MD Julian Jacob
email: julian.jacob@aphp.fr

N Reyns

C.-A Valéry

L Feuvret

J.-M Simon

J.-J Mazeron

C Jenny

M Cuttat

P Maingon

D Pasquier

Radiotherapy of non-tumoral refractory neurological pathologies

Keywords: radiotherapy, arteriovenous malformations, trigeminal neuralgia, tremor. Résumé radiothérapie, malformations artério-veineuses, névralgie faciale, tremblement

Intracranial radiotherapy has been improved, primarily because of the development of stereotactic approaches. While intracranial stereotactic body radiotherapy is mainly indicated for treatment of benign or malignant tumors, this procedure is also effective in the management of other neurological pathologies; it is delivered using GammaKnife ® and linear accelerators. Thus, brain arteriovenous malformations in patients who are likely to experience permanent neurological sequelae can be managed by single session intracranial stereotactic body radiotherapy, or radiosurgery, in specific situations, with an advantageous benefit/risk ratio. Radiosurgery can be recommended for patients with disabling symptoms, which are poorly controlled by medication, such as trigeminal neuralgia, and tremors, whether they are essential or secondary to Parkinson's disease. This literature review aims at defining the place of intracranial stereotactic body radiotherapy in the management of patients suffering from non-tumoral refractory neurological pathologies. It is clear that the multidisciplinary collaboration of experienced teams from Neurosurgery, Neurology, Neuroradiology, Radiation Oncology and Medical Physics is needed for the procedures using high precision radiotherapy techniques which deliver high doses to locations near functional brain areas.

Radiotherapy in functional Neurology -Jacob

Introduction

Stereotactic body radiotherapy (SBRT) is one of the major technical improvements in the field of Radiation Oncology. Intracranial SBRT, administered in one session using stereotactic radiosurgery (SRS) or several fractions using other approaches, plays a major role in the management of patients with malignant and benign tumors. However, little is known about its usefulness for non-tumoral refractory neurological pathologies which can seriously impair patients. This literature review presents the data published on intracranial SBRT in three chapters which treat the brain arteriovenous malformations (AVMs), trigeminal neuralgia (TN) and tremors. Whatever the pathology, these high precision radiotherapy techniques deliver elevated doses to target volumes that are both difficult to delineate and located close to functional brain areas. They therefore require the intervention of experienced multidisciplinary teams (Radiation Oncology, Neurosurgery, Neurology, Neuroradiology, and Medical Physics).

Brain arteriovenous malformations

Physiopathology and Epidemiology

AVMs are defined as the persistence of a direct link between an artery and a vein, without an intermediary capillary bed. The nidus is located where small arteries and veins are connected [START_REF] Fleetwood | Arteriovenous malformations[END_REF][START_REF] Barreau | Intracranial arteriovenous malformations[END_REF]. Brain AVMs are rare, with an incidence estimated between 1.12 and 1.42 cases per 100 000 person-years, and mostly occur in young patients [START_REF] Abecassis | Natural history of brain arteriovenous malformations: a systematic review[END_REF]. This article presents the literature data on adults treated for brain AVMs. Radiotherapy in functional Neurology -Jacob

Diagnosis

While brain arteriovenous malformation (AVM) diagnosis can be incidental, the most frequent clinical symptoms are related to intracranial hemorrhage and seizures. The respective annual risks of occurrence of an intracranial hemorrhage and seizures have been estimated at almost 3% and 1% [START_REF] Can | The natural history of cerebral arteriovenous malformations[END_REF]. The incidence of intracranial hemorrhagic events is higher in cases of deep AVM location, history of prior hemorrhage, exclusive deep venous drainage and of associated aneurysm [START_REF] Gross | Natural history of cerebral arteriovenous malformations: a metaanalysis[END_REF][START_REF] Stapf | Predictors of hemorrhage in patients with untreated brain arteriovenous malformation[END_REF]. Other symptoms, such as headaches, orbital congestive signs, and tinnitus secondary to venous hypertension, have also been reported [START_REF] Barreau | Intracranial arteriovenous malformations[END_REF].

Brain AVM diagnosis is also based on imaging; the computed tomography (CT) scan is the most useful for identifying an intracranial hemorrhage. Magnetic resonance imaging (MRI) is very effective in diagnosing this pathology because it detects old bleeding and, consequently, ruptured AVMs [START_REF] Barreau | Intracranial arteriovenous malformations[END_REF]. CT and MRI angiography are useful for the accurate definition of the nidus or suspected thrombophlebitis. Digital subtraction angiography is crucial not only for brain AVM diagnosis but also for grading, which can be determined using the surgical grading published by Spetzler et al. (Table 1) [START_REF] Spetzler | A proposed grading system for arteriovenous malformations[END_REF]. The morbidity and mortality of surgical resection are estimated by using this classification.

Therapeutic indications

There are three approaches for treating brain AVM patients: neurosurgery, embolization and intracranial SBRT. SRS, the intracranial SBRT approach mostly delivered in this pathology, is performed in locations not eligible for surgery, and, following a latency of several years, seems to reduce the risk of hemorrhage [START_REF] Maruyama | The risk of hemorrhage after radiosurgery for cerebral arteriovenous malformations[END_REF][START_REF] Seymour | Dose response and architecture in volume staged radiosurgery for large arteriovenous malformations: A multi-institutional study[END_REF].

European guidelines have been published regarding treatment of unruptured brain AVMs [START_REF] Cenzato | European consensus conference on unruptured brain AVMs treatment (Supported by EANS, ESMINT, EGKS, and SINCH)[END_REF]. Medical management has been suggested as a therapeutic option in patients with Radiotherapy in functional Neurology -Jacob unruptured brain AVMs according to two prospective studies, one of which is a randomized trial [START_REF] Salman | Outcome after conservative management or intervention for unruptured brain arteriovenous malformations[END_REF][START_REF] Mohr | Medical management with or without interventional therapy for unruptured brain arteriovenous malformations (ARUBA) : a multicentre, non-blinded, randomised trial[END_REF]. In this trial published by Mohr et al., patients with Spetzler-Martin grades II and III AVMs, the incidence of death or symptomatic stroke was significantly higher in the interventional therapy group, i.e. patients managed by radiotherapy, neurosurgery or embolization, than in the medically managed cohort. This trial was criticized by some authors mainly because of its short follow-up (mean time of 33.3 months) and its design, i.e. the heterogeneity of the treatments administered to the patients followed in the interventional arm [START_REF] Chan | Management of Unruptured AVMs: The Pendulum Swings[END_REF]. In patients with unruptured AVMs, the clinical benefits related to the obliteration induced by SRS appear after a latency period of three years or more, suggesting the need for a long-term follow-up to assess the treatment efficacy [START_REF] Tonetti | The benefit of radiosurgery for ARUBA-eligible arteriovenous malformations: a practical analysis over an appropriate follow-up period[END_REF][START_REF] Ding | Radiosurgery for Cerebral Arteriovenous Malformations in A Randomized Trial of Unruptured Brain Arteriovenous Malformations (ARUBA)-Eligible Patients: A Multicenter Study[END_REF][START_REF] Karlsson | The NASSAU (New ASSessment of cerebral Arteriovenous Malformations yet Unruptured) Analysis: Are the Results From the ARUBA Trial Also Applicable to Unruptured Arteriovenous Malformations Deemed Suitable for Gamma Knife Surgery?[END_REF]. Indeed, in these studies, following the completion of the latency period, the risk of local hemorrhage was lower in patients managed by SRS compared to groups with untreated AVMs. In a multicentric retrospective study, the annual postradiosurgery hemorrhage rate was calculated at 1.4% [START_REF] Ding | Radiosurgery for Unruptured Brain Arteriovenous Malformations: An International Multicenter Retrospective Cohort Study[END_REF].

Even though tissue modifications (hemorrhage, edema, stroke) can occur during the latency period, the long-term favorable outcomes of SRS and the risk of complications related to untreated AVMs argue for consideration of SRS. The treatment decision has to be taken in a multidisciplinary setting in which the situation of each patient is assessed individually.

Prognostic factors

The assessment of the AVM radiosensitivity can be difficult because of a prior embolization [START_REF] Flickinger | An analysis of the dose-response for arteriovenous malformation radiosurgery and other factors affecting obliteration[END_REF]. According to two meta-analyses, embolization performed before SRS provided significantly lower obliteration rates than SRS alone (at 3 years: 41% versus 59%, respectively; p < 0.00001) [START_REF] Xu | Stereotactic radiosurgery with and without embolization for intracranial arteriovenous malformations: a systematic review and meta-analysis[END_REF][START_REF] Russell | Stereotactic radiosurgery alone or combined with embolization for brain arteriovenous malformations: a systematic review and meta-analysis[END_REF]. Some AVM locations, e.g. the thalamus, are related to poorer outcomes and require multimodal management [START_REF] Madhugiri | Multimodal management of arteriovenous malformations of the basal ganglia and thalamus: factors affecting obliteration and outcome[END_REF]. Prognostic scores, mainly taking into account the AVM location (functional versus non-functional areas), volume, and history Radiotherapy in functional Neurology -Jacob of prior hemorrhage, have been published to assess the potential outcomes following SRS (Table 2) [START_REF] Pollock | A proposed radiosurgery-based grading system for arteriovenous malformations[END_REF][START_REF] Wegner | A modified radiosurgery-based arteriovenous malformation grading scale and its correlation with outcomes[END_REF][START_REF] Starke | A practical grading scale for predicting outcome after radiosurgery for arteriovenous malformations: analysis of 1012 treated patients[END_REF]. Starke et al. established the prognostic Virginia Radiosurgery AVM Scale score on 1012 patients with a mean follow-up time of 8 years [START_REF] Starke | A practical grading scale for predicting outcome after radiosurgery for arteriovenous malformations: analysis of 1012 treated patients[END_REF]. In this series, 80% of patients with a score of 0-1 points experienced a favorable outcome, defined as AVM obliteration and no post-SRS hemorrhage or permanent symptomatic complications induced by this treatment. Of the patients with scores of 2 points and 3-4 points, 70% and 45% experienced favorable outcomes, respectively [START_REF] Starke | A practical grading scale for predicting outcome after radiosurgery for arteriovenous malformations: analysis of 1012 treated patients[END_REF]. The yearly incidence of post-SRS hemorrhage was 1.14% in this study.

Stereotactic radiosurgery treatment planning

In this paragraph will be described the treatment planning procedure for brain AVM patients treated with GammaKnife ® SRS. In this indication, treatment planning requires specific imaging approaches, i.e. CT scan with thin slice images (1.2 mm, for example) and no contrast agent injection, T1-weighted three-dimensional spoiled gradient echo MRI with gadolinium infusion and square matrix (slice thickness: 1.2 mm, for example) and digital subtraction angiography with intravenous radiopaque iodine injection. For each technique, before radiological examinations, a fiducial box with markers is attached to the stereotactic head frame; these markers are then used to aid in matching the three-dimensional images. The target volume delineation requires multidisciplinary management involving teams from Neurosurgery, Neuroradiology and Radiation Oncology (Figure 1). Radiotherapy in functional Neurology -Jacob 1.6. Outcomes

GammaKnife ® stereotactic radiosurgery

A retrospective multicentric series involving more than 2236 patients treated with GammaKnife ® SRS described an overall obliteration rate of 64.7% with a mean follow-up of 7 years [START_REF] Starke | Stereotactic radiosurgery for cerebral arteriovenous malformations: evaluation of long-term outcomes in a multicenter cohort[END_REF]. In this series, the obliteration rate without hemorrhage or symptomatic neurological sequelae was 60.3% and the incidence of permanent morbidity due to local bleeding was 2.7% [START_REF] Starke | Stereotactic radiosurgery for cerebral arteriovenous malformations: evaluation of long-term outcomes in a multicenter cohort[END_REF]. The mean margin dose was 20.5 Gy (mean prescription isodose line: 54%). The factors linked with less favorable outcomes were mainly a lower prescribed dose and a higher number of isocenters, which could be explained by the larger target volume. A retrospective multicentric study presented favorable outcomes for SRS in grades I and II Spetzler-Martin AVM patients treated with a mean margin dose of 22.5 Gy (mean prescription isodose line: 54.5%) with obliteration rates of 72% and 87% at 5 and 10 years, respectively [START_REF] Ding | Stereotactic Radiosurgery for ARUBA (A Randomized Trial of Unruptured Brain Arteriovenous Malformations)-Eligible Spetzler-Martin Grade I and II Arteriovenous Malformations: A Multicenter Study[END_REF]. In this series, the multivariate analysis also highlighted that a lower margin dose was an independent predictive factor of unfavorable outcomes. In grade III Spetzler-Martin AVM patients, SRS outcomes appear encouraging with actuarial obliteration rates of 63% and 78% at 5 and 10 years, respectively [START_REF] Ding | Stereotactic radiosurgery for Spetzler-Martin Grade III arteriovenous malformations: an international multicenter study[END_REF]. The annual hemorrhage rate was estimated at 1.2% after the treatment completion. Outcomes improved in cases of higher margin doses and smaller nidus volume. The efficacy of GammaKnife ® SRS has been frequently confirmed for brain AVM patients several years after the treatment completion (Figure 2).

Given the limited efficacy of a single-session treatment in patients with large AVMs, several protocols of staged SRS, based mainly on dose and volume fractionations, have been proposed [START_REF] Patibandla | Stereotactic radiosurgery for Spetzler-Martin Grade IV and V arteriovenous malformations: an international multicenter study[END_REF][START_REF] Seymour | Volumestaged radiosurgery for large arteriovenous malformations : an evolving paradigm[END_REF][START_REF] Pollock | The rationale and technique of staged-volume arteriovenous malformation radiosurgery[END_REF][START_REF] Nagy | Staged-Volume Radiosurgery of Large Arteriovenous Malformations Improves Outcome by Reducing the Rate of Adverse Radiation Effects[END_REF][START_REF] Kano | How to improve obliteration rates during volume-staged stereotactic radiosurgery for large arteriovenous malformations[END_REF]. Ilyas et al. published a systematic review of patients treated with various SRS protocols for large AVMs, 44.6% of them having experienced prior hemorrhage [START_REF] Ilyas | Volume-staged versus dose-staged stereotactic radiosurgery outcomes for large brain arteriovenous malformations: a systematic review[END_REF]. In patients treated for brain AVMs larger than 10 cm 3 , volume-staged SRS provided higher obliteration rates than dose-staged SRS (40.3% versus 32.7%, respectively) but exposed Radiotherapy in functional Neurology -Jacob patients to greater risks of adverse events: post-therapeutic hemorrhage in 19.5% of patients versus 10.6%, mortality rate of 7.4% versus 4.6% [START_REF] Ilyas | Volume-staged versus dose-staged stereotactic radiosurgery outcomes for large brain arteriovenous malformations: a systematic review[END_REF]. In a retrospective multicentric study of 257 patients with large brain AVMs, a higher dose prescription level (higher than 17.5 Gy per volume stage) and a compact nidus architecture were associated with improved rates of near complete response in multivariate analysis [START_REF] Seymour | Dose response and architecture in volume staged radiosurgery for large arteriovenous malformations: A multi-institutional study[END_REF]. Improved rates of nidus obliteration and cures were also reported in patients treated with higher dose levels. Transient and permanent symptoms following the treatment completion were reported in 18.4% and 9.8% of the whole cohort, respectively. SRS also proved efficient in reducing the AVM volume in a preoperative setting [START_REF] Abla | A treatment paradigm for high-grade brain arteriovenous malformations: volume-staged radiosurgical downgrading followed by microsurgical resection[END_REF].

Linear accelerators

While many of the data published report the outcomes of patients treated using GammaKnife ® devices, SRS can also be administered using a linear accelerator. A French study of 179 patients treated with a linear accelerator delivering arctherapy or static beams at the mean peripheral dose of 17.7 Gy (prescription isodose: 70%) reported an obliteration rate of 82.7% observed at a mean time of 3 years following treatment [START_REF] Blanchard | Radiosurgery of cerebral arteriovenous malformations: a prescription algorithm[END_REF]. A peripheral dose higher than 18 Gy was associated with an improved outcome whereas a volume of the 12 Gy isodose larger than 11 cm 3 exposed the patients to a significant risk of radionecrosis. This was observed in 11.2% of the cohort at a mean time of 18 months following SRS. Prescription algorithms have also been developed to tailor the dose prescription in order to obliterate the nidus while reducing the risk of occurrence of adverse events [START_REF] Blanchard | Radiosurgery of cerebral arteriovenous malformations: a prescription algorithm[END_REF]. According to the authors, a reduction of the prescribed dose can be suggested if the risk of radionecrosis remains high (10%) despite treatment ballistics' optimization.

A Japanese retrospective study performed on 51 patients treated at the median marginal dose of 15 Gy (prescription isodose: 80%) delivered in one session reported an overall obliteration Radiotherapy in functional Neurology -Jacob rate of 72% with a median follow-up exceeding 15 years [START_REF] Matsuo | Linear accelerator-based radiosurgery alone for arteriovenous malformation : more than 12 years of observation[END_REF]. A dose-effect relationship was described in the univariate analysis but not in the multivariate one. Symptomatic radiation injury and cyst formation were observed in 17.6% and 9.8% of patients, respectively.

According to a French retrospective series describing the data of 90 patients, SRS, when administered using dynamic conformal arcs, provided an advantageous benefit/risk ratio with an obliteration rate of 69% in a median time of 30.4 months and a radionecrosis incidence estimated at 9% [START_REF] Clement-Colmou | Dynamic conformal arc radiosurgery for arteriovenous malformations: Outcome and influence of clinical and dosimetric data[END_REF]. The median prescribed dose was 22 Gy to the 80% isodose. In this series, the prescribed dose was predictive of the obliteration rate, with a cut-off defined at 22 Gy to the 80% isodose. A volume of healthy brain tissue exposed to a minimal dose of 12 Gy larger than 2 cm 3 showed a significant risk of adverse events.

A dosimetric study performed on 15 patients, 10 with AVM, assessed four SRS techniques: Novalis Tx ® dynamic conformal arctherapy and multileaf-collimation, GammaKnife ® and CyberKnife ® [START_REF] Gevaert | Dosimetric comparison of different treatment modalities for stereotactic radiosurgery of arteriovenous malformations and acoustic neuromas[END_REF]. In this series, the dose heterogeneity index was lower in the dynamic multileaf-collimation procedure compared to the other approaches: 0.18 versus 0.22, 0.3 and 0.84 for CyberKnife ® , dynamic conformal arctherapy and GammaKnife ® , respectively. These two last modalities, however, presented the highest Paddick conformity index. The longest treatment time measured was for GammaKnife ® SRS.

Fractionated intracranial stereotactic body radiotherapy

Fractionated intracranial SBRT is less used in patients with AVM, mainly due to the low obliteration and morbidity rates initially reported [START_REF] Karlsson | Longterm results after fractionated radiation therapy for large brain arteriovenous malformations[END_REF][START_REF] Veznedaroglu | Fractionated stereotactic radiotherapy for the treatment of large arteriovenous malformations with or without previous partial embolization[END_REF]. The α/β ratio for AVMs was globally estimated at 3.54 Gy, with between 4.6 and 6.4 Gy for AVMs smaller than 3 cm [START_REF] Kocher | Alpha/beta ratio for arteriovenous malformations estimated from obliteration rates after fractionated and single-dose irradiation[END_REF].

Two studies led on patients treated with intracranial SBRT between 25 and 35 Gy in four fractions reported obliteration rates of above 60% at 5 years with a lower incidence of Radiotherapy in functional Neurology -Jacob complications compared with single-session approaches [START_REF] Aoyama | Treatment outcome of single or hypofractionated single-isocentric stereotactic irradiation (STI) using a linear accelerator for intracranial arteriovenous malformation[END_REF][START_REF] Chang | Stereotactic irradiation for intracranial arteriovenous malformation using stereotactic radiosurgery or hypofractionated stereotactic radiotherapy[END_REF]. Nine patients were treated with intracranial SBRT using volumetric modulated arc therapy delivering 25 Gy in 5 fractions to large AVMs with interesting outcomes in terms of target volume coverage and organ at risk sparing [START_REF] Subramanian | Volumetric modulated arc-based hypofractionated stereotactic radiotherapy for the treatment of selected intracranial arteriovenous malformations: dosimetric report and early clinical experience[END_REF]. In this series, the clinical assessment of toxicities has to be considered with care: the median follow-up was short (9 months), and one case of intracranial bleeding was reported. Nevertheless, more recent data described an obliteration rate of 50% using fractionated intracranial SBRT delivering equivalent 2 Gy fraction doses higher than 70

Gy [START_REF] Knippen | Predictors for occlusion of cerebral AVMs following radiation therapy : Radiation dose and prior embolization, but not Spetzler-Martin grade[END_REF].

Repeat stereotactic radiosurgery

SRS can be repeated when no complete obliteration is reported after the initial treatment. This is likely to occur in patients with AVMs larger than 10 cm 3 and/or with high Spetzler-Martin grade [START_REF] Latorzeff | Radiosurgery for brain arteriovenous malformations[END_REF]. In a cohort of 105 patients treated with repeat GammaKnife ® SRS, the median dose prescribed was 18 Gy for the two treatment sessions [START_REF] Kano | Stereotactic radiosurgery for arteriovenous malformations, Part 3: outcome predictors and risks after repeat radiosurgery[END_REF]. The median time to complete obliteration following the second session was 39 months with actuarial rates of 77% and 80% at 5 and 10 years, respectively. A target volume reduction rate above 50% following the first procedure was associated with an improved outcome. The new hemorrhage rates were 4 % per year in the first two years and 1.8% per year between two and ten years after repeat SRS, with a significantly higher risk in patients with larger AVMs. Symptomatic adverse events were reported in 9.5% of patients after repeat SRS. Radiotherapy in functional Neurology -Jacob 1.7. Morbidity

GammaKnife ® stereotactic radiosurgery

After SRS in patients with AVM, tissue changes such as partial or complete thrombosis of the AVM vessels, thickening of the vessel walls, e.g. hyalinization, but also necrosis of the adjacent brain tissue impact the treatment efficacy and morbidity [START_REF] Chang | Stereotactic radiosurgery of arteriovenous malformations: pathologic changes in resected tissue[END_REF]. The histopathological and radiological modifications, e.g. edema, reported after SRS were correlated with the time elapsed since the treatment completion and the dose delivered to the healthy brain tissues [START_REF] Schneider | Histopathology of arteriovenous malformations after gamma knife radiosurgery[END_REF][START_REF] Levegrün | Radiation-induced changes of brain tissue after radiosurgery in patients with arteriovenous malformations: correlation with dose distribution parameters[END_REF].

The main secondary effects reported following GammaKnife ® SRS for AVMs are: hemorrhage, radionecrosis and, in the long term, occurrence of cysts, chronic encapsulated hematoma and secondary cancers [START_REF] Hasegawa | Gamma Knife Radiosurgery For Brain Vascular Malformations: Current Evidence And Future Tasks[END_REF]. Among 581 patients treated with GammaKnife ® and followed during a median time of 10.1 years, Hasegawa et al. reported cumulative incidence rates of cyst formation and encapsulated hematomas estimated at 2.8% at 10 years and 9.7% at 20 years, respectively [START_REF] Hasegawa | A Comprehensive Study of Symptomatic Late Radiation-Induced Complications After Radiosurgery for Brain Arteriovenous Malformations: Incidence, Risk Factors, and Clinical Outcomes[END_REF].

Linear accelerators

In a series of 129 patients treated with a linear accelerator by SRS (88.4% of patients, median dose: 18 Gy) or hypofractionated intracranial SBRT (11.6% of the cohort, median dose: 35 Gy, range: from 30 to 55 Gy delivered in 5 to 11 fractions), with a mean follow-up period of 43.4 months, the incidence of neurological symptoms, such as deficits, seizures, headaches, was 13.2% [START_REF] Boström | Treatment of Cerebral Arteriovenous Malformations with Radiosurgery or Hypofractionated Stereotactic Radiotherapy in functional Neurology -Jacob Radiotherapy in a Consecutive Pooled Linear Accelerator Series[END_REF]. Radionecrosis and edema were reported in 4% and 34% of patients, respectively. New bleeding was observed in 2 patients (1.7%) treated with hypofractionated intracranial SBRT for hemorrhagic AVMs. In a study of 50 patients managed by SRS using linear accelerators with the dose prescription adapted to the target volume (peripheral dose Radiotherapy in functional Neurology -Jacob between 15.0 and 27.5 Gy; prescription isodose: 90%), 34% of the cohort presented new neurological symptoms, among which 35.3 % were related to radionecrosis and/or severe edema [START_REF] Skjøth-Rasmussen | Complications following linear accelerator based stereotactic radiation for cerebral arteriovenous malformations[END_REF]. In a prospective series of 67 patients treated with a linear accelerator at the median marginal dose of 18 Gy delivered in one session, the incidence of neurological complications, including hemorrhage, was 15% [START_REF] Zacest | LINAC radiosurgery for cerebral arteriovenous malformations: a single centre prospective analysis and review of the literature[END_REF].

Predictive factors

Several studies reported the 12 Gy isodose volume and the target volume as predictive of the adverse event occurrence in patients treated with SRS using GammaKnife ® or a linear accelerator [START_REF] Hayhurst | Pretreatment predictors of adverse radiation effects after radiosurgery for arteriovenous malformation[END_REF][START_REF] Herbert | Factors predictive of symptomatic radiation injury after linear accelerator-based stereotactic radiosurgery for intracerebral arteriovenous malformations[END_REF]. According to Herbert et al., in a cohort of 39 patients treated at the median dose of 20 Gy administered in a single session using a linear accelerator, the actuarial symptomatic radiation injury rate was 23.0% at 4 years [START_REF] Herbert | Factors predictive of symptomatic radiation injury after linear accelerator-based stereotactic radiosurgery for intracerebral arteriovenous malformations[END_REF]. In this series, the actuarial incidences of symptomatic radiation injury were 34.1% and 53.2% when the 12 Gy isodose volumes were larger than 20 and 28 cm 3 , respectively. The 12 Gy isodose volume has frequently been associated with the risk of toxicities related to SRS and the target volume location could be another predictive factor [START_REF] Flickinger | Complications from arteriovenous malformation radiosurgery: multivariate analysis and risk modeling[END_REF][START_REF] Friedman | Analysis of factors predictive of success or complications in arteriovenous malformation radiosurgery[END_REF].

Prospects

The question of fractionated schemes is under debate in these patients presenting a long life expectancy. While a high dose per session being is likely to produce the nidus obliteration, it also induces neurologic disabilities. Ongoing feasibility studies of intensity-modulated radiotherapy planned in one to four fractions have produced interesting dosimetric results [START_REF] Sonier | Intensitymodulated stereotactic radiosurgery for arteriovenous malformations: guidance for treatment planning[END_REF][START_REF] Unkelbach | Spatiotemporal Fractionation Schemes for Irradiating Large Cerberal Arteriovenous Malformations[END_REF]. In young patients, promising outcomes of intracranial SBRT delivering charged particle beams, e.g. protontherapy, were reported with obliteration rates above 60% after several years Radiotherapy in functional Neurology -Jacob of follow-up [START_REF] Blomquist | Positive correlation between occlusion rate and nidus size of proton beam treated brain arteriovenous malformations (AVMs)[END_REF][START_REF] Hattangadi-Gluth | Single-fraction proton beam stereotactic radiosurgery for cerebral arteriovenous malformations[END_REF][START_REF] Vernimmen | Intracranial Stereotactic Radiation Therapy With Charged Particle Beams: An Opportunity to Regain the Momentum[END_REF]. Improvements in the imaging approaches based on angiography could help to enhance the accuracy of the target volume delineation [START_REF] Kang | Planning evaluation of C-arm cone beam CT angiography for target delineation in stereotactic radiation surgery of brain arteriovenous malformations[END_REF][START_REF] Steenbeke | Analysis of the targeting uncertainty of a stereotactic frameless radiosurgery technique for arteriovenous malformation[END_REF].

In brain AVM patient management, in carefully selected cases, intracranial SBRT plays an important role and provides favorable outcomes. Treatment timing is under debate in patients with unruptured AVMs. Several radiation devices can be used to perform SRS. The obliteration rates following single-session SRS can be higher than 75% at 10 years in Spetzler-Martin grades I to III AVMs. Because of the poorer outcomes observed in Grades IV and V AVMs, fractionated schemes, e.g. volume-staged or dose-staged SRS, are being developed. Linear accelerators are a relevant alternative technique with obliteration rates likely to reach 60% at 5 years with acceptable toxicities. Prognostic scores have been developed to predict the SRS outcomes. Symptomatic and permanent radiation-induced changes have been described in 8 to 11% and in 1 to 4% of patients treated with SRS, respectively. Although adverse events generally remain acceptable and transient, the risks of hemorrhage and radionecrosis have to be considered. Radiotherapy in functional Neurology -Jacob

Trigeminal neuralgia

Physiopathology and Epidemiology

Trigeminal nerve neuralgia is the most common physiopathology, and it predominantly affects the maxillary and mandibular branches, with a prevalence estimated between 0.03% and 0.3%, and involves more frequently women over 40 years of age [START_REF] Toledo | Prevalence of trigeminal neuralgia: A systematic review[END_REF]. Idiopathic or essential TN frequently occurs due to a vascular compression at the dorsal root entry zone (DREZ) [START_REF] Barker Fg 2 Nd | The long-term outcome of microvascular decompression for trigeminal neuralgia[END_REF].

Therapeutic indications

In terms of patients' management, intracranial SBRT (mainly SRS) is considered a secondline therapeutic option, along with surgical microvascular decompression, and percutaneous approaches based on physical or chemical processes, e.g. percutaneous glycerol rhizotomy, following the failure of, and/or significant adverse events related to, antidepressant or antiepileptic medications, e.g. carbamazepine [START_REF] Wiffen | Carbamazepine for chronic neuropathic pain and fibromyalgia in adults[END_REF][START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF].

Stereotactic radiosurgery treatment planning

In this paragraph will be described the treatment planning procedure for patients treated with

Outcomes

GammaKnife ® stereotactic radiosurgery

The International Stereotactic Radiosurgery Society has published specific guidelines based on a systematic review of 65 studies, mostly retrospective, including 6461 patients; 88% of whom were treated with GammaKnife ® (Figure 2), 8% with a linear accelerator and 4% with CyberKnife ® , respectively [START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF]. The mean maximal doses ranged from 71 to 90 Gy (prescription isodose: 100%), 70 to 90 Gy and 64.3 to 80.5 Gy for patients treated with GammaKnife ® , linear accelerators and CyberKnife ® , respectively (prescription isodoses: 100%, 80% and 90%, respectively). No statistically significant difference was reported between the three radiation techniques in terms of freedom from pain with or without medication, the mean rates ranging from 79.3% (CyberKnife ® ) to 87.3% (linear accelerators); 84.8% (GammaKnife ® ). The freedom from pain without medication mean rates ranged from 49.3% (linear accelerators) to 56.3% (CyberKnife ® ); 53% (GammaKnife ® ). No statistically significant differences were observed between the three radiation techniques. According to most authors of the reviewed studies, the maximum time to pain relief could be considered 180 days after SRS [START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF]. The sole prospective study in this review involved 100 patients treated with GammaKnife ® at the median dose of 85 Gy (prescription isodose: 100%) and followed for at least 12 months [START_REF] Régis | Prospective controlled trial of gamma knife surgery for essential trigeminal neuralgia[END_REF]. The freedom from pain with medication rate was 97% with a median time to pain relief estimated at 10 days (range: 1-180 days). Hypoesthesia was observed in 5% of patients, as well as paresthesia (incidence: 5%). Recurrence rate was 34%, the event occurring in a mean time of 6 months. Radiotherapy in functional Neurology -Jacob

Linear accelerators

Smith et al. published a retrospective study on 169 patients treated for essential or secondary

TN with SRS administered by a linear accelerator with various dose prescriptions, i.e. 70 and 90 Gy delivered to the normalization point [START_REF] Smith | Dedicated linear accelerator radiosurgery for trigeminal neuralgia: a single-center experience in 179 patients with varied dose prescriptions and treatment plans[END_REF]. The brainstem dose exposure varied from the 30% to the 50% isodose line depending on the treatment periods. Idiopathic TN was observed in 130 patients. In this subgroup, 94% of patients presented a significant pain relief at 12 months and 82.6% at 36 months, respectively. In patients with excellent results according to specific grading scales, no pain nor analgesic medications were reported for 87% of patients at 12 months and 60% at 36 months, respectively. Patients treated at the dose level of 90 Gy with a 50% isodose line to the brainstem, i.e. 35% of the whole cohort, presented a statistically significant higher rate of pain relief at 12 months than those exposed to a dose prescription of 70 Gy with a 30% isodose line to the brainstem; (16.5% of the cohort): 96.3% versus 75% (p=0.002) [START_REF] Smith | Dedicated linear accelerator radiosurgery for trigeminal neuralgia: a single-center experience in 179 patients with varied dose prescriptions and treatment plans[END_REF]. A similar trend was observed with the patients treated at 90 Gy with a 30% isodose line to the brainstem, i.e. 48.5% of the cohort, and whose pain relief rate at 12 months was 95% (p=0.052). Similarly, the median time to pain relief was significantly shorter in patients treated with higher dose levels: 0 months (90 Gy with a 50% isodose line to the brainstem) versus 1.5 months (90 Gy with a 30% isodose line to the brainstem) and 3.0 months (70 Gy with a 30% isodose line to the brainstem) (p<0.001). Pain recurrence was observed in 23% of patients who presented significant pain relief in a median time of 12.6 months. In terms of toxicities, 49.7% of the whole cohort presented paresthesia and/or facial numbness. According to the specific scale used by the authors, Grade IV or V facial numbness rates were higher in patients treated with increased dose levels: 17.8% and 10.6% in cases of dose prescriptions of 90 Gy with respective isodose lines of 50% and 30% to the brainstem, 3.6% when the dose was 70 Gy to the normalization point. Eye dryness and eye irritation were reported in 8.3% and 11.2% of patients, respectively. Radiotherapy in functional Neurology -Jacob Rashid et al. have published the data of 55 patients treated for TN with SRS delivered by a linear accelerator in two centers [START_REF] Rashid | LINAC stereotactic radiosurgery for trigeminal neuralgia -retrospective two-institutional examination of treatment outcomes[END_REF]. The prescription dose was 90 Gy to the 100% isodose point in most patients. The dose constraint to the brainstem was the 20% isodose line set at its anterior part. With a mean follow-up of 30.7 months, 69% of patients were pain free with or without medication (29% without medication). Target location at the DREZ was associated with statistically significant improved outcomes. In this cohort, when 40% of the trigeminal nerve volume was exposed to a dose higher than 70 Gy, the probabilities of pain relief two months after the completion of SRS and of hypoesthesia increased.

Another important study has reported the outcomes of using CyberKnife ® SRS administered to 527 patients; 343 (65%) were followed up with a minimum time of 36 months [START_REF] Romanelli | Cyberknife Radiosurgery for Trigeminal Neuralgia[END_REF]. Dose prescription was 60-65 Gy to the 80-90% isodose line. A brainstem volume equal to or less than 1 cm 3 was exposed at a dose of 10 Gy, with a maximum point dose (0.035 cm 3 ) of 30% of the prescription dose. The pain relief rates were 87%, 82% and 76% at 12, 24 and 36 months, respectively. Facial numbness occurred in 20% of patients and led to clinical disturbance in 6% of the cohort at 36 months.

Fractionated intracranial stereotactic body radiotherapy

A retrospective series of 45 patients compared SRS to hypofractionated intracranial SBRT delivered using a linear accelerator [START_REF] Fraioli | Preliminary results of 45 patients with trigeminal neuralgia treated with radiosurgery compared to hypofractionated stereotactic radiotherapy, using a dedicated linear accelerator[END_REF]. The prescribed doses to the isocenter were 40 Gy in a single session and 72 Gy in six fractions, with an α/β ratio at 2.5 Gy. With a median followup of 3.9 years, a higher pain recurrence rate was observed in patients treated with hypofractionated intracranial SBRT (27.3% versus 8.7% in the cohort managed by SRS).

Recurrences were reported in mean follow-up periods of 3 years and 4 years after the completion of hypofractionated intracranial SBRT and SRS, respectively. Facial numbness Radiotherapy in functional Neurology -Jacob was observed in 8.7% of patients treated in a single session whereas no adverse events were reported following hypofractionated intracranial SBRT.

Repeat stereotactic radiosurgery

According to a meta-analysis published by Tuleasca et al., the mean recurrence rates reported for SRS performed with GammaKnife ® , linear accelerator and CyberKnife ® were 24.6%, 32.2% and 25.8%, respectively [START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF]. No statistically significant difference was observed. The mean time to recurrence ranged from 6 to 48 months depending on the radiation technique.

Repeat SRS, especially using GammaKnife ® , is a relevant therapeutic option in case of pain recurrence. Two retrospective studies carried out on more than 100 patients each highlighted its efficacy with pain relief rates of above 50% at three years [START_REF] Park | Repeat gamma knife radiosurgery for trigeminal neuralgia[END_REF][START_REF] Helis | Repeat Radiosurgery for Trigeminal Neuralgia[END_REF]. In both studies, the second SRS session was delivered after median times ranging between 15.7 and 26.1 months from the first treatment. The median prescribed doses were 70 and 80 Gy respectively [START_REF] Park | Repeat gamma knife radiosurgery for trigeminal neuralgia[END_REF][START_REF] Helis | Repeat Radiosurgery for Trigeminal Neuralgia[END_REF]. The most frequently reported adverse events after the second SRS session were: facial sensory dysfunction (21.0%) and mild corneal dryness (6.58%) [START_REF] Park | Repeat gamma knife radiosurgery for trigeminal neuralgia[END_REF][START_REF] Helis | Repeat Radiosurgery for Trigeminal Neuralgia[END_REF]. The clinical response, the facial numbness which occurred following the first treatment session and the facial sensory dysfunction which was reported after retreatment could be predictive of the repeat SRS outcomes. According to Park et al., a cumulative dose higher than or equal to 44 Gy to the brainstem edge increased the risk of sensory loss following retreatment [START_REF] Park | Repeat gamma knife radiosurgery for trigeminal neuralgia[END_REF].

Morbidity

The main complication post-SRS indicated in patients with TN was local hypoesthesia with an incidence estimated at 21.7%, 27.6% and 29% of patients treated with GammaKnife ® , Radiotherapy in functional Neurology -Jacob linear accelerator and CyberKnife ® , respectively [START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF]. Other secondary events were: deafferentation pain, dysesthesia, keratitis or corneal anesthesia, and dry eye [START_REF] Smith | Dedicated linear accelerator radiosurgery for trigeminal neuralgia: a single-center experience in 179 patients with varied dose prescriptions and treatment plans[END_REF][START_REF] Marshall | Predictive variables for the successful treatment of trigeminal neuralgia with gamma knife radiosurgery[END_REF][START_REF] Aykol | Results of radiosurgery for trigeminal neuralgia: Ankara experience[END_REF].

Predictive factor

Brainstem exposure is predictive of the toxicities related to SRS in patients with TN [START_REF] Matsuda | Gamma knife radiosurgery for trigeminal neuralgia: the dry-eye complication[END_REF][START_REF] Goss | Linear accelerator radiosurgery using 90 gray for essential trigeminal neuralgia : results and dose volume histogram analysis[END_REF][START_REF] Régis | Long-term safety and efficacy of Gamma Knife surgery in classical trigeminal neuralgia: a 497-patient historical cohort study[END_REF].

A maximum brainstem dose of 25 Gy and an isocenter position closer than 4.5 mm to the exit of the trigeminal nerve is predictive of the dry eye complication; this has been reported in a series of 41 patients treated with GammaKnife ® SRS at the dose of 80 Gy [START_REF] Matsuda | Gamma knife radiosurgery for trigeminal neuralgia: the dry-eye complication[END_REF]. In a retrospective study conducted on 25 patients treated at the dose of 90 Gy using a linear accelerator, a statistically significant correlation was observed between the mean brainstem volume included in the 50% isodose line and the occurrence of facial numbness [START_REF] Goss | Linear accelerator radiosurgery using 90 gray for essential trigeminal neuralgia : results and dose volume histogram analysis[END_REF]. In a large retrospective series, GammaKnife ® SRS delivered at the median maximum dose of 85 Gy with the brainstem volume of 10 mm 3 exposed to a maximal dose of 15 Gy yielded interesting outcomes: a 45.3% probability for patients to still be pain free from medication at 10 years with a hypoesthesia rate of 21% at 7 years [START_REF] Régis | Long-term safety and efficacy of Gamma Knife surgery in classical trigeminal neuralgia: a 497-patient historical cohort study[END_REF].

Technical issues

One important issue is that of target volume delineation, because the choice made must provide as much pain relief as possible and reduce the dose delivered to the healthy structures, namely the brainstem and ipsilateral temporal lobe [START_REF] Dupic | Functional stereotactic radiosurgery: Indications and perspectives[END_REF]. The DREZ, which is close to the brainstem, seems to be more radiosensitive than the retrogasserian portion of the trigeminal nerve [START_REF] Tuleasca | Stereotactic radiosurgery for trigeminal neuralgia: a systematic review[END_REF][START_REF] Dupic | Functional stereotactic radiosurgery: Indications and perspectives[END_REF]. Two retrospective studies have assessed the SRS efficacy and toxicities as they relate to target volume location [START_REF] Park | The retrogasserian zone versus dorsal root entry zone: comparison of two targeting techniques of gamma knife radiosurgery for trigeminal neuralgia[END_REF][START_REF] Xu | Impact of target location on the response of trigeminal neuralgia to stereotactic radiosurgery[END_REF]. Park et al. have reported the data of 39 patients treated with GammaKnife ® SRS at the dose of 80-90 Gy delivered to the 100% isodose line in Radiotherapy in functional Neurology -Jacob a single session using a 4-mm isocenter [START_REF] Park | The retrogasserian zone versus dorsal root entry zone: comparison of two targeting techniques of gamma knife radiosurgery for trigeminal neuralgia[END_REF]. Whatever the isocenter position was, i.e. the retrogasserian zone or the DREZ, the 20% isodose line had to be located at the anterior edge of the brainstem. With a mean follow-up of 26 months, pain was relieved in 93.8% and 87% of patients treated with anterior and posterior target volumes, respectively, with no statistically significant difference. Response time was significantly shorter in patients treated at the retrogasserian zone (4 weeks following SRS versus 6.4 weeks, p=0.044). Pain recurrence and impairing facial numbness rates were higher in patients treated at the DREZ (30.4% versus 6.2%; 13.1% versus 0% in case of anterior target volume, respectively), with no statistically significant differences.

In a cohort of 99 patients treated with GammaKnife ® SRS at the dose of 80 Gy in one session, Xu et al. respectively treated 36 of them (36.4%) at the DREZ, the 50% isodose line having overlapped the brainstem, and 63 (63.6%) at a more distal target, setting the 20% isodose line close to the brainstem [START_REF] Xu | Impact of target location on the response of trigeminal neuralgia to stereotactic radiosurgery[END_REF]. The median follow-up period was 33 months. The pain relief with or without medication was observed in 81% of patients at 2 years and 69% at 4 years, respectively. Targeting the more proximal part of the trigeminal nerve was statistically associated with a longer durability of pain relief (p=0.018), but this location also exposed the treated patients to a higher risk of facial numbness classified using the Barrow Neurological Institute score (score II or III in 53% of patients versus 25% in the distal group, p=0.015) [START_REF] Xu | Impact of target location on the response of trigeminal neuralgia to stereotactic radiosurgery[END_REF][START_REF] Rogers | Gamma knife radiosurgery for trigeminal neuralgia: the initial experience of The Barrow Neurological Institute[END_REF].

A randomized study by Flickinger et al. on 87 patients treated with GammaKnife ® SRS at the maximum dose of 75 Gy in one session used one or two isocenters to assess the impact that the treated length of the trigeminal nerve had on treatment outcomes [START_REF] Flickinger | Does increased nerve length within the treatment volume improve trigeminal neuralgia radiosurgery? A prospective double-blind, randomized study[END_REF]. For all patients one isocenter was set close to the DREZ. The median follow-up was 26 months. The actuarial rate of pain relief with or without medication was 67.7% with no statistically significant difference between the two groups. Patients treated with two isocenters experienced fewer relapses, with Radiotherapy in functional Neurology -Jacob no statistically significant difference (29.5% versus 37.8% in patients treated with one isocenter, p=0.45). As regards the adverse events, 32.6% of patients treated with two isocenters presented facial numbness, and mild or severe paresthesia compared with 15.9% in the group treated with one isocenter; no statistically significant difference was reported. The multivariate analysis highlighted that the length of the irradiated trigeminal nerve was significantly associated with the risk of facial numbness (p=0.018) and not with the occurrence of pain relief. According to this same analysis, improved outcomes were more frequently observed in younger patients (p=0.025) and/or when patients underwent fewer procedures (p=0.039).

In a prospective series of 50 patients treated with GammaKnife ® at dose levels ranging from 80 to 86 Gy to the 100% isodose line in one session and using an isocenter located at the DREZ, the multivariate analysis highlighted a statistically significant correlation between a higher delivered dose (86 Gy) and the quality of life improvement assessed using a specific validated questionnaire [START_REF] Kotecha | Stereotactic Radiosurgery for Trigeminal Neuralgia Improves Patient-Reported Quality of Life and Reduces Depression[END_REF]. A significantly faster quality of life improvement was also reported in patients treated at 86 Gy. A dose-effect relationship was also observed in a retrospective multicentric series of 870 patients treated with GammaKnife ® SRS at the median dose of 86 Gy to the 100% isodose line with an isocenter located at the DREZ [START_REF] Kotecha | Trigeminal Neuralgia Treated With Stereotactic Radiosurgery: The Effect of Dose Escalation on Pain Control and Treatment Outcomes[END_REF]. In this study, a dose level higher than 90 Gy was significantly correlated with a higher probability of pain relief but also with the risk of developing facial numbness requiring specific medication [START_REF] Kotecha | Trigeminal Neuralgia Treated With Stereotactic Radiosurgery: The Effect of Dose Escalation on Pain Control and Treatment Outcomes[END_REF].

A nomogram taking into account pre-(e.g. age, prior therapeutic procedures) and posttreatment (pain relief at 6 months, posttreatment facial numbness or paresthesia) features was developed by Lucas et al. to estimate the durability of pain relief following GammaKnife ® SRS [START_REF] Lucas | Predictive normogram for the durability of pain relief from gamma knife radiation surgery in the treatment of trigeminal neuralgia[END_REF]. In a retrospective study performed on 200 patients treated with SRS at a maximal dose of 80 Gy, Flickinger et al. reported no statistically significant effect of Radiotherapy in functional Neurology -Jacob medications (anticonvulsants, neuroleptics) on pain relief after treatment [START_REF] Flickinger | Do carbamazepine, gabapentin, or other anticonvulsants exert sufficient radioprotective effects to alter responses from trigeminal neuralgia radiosurgery?[END_REF]. However, the uni-and multivariate analyses showed a significantly reduced risk of post-SRS trigeminal neuropathy (new facial numbness or paresthesia) in patients treated with gabapentin.

Prospects

With sector beam intensity modulation intracranial SBRT performed with GammaKnife ® , a procedure under evaluation, the maximum dose to the brainstem could be reduced but there would also be an increase in the integral dose to the trigeminal nerve [START_REF] Ma | Clinical realization of sector beam intensity modulation for Gamma Knife radiosurgery: a pilot treatment planning study[END_REF].

To sum up, intracranial SBRT, mainly SRS, is one therapeutic option for patients suffering from TN poorly controlled by pharmacological approaches. The clinical outcomes are quite satisfying with pain relief reported in more than 60% of patients over several years following the completion of treatment delivered using several different devices. The location of the target volume is a key point; the choice of the DREZ yields improved outcomes for treated patients but also exposes them to a higher risk of adverse events. The most frequent adverse event related to SRS is facial hypoesthesia with an incidence which has been estimated between 20 and 30% of the treated patients. Radiotherapy in functional Neurology -Jacob 3. Thalamotomy performed using GammaKnife ® stereotactic radiosurgery

Therapeutic indications

GammaKnife ® SRS can be proposed in patients presenting tremors related to Parkinson's disease and essential tremor. SRS is a therapeutic option in patients presenting tremors which are poorly controlled by medication and those not responsive to, or eligible for, surgical deep brain stimulation or radiofrequency thalamotomy [START_REF] Campbell | Gamma knife stereotactic radiosurgical thalamotomy for intractable tremor: a systematic review of the literature[END_REF][START_REF] Niranjan | Leksell Radiosurgery for Movement Disorders[END_REF].

The International Stereotactic Radiosurgery Society recognizes thalamotomy performed using GammaKnife ® as a relevant therapeutic option, even in patients eligible for surgery, due to its clinical efficacy and satisfactory tolerance profile [START_REF] Ne | Stereotactic radiosurgery for tremor: systematic review[END_REF]. Nevertheless, prospective randomized multicentric studies are needed, due to the limitations of the available literature data: singleinstitution series, heterogeneity in terms of evaluation scales and follow-up [START_REF] Ne | Stereotactic radiosurgery for tremor: systematic review[END_REF]. The target is the thalamic ventrointermediate nucleus contralateral to the affected limb, usually defined by experienced neurosurgical teams using stereotactic procedures with the Radiotherapy in functional Neurology -Jacob anterior commissure/posterior commissure line as reference (Figure 4). The dose levels administered by single-session SRS ranged from 100 to 180 Gy (most frequently 130 Gy) according to a literature review [START_REF] Campbell | Gamma knife stereotactic radiosurgical thalamotomy for intractable tremor: a systematic review of the literature[END_REF].

Stereotactic radiosurgery treatment planning

Outcomes

A prospective multicentric trial with 72 patients treated for tremors secondary to Parkinson's disease or essential tremor at the dose of 130 Gy delivered in one session to the isocenter showed a significant improvement of the symptoms 24 months after treatment completion [START_REF] Ohye | Gamma knife thalamotomy for Parkinson disease and essential tremor: a prospective multicentric study[END_REF]. The majority of patients (almost 80%) continued with specific medications at the same doses after SRS. The clinical benefits were delayed and observed with a median time of almost 4 months after SRS completion in patients with essential tremor, the maximum therapeutic effect mostly occurring at 9 months posttreatment [START_REF] Dallapiazza | Outcomes from stereotactic surgery for essential tremor[END_REF]. A retrospective series of 172 patients treated for essential tremor reported a statistically significant improvement in the writing and drawing tasks following unilateral or bilateral thalamotomies performed using GammaKnife ® with a mean postoperative follow-up of 44 months [START_REF] Young | Gamma Knife thalamotomy for treatment of essential tremor: long-term results[END_REF]. Two prospective series highlighted a clinical improvement in the activities of daily living [START_REF] Lim | Gamma knife thalamotomy for disabling tremor: a blinded evaluation[END_REF][START_REF] Witjas | A prospective single-blind study of Gamma Knife thalamotomy for tremor[END_REF]. In selected patients, staged bilateral SRS thalamotomy can be proposed with a satisfactory benefit/risk ratio [101].

Morbidity

The main complications reported in patients treated with this approach were contralateral sensitive or motor impairments (e.g. hypoesthesia, hemiparesis contralateral to the treated side, dysarthria and dysphagia) which were mostly transient [START_REF] Young | Gamma Knife thalamotomy for treatment of essential tremor: long-term results[END_REF][START_REF] Kooshkabadi | Gamma Knife thalamotomy for tremor in the magnetic resonance imaging era[END_REF]. Radiotherapy in functional Neurology -Jacob Radiosurgical thalamotomy can be proposed to patients presenting different kinds of tremors (essential tremor, tremor secondary to Parkinson's disease) who are not eligible for surgical procedures such as deep brain stimulation of the ventrointermediate nucleus. Patient quality of life seems to be improved while the side effects are in the main mild and transient.

Conclusion

The efficacy of intracranial SBRT has been proved in the management of benign and malignant tumors, and it can be proposed to patients treated for non-tumoral refractory neurological pathologies. The incidence of adverse events is quite low making this approach viable for patients ineligible for an invasive procedure. The quality of life was preserved or improved following SRS administered in these indications according to prospective studies [START_REF] Lim | Gamma knife thalamotomy for disabling tremor: a blinded evaluation[END_REF][START_REF] Witjas | A prospective single-blind study of Gamma Knife thalamotomy for tremor[END_REF][START_REF] Thakkar | Quality of life after gamma knife radiosurgery for benign lesions: a prospective study[END_REF]. If there were more data regarding GammaKnife ® SRS, proving its efficacy, linear accelerators could also be used. Randomized trials would help to more accurately define the place of intracranial SBRT among the different therapies but are difficult to carry out due to the low prevalence of each of these pathologies and the lack of standardized functional data. More prospective studies involving cohorts treated for various intracranial benign indications are needed as well as compilations of patient and caregiver-reported outcomes over the long term following intracranial SBRT.

In these non-tumoral refractory neurological pathologies, intracranial SBRT, based on approaches delivering high doses near healthy brain tissues, has to be managed by experienced teams in Radiation Oncology, Neurosurgery, Neuroradiology and Medical Physics. The potential long-term adverse events attributable to intracranial SBRT (radionecrosis, hemorrhage, neurological disabilities) make multidisciplinary decision-making and follow-up essential.

Figure 1A

  GammaKnife ® SRS for TN. In this pathology, SRS treatment planning is based on CT scan with thin slice images (1.2 mm, for instance) and no contrast agent injection. Brain MRI sequences recommended are: T1-weighted three-dimensional spoiled gradient echo images with gadolinium infusion and square matrix (slice thickness: 1.2 mm, for instance), T2weighted three-dimensional Fast Imaging Employing Steady State Acquisition or Constructive Interference Steady State for more accurate the trigeminal nerve delineation. Before the acquisition of the CT scan and MRI, a fiducial box is attached to the stereotactic Radiotherapy in functional Neurology -Jacob head frame to aid in the three-dimensional image registration with markers. The target volume delineation is performed by teams from Neurosurgery, Neuroradiology and Radiation Oncology (Figure 3).

  GammaKnife ® SRS treatment planning for thalamotomy requires the acquisition of CT scan with thin slice images (1.2 mm, for instance). No contrast agent injection is injected. The brain MRI sequences useful in this indication are: T1-weighted three-dimensional spoiled gradient echo images with gadolinium infusion, square matrix (slice thickness: 1.2 mm, for instance), T2-weighted three-dimensional Fast Imaging Employing Steady State Acquisition or Constructive Interference Steady State. Diffusion tensor imaging is performed to identify the corticospinal tract, the internal capsule and the external edge of thalamus. The fiducial box with markers is attached to the stereotactic head frame; these markers aid in the threedimensional image registration with markers.

Figures Figure 1

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Figure 2

 2 Figure 2

Figure 3

 3 Figure 3

  Figure 1C

Figure

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  Figure 3B

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Figure 4B

  Figure 4B

Radiotherapy in functional Neurology -Jacob

Tableau/Table

The authors would like to thank the Société Française de Radiothérapie Oncologique (SFRO), the and Professor Juliette Thariat for their invitation and help. We are also grateful to Ms. Lois Rose for her help with the English version.

Tables Table 1: Surgical grading of brain arteriovenous malformations according to Spetzler and Martin [START_REF] Spetzler | A proposed grading system for arteriovenous malformations[END_REF].

Features Points

AVM size Small (< 3 cm) Medium (3-6 cm) Large (> 6 cm) The total score is based on the addition of the points assigned to each criterion. The higher the score, the poorer the outcome.

Tableau/Table An elevated score was significantly correlated with poorer outcomes in this study. In the article published by Wegner et al. [START_REF] Aoyama | Treatment outcome of single or hypofractionated single-isocentric stereotactic irradiation (STI) using a linear accelerator for intracranial arteriovenous malformation[END_REF], the AVM location item was separated in deep (i.e. basal ganglia, thalamus or brainstem; 1 point) versus other ones (0 point). A coefficient of 0.5 was attributed to this criterion.

(2) : A higher score was significantly associated with poorer outcomes. Three groups of patients were defined according to the score: 0-1 point; 2 points; 3-4 points.


Keywords: Battery Management Systems (BMS), Control methods for electrical systems, MPC (Model-based Predictive Control), Device modelling, Microgrid

Energy storage systems are key elements for enabling the design of MicroGrids in buildings, specially to deal with stochastic renewable energy resources and to promote peak shifting. However, inaccuracies in the batteries' mathematical models due to temperature and ageing effects can reduce the performance of a MicroGrid system. To tackle these uncertainties, this article presents a two-level hierarchical model predictive controller empowered with a data-driven algorithm for real-time model identification of Lithium-ion batteries. The objective is to enhance their state of charge estimation and to make their maximum use without damaging them. The results demonstrate that it improves up to three times the accuracy of state-of-charge estimation and increases about 3% the annual building MicroGrid selfconsumption rate. Furthermore, the division of the building MicroGrid energy management system into two hierarchical levels soften the drawbacks arise from the inaccuracies of day-ahead data prediction while reducing the computational cost. The proposed architecture guarantees higher energetic autonomy indexes than a conventional rule-based controller in all scenarios under study.

Real-time Parameters Identification of Lithium-ion Batteries Model to Improve the Hierarchical Model Predictive Control of Building MicroGrids

Daniela Yassuda Yamashita a,b , Ionel Vechiu a and Jean-Paul Gaubert b a ESTIA INSTITUTE OF TECHNOLOGY Technopole Izarbel -92, Allée Théodore Monod -Bidart, France b LABORATOIRE D'INFORMATIQUE ET D'AUTOMATIQUE POUR LES SYSTEMES (LIAS), UNIVERSITE DE POITIERS 4, Rue Pierre Brousse -Poitiers, France d.yamashita@estia.fr, i.vechiu@estia.fr, jean.paul.gaubert@univ-poitiers.fr

Introduction

Building MicroGrids (BMGs) are an attractive alternative to foster the integration of renewable energy sources (RESs) into the electrical grid [START_REF] Lawrence | Ten questions concerning integrating smart buildings into the smart grid[END_REF]. However, their wide implementation is restrained by the difficulty of designing a generic Energy Management System (EMS) capable of operating under stochastic variations in the power plant. Unpredictability in power generation and power consumption, along with inaccuracy of mathematical models of BMG components lead to the under exploitation of BMG resources or equipment damaging [START_REF] Cardoso | Battery aging in multi-energy microgrid design using mixed integer linear programming[END_REF].

In the literature, there are various strategies to implement a reliable and efficient EMS [START_REF] Zia | Microgrids energy management systems: A critical review on methods, solutions, and prospects[END_REF]. Among the existent strategies, the hierarchical control structure enables to embed algorithms with different complexities at the same time, thanks to the parallel coordination of multiple control layers deployed at different sample time [START_REF] Cheng | To Centralize or to Distribute: That Is the Question: A Comparison of Advanced Microgrid Management Systems[END_REF]. Concerning the EMS algorithms, those based on model predictive control (MPC) have proved their robustness against environmental disturbances, even with simplified plant model [START_REF] Mendes | Energy management of an experimental microgrid coupled to a V2G system[END_REF]. Nonetheless, there is a lack in evaluating the MPC performance under environmental changes, such as temperature, electric devices ageing and batteries model parameters inaccuracy.

In order to face these uncertainties, there are several techniques to estimate better the intrinsic parameters of batteries, such as the Arrhenius equation [START_REF] Su | Path dependence of lithium ion cells aging under storage conditions[END_REF], or models devised from technical specifications [START_REF] Cardoso | Battery aging in multi-energy microgrid design using mixed integer linear programming[END_REF]. However, they require beforehand model calibration, which can lead to uncertainties throughout the batteries' life. In this context, strategies based on data analysis are increasingly implemented. The most pertinent algorithms are incremental analysis of the voltage and capacity to estimate the batteries' state of health [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF], Kalman filter estimator [START_REF] Song | The sequential algorithm for combined state of charge and state of health estimation of lithium-ion battery based on active current injection[END_REF] and other machine learning methods [START_REF] Li | Data-driven health estimation and lifetime prediction of lithium-ion batteries: A review[END_REF]. Therefore, this article proposes a two-level Hierarchical MPC (HMPC) with a Real-Time Model Identification (RTMI) module to deal with batteries' parameter inaccuracy. Equipped with photovoltaic (PV) panels and Li-ion batteries, the purpose of the proposed HMPC is to maximise the MicroGrid selfconsumption rate (𝜏 𝑎 ) [START_REF] Luthander | Photovoltaic self-consumption in buildings: A review[END_REF]. Relying on continuous data measurement and forecast data, the strategy is to optimise its internal power flow by promoting the use of renewables and reducing the energy dependency on the external grid. The results show that the RTMI enhances the State-of-Charge (SoC) estimation, leveraging 𝜏 𝑎 and leading HMPC to outperform the traditional rule-based strategy.

The remaining of this paper is organised as follows. In the second section, the overview of the underlying hierarchical control structure associated with the proposed algorithm for identifying the battery parameters is described. The third section details the new model developed for the estimation of the SoC and the RTMI algorithm. The fourth section presents the performance of the proposed control strategy, by comparing the HMPC with RTMI module with both the conventional MPC under hierarchical and non-hierarchical architecture and a traditional rule-based controller. In the last section, the conclusions on the advantages and disadvantages of the proposed approach are summarized.

Overview of the hierarchical energy management system

The designed hierarchical EMS is a centralised controller for optimising the power flow of a gridconnected BMG that interacts with the external grid throughout a community aggregator, as illustrated in Fig. 1. Composed mainly by three control units, the objective of the proposed control architecture is to assure the building internal power balance with minimum dependency on the main grid. The following two subsections describe the hierarchical MPC and the RTMI module, respectively.

Grid AC DC

Measurements [vbat,ibat] Day-ahead planning

Weather forecast

EMPC

Nh: 48h

Ts: 24h

RTMI module

SoC ^

Fig. 1: Two-level hierarchical model predictive control architecture empowered with the batteries' realtime model identification module for energy management of a grid-connected building MicroGrid.

Hierarchical model predictive controller

The proposed HMPC disposes of two control levels, namely Economic MPC (EMPC) and Tracking MPC (TMPC). The upper control level performs the economic power dispatch through an EMPC and determines both the day-ahead electricity trading planning to be sent toward the community aggregator and the batteries' State-of-Charge references (𝑆𝑜𝐶 𝑟𝑒𝑓 ) to be forward to the TMPC. Simultaneously, the lower level determines the batteries' power references (𝑃 𝑏𝑎𝑡 ) based on the updated prediction data and measurements, by performing a TMPC to follow 𝑆𝑜𝐶 𝑟𝑒𝑓 .

Considering the fluctuations of the data prediction of both PV power generation and building power consumption (𝑃 𝑐𝑜𝑛𝑠 𝑝𝑟𝑒𝑑 ), the EMPC determines 𝑆𝑜𝐶 𝑟𝑒𝑓 that, within a horizon of 𝑁 ℎ 𝐸𝑀𝑃𝐶 48ℎ, minimises the external grid energy dependency. Therefore, the EMPC maximises 𝜏 𝑎 defined by (1) and minimises additional energy imports to reduce the cost of purchasing electricity. For this, the cost function defined by ( 2) is optimised daily (𝑇 𝑠 𝐸𝑀𝑃𝐶 24ℎ), in which the total power injected (𝑃 𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 ) and the total power purchased (𝑃 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑 ) are minimised. It is noteworthy that the horizon (𝑁 ℎ 𝐸𝑀𝑃𝐶 ) twice bigger than the updating time (𝑇 𝑠 𝐸𝑀𝑃𝐶 ) is to guarantee the most suitable 𝑆𝑜𝐶 at the end of the day. Doubling the horizon avoids unnecessary completely discharge of batteries at the end of the day since the last half of control variables determined by the EMPC is not implemented.

Parallelly to the EMPC, the TMPC is triggered at each 𝑇 𝑠 𝑇𝑀𝑃𝐶 1ℎ to follow the 𝑆𝑜𝐶 𝑟𝑒𝑓 as much as possible, considering the updated data prediction and the 𝑆𝑜𝐶 ̂ estimated from the last measurements. To achieve this, the TMPC minimises hourly the cost function defined by [START_REF] Zia | Microgrids energy management systems: A critical review on methods, solutions, and prospects[END_REF]. The purpose of the TMPC is to reduce the stochastic variation of the day-ahead power imbalance and battery model inaccuracies, by using a shorten horizon, simplifying the optimisation process.

𝑃 𝑏𝑎𝑡 arg (min 𝑃 𝑏𝑎𝑡 ∑ (𝑆𝑜𝐶 𝑟𝑒𝑓 𝑘 -𝑆𝑜𝐶 ̂𝑘) 2 𝑁 ℎ 𝑇𝑀𝑃𝐶 =6 𝑘=1 ) (3)
Meanwhile, in the background, the EMPC supervises the performance of the TMPC. Every hour, the EMPC compares the accuracy of day-ahead power planning sent to the aggregator (𝑃 𝑔𝑟𝑖𝑑 𝑟𝑒𝑓 ) and 𝑆𝑜𝐶 𝑟𝑒𝑓 with real measurements. As soon as the absolute difference between them is higher than a predefined thresholdnamed Δ 𝑆𝑜𝐶 𝑡ℎ𝑟 or Δ 𝑃 𝑔𝑟𝑖𝑑 𝑡ℎ𝑟 at time 𝑡 𝑟𝑒𝑂𝑝𝑡the EMPC determines new 𝑆𝑜𝐶 𝑟𝑒𝑓 using the updated prediction data, but with a reduced horizon, as illustrated in Fig. 2. The reduced horizon comprehends the data prediction of the remaining period of the first EMPC optimisation and is equal to 𝑁 ℎ -𝑡 𝑟𝑒𝑂𝑝𝑡 . The main constraints embedded in both EMPC and TMPC are those to respect the recommendations of the French Energy Regulation Commission (ERC) for grid-connected BMGs with PV capacity over 100 kWc [START_REF]Enedis l'électricité en réseau[END_REF]. To prevent energy speculation with the local energy storage systems (ESSs) when trading electricity, the ERC allows charging the ESSs only from renewable power generated locally.

Additionally, it imposes that 𝜏 𝑎 ≥ 50% at the end of the year [START_REF]la réalisation et l'exploitation d'Installations de production d'électricité à partir d'énergies renouvelables en autoconsommation et situées en métropole continentale[END_REF]. As a result, batteries can be charged (𝑃 𝑏𝑎𝑡 𝑐ℎ ) as soon as there is an internal energy surplus, while they can be discharged (𝑃 𝑏𝑎𝑡 𝑑𝑖𝑠 ) when there is an energy deficit, as detailed in [START_REF] Cheng | To Centralize or to Distribute: That Is the Question: A Comparison of Advanced Microgrid Management Systems[END_REF]. Moreover, aiming to extend the life of batteries, the maximum power rate (𝑃 𝑏𝑎𝑡 𝑚𝑎𝑥 ) and the SoC limits defined by their manufacturer must be supervised [START_REF] Cardoso | Battery aging in multi-energy microgrid design using mixed integer linear programming[END_REF], [START_REF] Tremblay | Experimental Validation of a Battery Dynamic Model for EV Applications[END_REF]. Consequently, it is important to well estimate future SoC (𝑆𝑜𝐶 𝑘+1 ) and to limit them. Similarly, to maximize 𝜏 𝑎 beyond the MPC horizon, the SoC at 𝑘 48ℎ is forced to be higher than 40%.

-|max(𝑃 𝑏𝑎𝑡 𝑀𝐴𝑋 𝑃 𝑠𝑢𝑟𝑝𝑙𝑢𝑠 )| ≤ 𝑃 𝑏𝑎𝑡 𝑘 𝑐ℎ ≤ 0 ; 0 ≤ 𝑃 𝑏𝑎𝑡 𝑘 𝑑𝑖𝑠 ≤ |max(𝑃 𝑏𝑎𝑡 𝑀𝐴𝑋 𝑃 𝑑𝑒𝑓𝑖𝑐𝑖𝑡 )| (4)

Real-time model identification algorithm

Most of the scientific studies [START_REF] Mendes | Energy management of an experimental microgrid coupled to a V2G system[END_REF] estimate 𝑆𝑜𝐶 𝑘+1 through a model composed of time-invariant parameters derived from batteries' technical specification, as formulated in [START_REF] Mendes | Energy management of an experimental microgrid coupled to a V2G system[END_REF]. However, based on more realistic models of Li-ion batteries [START_REF] Tremblay | Experimental Validation of a Battery Dynamic Model for EV Applications[END_REF], the efficiency during its charge (𝜂 𝑐ℎ ) or discharge (𝜂 𝑑𝑖𝑠 ), the nominal capacity (𝑄 𝑛𝑜𝑚 ) and the nominal voltage (𝑣 𝑛𝑜𝑚 ) change according to the intensity of current, battery age and cell temperature (𝑇 𝑐𝑒𝑙𝑙 ). According to [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF], [START_REF] Li | Data-driven health estimation and lifetime prediction of lithium-ion batteries: A review[END_REF] the voltage variation can be around 10% of the nominal voltage when they are fully charged and discharged. Moreover, the nominal capacity can reach, at the end of their life, up to 80% of its initial value [START_REF] Li | Data-driven health estimation and lifetime prediction of lithium-ion batteries: A review[END_REF]. Consequently, additional uncertainties on 𝑆𝑜𝐶 𝑘+1 estimation arise, which may result in under or overuse of the batteries. To tackle this issue, the authors designed the algorithm RTMI to improve the estimation of both 𝑆𝑜𝐶 𝑘+1 and their upper (𝑆𝑜𝐶 𝑚𝑎𝑥 𝑘+1 ) and lower boundaries (𝑆𝑜𝐶 𝑚𝑖𝑛 𝑘+1 ). The main objective of this new algorithm is to reduce human intervention for model calibration and BMG maintenance while taking full advantages of batteries. Therefore, the RTMI module updates the parameters of 𝑆𝑜𝐶 𝑘+1 , as illustrated in Fig. 3. The limits 𝑆𝑜𝐶 𝑚𝑎𝑥 𝑘+1 and 𝑆𝑜𝐶 𝑚𝑖𝑛 𝑘+1 are determined to guarantee the operation of the batteries in the linear region, as shown in Fig. 4a. The RTMI module determines the parameters , and of the 𝑆𝑜𝐶 model defined by ( 6) and ( 7) by implementing an unconventional linear regression based on previous data measurements. The variable 𝛿 𝑘 is a Boolean variable which is worth 1 when the battery is active and 0 otherwise. Remarkably, equation [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF]

𝑆𝑜𝐶 𝑘+1 𝑆𝑜𝐶 𝑘 + 𝜂 𝑐ℎ 𝑇 𝑠 𝑣 𝑛𝑜𝑚 • 𝑄 𝑛𝑜𝑚 • 𝑃 𝑐ℎ 𝑘 + 𝑇 𝑠 𝑣 𝑛𝑜𝑚 • 𝑄 𝑛𝑜𝑚 𝜂 𝑑𝑖𝑠 • 𝑃 𝑑𝑖𝑠 𝑘 (5)

Details of the RTMI algorithm

Synchronized with both EMPC and TMPC, the RTMI module acquires a new measurement point 𝑀, composed by the integral of the current (𝑖 𝑡 ), the reference power (𝑃 𝑏𝑎𝑡 ) and the battery voltage (𝑣) at each TMPC sampling time (i.e. 𝑇 𝑠 𝑇𝑀𝑃𝐶 ). As detailed in the following sub-sections, the RTMI algorithm is divided into three steps, namely: classification, updating and identification of the limits for charging and discharging the batteries.

Step 1: Classification of data measurements by temperature interval As depicted in Fig. 4a, 𝑣 is directly correlated to the 𝑇 𝑐𝑒𝑙𝑙 . This effect in the battery voltage impacts the batteries round-trip efficiency, reducing the accuracy of the classical model defined by [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF]. Higher voltage when charging the batteries or lower voltages when discharging them implies a loss of efficiency, once the storage energy variation is mainly dependent on the current flowing through the batterie cells. Therefore, aiming to improve the robustness against 𝑇 𝑐𝑒𝑙𝑙 disturbance, the RTMI fits a linear model for each temperature ranges using classified measurements points. Without using any temperature sensor, the RTMI algorithm estimates the temperature interval from 𝑣 measurements. The Fig. 4a and Fig. 5a show that the 𝑇 𝑐𝑒𝑙𝑙 mainly involves a vertical offset in the batteries discharge curve, but almost does not affect the slope of 𝑣 with respect to 𝑖 𝑡 . In this way, the maximum amplitude of 𝑣 (𝛥𝑣 𝑀𝐴𝑋 ) due to the full charges and discharges of the batteries is almost unchanged, resulting in a quasiconstant slope along with different 𝑇 𝑐𝑒𝑙𝑙 , as depicted in Fig. 5b and Fig. 5c.

Consequently, when the BMG operates for the first time, the EMS will force two full charges and discharges of the batteries to determine 𝛥𝑣 𝑀𝐴𝑋 , and therefore, the absolute value of the slope of the discharge curve (|Δ𝑣/𝑖 𝑡 |). After this preliminary step, the temperature intervals are defined by identifying the lines 𝑠 ⃗ 𝑇 𝑐𝑙𝑎𝑠𝑠 , as shown in Fig. 5c, where 𝑇 𝑐𝑙𝑎𝑠𝑠 is the identification number for a specific temperature class. The lines 𝑠 ⃗ 𝑇 𝑐𝑙𝑎𝑠𝑠 are outspread from a voltage interval which is chosen a priori regarding the desired precision, the computing resources and the RTMI convergence time. Typically, this voltage offset can be set as 1% of 𝑣 𝑛𝑜𝑚 . As a result, a measurement 𝑀 belongs to 𝑇 𝑐𝑙𝑎𝑠𝑠 , if and only if, the point 𝑀 is between the lines 𝑠 ⃗ 𝑇 𝑐𝑙𝑎𝑠𝑠 and 𝑠 ⃗ 𝑇 𝑐𝑙𝑎𝑠𝑠 +1 .

Step 2: Updating the values of , and

Once determined which class of temperature the measurement 𝑀 belongs to, it is possible to adjust the values of and of these respective 𝑇 𝑐𝑙𝑎𝑠𝑠 . The determination of and is implemented based on an iterative and real-time process. As soon as a new measurement 𝑀 is acquired, the parameters of and are updated. Remarkably, and are vectors of size equals to the number of 𝑇 𝑐𝑙𝑎𝑠𝑠 , where each element of these vectors is the fitted values of the model defined by ( 8) over all measurements 𝑀. The upmost advantage of the RTMI algorithm is the non-dependency on storage of the past measurements. The knowledge of the previous measures is stored in dynamic confidence weights, named 𝜔 𝐴 𝑘 , 𝜔 𝐵 𝑘 and 𝜔 𝐶 𝑘 .

Starting from minimal valuestypically between 0.1 and 1.0these confidence weights grow with the acquisition of new measurements. Since the pivot points possess fixed values in the plan 𝑥𝑦 [𝑃 𝑖 𝑡 ], the fitting process only consists of calculating the values of z-axis (i.e. 𝛥𝑖 𝑡 𝑘 𝐴 , 𝛥𝑖 𝑡 𝑘 𝐵 and 𝛥𝑖 𝑡 𝑘 𝐶 ). For the first iteration, the pivot points are initialized according to [START_REF] Song | The sequential algorithm for combined state of charge and state of health estimation of lithium-ion battery based on active current injection[END_REF], where it is used the parameters given by the technical specification of the batteries and using the classical model defined in [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF]. 9) are calculated with regard the normalized distance between the measure 𝑀 and the respective intermediate pivot point projected on the plan 𝑥𝑦 (i.e. operator ‖•‖ 𝑥𝑦 ), as defined in [START_REF] Luthander | Photovoltaic self-consumption in buildings: A review[END_REF] for the pivot point 𝐴 ′ .

𝛥𝑖
𝑛 𝐴 𝑘 ‖𝑀𝐴 ̅̅̅̅̅ ‖ 𝑥𝑦 ‖𝑀𝐴 ̅̅̅̅̅ ‖ 𝑥𝑦 + ‖𝑀𝐵 ̅̅̅̅̅ ‖ 𝑥𝑦 + ‖𝑀𝐶 ̅̅̅̅̅ ‖ 𝑥𝑦 (10)
Once determined the intermediary plan 𝜙 , Δ𝑖 𝑡 𝑘 𝐴 , Δ𝑖 𝑡 𝑘 𝐵 and Δ𝑖 𝑡 𝑘 𝐶 and the confidence weights 𝜔 𝐴 𝑘 , 𝜔 𝐵 𝑘 and 𝜔 𝐶 𝑘 are updated following equations ( 11) -( 14), in the order of compilation. For the sake of simplicity, in this paper, it will only be detailed the updating process of the pivot point 𝐴 , but it is important to highlight that similar equations are used for 𝐵 ′ and 𝐶 ′ .

𝜔 𝐴 𝑘+1 𝜔 𝐴 𝑘+1 -𝜏 𝑑𝑒𝑐𝑎𝑦 • (𝜔 𝐴 𝑘+1 -𝜔 𝑚𝑖𝑛 ) (13)
Firstly, the confidence weight 𝜔 𝐴 𝑘 is updated using [START_REF]Enedis l'électricité en réseau[END_REF]. The closer 𝑀 is to the 𝐴 , the more reliable the value of 𝑧 𝐴 calculated from ( 9) is. Consequently, 𝜔 𝐴 𝑘+1 is inversely proportional to the distance between the pivot point 𝐴 and the measurement 𝑀, and it is normalised according to the distance from other pivot points (i.e. 𝐵 and 𝐶 ). To improve the robustness against measurement noise, the updated Δ𝑖 𝑡 𝑘+1 𝐴 is a weighted value between the previous Δ𝑖 𝑡 𝑘 𝐴 and the new fitted 𝑧 𝐴 . The inertial factor handles the ponderation among these two variables 𝜔 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑘 𝐴 and the confidence weight 𝜔 𝐴 𝑘+1 as defined in [START_REF] Tremblay | Experimental Validation of a Battery Dynamic Model for EV Applications[END_REF]. The equilibrium amongst 𝜔 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑘 𝐴 and 𝜔 𝐴 𝑘+1 control the convergence time of Δ𝑖 𝑡 𝑘+1 𝐴 to z A . This balance can be manually set by tuning the value of 𝜎, which indicates the importance of the new measurement regarding the previous value. Notably, 𝜔 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑘 𝐴 is upper limited by 𝜔 𝐴 𝑘+1 , to restrict the convergence time to a scale of two. Since the charge/discharge curve of batteries changes with age, the confidence weight 𝜔 𝐴 𝑘 + 1 gradually decreases with a time constant 𝜏 𝑑𝑒𝑐𝑎𝑦 until a minimum value 𝜔 𝑚𝑖𝑛 which is equal to the starting value, as specified in [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF].

After updating the plan Σ , the coefficients , and are calculated using the principles of analytical geometry. As the last step, it is still necessary to refine the developed model, because according to [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF] and Fig. 4b, the plan Σ does not cross the 𝑖 ⃗ 𝑡 axis, which makes batteries model inaccurate for low powers. Therefore, the model polishing consists of forcing the plan Σ cross the 𝑖 ⃗ 𝑡 axis by using [START_REF] Yu | An integrated and reconfigurable hybrid AC/DC microgrid architecture with autonomous power flow control for nearly/net zero energy buildings[END_REF].

∆𝑖 𝑡 𝑘 ∑ 𝜃 𝑃 𝑘 ={𝑐𝑑 𝑑𝑑 𝑐𝑐 𝑑𝑐} ∑ 𝑃 𝑘 ( + 1 𝑃′ 𝑏𝑎𝑡 𝑀𝐴𝑋 • ( + (𝑖′ 𝑡 𝑀𝐴𝑋 + 𝑖′ 𝑡 𝑀𝐼𝑁 ) • 2 )) ={𝑐𝑑 𝑑𝑑 𝑐𝑐 𝑑𝑐} (15)
Step 3: Identification of the limits for charging and discharging the batteries

As mentioned before and illustrated in Fig. 6a, the batteries must operate between points 𝑄(𝑣 𝑒𝑥𝑝 ) and 𝑄(𝑣 𝑛𝑜𝑚 ). The strategy consists in adjusting 𝑖 𝑡 𝑚𝑎𝑥 and 𝑖 𝑡 𝑚𝑖𝑛 to reduce the variations of the slope |Δ𝑣/Δ𝑖 𝑡 |. According to Fig. 6a, while 𝑖 𝑡 is inside the linear zone (zone 1), the slope |Δ𝑣/Δ𝑖 𝑡 | is quasiconstant, because 𝑣 is linearly dependent on 𝑖 𝑡 . However, when operating outside this zone (zone 2), Based on this phenomenon, the developed algorithm for identifying the actual boundaries of 𝑖 𝑡 divides the temporal graphs into two zones, named 𝑧𝑜𝑛𝑒 𝐴 and 𝑧𝑜𝑛𝑒 𝐵, as illustrated in Fig. 6b and Fig. 6c. The 𝑧𝑜𝑛𝑒 𝐴, indicated by the red dots, represents the range where the batteries certainly operates in the linear zone. On the other hand, the 𝑧𝑜𝑛𝑒 𝐵, outlined by the blue dots, is the zone for which the battery can be either in the linear or non-linear region. Therefore, 𝑧𝑜𝑛𝑒 𝐴 comprehends the range of 20% to 80% of the predefined limits of 𝑖 𝑡 𝑚𝑖𝑛 and 𝑖 𝑡 𝑚𝑎𝑥 , whereas 𝑧𝑜𝑛𝑒 𝐵 is its complementary region.

|Δ𝑣/Δ𝑖 𝑡 | is not constant because 𝑣 is non-linear regarding 𝑖 𝑡 . (a) (b) (c)
Remarkably, the oscillation of the slope is more intense when zone B is in the non-linear zone (Fig. 6c) than when it is inside the linear range (Fig. 6b). As a result, to determine whether zone B corresponds to the linear or non-linear range, the average of the absolute difference between each |Δ𝑣/Δ𝑖 𝑡 | measured inside 𝑧𝑜𝑛𝑒 𝐵 (𝑚 𝑢𝑝 ) and those measured inside 𝑧𝑜𝑛𝑒 𝐴 (𝑚 𝑑𝑜𝑤𝑛 ) are constant compared, through the mean deviations 𝜗 𝑢𝑝 and 𝜗 𝑑𝑜𝑤𝑛 calculated as detailed in Fig. 7. These deviations 𝜗 𝑢𝑝 and 𝜗 𝑑𝑜𝑤𝑛 are monitored by means of a proportional controller with a hysteresis at its input to maintain them within the range 𝜗 𝑢𝑝 𝑑𝑜𝑤𝑛 𝑟𝑒𝑓 ± 𝑇𝑜𝑙. The values of 𝑇𝑜𝑙 ≅ 2% and 𝜗 𝑢𝑝 𝑑𝑜𝑤𝑛 𝑟𝑒𝑓 ≅ 10% were manually regulated, but they can be re-adjusted to reduce the oscillations or increase the response time of 𝑖 𝑡 𝑚𝑖𝑛 and 𝑖 𝑡 𝑚𝑎𝑥 . If the 𝑖 𝑡 boundaries are modified more than 10% of the previous values, the feasibility zone of Fig. 4b is readjusted accordingly and the confidence values 𝜔 𝐴 , 𝜔 𝐵 , 𝜔 𝐶 are reset to 𝜔 𝑚𝑖𝑛 . Fig. 7: Algorithm for identifying the 𝑖 𝑡 limits to guarantee the batteries' operation inside the linear zone.

Simulation results

Aiming to evaluate the performance of the proposed control architecture, a BMG equipped with PV arrays with 107kWc and Li-ion batteries with nominal voltage of 700V and nominal capacity of 167Ah was simulated for 365 days in MATLAB Simulink ® under several scenarios. The simulations were carried out using real solar radiation data [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF] and the estimated annual building energy consumption, resulting in 135.95 MWh energy generation and 241.85 MWh energy consumption per year. The 𝜏 𝑎 and the total power exchanged with the main grid was used for assessing the proposed HMPC with RTMI module, a non-hierarchical MPC, and a conventional rule-based (RB) strategy with and without data prediction inaccuracy.

The RB was adapted from [START_REF] Yu | An integrated and reconfigurable hybrid AC/DC microgrid architecture with autonomous power flow control for nearly/net zero energy buildings[END_REF], where, using no data prediction, the batteries are charged when there is an energy surplus and discharged when there is an energy deficit. The simulation results are divided into two subsections. The first one is to validate the RTMI algorithm for SoC and batteries' capacity estimation, while the second one is for assessing the benefits of dividing the EMS hierarchically.

Performance of the algorithm for batteries' parameter identification

To show the robustness of the RTMI algorithm against inaccuracies of the parameters coming from the technical specifications, three initial values of 𝑄 were considered, namely 𝑄 80 , 𝑄 100 and 𝑄 120 , corresponding to 80%, 100% and 120% of the actual capacity (167 Ah), respectively. To verify the error between the day-ahead 𝑆𝑜𝐶 𝑟𝑒𝑓 calculated by the EMPC and the real one, the TMPC in these scenarios was considered as a perfect router. Consequently, instead of optimising (3), it does implement the control variables determined by EMPC. In this manner, it is possible to decouple the effect of TMPC and highlight only the impact of the errors in EMPC state of charge estimation on the BMG performance. The graph in Fig. 8a shows that, in all study cases, the cumulative error in predicting 𝑆𝑜𝐶 𝑘+1 using the RTMI module is about 3 times lower than using the conventional model with static parameters. Furthermore, the graphs in Fig. 8b shows that the EMPC empowered by RTMI module assures 𝜏 𝑎 about 3% higher than the EMPC without RTMI module with 𝑄 80 and between 2% and 4% higher than RB controller. The increase in 𝜏 𝑎 is mainly due to the enlargement of the 𝑖 𝑡 boundaries closed to the real frontiers (𝑄 𝑒𝑥𝑝 and 𝑄 𝑛𝑜𝑚 ), as shown in Fig. 8c. Accurately estimating the battery capacity reinforces the potential of batteries of shifting the load toward the periods of energy surplus, resulting in an enhanced internal load matching and lower grid dependency. For this reason, the relative difference between 𝜏 𝑎 is more remarkable when the batteries' capacity is underestimated (i.e. 𝑄 80 ). Nonetheless, there is a tradeoff between fostering 𝜏 𝑎 and the batteries' state of heathy. Fig. 8c shows that expanding 𝑖 𝑡 boundaries, the attained depth of discharge (DoD) is higher when employing RTMI module than not using it (at the beginning of the simulation), accelerating the batteries' degradation [START_REF] Li | State of health estimation for Li-Ion battery using incremental capacity analysis and Gaussian process regression[END_REF]. In addition, as shown in Fig. 8d employing the RTMI module and processing of 𝑣 and 𝑖 𝑡 measurements, 𝑇 𝑐𝑒𝑙𝑙 can be supervised without needing any thermal sensor, because 𝑇 𝑐𝑙𝑎𝑠𝑠 is an image of the temperature variation. This can provide further information to the HMPC for preserving the batteries state of healthy [START_REF] Cardoso | Battery aging in multi-energy microgrid design using mixed integer linear programming[END_REF], [START_REF] Song | The sequential algorithm for combined state of charge and state of health estimation of lithium-ion battery based on active current injection[END_REF].

Performance of the proposed two-level hierarchical control structure

Since the purpose of the hierarchical EMS is to soften the drawbacks provoked by stochastic variations in the internal power imbalance, the prediction data were multiplied by a random time-dependent factor (𝜌), as detailed in Fig. 9a. In this manner, the error in the estimate power imbalance grows according to the horizon, attaining up to 60% at 𝑁 ℎ 𝐸𝑀𝑃𝐶 48, as shown in Fig. 9b and Fig. 9c. Aiming to assess the proposed HMPC, five control architectures were investigated. The first and second ones are the HMPC with and without RTMI modules. The third and fourth control structures are the non-hierarchical MPC with and without RTMI module, in which only EMPC is updated hourly with the full horizon. Finally, the fourth control disposal is the simple RB. Fig. 10a and Fig. 10b show that both HMPC and the non-hierarchical MPC are robust against data prediction uncertainties, because with and without errors, the BMG imported and exported almost the same amount of energy in a year. Remarkably, when using HMPC with RTMI, it imported about 1% less energy and exported 8% less than without RTMI. Even though the non-hierarchical MPC optimises the cost function (2) around 11 times more than the HMPC, the results are very similar. Indeed, the nonhierarchical structure triggers EMPC at each hour, which means 8760 optimizations in a year, whereas only up to 631 times (number of re-optimisations plus once per day) using the proposed control structure, as detailed in Fig. 10c. The cooperation of two control layersone with long and another with a short horizonenables to handle prediction data variability without needing to optimize the laborious cost function (2) every hour, but only when the error in either 𝑆𝑜𝐶 𝑟𝑒𝑓 or 𝑃 𝑔𝑟𝑖𝑑 𝑟𝑒𝑓 is greater than Δ 𝑆𝑜𝐶 𝑡ℎ𝑟 Δ 𝑃 𝑔𝑟𝑖𝑑 𝑡ℎ𝑟 7%. Fig. 10d shows that errors in 𝑃 𝑔𝑟𝑖𝑑 𝑟𝑒𝑓 are higher when not using the RTMI module, because of imprecisions in SoC estimation, which result in 391 more re-optimisations in the scenario without errors and 135 with errors.

Conclusions

This paper aims to develop a generic Building MicroGrid Energy Management System capable of adapting to external changes, such as Li-ion batteries modelling inaccuracy and inherent power imbalance uncertainties. The hierarchical MPC empowered with the proposed real-time parameter identification module increases the self-consumption rate regarding a well-established rule-based controller and the conventional MPC. This data-driven algorithm enables to identify the original batteries capacity and the cell temperature without any previous modelling step and without any thermal sensor, which simplify the energy management system design. Moreover, the simulation results demonstrated that the division in two control layers reduces the number of optimizations while maintaining the building less energetic dependent on the external grid even under imprecisions in data predictions.

Fig. 2 :

 2 Fig. 2: Re-optimization process of EMPC with reduced horizon.

Fig. 3 :

 3 Fig. 3: Real-time parameters identification algorithm for a charge and discharge sequence ( 𝑘 ).

  Fig. 4: RTMI module operation. (a) Discharge curves of a Li-ion battery under temperature cell variations. (b) Model perspective with the three pivot points for predicting the variation of SoC (𝛥𝑖 𝑡 ).

Fig. 5 :

 5 One-year simulation of the BMG with SoCmax = 80% and SoCmin = 20%. (a) 𝑣 and 𝑇 𝑐𝑒𝑙𝑙 correlation. (b) Slope of battery discharge curve. (c) Correlation between voltage measurements, current integral and 𝑇 𝑐𝑒𝑙𝑙 .

Fig. 6 :

 6 Simulation of Li-ion batteries. (a) The slope of the charge/discharge curves as a function of 𝑖 𝑡 . (b) and (c) Temporal evolution of the slope when operating in linear and non-linear zones, respectively.

Fig. 8 :

 8 Fig. 8: Evaluation of the EMPC robustness against parameter imprecision. (a) The cumulative error of the SoC estimation. (b) Self-consumption rate comparison. (c) Maximum (in blue lines) and minimum (in red lines) boundaries of the 𝑖 𝑡 when using RTMI. (d) Estimation of 𝑇 𝑐𝑒𝑙𝑙 variation through 𝑇 𝑐𝑙𝑎𝑠𝑠 .

Fig. 9 :

 9 Real power and its 48-hours data prediction of a summer day. (a) Time-variant factor. (b) Power consumption. (c) Power generation.

Fig. 10 .

 10 Fig. 10. Comparison between different control architectures. (a) Total power injected. (b) Total power purchased. (c) Number of EMPC re-optimization. (d) Moving average error in 𝑃 𝑔𝑟𝑖𝑑 𝑟𝑒𝑓 of the last 10 days

  can be interpreted as four surfaces 𝛴 of axis 𝑥𝑦𝑧 [𝑃 𝑖 𝑡 𝛥𝑖 𝑡 ] , as illustrated in Fig.4b. Then, each surface 𝑥𝑦𝑧 can be represented by three pivot points, namely 𝐴 Consequently, the RTMI sends daily to the HMPC only the values of , and corresponding to the most likely temperature for the next day, which is the average temperature of the previous day.

								[ 𝑖 𝑡 𝑀𝐼𝑁 𝑃 𝑏𝑎𝑡 𝑀𝐼𝑁 𝛥𝑖 𝑡 𝐴 ] ,
	𝐵	[ 𝑖 𝑡 𝑀𝐴𝑋 𝑃 𝑏𝑎𝑡 𝑀𝐼𝑁 𝛥𝑖 𝑡 𝐵 ] and 𝐶	[ 𝑖 𝑡 𝑀𝐴𝑋 𝑃 𝑏𝑎𝑡 𝑀𝐴𝑋 𝛥𝑖 𝑡 𝐶 ] , which cover the feasibility zone which
	corresponds to the limits recommended by the manufacturer extended in 20%. To reduce the impact of
	voltage variation, the surfaces 𝛴 are classified into four main categories, named	{	},
	which indicate the sequence of batteries' charge ( ) and discharge ( ), at time 𝑘 -1 and 𝑘.
		𝑆𝑜𝐶 𝑘+1 1 -	𝑖 𝑡 𝑘 + ∆𝑖 𝑡 𝑘 𝑄 𝑛𝑜𝑚 𝑘	; 𝑃 𝑏𝑎𝑡 𝑘 𝑃 𝑏𝑎𝑡 𝑘 𝑐ℎ	+ 𝑃 𝑏𝑎𝑡 𝑘 𝑑𝑖𝑠	={𝑐𝑑 𝑑𝑑 𝑐𝑐 𝑑𝑐} ∑	𝑃 𝑘	(6)
		∆𝑖 𝑡 𝑘	∑	𝑃 𝑘 + 𝑖 𝑡 𝑘-1 + 𝛿 𝑘
				={𝑐𝑑 𝑑𝑑 𝑐𝑐 𝑑𝑐}

  To achieve this, an intermediate surface 𝜙 defined by the intermediary pivot points 𝐴′ , 𝐵′ and 𝐶′ , with z-axis values equal to 𝑧 𝐴 , 𝑧 𝐵 and 𝑧 𝐶 , are calculated through the optimisation of the cost function defined by[START_REF] Li | Data-driven health estimation and lifetime prediction of lithium-ion batteries: A review[END_REF]. The surface 𝜙 contains the point 𝑀 and is determined to make 𝛥𝑖 𝑡 𝑘 + 1 𝐴 𝐵 𝐶 of the next iteration as close as possible of 𝛥𝑖 𝑡 𝑘 𝐴 𝐵 𝐶 of the previous period. Intuitively, high value of 𝑛 𝐴 𝑘 and low values of 𝑛 𝐵 𝑘 and 𝑛 𝐶 𝑘 lead 𝑧 𝐴 approach to Δ𝑖 𝑡 𝑘 𝐴 faster than 𝑧 𝐵 and 𝑧 𝐶 to Δ𝑖 𝑡 𝑘 𝐵 and Δ𝑖 𝑡 𝑘 𝐶 , whereas balanced values of 𝑛 𝐴 𝑘 , 𝑛 𝐵 𝑘 and 𝑛 𝐶 𝑘 result in a fair variation among 𝑧 𝐴 𝑧 𝐵 and 𝑧 𝐶 . The weights of (

	𝑡 𝑘=0 𝐴	𝛥𝑖 𝑡 𝑘=0 𝐵	𝜂 𝑐ℎ 𝑇 𝑠 𝑄 𝑛𝑜𝑚 𝑣 𝑛𝑜𝑚	• 𝑃 𝑏𝑎𝑡 𝑀𝐼𝑁 ; 𝛥𝑖 𝑡 𝑘=0 𝐶	𝑇 𝑠 𝑄 𝑛𝑜𝑚 𝑣 𝑛𝑜𝑚 𝜂 𝑑𝑖𝑠	• 𝑃 𝑏𝑎𝑡 𝑀𝐴𝑋	(8)
	Thereafter, the values of Δ𝑖 𝑡 𝑘 𝐴 , Δ𝑖 𝑡 𝑘 𝐵 and Δ𝑖 𝑡 𝑘 𝐶 are updated so that to approach the surface Σ to the new
	measurement 𝑀. 𝑧 𝐴 z B z	arg ( min 𝑧 𝐴 z B z C	𝑛 𝐴 𝑘 (Δ𝑖 𝑡 𝑘 𝐴 -𝑧 𝐴 )	2 + 𝑛 𝐵 𝑘 (Δ𝑖 𝑡 𝑘 𝐵 -z B )	2 + 𝑛 𝐶 𝑘 (Δ𝑖 𝑡 𝑘 𝐶 -z )	2 )	(9)
	Subject to:
			𝑀 𝜖 𝐴 ′ 𝐵 ′ 𝐶 ′ ; 𝑧 𝐴 𝑧 𝐵 𝑧 𝐶 ≥ 0

Acknowledgements

The authors would like to thank the New Aquitaine Region for their financial support.