=9493
			1 year before RRT 2 years before RRT
			n (%)	n (%)
	Physician claims (visit)	Certain	6410 (67.5)	4088 (43.1)
		Probable	1262 (13.3)	1323 (13.9)
		Possible	533 (5.6)	1161 (12.2)
		Undetected	1288 (13.6)	2921 (30.8)
	Medication deliverance	Certain	4925 (51.9)	2268 (23.9)
		Probable	2167 (22.8)	2124 (22.4)
		Possible	1563 (16.5)	3579 (37.7)
		Undetected	838 (8.8)	1522 (16)
	Biological tests	Certain	2374 (25)	1613 (17)
		Probable	3351 (35.3)	2252 (23.7)
		Possible	3111 (32.8)	4113 (43.3)
		Undetected	657 (6.9)	1515 (16)
	Diagnoses-related groups Certain	4576 (48.2)	642 (6.8)
		Probable	1232 (13)	333 (3.5)
		Undetected	3685 (38.8)	8518 (89.7)
	Medical acts	Certain	6307 (66.4)	701 (7.4)
		Probable	1119 (11.8)	270 (2.8)
		Possible	481 (5.1)	988 (10.4)
		Undetected	1586 (16.7)	7534 (79.4)
	Hospitalization diagnoses Certain	6778 (71.4)	1494 (15.7)
		Probable	50 (0.5)	32 (0.3)
		Possible	22 (0.2)	24 (0.3)
		Undetected	2643 (27.8)	7943 (83.7)
	Total	Certain	8885 (93.6)	5226 (55.1)
		Probable	240 (2.5)	959 (10.1)
		Possible	261 (2.7)	2381 (25.1)
		Undetected	107 (1.1)	927 (9.8)

Abbreviations: CKD: chronic kidney disease; ESRD: end-stage renal disease Accepted manuscript

Accepted manuscript


Vinh Le-Van
email: vinh.le-van@g2elab.grenoble-inp.fr

Gérard Meunier

Olivier Chadebec

Jean-Michel Guichon

A Magnetic Vector Potential Volume Integral Formulation for Nonlinear Magnetostatic Problems

Keywords: edge elements, integral formulation, nonlinear magnetostatic, volume integral method

This paper presents a novel volume integral formulation based on the interpolation of the magnetic vector potential on edge elements in order to deal with 3D nonlinear magnetostatic problems. The formulation ensures rigorously the solenoidality of magnetic induction. A strong point is that the convergence of the nonlinear resolution is easily reached after a few iterations without any relaxation. Computed results for the TEAM Workshop problem 13 and for an actuator demonstrate the efficiency and accuracy of this new formulation.

I. INTRODUCTION

OLUME integral method (VIM) is known today as an interesting alternative to classical finite element method (FEM) for solving of nonlinear magnetostatic problems. The main advantage of VIM over FEM is that neither free space mesh nor boundary conditions are required, only active regions have to be meshed.

The integral formulations are the core of magnetostatic VIM. Different kinds of formulations have already been presented in the literature [START_REF] Chadebec | A review of magnetostatic moment method[END_REF]- [START_REF] Le-Van | A Volume Integral Formulation Based on Facet Elements for Nonlinear Magnetostatic Problems[END_REF]. Although these formulations are able to yield accurate results in variety numerical examples, their applications to general magnetostatic problems are not without any difficulties. The magnetic moment method (MMM) [START_REF] Chadebec | A review of magnetostatic moment method[END_REF] [START_REF] Takahashi | Large-scale and fast nonlinear magnetostatic field analysis by the magnetic moment method with the ACA[END_REF], which is the oldest and also the most popular formulation, cannot accurately deal with problems with high susceptibility material and can suffer from the well-known "looping pattern". More recently, H-edge and φ-nodal formulations were proposed in [START_REF] Kettunen | A volume integral formulation for nonlinear magnetostatics and eddy currents using edge elements[END_REF] [4] [START_REF] Carpentier | Resolution of nonlinear magnetostatic problems with a volume integral method using the magnetic scalar potential[END_REF]. The H-edge formulation [START_REF] Kettunen | A volume integral formulation for nonlinear magnetostatics and eddy currents using edge elements[END_REF] is based on the interpolation of the magnetic field with edge elements. The φ-nodal formulation is on the other hand [START_REF] Kalimov | Three-Dimensional Magnetostatic Field Calculation Using Integro-Differential Equation for Scalar Potential[END_REF] [START_REF] Carpentier | Resolution of nonlinear magnetostatic problems with a volume integral method using the magnetic scalar potential[END_REF] established by an interpolation of the scalar potential on nodal elements. They seem to be more flexible and robust than the MMM. However, these formulations present a quite poor convergence rate for nonlinear resolution due to the use of a B(H) curve. Thus, a relaxation process, which increases the computation time, is required in order to achieve the convergence. Most recently, the B-facet formulation based on interpolation of magnetic flux on facet elements has been proposed in [START_REF] Le-Van | A Volume Integral Formulation Based on Facet Elements for Nonlinear Magnetostatic Problems[END_REF]. This formulation is a very interesting approach and ensures a good convergence rate due to the use of a H(B) curve and results in an excellent accuracy even with very coarse meshes. Nevertheless, the B-facet formulation requires the use a face-tree in order to impose the solenoidality of B. This procedure can be time consuming and decreases the convergence rate of linear system resolutions.

We propose in this paper a useful formulation based on the interpolation of the magnetic vector potential on edge elements. We will demonstrate that this formulation can overcome the limits of the aforementioned integral formulations. The convergence of the nonlinear resolution is easily reached after a few iterations without any relaxation. Moreover, the fully dense integral matrix does not need to be recomputed during the nonlinear resolution and can be compressed thanks to efficient matrix compression algorithms such as the fast multipoles method or adaptive cross approximation.

II. FORMULATIONS

For a magnetostatic problem, the governing equations are divB = 0 and rotH = J 0 . By introducing the magnetization M so that B = µ 0 (H + M), the constitutive equation is expressed either as H = υB or M = (υ 0 -υ)B. The Maxwell-Ampère equation allows the writing of the magnetic field H as

r ϕ grad H H - = 0 ,
(1) where H 0 is the magnetic field created by J 0 in the vacuum, and φ r is the reduced magnetic scalar potential created by magnetic material. The potential φ r is determined by

∫ Ω Ω       ⋅ = d 1 4 1 r r grad M π ϕ , ( 2
)
where r is the distance between the observation and integration points, and Ω the magnetic domain. The magnetic vector potential A defined by B = rotA can be approximated by the interpolation of tangential components A j on edge elements as = . in this expression; N is the number of edge elements and the w j are first order shape functions of edge elements. Projecting (1) on rotw i in domain Ω, we get

Ω ⋅ = Ω ⋅ + ⋅ ∫ ∫ ∫ Ω Ω Ω d d dΩ 0 H rotw grad rotw H rotw i r i i ϕ .
(

) 3
The two terms on the left hand can be developed as follows: ) ( ) Hence, using ( 4) and ( 6), we write (3) in matrix form as ( )

; d d dΩ 1 1 j N j j i i i A ∑ ∫ ∫ ∫ = Ω Ω Ω           Ω ⋅ = Ω ⋅ = ⋅ = rotw rotw B rotw H rotw I ν ν (4) V ( ) Γ ⋅ = Ω ⋅ = ⋅ = ∫ ∫ ∫ Γ Ω Ω d d dΩ
j N j j i A r ∑ ∫ ∫ = Ω Γ           Γ           Ω       ⋅ - ⋅ =
0 U A = + L R , ( 7
)
where matrices R, L, and vector U 0 are defined by ( ) ( )

. d d d 1 4 1 d 0 0 0 ∫ ∫ ∫ ∫ Ω Ω Γ Ω Ω ⋅ = Γ           Ω       ⋅ - ⋅ = Ω ⋅ = H rotw grad rotw n rotw L rotw rotw R i i j i ij j i ij U r ν ν π ν (8)
Since edge elements of first order are used, the magnetic field B is constant in each tetrahedral element. Thus, υ is constant piecewise and matrix L is determined by

( ) Γ           Γ ⋅ ⋅ = ∑ ∫ ∫ = Γ Γ d d 4 1 1 f f N k f j f i ij r n rotw n rotw L δν π , ( 9
)
where N f is number of facets and δυ f the difference of reluctivity between two elements sharing the facet f. Let us notice that rotw i •n is a scalar quantity defined by ±1/S f where S f is the surface of a facet element containing edge i. Thus, rotw i •n is taken out of the integrand (9) and matrix L is decomposed into the product of two terms. The first is the vector δυ dealing with the material properties. The second is the fully populated matrix L 0 which depends only on the double surface integration over faces i.e. properties of elements geometry. Thus, matrix L 0 can be computed only once and during the nonlinear resolution matrix L is updated by the product of L 0 and vector δυ, which is recomputed after each iteration. Matrix R is a FEM matrix structure.

The singularity of Green's function in (9) is computed exactly by an analytical integration. Due to the use of the vector potential A, the solenoidality of magnetic induction is naturally verified. System ( 7) is compatible according to [START_REF] Ren | on the Convergence Behaviour[END_REF]. Consequently, it can be straightforwardly solved by an iterative solver without any gauge condition and the obtained solution is unique.

III. NUMERICAL RESULTS

The TEAM Workshop problem 13 [START_REF] Nakata | Summary of results for TEAM workshop problem 13[END_REF] (Fig. 1.a) and an actuator (Fig. 2.a) were studied in order to demonstrate the efficiency of the proposed formulation. The classic Newton Raphson method was used for solving of the nonlinear system.

For the problems13, the magnetomotive force is 1000 AT. The goal of the second example was to compute the global force acting on the pallet of the actuator. Once (7) was solved, both the method of equivalent magnetic charges and virtual work can be applied. They provide the same results. Fig. 2.b shows the comparaison of the convergence with FEM. In order to obtain a result deviated by 5% from the reference value, VIM needs only a mesh of 100 tetrahedra whereas the FEM requires a mesh containing 12,000 elements on active regions.

  which Γ is the boundary of domain Ω, n is normal of Γ. From (2) and the constitutive equation M = (υ 0 -υ)B, we express the potential φ r as a function of the A j . The term I 2 is then written as

(

Fig. 1 .

 1 Fig.1. TEAM 13. a) Geometry. b) Average flux density inside steel plate.

  value of FEM (3.8 million elts) Force computed by VIM Force computed by FEM


L Blattner Martinho

J Siau

B Bannwarth

J.-M Guichon

O Chadebec
email: olivier.chadebec@g2elab.grenoble-inp.fr

G Meunier
email: gerard.meunier@g2elab.grenoble-inp.fr

V C Silva

A generalized PEEC Analysis of Inductive Coupling Phenomena in a Transmission Line Right-of-Way

Keywords: Eddy Currents, electromagnetic coupling, generalized partial element equivalent circuit (PEEC) method, integral equation, transmission lines

A 3-D inductive coupling problem in a transmission line right-of-way is analysed with the generalized Partial Element Equivalent Circuit (PEEC) method. This new approach does not rely on the parallelism with the transmission line and permits the determination of the induced current density in underground objects at arbitrary positions and orientations.

I. INTRODUCTION

T HE inductive coupling between high voltage transmission lines and other structures sharing a common right-ofway is unavoidable. The numerical techniques used for the modelling of this class of problems frequently take advantage of the parallelism between the transmission line and elongated structures, as in the case of buried pipelines or transportation rails placed in a transmission line right-of-way [START_REF] Dawalibi | Analysis of electrical interference from power lines to gas pipelines. I. computation methods[END_REF] [START_REF] Christoforidis | Inductive interference on pipelines buried in multilayer soil due to magnetic fields from nearby faulted power lines[END_REF].

This paper analyses the modelling of an inductive coupling situation in which the referred hypothesis of parallelism between the transmission line and the object of interest is not required. The generalized Partial Element Equivalent Circuit (PEEC) integral method [START_REF] Nguyen | An integral formulation for the computation of 3-D eddy current using facet elements[END_REF] is employed to determine the induced current density in an object beneath the soil surface and in the vicinities of a three-phase transmission line.

II. PEEC ANALYSIS OF AN INDUCTIVE COUPLING

A schematic view of the right-of-way under analysis is available in Fig. 1. The buried object under investigation has a prismatic and elongated shape and is much more conductive than the surrounding soil, which is supposed to have a uniform and isotropic resistivity. The PEEC approach arises from an application of the Galerkin Residual method to the integral equation

J σ + jω µ 0 4π Ω J r dΩ = -∇V, (1)
which states the frequency domain relationship between the current density J and the electrical potential V in a three dimensional domain Ω [START_REF] Nguyen | An integral formulation for the computation of 3-D eddy current using facet elements[END_REF]. The media in Ω are supposed to be non-magnetic, non-dielectric and with conductivity σ.

The procedure requires a finite element approximation for J.

Its particular choice and the assembly of the corresponding system of equations are detailed in the sequence.

A. Current density discretization and external circuit

Fig. 2 shows the delimitation of the domain Ω and its decomposition into two subregions. A sufficiently large soil volume Ω V bounded by the earth surface and containing the embedded underground object is defined. In Ω V , J is approximated by vector facet elements [START_REF] Bossavit | Whitney forms: a class of finite elements for threedimensional computations in electromagnetism[END_REF]. The transmission line Ω L is represented by line elements, each one carrying a constant complex current and with one long element per phase conductor. This corresponds to the choice of zeroorder interpolation functions with a pre-defined direction, limiting the number of additional degrees of freedom. After the Galerkin projection, this interpolation scheme yields the following system of equations:

( [R] + jω [L] ) [I f ] = [∆V ] , (2)
in which [R] and [L] may be regarded as equivalent resistance and inductance matrices. This system gives an equivalent circuit interpretation of (1) in terms of both the facet and line currents in [I f ], and as a consequence (2) may be coupled to an external network and analysed with an electrical circuit solver [START_REF] Nguyen | An integral formulation for the computation of 3-D eddy current using facet elements[END_REF]. Fig. 2 also shows the required external circuit connections to provide current excitation and to establish an underground path for the flow of zero-sequence current components (if an unbalanced operation condition was to be considered). B. Block integration and assembly of the system of equations Matrices [R] and [L] have a 2X2 block structure due to the interpolation scheme adopted for J. While [R] is sparse, [L] is densely populated and two of its blocks are dealt with in a particular way. The block corresponding to the Ω V × Ω V interaction is compressed using the H-matrix representation and treated with the HCA technique [START_REF] Börm | Hybrid cross approximation of integral operators[END_REF]. The block associated to the Ω L × Ω L interaction is equivalent to the inductance matrix of the transmission line. This block is substituted by analytical computations of the conductor's self inductances and of the mutual inductances between phases. The two remaining blocks correspond to the inductive coupling between Ω L and Ω V . Only one of these blocks needs to be computed since the other may be obtained by matrix transposition.

III. APPLICATION AND RESULTS

Two relative positions between the line and the object are considered. For each one of them, the complete equivalent network arising from the numerical scheme has approximately 15000 branches and 9500 independent current loops.

First, the longest dimension of the object is placed in parallel to the line, in a configuration that permits the validation of the PEEC numerical scheme by comparison with the 2-D FEM solution of a related problem. In this latter approach the buried object is supposed to have an infinite length and J has a single component in the direction parallel to the transmission line. Due to a symmetry argument, this solution may be compared with the PEEC solution obtained in the midsection of the buried object, which is highlighted in Fig. 1

(b).

A very good agreement between the two solutions is verified. For instance, if the mean current density is computed on a 0.25δ × 0.25δ patch in the upper corner of the referred midsection, an error lower than 2.15% is verified between both solutions (|

J PEEC | = 17.39 A/m 2 , | J FEM 2-D | = 17.77 A/m 2 , δ = skin depth ≈ 0.205 m).
In a second analysis, the object is rotated and positioned with its largest dimension along a direction orthogonal to the transmission line. The resulting configuration cannot be accurately modeled by a 2-D approach as previously. It can be nevertheless handled by the PEEC technique as before, and the current density distribution inside the object was once again determined by this procedure. Fig. 3 provides a sample of this distribution by showing the absolute value of each component of J on the horizontal mid-section of the object (z = -0.3 m).

IV. CONCLUSION

A PEEC approach for the analysis of an inductive coupling problem involving a three-phase line and an underground object was presented. Arbitrary relative positions between the transmission line and the object could be considered, and the method also avoids the discretization of inactive air regions.

The current density distribution information obtained provides base data for the study of AC corrosion phenomena. Additionally, the electric potential solution obtained with the equivalent circuit interpretation may also be employed for the evaluation of dangerous induced overvoltages.

The previously discussed applications considered only the case of a balanced system of three-phase currents flowing in the phase conductors of the transmission line. Future work on this subject intend to investigate the case of unbalanced operation and the consequent superposition of conductive coupling phenomena, resulting from the flow of zero-sequence current components in the soil.

Fig. 1 .

 1 Fig. 1. The right-of-way (a) and the conductive object underground (b).

Fig. 2 .

 2 Fig. 2. The computational domain and its connections to the external circuit.

Fig. 3 .

 3 Fig. 3. Current density in the horizontal mid-section of the buried object.