domain: stat sdu

text:
Arthur Cuvier

Éric Beucler

Raphaël Mickaël Bonnin

Raphaël Garcia

Ludovic Margerin

Martin Schimmel

Quantifying the Non-Gaussian Perturbations in the Continuous Seismic Signal

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Several methods classically exist for capturing perturbations (earthquake, explosion...) in the continuous seismic signal. Most of them, like the famous STA/LTA, detect seismic events by taking advantage of the impulsive nature of their waveform compared to the background noise. In contrast, very few are able to point out long period transient signals (typically greater to 5 s period). However, these long-period waveforms often exist in the continuous seismic signal due to technical problems (as the so called glitches) or to external forcings (e.g. pressure drops, acoustic waves). We present here a novel method that quantifies the strength of the Gaussianity of the seismic signal in a given time window. It relies on the basic assumption that the ground motion oscillations balance around an equilibrium position. After a sorting operation, the localisation of the expected Gaussian distribution enables us to extract the mean and the standard deviation of its background noise which allows, in turn, an analysis of the signal' statistical deviations through the Z-score parameter. We show that this method is able to quantify the Gaussianity of various types of smooth perturbations in a noisy background signal. Some examples using both terrestrial and martian seismic data are shown. Finally, based only on statistical thresholds we propose a new way to detect non-Gaussian perturbations.

-Motivations: Gaussianity of the seismic signal

The seismic signals can be characterised by the following associated amplitude's histograms : The good fit between a seismic signal's histograms and a Gaussian probability density function seems to be caused by the Gaussianity of the seismic signal.

-Quantifying the deviation to the Gaussianity

• After sorting the data by increasing order of amplitude, the localisation of the expected Gaussian distribution enables us to extract the mean µ and standard deviation σ of the background noise

• An analysis of the signal' statistical deviations through the Z-score parameter allows to locate each point altered by a perturbation.

-Applications -Earth data

In Fig 3, we apply the method presented in section 2 on a seismic data set altered by two perturbations: one quarry blast around 9h59:30s and one other minor perturbation at 9h58. The perturbed points are localised, as testified by the large values of the Z-score.

-Applications -Martian data

In Fig 4, the application of our method on the martian seismic signal recorded from the SEIS instrument of the InSight mission allows the detection of the "glitches", which are perturbations characterised by long period transients signals.

Conclusion and perspectives

• We developed a new approach to detect any non-Gaussian perturbations

• This method can find applications on any signal with Gaussian noise

• Finally, the next step will be to perform a statistical study of all the perturbations detected during the whole InSight mission (4 years of data).

This project is funded by ANR-MAGIS-19-CE31-0008-02

Fig. 1 :

 1 Fig. 1: Raw seismic signal (top) and its associated histogram amplitude distribution, bins=100 (bottom)

Fig. 2 :

 2 Fig. 2: Quantifying the points which have deviated from the Gaussianity on two examples. Balanced data (left), unbalanced data (right)

Fig. 3 :

 3 Fig. 3: Extracting the perturbed points from the Gaussian noise using the Z-score.

Fig. 4 :

 4 Fig.4: Extracting the perturbed points from the Gaussian noise using the Z-score.


Nunzia Santa D'angeli

Cinzia Paladino

Christelle Fargette

Cendrine Archange

Wassim M Essamet

Zeinab Hamidou

Fausto Palazzo

Karine Baumstarck

Frédéric Sebag

David Taïeb

Panelsanta D'angeli

Nunzia Cinzia Paladino

Lymph node thyroglobulin in the diagnosis of metastases of thyroid carcinoma with the thyroid in situ: A prospective intraoperative study

Keywords:

Introduction

Papillary thyroid cancer (PTC) is the commonest form of thyroid cancer and is characterized by frequent spread to cervical lymph nodes. Persistent and recurrent lymph node (LN) disease occurs primarily in patients where the disease at presentation has been underestimated with suboptimal initial surgery or in whom extensive nodal involvement (high metastatic LN ratio and/or presence of extracapsular extension) is present. In PTC patients with elevated serum thyroglobulin (Tg) levels after initial management, neck ultrasound (US) is highly sensitive and specific for the detection of regional lymphadenopathy. It is of fundamental importance to distinguish LN metastasis from benign reactive LN enlargement in order to avoid unnecessary intervention and the associated risks of morbidity. Whilst fine-needle aspiration cytology (FNAC) accurately complements neck US, its sensitivity varies from 75 to 85% with a high rate of nondiagnostic aspirates [START_REF] Pacini | Detection of thyroglobulin in fine needle aspirates of nonthyroidal neck masses: a clue to the diagnosis of metastatic differentiated thyroid cancer[END_REF][START_REF] Frasoldati | Role of thyroglobulin measurement in fine-needle aspiration biopsies of cervical lymph nodes in patients with differentiated thyroid cancer[END_REF]. The measurement of needle aspirate washout (FNA-Tg) have been reported to increase the sensitivity of FNA cytology alone in identifying LN metastases from differentiated thyroid cancer (DTC), especially for small LN [ [START_REF] Zhu | Diagnostic values of thyroglobulin in lymph node fine-needle aspiration washout: a systematic review and meta-analysis diagnostic values of FNA-Tg[END_REF], [START_REF] Xu | Diagnostic value of cytology, thyroglobulin, and combination of them in fine-needle aspiration of metastatic lymph nodes in patients with differentiated thyroid cancer Medicine[END_REF], [START_REF] Grani | Thyroglobulin in lymph node fine-needle aspiration washout: a systematic review and meta-analysis of diagnostic accuracy[END_REF]] and it is a recommendation found in both the American and European Thyroid Association Guidelines [ [START_REF] Haugen | American thyroid association management guidelines for adult patients with thyroid nodules and differentiated thyroid cancer: the American thyroid association guidelines task force on thyroid nodules and differentiated thyroid cancer[END_REF], [START_REF] Leenhardt | European thyroid association guidelines for cervical ultrasound scan and ultrasound-guided techniques in the postoperative management of patients with thyroid cancer[END_REF], [START_REF] Luster | European perspective on 2015 American thyroid association management guidelines for adult patients with thyroid nodules and differentiated thyroid cancer: proceedings of an interactive international symposium[END_REF]]. In the 2015 ATA guidelines, a FNA-Tg concentration above 10 ng/ml is considered the threshold of highly suspicious of malignancy [START_REF] Haugen | American thyroid association management guidelines for adult patients with thyroid nodules and differentiated thyroid cancer: the American thyroid association guidelines task force on thyroid nodules and differentiated thyroid cancer[END_REF]. There are however potential confounders in the FNA-Tg, including the presence of thyroid tissue in situ at the time of the FNA [START_REF] Grani | Thyroglobulin in lymph node fine-needle aspiration washout: a systematic review and meta-analysis of diagnostic accuracy[END_REF]. With the increasing use of active follow-up or lobectomy alone for low-risk thyroid cancers, the determination of the optimal diagnostic threshold for FNA-Tg that confirms metastatic LN from PTC has assumed even greater importance. The purpose of this study is to address the potential discordance between the FNA-Tg and final pathological report in suspected metastatic cervical LN from PTC. The study took the form of a prospective analysis of intraoperative aspiration of cervical nodes for Tg and the comparison of the Tg values with the histological analysis. A head-to-head comparison between FNA-Tg and final diagnosis in each individual nodes was performed with estimation of the optimal FNA-Tg cutoff value for determining malignancy when the thyroid is still all or partially in situ.

Methods

Population

The inclusion criteria were: age ≥18 years, patients undergoing primary total thyroidectomy for papillary thyroid carcinoma (PTC) or benign disorder or reoperation for metastatic LN. All patients gave their signed informed consent for participation in the study. The exclusion criteria were presence of distant metastases.

The study was approved by the local ethics committee (Comité de Protection des Personnes Sud-Méditerranée II). The study had been registered with ClinicalTrials.gov (identifier: NCT02602717).

Study design

This was a prospective single centre study with consecutive patients included based on the aforementioned inclusion and exclusion criteria. All patients were investigated for preoperative serum Tg, TSH and Tg antibody (TgAb).

Assessment of Tg in the washout fluid was obtained intraoperatively according to the followup protocol:

1

At the initial surgery performed for cytologically proven PTC but without suspected LN disease on neck US or surgery for benign thyroid disease, FNAC precedes thyroidectomy and is performed on an identified lateral LN (near the area of dissection of the carotid sheath for the implantation of a vagus nerve stimulator for intraoperative neuromonitoring) and if present an identifiable central LN.

2 At the initial surgery for PTC with suspected LN on neck US, FNAC precedes thyroidectomy and is performed on at least one macroscopically abnormal LN and if possible on one apparently normal LN.

3 At reoperative surgery for PTC, FNAC is performed on at least one abnormal LN and if possible one macroscopically normal LN.

Fnac was always performed by an experienced surgeon on isolated lymph node in order to avoid its contact with thyroid parenchyma and thus possible aspiration from the thyroid gland.

All aspirated LN and their respective samples were rigorously isolated in appropriately labelled boxes describing the LN and its precise location. Aspiration samples were subsequently dispersed onto slides (2-3 slides/node) and fixed by air-drying. The same needles were rinsed with 1 ml of normal saline stored at room temperature, and the washout was submitted for FNA-Tg measurement.

Gold standard

Diagnosis of malignancy and benignity was based histological examination of surgically resected LNs. The analysis way performed by a trained pathologist (WE) who was blinded to the Tg values.

Biochemical analysis

Tg was assayed with an immunoradiometric assay (RIA Tg-plus; BRAHMS GmbH, Hennigsdorf, Germany). The analytical sensitivity read at the optimal curve is 0.08 ng/mL, and the functional assay sensitivity (20% interassay coefficient of variation) is 0.2 ng/mL. For comparison with CRM-457-calibrated assay, the values should be multiplied by 1.92 (according to the manufacturer's recommendation). TSH was assayed with an immunoradiometric assay (Elecsys ECLIA assay; Roche Diagnostics). Anti-Tg antibody (TgAb) was assayed with RIA (anti-Tgn; BRAHMS GmbH). The analytical sensitivity is 5.5 U/mL, and the functional assay sensitivity is < 20 U/mL.

Data analysis

To determine the performance of FNA-Tg for the diagnosis of LNs malignancy, Receiveroperating characteristic (ROC) curve and area under the curve (AUC) estimates. The optimal threshold to better discriminate between benign and malignant LNs was calculated. Sensitivity (Se), specificity (Sp), and accuracy were provided with their 95% confidence intervals (95%CI). Negative and positive predictive values (NPV and PPV) were only provided for information, Statistical analyses were performed with SPSS software (version 18.0; SPSS, Chicago, Illinois). Data with a P value < 0.05 were considered significant.

Results

Patients' characteristics

Thirty-two patients were included in the study, 27 with thyroid cancer (25 at presentation, 2 with recurrence) on pathological analysis and the remaining 5 cases had benign tumors (follicular adenoma). Table 1 summarizes the patient and primary tumors characteristics and preoperative baseline serum Tg and Tg antibodies.

Optimal cutoff values of FNA-Tg in determining LNs malignancy

A total of 58 lymph nodes (LN) were analyzed with comparisons of FNA-Tg and pathological analysis for each individual nodes. For 54 LN, FNA was performed at initial surgery prior to thyroidectomy: 46 in patients operated for thyroid cancer group and the latter 8 nodes in patients operated for benign disorder. In malignant (n = 34) and benign (n = 24) LNs, median [Interquartile, IQR] FNA-Tg were 629.5 [47.5-1503.5] and 1.5[0-28], respectively (p=<0.001). The optimal cutoff values of FNA-Tg in determining malignant LNs from ROC analysis was 60 ng/ml with sensitivity and specificity of 75% (95%CI: 57.89-86.75) and 87.5% (95%CI: 69-95.66), respectively. When evaluating Tg-FNA in the thyroid cancer group only, the cutoff value of FNA-Tg was 34 ng/ml with sensitivity and specificity of 78.1% (95%CI: 61.24, 88.98) and 87.5% (95%CI: 64.15, 93.32), respectively (Fig. 1). All the details are provided in Table 2.

Discussion

The use of Tg concentration in the washout fluid from fine-needle aspiration (FNA-Tg) of cervival LN in thyroid cancer is an established technique of diagnosing LN metastases. However, there is much heterogeneity across studies that may influence the values of FNA-Tg found (e.g., serum TSH and Tg, presence of a thyroid gland in situ, gold standard, sampling bias, Tg assays etc.) [START_REF] Grani | Thyroglobulin in lymph node fine-needle aspiration washout: a systematic review and meta-analysis of diagnostic accuracy[END_REF]. In the largest study from Moon et al. which included 528 LNs (190 metastatic, 338 benign), the optimal cutoff value of FNA-Tg in distinguishing malignant LNs from benign LNs was 1.0 ng/mL with high sensitivity and specificity, 93.2% and 95.9% respectively [START_REF] Moon | Thyroglobulin in washout fluid from lymph node fine-needle aspiration biopsy in papillary thyroid cancer: large-scale validation of the cutoff value to determine malignancy and evaluation of discrepant results[END_REF]. The cutoff value of FNA-Tg was 2.24 ng/mL in the subgroup (n = 87) of patients who did not undergo thyroidectomy (sensitivity = 95.0%, specificity = 96.3%). In this study, the gold standard comparator was adopted (i.e. histological examination of surgically resected LNs or cytological evidence of recurrence at follow-up or negative histological examination of surgically resected LNs; or in the case of a benign diagnosis imaging showing disappearance or absence of evolution without cytological evidence of malignancy at 12 months or more follow-up). However, as in other studies, there is still uncertaincy regarding the concordance between the biopsied node for FNA-Tg assessement and the final pathological report. For example, a low preoperative FNA-Tg can be obtained in a LN with inconclusive cytology (possibly benign) and final report can describe a metastatic LN found in one of multiple resected lymph nodes, leading to a potential misleading interpretation of the result. It should be emphasized that in PTC, where lymph node metastasis may predict recurrence but not patient survival the priority is to minimise false positive results in order to avoid unnecessary surgery. In our study, the higher Tg-FNA threshold can be attributed to the large proportion of patients with a thyroid in situ.

Very few studies have investigated the difference in the cutoff values for LN aspirates between patients with a thyroid in situ and those where the thyroid has already been removed [ [START_REF] Moon | Thyroglobulin in washout fluid from lymph node fine-needle aspiration biopsy in papillary thyroid cancer: large-scale validation of the cutoff value to determine malignancy and evaluation of discrepant results[END_REF], [START_REF] Boi | The diagnostic value for differentiated thyroid carcinoma metastases of thyroglobulin (Tg) measurement in washout fluid from fine-needle aspiration biopsy of neck lymph nodes is maintained in the presence of circulating anti-Tg antibodies[END_REF], [START_REF] Pak | Diagnostic values of thyroglobulin measurement in fine-needle aspiration of lymph nodes in patients with thyroid cancer[END_REF]]. In one large study that has specifically addressed this issue in the chinese population operated with various surgical procedures, the cutoff value for Tg-FNA was higher in patients with thyroid tissue (before initial surgery or after lobectomy) compared to those without thyroid (55.99 vs 9.7 ng/mL, respectively) [START_REF] Zhao | Influence of presence/absence of thyroid gland on the cutoff value for thyroglobulin in lymph-node aspiration to detect metastatic papillary thyroid carcinoma[END_REF]. One of the hypotheses is that it could be related to secretion and circulation of small amounts of Tg in the lymphatic vessels and nodes. The influence of intact thyroid tissue should be also considered for example for thyroid cancer patients who have been treated by lobectomy alone, a situation that may become more frequent for low-risk thyroid cancers. One scenario which still remains unresolved is the value of Tg in washout fluid from LN in the presence of contralateral thyroid tissue. Pak et al. also recommended to raise the Tg cutoff value to 32.04 ng/mL for preoperative patients [START_REF] Pak | Diagnostic values of thyroglobulin measurement in fine-needle aspiration of lymph nodes in patients with thyroid cancer[END_REF].

Finally, another study has evaluated the value of intraoperative measurement of thyroglobulin in lymph node aspirates for guiding the proper surgical procedure. In this study, only one patient had previous history of thyroidectomy. Although the purpose of this study was different, the optimal Tg-FNA cutoff value for malignancy was high (147.5 ng/mL, sensitivity of 81.1% and specificity of 100%, 95%CI: 0.863-0.99) [START_REF] Wang | He Intraoperative measurement of thyroglobulin in lymph node aspirates for the detection of metastatic papillary thyroid carcinoma[END_REF].

We could also argue whether the manipulation of the thyroid gland during surgery might induce Tg release. Of note, LN aspiration was performed prior thyroidectomy. A recent study by Senese et al. has analyzed Tg preoperatively (T0), intraoperatively (T1), and postoperatively (T2) in a series of 67 patients. The authors found that Tg levels increased significantly from T0 to T1 and decreased significantly from T1 to T2. These results support that manipulation of the thyroid leads to changes in thyroid hormone levels but that study only analyzed changes in serum Tg and did not analyse Tg in washout fluid [START_REF] Senese | Dequanter Changes in TSH, T4, T3 and thyroglobulin levels throughout total thyroidectomy[END_REF].

It is well known that serum Tg concentrations can be affected by the presence of TgAb but this seems to not interfere with Tg in washout fluid [START_REF] Trimboli | Giovanella Measurement of thyroglobulin, calcitonin, and PTH in FNA washout fluids[END_REF]. In our study, the optimal cutoff value of FNA-Tg in determining malignant LNs is 60 ng/ml in all enrolled cases and 34 ng/ml in the thyroid cancer group only. We acknowledge several limitations of the present study such as limited number of cases mainly due to the design of the study.

In conclusion, our results support the hypothesis that presence of thyroid increases the Tg-FNA level. In this setting, the use of lower thresholds could result in false positive results and lead to unnecessary surgical intervention. Our results would need to be evaluated in patients under active surveillance or being managed after thyroid lobectomy.

Fig. 1 .

 1 Fig. 1. ROC curves for FNA-Tg in the diagnosis of malignant LN A. All cases. B. Thyroid cancer patients only.

Table 1 .

 1 Patient characteristics according to the final diagnosis of pathology thyroid. IQR Interquartile Range.

	Empty Cell

Table 2 .

 2

	Empty Cell	All cases	Thyroid cancers
	AUC (95% CI)	0.85 (0.74-0.96)	0.86 (0.72-0.99)
	Se (95% CI)	0.75 (0.58-0.87)	0.78 (0.61-0.88)
	Sp (95% CI)	0.88 (0.70-0.96)	0.83 (0.64-0.93)
	PPV (95% CI)	0.89 (0.72-0.96)	0.86 (0.69-0.94)
	NPV (95% CI)	0.72 (0.54-0.85)	0.74 (0.55-0.86)
	Accuracy (95% CI)	0.80 (0.68-0.89)	0.80 (0.68-0.88)

FNA-Tg in determining LNs malignancy AUC area under curve; Se sensitivity; Sp specificity; PPV positive predictive value; NPV negative predictive value; 95% CI 95% confidence interval.

Source of funding

This work was supported by intramural Research Program of Assistance Publique des Hôpitaux des Marseille (APHM).

Conflict of interest

None.

CRediT authorship contribution statement


Johannes Schiffer
email: schiffer@b-tu.de

Denis Efimov
email: denis.efimov@inria.fr.

Strong and weak Leonov functions for global boundedness of state periodic systems

We present new results for the analysis of global boundedness of state periodic systems. Thereby, we address both the case of systems, whose dynamics is periodic with respect to a part of the state vector and the case of systems, whose dynamics is periodic with respect to all state variables. To derive the results the notion of strong Leonov functions is introduced. The main results are complemented by a number of relaxations based on the concept of weak Leonov functions.

I. INTRODUCTION

Stability is one of the most fundamental properties in the analysis, design and engineering of dynamical systems [START_REF] Bhatia | Stability theory of dynamical systems[END_REF]- [START_REF]Stability and feedback stabilization[END_REF]. It is, hence, not surprising that the development of methods and tools to assess stability is a highly active research domain in the control systems community and beyond. As of today, the prevalent theory underpinning many modern approaches, e.g., [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Branicky | Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[END_REF]- [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], is Lyapunov theory [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]. The importance of Lyapunov-based stability methods rests upon the fact that they permit to assess boundedness and convergence properties of a system's solutions without explicit computation of the latter.

Different kinds of stability properties may be of relevance in characterizing the behavior of dynamical systems [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF], [START_REF] Khalil | Nonlinear systems[END_REF]. The classical notion of stability concerns properties of a single isolated equilibrium point [START_REF] Lyapunov | The general problem of the stability of motion[END_REF], [START_REF] Khalil | Nonlinear systems[END_REF]. However, many dynamical systems appearing in (engineering) applications exhibit multiple equilibrium points. Paramount examples comprise biological systems [START_REF] Angeli | Detection of multistability, bifurcations, and hysteresis in a large class of biological positivefeedback systems[END_REF]- [START_REF] Santos-Moreno | Multistable and dynamic CRISPRi-based synthetic circuits[END_REF], power systems and microgrids [START_REF] Schiffer | A survey on modeling of microgrids -from fundamental physics to phasors and voltage sources[END_REF]- [START_REF] Hellmann | Network-induced multistability through lossy coupling and exotic solitary states[END_REF] as well as complex networks of coupled oscillators [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF]- [START_REF] Ahmed | Robust global synchronization of Brockett oscillators[END_REF]. In recent years the classical Lyapunov theory has been extended to the case of systems with several disjoint invariant sets [START_REF] Angeli | Detection of multistability, bifurcations, and hysteresis in a large class of biological positivefeedback systems[END_REF], [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF], [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF]- [START_REF] Efimov | Global Lyapunov analysis of multistable nonlinear systems[END_REF]. A characterization of the input-to-state stability (ISS) property in the presence of multiple equilibria is provided in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], [START_REF]Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] and extended to cascaded systems in [START_REF] Forni | Input-to-state stability for cascade systems with multiple invariant sets[END_REF] to delay systems in [START_REF] Efimov | Oscillatority conditions for nonlinear systems with delay[END_REF], [START_REF] Efimov | Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids[END_REF] as well as to the integral ISS (iISS) property in [START_REF] Forni | Characterizations of integral input-to-state stability for systems with multiple invariant sets[END_REF], see also the overview given in [START_REF]The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview[END_REF]. Additional related results on robust stability analysis are given in [START_REF] Chaves | Bistable biological systems: A characterization through local compact input-to-state stability[END_REF], [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF], [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF].

In the present work, we focus on two special classes of systems, which inherently exhibit multiple equilibria. The first class are systems, whose dynamics is periodic with respect to a part of the state vector. The second class are systems, whose dynamics is periodic with respect to all state variables. In both cases, the state evolves in R n . Applications in which the former type of dynamics arises are power systems and microgrids [START_REF] Hellmann | Network-induced multistability through lossy coupling and exotic solitary states[END_REF], [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF]- [START_REF] Kuznetsov | Power supply system for aircraft with electric traction[END_REF], mechanical systems, such as multilink pendulums [START_REF] Zhong | Energy and passivity based control of the double inverted pendulum on a cart[END_REF], and phase-locked loops [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF], [START_REF] Leonov | Analytical method for computation of phase-detector characteristic[END_REF]- [START_REF] Kuznetsov | On the generalized Gardner problem for phase-locked loops in electrical grids[END_REF]. Paramount examples for the latter type of dynamics are complex networks of coupled oscillators [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF], [START_REF] Tang | Synchronization in complex networks and its application-a survey of recent advances and challenges[END_REF] and, in particular, Kuramoto oscillators [START_REF] Dörfler | Synchronization in complex networks of phase oscillators: A survey[END_REF], [START_REF] Strogatz | From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators[END_REF].

For both classes of systems, a direct application of the stability results reviewed above is highly challenging. This originates from the fact that the periodicity of the dynamics significantly complicates the construction of a Lyapunov function. More precisely, an application of the approach presented in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], [START_REF]Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] to a state periodic system would require to construct a Lyapunov function, which respects the system periodicity, i.e., it also needs to verify the definiteness requirements with respect to the equilibria (or invariant sets) as well as of its time derivative along the system's solutions. This also applies to the periodic Lyapunov functional method [START_REF] Smirnova | Stability of systems with periodic nonlinearities: a method of periodic Lyapunov functionals[END_REF]. These stringent requirements can often not be met in applications and, consequently, the majority of stability results for state periodic systems are limited to local stability analysis, see [START_REF] Dörfler | Synchronization in complex networks of phase oscillators: A survey[END_REF], [START_REF] Liu | Stability and control of power grids[END_REF].

In this context, usually the main difficulty resides in establishing global boundedness of solutions, i.e., Lagrange stability [START_REF] Bhatia | Stability theory of dynamical systems[END_REF]. For systems with infinitely many equilibria, the property of boundedness of trajectories is the natural counterpart to stability of a single equilibrium point in the classical Lyapunov theory. Once the existence of compact forward invariant sets for the system's solutions is established, there are a variety of methods available to assess their asymptotic behavior. These comprise LaSalle's invariance principle [START_REF] Khalil | Nonlinear systems[END_REF], contraction [START_REF] Lohmiller | On contraction analysis of nonlinear systems[END_REF], [START_REF] Forni | A differential Lyapunov framework for contraction analysis[END_REF] and convergence [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF], or differential positivity [START_REF] Forni | Differential positivity on compact sets[END_REF]. Hence, our main focus in the present paper is on the provision of relaxed conditions to infer global boundedness of solutions.

The path we take to address the above objective is to explicitly exploit the periodicity of the system dynamics in order to relax the (definiteness and periodicity) requirements compared to standard Lyapunov functions, while still ensuring boundedness of solutions. This approach was pioneered by Leonov [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF] and Noldus [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] for systems, whose dynamics are periodic with respect to a scalar state variable. Their main observation is as follows. In a first step and by using the periodicity of the system dynamics, one constructs unbounded, but forward invariant and globally attractive sets via a function, which is sign-indefinite in the periodic state. The second step consists of noting that the intersection of these unbounded forward invariant sets consists of globally attractive compact forward invariant and isolated sets, i.e., cells. Consequently, this method is termed cell structure approach.

The original derivations of Leonov and Noldus critically rely on the restriction that the periodic variable is scalar. In such a setting, the individual cells are only connected via equilibria, which is instrumental for the proofs in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF]. Yet, obviously, this restriction significantly limits the application of these results to networks and interconnected systems with periodic dynamics.

Motivated by the successful application of the original cell structure approach to establish global stability properties of a synchronous machine infinite bus system [START_REF] Barabanov | Almost global attractivity of a synchronous generator connected to an infinite bus[END_REF], [START_REF]Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF] as well as first extensions of the ISS theory to state periodic systems [START_REF] Efimov | Relaxing the conditions of ISS for multistable periodic systems[END_REF]- [START_REF] Schiffer | An input-to-state stability approach to verify almost global stability of a synchronousmachine-infinite-bus system[END_REF], in our previous work [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF] a multivariable cell structure framework is derived. The derivation is built upon the concept of a Leonov function and it yields sufficient conditions for boundedness of solutions of systems whose dynamics are periodic with respect to several state variables.

In a nutshell, a Leonov function is sign-indefinite with respect to the periodic states and radially unbounded with respect to the nonperiodic states. Moreover, it is negative definite with respect to the distance to a set, which separates the equilibria of the system. This last requirement is a major difference compared to the functions employed in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] and is key to establish the main result in the multivariable case, which -as already pointed out in [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] -is far from being trivial. First applications of these results to establish almost global stability properties of power systems and microgrids have been reported in [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF], [START_REF]Global synchronization analysis of droop-controlled microgridsa multivariable cell structure approach[END_REF]. More recently, also the iISS theory has been adapted to state periodic systems [START_REF] Mendoza-Avila | On relaxed conditions of integral ISS for multistable periodic systems[END_REF].

In continuation of the results given in [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], the main contributions of the present paper are three-fold.

1) The notion of a strong Leonov function (sLeF) is introduced, based on which sufficient conditions for global boundedness of systems, whose dynamics are periodic with respect to a part of the state vector are presented. Compared to our previous, related work [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], the required properties of an sLeF are refined. These refinements are instrumental for improving the proof of the main result of [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], which in turn is instrumental for the derivations in item 2) below. 2) Based on the refinements in item 1), we introduce several relaxations of our main result. These are formalized via the concept of a weak Leonov function (wLeF).

3) The properties established in items 1) and 2) are extended to the case of systems, whose dynamics are periodic with respect to the full state vector. This type of systems has not been considered previously neither in [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] nor in [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF]. The remainder of the paper is structured as follows. Preliminaries are given in Section II. Sufficient conditions for boundedness of systems, whose dynamics are periodic with respect to part of the state vector are presented in Section III based on the notions of sLeFs and wLeFs. Their counterparts for systems with purely periodic dynamics are given in Section IV. The paper is wrapped-up in Section V, in which also an outlook on future work is provided.

II. PRELIMINARIES

Denote the sets of real and integer numbers by R and Z, respectively, and let R + = {s ∈ R : s ≥ 0} as well as

Z + = Z ∩ R + . Let Z k denote a vector of k > 0 integers.
Let the map f (x) : R n → R n be of class C 0 , f (0) = 0, and consider a nonlinear system of the following form:

ẋ(t) = f (x(t)) ∀t ≥ 0 (II.1)
with the state x(t) ∈ R n . We assume that (II.1) admits uniquely defined solutions and denote by X(t, x) the solution of (II.1) at time t fulfilling X(0, x) = x. In the sequel, we also refer to X(t, x) as a trajectory of (II.1).

A set S ⊂ R n is forward invariant for the system (II.1) if for any t 0 ∈ R, X(t 0 , x) ∈ S implies that X(t, x) ∈ S for all t ≥ t 0 ∈ R and for all x ∈ R n . Likewise, a set S ⊂ R n is backward invariant for the system (II.1) if for any t b ∈ R, X(t b , x) ∈ S implies that X(t b , x) ∈ S for all t ≤ t b ∈ R and for all x ∈ R n . Finally, a set S ⊂ R n is invariant for the system (II.1) if it is both forward and backward invariant. For x ∈ R n the point y ∈ R n belongs to its ω-limit (αlimit) set if there is a sequence t i , lim i→∞ t i = +∞, such that lim i→+∞ X(t i , x) = y (lim i→+∞ X(-t i , x; 0) = y). For any x ∈ R n its αand ω-limit sets are invariant [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems, and bifurcations of vector fields[END_REF]. We define the distance from a point x ∈ R n to the set S ⊂ R n as |x| S = inf a∈S |x -a|, where |x| = |x| 0 for x ∈ R n is the usual Euclidean norm of a vector x ∈ R n . Furthermore, we introduce the vector norm |x| ∞ = max 1≤i≤n |x i |. A continuous function α : R + → R + belongs to class K if α(0) = 0 and the function is strictly increasing. The function

α : R + → R + belongs to the class K ∞ if α ∈ K and its supremum is infinity. A continuous function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R + and lim t→∞ β(s, t) = 0 for each fixed s ∈ R + .

III. BOUNDEDNESS OF DYNAMICAL SYSTEMS THAT ARE PERIODIC WITH RESPECT TO PART OF THE STATE

VARIABLES

In this section, we make the following assumption on the dynamics (II.1). Assumption III.1. Let x = (z, θ) ∈ R n , where z ∈ R k and θ ∈ R q are two components of the state vector, n = k + q, k > 0 and q > 0. The vector field f in (II.1) is 2π-periodic with respect to θ.

In other words, we suppose that the dynamics (II.1) are periodic with respect to the part of the state vector denoted by θ. Thus, the system (II.1) can be embedded into a manifold M = R k × S q by a simple projection of the variables θ on the set S q , where S = [0, 2π).

A. Strong Leonov functions and main result

We introduce the notion of a sLeF based on our previous work [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF]. For our subsequent derivations, it is convenient to introduce some auxiliary sets. Let W ⊆ R k × S q , such that for some c ∈ (0, 2π) and c ∈ (π, 2π), c < c,

{x ∈ R n : c ≤ |θ| ∞ ≤ c} ⊆ W, U = r∈Z+ U r , (III.1) U r = {x = (z, θ) ∈ R n : |θ| ∞ = 2rπ, f (x) = 0}.
Thus, the set U includes all equilibria obtained by shifting the one at the origin using the property that, by assumption, f is 2π-periodic in θ (due to continuity of f there are also other equilibria of (II.1) not included in U).

Compared to [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], in (III.1) the properties of the set W in (III.1) are made more precise. With this refinement we can present an improved proof of the main result of [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], which paves the path for the subsequently derived relaxations. To this end, we employ the following definition of a sLeF.

Definition III.2. A C 1 function V : R n → R is a strong
Leonov function (sLeF) for (II.1) with Assumption III.1 if there exist functions α ∈ K ∞ , ψ ∈ K, and a continuous function λ : R → R, λ(0) = 0 and λ(s)s > 0 for all s ̸ = 0 as well as a scalar g ≥ 0, such that for all

x = (z, θ) ∈ R n α(|z|) -ψ(|θ|) -g ≤ V (x) ∀x ∈ R n , inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0 (III.2)
and the following dissipation holds:

DV (x)f (x) ≤ -λ (V (x)) . (III.3)
We are not asking the function V to be 2π-periodic in θ. Hence, any analysis involving such a sLeF has to be carried out in R n . This observation guides the derivations of the following result.

Theorem III.3. If for the system (II.1) under Assumption III.1 there exists a sLeF as in Definition III.2, then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. Denote the set of negative values of V as

Ω = {x ∈ R n : V (x) ≤ 0}.
From the definition of the set Ω and the lower bound of the function V in (III.2) we obtain:

α(|z|) ≤ g + ψ(|θ|) ∀x = (z, θ) ∈ Ω.
Recall the sets U and W defined in (III.1), and that |θ| ≤ q|θ| ∞ . Then from the definition of a sLeF, see (III.2), the set U belongs to Ω, while W / ∈ Ω. Furthermore, the set Ω can be decomposed into two parts:

Ω = Ω ′ ∪ Ω ′′ .
The first of these sets,

Ω ′ ⊂ {x ∈ R n : |z| ≤ α -1 (g), |θ| ∞ < c}
is a compact set containing the origin, see the definition of the set W in (III.1), i.e., U 0 ⊂ Ω ′ , with g ≤ g ≤ g + ψ(qc) (the constant g evaluates the upper bound on |z| after which the growing terms in z will force V to take positive values for any |θ| ∞ < c). The second set is defined as

Ω ′′ ⊂ {x ∈ R n : |θ| ∞ > c}.
Both these sets are non-empty since 0 ∈ Ω ′ and U r ∈ Ω ′′ for r ≥ 1 due to (III.2). Furthermore, the set Ω ′′ may be unbounded. In addition, the subsets Ω ′ and Ω ′′ are separated by the set W.

Denote V (t) = V (X(t, x 0 )) for any x 0 ∈ R n , then under the conditions of the theorem we have:

V (t) + λ (V (t)) ≤ 0 ∀t ≥ 0.
Thus, clearly, V (t) is strictly decreasing while X(t, x 0 ) ∈ R n \ Ω, i.e., while V (t) > 0. Therefore, if x 0 ∈ Ω then X(t, x 0 ) ∈ Ω for all t ≥ 0 and the set Ω is forward invariant for (II.1). Conversely, if x 0 ∈ R n \ Ω then there exists 0 ≤ T x0 ≤ +∞ such that X(T x0 , x 0 ) ∈ Ω and, by invariance, the trajectory remains in this set for all t ≥ T x0 . Thus, the set Ω is globally attractive and forward invariant. Note that this property does not imply any kind of stability since Ω may be unbounded (it could also be interpreted as instability of a set containing W with the domain of repulsion R n \ Ω).

To establish stability, we exploit the periodicity of (II.1). Denote by j = [j 1 , . . . , j q ] ∈ Z q a multi-index vector.

Introduce the new variable x j = x-0 k 2πj and the function

V j (x) = V x - 0 k 2πj
for any such multi-index vector j (i.e., V 0 (x) = V (x)), where 0 k is the zero vector of dimension k. Hence

V j (x) = V j x j + 0 k 2πj = V (x j ).
Furthermore, by 2π-periodicity of f in θ,

ẋj (t) = f (x(t)) = f (x j (t)) ∀t ≥ 0.
Therefore, taking into account the properties substantiated for (II.1) and V , we obtain that the set {x j ∈ R n : V (x j ) ≤ 0} is globally attractive and forward invariant, which in the original coordinates x implies these properties for the set

Ω j = {x ∈ R n : V j (x) ≤ 0} = x ∈ R n : V x - 0 k 2πj ≤ 0 .
Using similar arguments as for Ω it is possible to show that

Ω j = Ω ′ j ∪ Ω ′′ j ,
where

Ω ′ j ⊂ {x ∈ R n : |z| ≤ α -1 (g), |θ -2πj| ∞ < c}, Ω ′′ j ⊂ {x ∈ R n : |θ -2πj| ∞ > c}.
Since the sets Ω j are forward invariant, any trajectory X(t, x 0 ) has to belong asymptotically to the intersection of these sets:

j∈Z q Ω j = j∈Z q Ω ′ j ∪ Ω ′′ j .
Next, we show that j∈Z q Ω j is composed of compact isolated subsets. At first, we note that since c < 2π, then j∈Z q Ω ′ j =

∅. Furthermore, by definition j∈Z q Ω ′′ j = ∅ due to the fact that

Ω ′′ j ⊂ R n \ Υ j with Υ j = {x ∈ R n : |θ -2πj| ∞ ≤ c} and c > π, then j∈Z q Ω ′′ j ⊂ j∈Z q R n \ Υ j = R n \ j Υ j = R n \ R n = ∅.
Recall that Ω j = Ω ′ j ∪ Ω ′′ j with j ∈ Z q and j∈Z q Ω ′ j = j∈Z q Ω ′′ j = Ω ′ j ∩ Ω ′′ j = ∅. Then,

j∈Z q Ω j = j∈Z q Ω ′ j ∪ Ω ′′ j = j∈Z q Ω ′ j ∪ j∈Z q Ω ′′ j ∪ j∈Z q   Ω ′ j ∩ k∈Z q ,k̸ =j Ω ′′ k   ∪ i,j∈Z q ,i̸ =j   Ω ′ j ∩ Ω ′ i ∩ k∈Z q ,k̸ =j,k̸ =i Ω ′′ k   ∪ i,j,s∈Z q ,i̸ =j,s̸ =j,i̸ =s   Ω ′ j ∩ Ω ′ i ∩ Ω ′ s ∩ k∈Z q ,k̸ =j,k̸ =i,k̸ =s Ω ′′ k   ∪ . . .
If Ω ′ j ∩ Ω ′ k = ∅ for j, k ∈ Z q and j ̸ = k, then all intersections in the second and other lines disappear (they all are empty sets). This is in particular the case if c ∈ (0, π), see (III.1).

If c ∈ [π, 2π), then some of the elementary intersections

Ω ′ j ∩ Ω ′ i or Ω ′ j ∩ Ω ′ i ∩ Ω ′ s , etc.
, as above, may not be empty. Yet, then they belong to Ω ′ j (by construction, as Ω ′ j is the first element of the intersections) and they form disjoint (positively separated) sets in Ω ′ j . To clarify the latter claim consider two terms:

T 1 = Ω ′ j ∩ k∈Z q ,k̸ =j Ω ′′ k and T 2 = Ω ′ j ∩ Ω ′ i ∩ k∈Z q ,k̸
=j,k̸ =i Ω ′′ k for any i, j ∈ Z q and i ̸ = j, then

T 1 ∩ T 2 = ∅ since Ω ′ i ∩ Ω ′′ i = ∅ by definition and T 1 ⊂ Ω ′′ i , while T 2 ⊂ Ω ′
i (and there is a non-zero distance between sets Ω ′ i and Ω ′′ i ). The same verification can be repeated for any other terms.

Hence,

j∈Z q Ω j = j∈Z q Ω ′ j ∪ j∈Z q Ω ′′ j ∪ j∈Z q Ω ′ j ∩ R j = j∈Z q R j ,
where R j ⊆ Ω ′ j is composed by the disjoint sets representing all non-empty intersections of Ω ′ j with other Ω ′ k and Ω ′′ i for i, j, k ∈ Z q , k ̸ = j and i ̸ = j.

Thus, the compact sets in j∈Z q Ω j form a kind of cell cover of {x ∈ R n : |z| ≤ α -1 (g)}. Recall that for all multi-index vectors j the corresponding sets Ω j are globally attracting and forward invariant. Take any x 0 ∈ R n , then X(t, x 0 ) asymptotically enters and stays in a cell R j for some j. Hence, for any x 0 ∈ R n the corresponding solution X(t, x 0 ) is bounded for all t ≥ 0.

We have the following corollary to Theorem III.3. For its presentation, we introduce the equilibrium set of the dynamics (II.1), i.e.,

D = {x ∈ R n | f (x) = 0}. (III.4)
Clearly, U ⊆ D.

Corollary III.4. If for the system (II.1) under Assumption III.1 there exists a sLeF as in Definition III.2 with

V (0) ≤ 0, inf x∈U \U0 V (x) ≤ 0, inf x∈W V (x) > 0, (III.5)
and with the following dissipation inequality

DV (x)f (x) < 0 ∀x ∈ R n \ {D ∪ Ω}, (III.6)
then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. The set

Ω = {x ∈ R n : V (x) ≤ 0}
has the same structure as in Theorem III.3. That is,

Ω = Ω ′ ∪ Ω ′′ ,
where, due to (III.5), U 0 ⊆ Ω ′ (and it is compact by definition) and there is at least one k ∈ Z + , such that U k ⊆ Ω ′′ . Thus, both Ω ′ and Ω ′′ are non-empty, but Ω ′′ may be unbounded. Clearly,

DV (x)f (x) = 0 ∀x ∈ D.
Hence, from (III.6) it follows that Ω is forward invariant and the set Ω ∪ D is globally attractive. Recall that D only contains equilibria of (II.1). This together with the fact that, due to (III.6), V is strictly decreasing for all x ∈ R n \ {D ∪ Ω}, implies that any trajectory that enters D cannot move inside D and, consequently, is bounded (even if D itself may contain an unbounded continuum of equilibria). Next, the proof of Theorem III.3 can be repeated since asymptotically all trajectories have to be in D ∪ j∈Z q (Ω ′ j ∪ Ω ′′ j ).

B. Further relaxations via weak Leonov functions

In this section, we consider the following special case of (II.1) under Assumption III.1 θ = p(x), ż = g(z, h(θ)), (III.7)

with the maps p : R n → R q , p ∈ C 0 , p(0) = 0, g : R k+w → R k , g(0, 0) = 0, and h : R q → R w , h ∈ C 0 , h(0) = 0. Since h ∈ C 0 and, under Assumption III.1, h is periodic in θ, there exists a constant h, such that

h > sup θ∈R q |h(θ)|.
(III.8)

In addition, we make the following assumption.

Assumption III.5. The dynamics ż = g(z, h(θ)) are input-tostate stable (ISS) with respect to the input h(θ). Under Assumption III.5, there exist functions β ∈ KL and γ ∈ K ∞ , such that for all z 0 ∈ R k [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]

|z(t)| ≤ β(|z 0 |, t) + γ( h), t ≥ 0, (III.9)
which implies that z is bounded.

Next the notion of a wLeF is introduced as follows.

Definition III.6. A C 1 function V : R n → R is a weak Leonov function (wLeF) for (III.7) with Assumptions III.1 and III.5 if there exist functions α i ∈ K ∞ , ψ i ∈ K, i = 1, 2, and a continuous function λ : R → R, λ(0) = 0 and λ(s)s > 0 for all s ̸ = 0 as well as constants g i ≥ 0, i = 1, 2, such that

α 2 (γ( h)) + g 2 < ψ * 2 , ψ * 2 = sup s>0 ψ 2 (s), (III.10)
where h is given in (III.8) and γ is given in (III.9), it holds for all x = (z, θ) ∈ R n that

α 1 (|z|) -ψ 1 (|θ|) -g 1 ≤ V (x) ≤ α 2 (|z|) -ψ 2 (|θ|) + g 2 inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0, (III.11) and DV (x)f (x) ≤ -λ (V (x)) (III.12)
is satisfied for either one of these three options:

a) ∀x ∈ Ω ϵ \ Ω, b) ∀x ∈ Ω ′ ϵ \ Ω ′ and α 2 (g( h)) + g 2 < ψ 2 (c), 0 < c < c, c) ∀x ∈ Ω ′′ ϵ \ Ω ′′ , where 1 Ω ϵ = Ω ′ ϵ ∪ Ω ′′ ϵ = {x ∈ R n : V (x)
≤ ϵ} for some sufficiently small ϵ > 0.

With Definition III.6, we relax the conditions of Theorem III.3, which may be important in applications. In addition, the result below corrects some flaws in its previous counterparts [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF]Corollaries 2 and 3].

Theorem III.7. If for the system (III.7) under Assumptions III.1 and III.5 there exists a wLeF as in Definition III.6, then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. Under the imposed restrictions, the solutions X(t, x 0 ) are defined for all x 0 ∈ R n and all t ≥ 0 (i.e., the system (III.7) is forward complete). Indeed, z(t) is bounded due to (III.9), while p is a continuous periodic function of θ, with bounded z. Hence, p(X(t, x 0 )) ≤ p * |z0|,

h for some finite p * |z0|, h ∈ R >0 .
According to (III.11), Ω = Ω ′ ∪ Ω ′′ as in Theorem III. [START_REF] Lyapunov | The general problem of the stability of motion[END_REF], where Ω ′ is a bounded set. Also, from (III.11), for all x 0 ∈ R n while V (X(t, x 0 )) > 0, we have

ψ 2 (|θ(t)|) ≤ α 2 (β(|z 0 |, t) + γ( h)) + g 2 , (III.13)
which in light of (III.10) together with the fact that β ∈ KL shows that for any x 0 = (z 0 , θ 0 ) the set V ⊂ {x ∈ R n : V (x) > 0}, which contains the solutions X(t, x 0 ) under the constraint V (t) > 0, is a bounded set for t → ∞. Case a). Since (III.12) is verified on Ω ϵ \ Ω, then Ω is locally attractive and forward invariant. For any x 0 ∈ R n there are two options. First, X(t, x 0 ) stays in V for all t ≥ 0 and, consequently, the trajectory is bounded (i.e., the set V contains forward invariant subsets for (III.7)). Alternatively, at some point in time 0 ≤ T x0 ≤ +∞, X(t, x 0 ) crosses to Ω and, by invariance, the trajectory remains in Ω for all t ≥ T x0 . In this case, boundedness of the trajectory has to be analyzed following the arguments based on the periodicity of the system (III.7) from Theorem III.3. With V j (x) as in the proof of Theorem III.3, we have the sets

Ω j = Ω ′ j ∪ Ω ′′ j = {x ∈ R n : V j (x) ≤ 0}, V j ⊂ {x ∈ R n : V j (x) > 0}.
(III.14) 1 Recall from the proof of Theorem III.3 that Ω = {x ∈ R n : V (x) ≤ 0}. Hence, by definition Ω ⊂ Ωϵ and since

Ω = Ω ′ ∪ Ω ′′ , then Ωϵ = Ω ′ ϵ ∪ Ω ′′ ϵ , where Ω ′ ⊂ Ω ′
ϵ and Ω ′′ ⊂ Ω ′′ ϵ are separated sets for a sufficiently small ϵ > 0.

By using the same arguments as for Ω and V it is possible to show that Ω ′ j and V j are bounded sets and that Ω j is locally attractive and forward invariant and that V j may contain forward invariant subsets.

From the above considerations, it follows that for each j ∈ Z q , all trajectories of (III.7) asymptotically belong to the union

Ω ′ j ∪ Ω ′′ j ∪ V j = Ω ′ j ∪ Ω ′′ j ∪ V j , (III.15)
which implies that asymptotically all trajectories belong to the intersection of the sets in (III.15), i.e., to

j∈Z q Ω ′ j ∪ Ω ′′ j ∪ V j .
For each j ∈ Z q , if a trajectory is asymptotically settled either in Ω ′ j or in V j , then it has to be bounded by boundedness of the respective set. If the trajectory is in Ω ′′ j , then the periodicity arguments can be used by repeating the same consideration for other indexes j ∈ Z q and we recall that j Ω ′′ j = ∅. This shows that no trajectory can be in Ω ′′ j for all j ∈ Z q , which implies that any trajectory is captured by some V j or Ω ′ j . Hence, it is bounded. Case b). In this case the set Ω ′ is forward invariant and locally attractive and the properties of Ω ′′ are not specified by the conditions of the theorem. Since α 2 (g( h)) + g 2 -ψ 2 (c) < 0, with (III.10) we have asymptotically that any X(t, x 0 ) ∈ V satisfies

|θ| ∞ ≤ |θ| ≤ ψ -1 2 α 2 (γ( h)) + g 2 < c.
This implies that the set Ω ′ ∪ V has similar properties to Ω ′ in Theorem III.3. Most importantly, since

V ⊂ {x ∈ R n : |θ| ∞ < c}, Ω ′′ ⊂ {x ∈ R n : |θ| ∞ > c},
V and Ω ′′ are separated sets, which means that there can be no trajectory cycling between V and Ω ′′ . The rest of the proof repeats the arguments of case a).

Case c). In this case the set Ω ′′ is forward invariant and locally attractive and the properties of Ω ′ are not specified by the conditions of the theorem. For any x 0 ∈ R n there are two options. First, X(t, x 0 ) stays in V ∪ Ω ′ for all t ≥ 0 and, consequently, the trajectory is bounded (i.e., the set V ∪ Ω ′ contains forward invariant subsets for (III.7)). Alternatively, at some point in time 0 ≤ T x0 ≤ +∞, X(t, x 0 ) crosses to Ω ′′ and, by invariance, the trajectory remains in Ω ′′ for all t ≥ T x0 . If the trajectory is in Ω ′′ , then the periodicity arguments can be used by repeating the same consideration for other indexes j ∈ Z q and we recall that j Ω ′′ j = ∅. This shows that no trajectory can be in Ω ′′ j for all j ∈ Z q , which implies that any trajectory is captured by some V j or Ω ′ j . Hence, it is bounded. The same reasoning applies for the special case Ω ′ = ∅ (which only requires a slight redefinition of the set U).

IV. BOUNDEDNESS OF DYNAMICAL SYSTEMS THAT ARE PERIODIC WITH RESPECT TO ALL STATE VARIABLES

Next, we focus on systems, whose dynamics are periodic with respect to all state variables. This is formalized in the assumption below.

Assumption IV.1. The vector field f in (II.1) is 2π-periodic, i.e., f (x+j2π) = f (x), where j ∈ Z n is a multi-index vector, for all x ∈ R n .

Clearly, Assumption IV.1 implies that x = θ ∈ R n .

A. Strong Leonov functions and main result

As in the previous section, it is convenient to introduce the auxiliary sets W and U, which under Assumption IV.1 become

{x ∈ R n : c ≤ |x| ∞ ≤ c} ⊆ W, U = r∈Z+ U r , (IV.1) U r = {x ∈ R n : |x| ∞ = 2rπ, f (x) = 0}.
We employ the following definition.

Definition IV.2. We say that a C 1 function V : R n → R is a strong Leonov function (sLeF) for (II.1) with Assumption IV.1 if

inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0 (IV.2)
and there exists a continuous function λ : R → R satisfying λ(0) = 0 and λ(s)s > 0 for all s ̸ = 0, such that

∂V (x) ∂x f (x) ≤ -λ (V (x)) ∀x ∈ R n . (IV.3)
As in Definition III.2, we are not asking the function V to be 2π-periodic, hence, any analysis involving such a sLeF has to be carried out in R n . Moreover, compared to Definition III.2 no requirements on lower bounds of V (x) are made, since x = θ, i.e., the dynamics (II.1) are assumed to be periodic with respect to all state variables x = θ.

Theorem IV.3. If for the system (II.1) under Assumption III.1 there exists a sLeF as in Definition IV.2, then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0. Proof. Denote the set of negative values of V as

Ω = {x ∈ R n : V (x) ≤ 0}.
Recall the sets U and W defined in (IV.1). Then from the definition of a sLeF, see (IV.2), the set U belongs to Ω, while W / ∈ Ω. Furthermore, the set Ω can be decomposed into two parts: Ω = Ω ′ ∪ Ω ′′ . The first of these sets,

Ω ′ ⊂ {x ∈ R n : |x| ∞ < c}
is a compact set containing the origin, see the definition of the set W in (IV.1), i.e. U 0 ⊂ Ω ′ . The second set is defined as

Ω ′′ ⊂ {x ∈ R n : |x| ∞ > c}.
Both these sets are non-empty since 0 ∈ Ω ′ and U r ∈ Ω ′′ for r ≥ 1 due to (IV.2). Furthermore, the set Ω ′′ is unbounded. In addition, the subsets Ω ′ and Ω ′′ are separated by the set W.

Denote V (t) = V (X(t, x 0 )) for any x 0 ∈ R n , then under the conditions of the theorem we have:

V (t) + λ (V (t)) ≤ 0 ∀t ≥ 0.
Thus, clearly, V (t) is strictly decreasing while X(t, x 0 ) ∈ R n \ Ω, i.e., while V (t) > 0. Therefore, if x 0 ∈ Ω then X(t, x 0 ) ∈ Ω for all t ≥ 0 and the set Ω is forward invariant for (II.1). Conversely, if x 0 ∈ R n \ Ω then there exists 0 ≤ T x0 ≤ +∞ such that X(T x0 , x 0 ) ∈ Ω and, by invariance, the trajectory remains in this set for all t ≥ T x0 . Thus, the set Ω is globally attractive and forward invariant. Note that this property does not imply any kind of stability since Ω may be unbounded (it could also be interpreted as instability of a set containing W with the domain of repulsion R n \ Ω).

From here on, the claim is completed by following the steps of the proof of Theorem III.3.

We have the following corollary to Theorem IV.3. For its presentation, we recall the equilibrium set D of the dynamics (II.1) defined in (III.4).

Corollary IV.4. If for the system (II.1) under Assumption IV.1 there exists a sLeF as in Definition IV.2 with

V (0) ≤ 0, inf x∈U \U0 V (x) ≤ 0, inf x∈W V (x) > 0, (IV.4)
and with the following dissipation inequality

DV (x)f (x) < 0 ∀x ∈ R n \ {D ∪ Ω}, (IV.5)
then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. The claim is established by following the steps of the proof of Corollary III.4.

B. Further relaxations via weak Leonov functions

With the aim of relaxing the conditions of Theorem IV.3, we introduce the following notion.

Definition IV.5. Consider a C 1 function V : R n → R and let

Ξ := {x ∈ R n : V (x) > 0}, Ω := R n \ Ξ. (IV.6)
Then, V is a weak Leonov function (wLeF) for (II.1) with Assumption IV.1 if there exist constants c ∈ (0, 2π) and c ∈ (π, 2π), c < c, such that

{x ∈ R n : c < |x| ∞ < c} ⊂ Ξ ⊂ {x ∈ R n : |x| ∞ < 2π}, (IV.7) and ∂V (x) ∂x f (x) ≤ 0 ∀x ∈ ∂Ξ, (IV.8)
where ∂Ξ denotes the boundary of Ξ.

The function V takes positive values on the compact set Ξ and nonpositive values on the set Ω. Thus, roughly speaking, the function V is negative definite with respect to the distance to the set Ξ (it takes positive values on the set Ξ and negative values on its complement Ω). Compared to a sLeF, see Definition IV.2, in the case of a wLeF Ω does not necessarily contain the origin. Also, its time derivative on the interior of the set Ξ is not further specified. Yet, since V is a C 1 function and Ξ is a bounded set, we know that sup x∈Ξ V (x) < +∞.

Theorem IV.6. If for the system (II.1) under Assumption IV.1 there exists a wLeF as in Definition IV.5, then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

Proof. In general, the set Ω defined in (IV.6) can be decomposed into two parts:

Ω = Ω ′ ∪ Ω ′′ .
The first of these sets,

Ω ′ ⊂ {x ∈ R n : |x| ∞ < c}, maybe empty (if 0 ∈ Ξ).
If it is non-empty, then Ω ′ is a compact set for some c ∈ (0, 2π), where the constant c is determined by the properties of the set Ξ stated in (IV.7). The second set is defined as

Ω ′′ ⊂ {x ∈ R n : |x| ∞ > c},
where again the constant c is determined by the properties of the set Ξ. Furthermore, the set Ω ′′ is non-empty and unbounded. In addition, if Ω ′ is non-empty, the subsets Ω ′ and Ω ′′ are separated by the set Ξ.

If x 0 ∈ Ω, then under the conditions of the corollary X(t, x 0 ) ∈ Ω for all t ≥ 0 since

sup x∈Ω V (x) ≤ 0, inf x∈Ξ V (x) ≥ 0, Ξ = R n \ Ω and ∂V (x) ∂x f (x) ≤ 0 ∀x ∈ ∂Ξ.
Thus, the set Ω is forward invariant for (II.1). Conversely, suppose that x 0 ∈ Ξ = R n \ Ω. Then, the behavior of V (t) = V (X(t, x 0 )) in the interior of Ξ is not specified by the conditions of the theorem. Thus, there are two possibilities. Either X(t, x 0 ) stays in the compact set Ξ for all t ≥ 0 or at some point in time 0 ≤ T x0 ≤ +∞ it crosses to Ω and, by invariance, the trajectory remains in Ω for all t ≥ T x0 . Note that these properties do not imply any kind of stability since Ω is unbounded.

To establish stability, we exploit the periodicity of (II.1). Denote by j = [j 1 , . . . , j n ] ∈ Z n a multi-index vector. Introduce the new variable x j = x -2πj and the function V j (x) = V (x -2πj) for any such multi-index vector j (i.e., V 0 (x) = V (x)). Hence V j (x) = V j (x j + 2πj) = V (x j ).

Furthermore, by 2π-periodicity of f , ẋj (t) = f (x(t)) = f (x j (t)) ∀t ≥ 0.

Therefore, taking into account the properties substantiated for (II.1) and V , we obtain that the set

Ω j = {x ∈ R n : V j (x) ≤ 0} = {x ∈ R n : V (x -2πj) ≤ 0} is forward invariant. By letting {x ∈ R n : c < |x -2πj| ∞ < c} ⊂ Ξ j = {x ∈ R n : V j (x) > 0} ⊂ {x ∈ R n : |x -2πj| ∞ < 2π}
and using similar arguments as for Ω it is possible to show that Ω j = Ω ′ j ∪ Ω ′′ j , where

Ω ′ j ⊂ {x ∈ R n : |x -2πj| ∞ < c}, Ω ′′ j ⊂ {x ∈ R n : |x -2πj| ∞ > c}.
In order to show that under the conditions of the corollary, all solutions of (II.1) are bounded, we make the following observations. As it has been shown above for any x 0 ∈ R n the respective trajectory X(t, x 0 ) has at most three possibilities: 1) If Ω ′ j ̸ = ∅, it reaches Ω ′ j for some j ∈ Z n . Then it has to stay there by invariance of Ω ′ j , which also implies boundedness of the trajectory due to compactness of Ω ′ j .

2) It stays in an invariant subset of Ξ j . Then the trajectory is also bounded due to the same property of Ξ j . 3) It is captured by Ω ′′ j , which is unbounded and this case has to be analyzed. Note that ∩ j∈Z n Ω ′′ j = ∅ (the proof is given in Theorem IV.3). Thus, it has to be in an intersection of Ω ′′ j with Ω ′ k or Ξ k for some j, k ∈ Z n with k ̸ = j. This implies boundedness of the trajectory X(t, x 0 ).

V. CONCLUSIONS

We have addressed the problem of establishing global boundedness of solutions for two classes of state periodic systems -namely, systems, whose dynamics are periodic either with respect to a part of the state vector or with respect to the full state vector. Both classes of systems are frequently encountered in diverse applications, such as AC power systems or networks of oscillators. Due to their periodicity, such systems always exhibit multiple equilibria. This significantly complicates the derivation of global system properties.

This challenge has been tackled by introducing the notion of sLeFs, based on which we have provided sufficient conditions for global boundedness of solutions. Several relaxations of this main result have been derived via the concept of wLeFs.

Our current work is geared towards the application of the presented methodical framework to asses global stability properties of prominent engineering examples. These comprise, in particular, low-inertia power systems with dynamical power line models [START_REF] Schiffer | A survey on modeling of microgrids -from fundamental physics to phasors and voltage sources[END_REF], [START_REF] Liu | Stability and control of power grids[END_REF] and networks of coupled Kuramoto oscillators [START_REF] Dörfler | Synchronization in complex networks of phase oscillators: A survey[END_REF]. Furthermore, we seek to develop techniques to explicitly construct Leonov functions for particular classes of systems, which are periodic with respect to all or part of the states.

This work is partially supported by the ANR-DFG project SyNPiD -Project number 446182476.


Panelphilippe Thuillier

Danielle Benisvy

Catherine Ansquer C , Ber Nard Corvilain

Eric Mirallié

David Taïeb

Françoise Borson- Chazot

Charlotte Lussey-Lepoutre

SFE-AFCE-SFMN 2022 Consensus on the management of thyroid nodules : What is the role of functional imaging and isotopic treatment? Author links open overlay

Keywords:

SFE-AFCE-SFMN 2022 Consensus on the management

of thyroid nodules : What is the role of functional imaging and isotopic treatment?

1. Thyroid scintigraphy 1.1. When should thyroid scan be performed in case of thyroid nodule?

Thyroid scintigraphy (TS) is a prime example of molecular imaging. TS is the only technique that provides functional information regarding a nodule. It is systematically indicated in the assessment of thyroid nodules associated with clinical or subclinical hyperthyroidism (below the lower limit of normal, generally around 0.4 mIU/L). It determines whether the nodule is autonomous or not and possibly associated with diffuse functional autonomy. The risk of malignancy is very low in autonomous nodules [1]. TS is classically not recommended if serum TSH is in the normal range. However, TSH may remain normal in autonomous nodules when iodine intake is suboptimal, suggesting a slightly higher TSH threshold for TS [2], [START_REF] Rosario | Calsolari Is radioiodine scintigraphy still of value in thyroid nodules with indeterminate cytology?: a prospective study in an iodine-sufficient area[END_REF], especially when fine-needle cytology (FNAC) is indicated. The prevalence of doubtful FNAC results (Bethesda III and, more rarely, IV) is higher for autonomous than for non-functional nodules, and FNAC is not indicated in this setting [1], as it could lead to unnecessary surgeries.

What type of radiotracer should be used?

TS can be performed with iodine-123 (123I) or technetium-99 m ( 99m TcO4-, 99m Tc). Qualitatively, images with 99m Tc and 123 I are grossly equivalent but 123 I offers additional quantitative information. 123 I uptake is proportional to the NIS protein density and the integrity of the organification process. By contrast, 99m Tc is taken up by thyrocytes via the same iodine symporter, but rapidly released from the cells [START_REF] Giovanella | Clerc Molecular imaging for thyrotoxicosis and thyroid nodules[END_REF]. Elimination is mainly urinary. These isotopes are chosen for their short half-life, their imaging quality and lower radiation exposure compared to iodine-131 ( 131 I). Low-activity iodine-131 (2 to 5 MBq) is no longer used for thyroid nodule characterization, but can be used for dosimetric purposes. The advantages and disadvantages of 99m Tc and 123 I are presented in Table 1. Although both are readily available, 123 I has to be ordered specifically and at higher cost, which may limit its use in daily practice. ATA guidelines favor 123 I [1], in contrast to the EANM and the AACE/ACE/AME guidelines [START_REF] Gharib | American Association of Clinical Endocrinologists, American College of Endocrinology, and Associazione Medici Endocrinologi Medical Guidelines for Clinical Practice for the Diagnosis and Management of Thyroid Nodules--2016[END_REF], [START_REF] Verburg | Why the European Association of Nuclear Medicine has declined to endorse the 2015 American Thyroid Association management guidelines for adult patients with thyroid nodules and differentiated thyroid cancer[END_REF]. [START_REF] Rosario | Calsolari Is radioiodine scintigraphy still of value in thyroid nodules with indeterminate cytology?: a prospective study in an iodine-sufficient area[END_REF]. Their study was conducted in a geographical region where iodine intake is considered sufficient; only 78 patients with >1 cm nodules (84 nodules), without highly suspicious ultrasound criteria and without history of childhood irradiation, increased calcitoninemia or family history of thyroid cancer, all operated on, were included. Twenty-one nodules (25%) were hot on scintigraphy performed at 24 hours post-injection, with higher frequency when TSH was lower: 50%, 33.3%, 12% and 0% for TSH respectively < 1 mIU/l (n = 16), 1-2 mIU/l (n = 24), 2-3 mIU/l (n = 25) and > 3 mIU/l (n = 13). None of the hot nodules were malignant (20 benign and 1 tumor of uncertain malignancy), confirming the good NPV of nodular hyperfixation on thyroid scintigraphy to rule out malignancy. Of the 63 nodules without iodine uptake, only 7 (11%) were malignant, the others being uncertain (n = 3), NIFT-P (n = 9) or benign (n = 64). Thyroid scintigraphy may be useful to characterize of Bethesda III or IV nodules in some cases. In a recent meta-analysis, including 8 studies and 2761 hot nodules on thyroid scan, 50% (95% CI: 32-68%) of patients had normal TSH. This meta-analysis included series mainly from Europe, where moderate iodine deficiency is common [START_REF] Treglia | Prevalence of normal TSH value among patients with autonomously functioning thyroid nodule[END_REF]. The study by Chami et al. also showed that nodule size in relation to normal TSH is an imperfect criterion despite the inverse relationship between nodule size and TSH level in autonomous nodules [2]. The contribution of thyroid scintigraphy therefore needs to be assessed by comparing TSH level with nodule size and ultrasound characteristics, and should be validated prospectively. Recommendation 5.5

In Bethesda III nodules, when TSH is < 1 mIU/l, thyroid scintigraphy, ideally with iodine 123, may be considered to search for an autonomous nodule. However, the added value of combined ultrasound/cytological classification approach has not been assessed. Grade C ++ 2.2. 99m Tc-methoxyisobutylisonitrile ( 99m Tc-MIBI) scintigraphy for the diagnosis of indeterminate nodules (Bethesda III and IV),

MIBI is a lipophilic monovalent cation of the isonitrile family, which crosses the cell membrane and concentrates in the mitochondria due to the electrical potential. In thyroid nodules, 99m Tc-MIBI uptake reflects mitochondrial activity and consequently cellular oxidative metabolism. Intense 99m Tc-MIBI uptake is thus expected in cancer cells, but is not specific for malignancy.

There is a large body of literature, mainly before 2016, on the role of 99m Tc-MIBI scintigraphy in the characterization of thyroid nodules. Nodule selection criteria were heterogeneous, and imaging procedures and interpretation criteria were not standardized. Probably because of these disparities, and the lack of specificity of 99m Tc-MIBI fixation, the technique has not become widespread, despite an excellent NPV of 80 to 100%, depending on the study [START_REF] Rager | Radioisotope imaging for discriminating benign from malignant cytologically indeterminate thyroid nodules[END_REF].

False positives are numerous with 99m Tc-MIBI, including situations as diverse as hyperplastic nodular goiter, macro-or micro-follicular adenoma, oncocytic adenoma or autoimmune or subacute nodular thyroiditis.

Recent reports indicate that the performance of 99m Tc-MIBI scintigraphy can be improved by coupling the examination to a thyroid scan (to avoid study of usually benign hot nodules), by performing a tomoscintigraphy (3D scan) coupled to the CT scan, and by studying the retention of the tracer [START_REF] Benderradji | Quantitative dual isotope 123iodine/99mTc-MIBI scintigraphy: A new approach to rule out malignancy in thyroid nodules[END_REF], [START_REF] Campennì | 99 m)Tc-Methoxy-Isobutyl-Isonitrile scintigraphy is a useful tool for assessing the risk of malignancy in thyroid nodules with indeterminate fineneedle cytology[END_REF], [START_REF] Giovanella | Molecular imaging with (99 m)Tc-MIBI and molecular testing for mutations in differentiating benign from malignant follicular neoplasm: a prospective comparison[END_REF]. The results of two prospective studies by the same team [START_REF] Campennì | 99 m)Tc-Methoxy-Isobutyl-Isonitrile scintigraphy is a useful tool for assessing the risk of malignancy in thyroid nodules with indeterminate fineneedle cytology[END_REF], [START_REF] Giovanella | Molecular imaging with (99 m)Tc-MIBI and molecular testing for mutations in differentiating benign from malignant follicular neoplasm: a prospective comparison[END_REF], involving 105 and 61 operated patients, showing excellent results in terms of sensitivity and NPV, are presented in Table 2. Semi-quantitative analysis ( 99m Tc-MIBI washout analysis) also improved the performance of the examination, which was superior to that of molecular analysis (KRAS, HRAS, NRAS, BRAF mutations and translocations of PAX8/PPARγ, RET/PTC1 and RET/PTC3) [START_REF] Giovanella | Molecular imaging with (99 m)Tc-MIBI and molecular testing for mutations in differentiating benign from malignant follicular neoplasm: a prospective comparison[END_REF]. In conclusion, 99m Tc-MIBI scintigraphy has an excellent NPV for ruling out malignancy in thyroid nodules, but low specificity and PPV due to tracer uptake by many benign tumor types. Recommendation 5.6

99m Tc-MIBI scintigraphy can be performed for thyroid nodules >15 mm, not hot on thyroid scintigraphy ( 99m Tc or 123 I) and indeterminate on FNAC (Bethesda III-IV), due to its high NPV. Grade C ++. The level of evidence and standardization of imaging procedures need to be validated prospectively in a multi-center setting.

2.3. 18 F-FDG-PET/CT for diagnosis of indeterminate nodules (Bethesda III and IV), and cost-benefit ratio 18 F-FDG is a labeled glucose analogue. It is used primarily to assess the metabolic characteristics of tissues. Like for 99m Tc-MIBI, intense uptake of 18 F-FDG is observed in many cancer cells, but is not always specific enough to differentiate a benign process from malignant pathology. The results of major studies and meta-analyses on the value of 18 F-FDG-PET/CT in cytologically indeterminate thyroid nodules (Bethesda III-IV) are presented in Table 3. A review of the literature and meta-analysis published in 2011 [START_REF] Vriens | Oei The role of [18F]-2-fluoro-2-deoxy-d-glucose-positron emission tomography in thyroid nodules with indeterminate fine-needle aspiration biopsy: systematic review and meta-analysis of the literature[END_REF] found excellent diagnostic performance, with a sensitivity of 96% and NPV of 95%, partly in contradiction with a more recent meta-analysis of 2019, including 8 studies (n = 8) [START_REF] Castellana | Performance of 18F-FDG PET/CT in selecting thyroid nodules with indeterminate fine-needle aspiration cytology for surgery. A systematic review and a meta-analysis[END_REF], where sensitivity and NPV were only 74%. Several studies showed improvement in both parameters in patients with >1 cm thyroid nodules [START_REF] Muñoz Pérez | Could 18F-FDG-PET/CT avoid unnecessary thyroidectomies in patients with cytological diagnosis of follicular neoplasm?[END_REF], [START_REF] Piccardo | Thyroid nodules with indeterminate cytology: prospective comparison between 18F-FDG-PET/CT, multiparametric neck ultrasonography, 99mTc-MIBI scintigraphy and histology[END_REF]. In 2011, the meta-analysis by Vriens et al. found 100% sensitivity and NPV in the subgroup with >15 mm lesion [START_REF] Vriens | Oei The role of [18F]-2-fluoro-2-deoxy-d-glucose-positron emission tomography in thyroid nodules with indeterminate fine-needle aspiration biopsy: systematic review and meta-analysis of the literature[END_REF]. Again, these excellent performances do not seem to have been confirmed in more recent studies [START_REF] Deandreis | Is (18)F-fluorodeoxyglucose-PET/CT useful for the presurgical characterization of thyroid nodules with indeterminate fine needle aspiration cytology[END_REF], [START_REF] Nguyen | PET/CT and prediction of thyroid cancer in patients with follicular neoplasm or atypia[END_REF] and data for 18 F-FDG-PET/CT performance according to nodule size are finally quite limited [START_REF] Castellana | Performance of 18F-FDG PET/CT in selecting thyroid nodules with indeterminate fine-needle aspiration cytology for surgery. A systematic review and a meta-analysis[END_REF].

Specificity and PPV values are also rather disappointing. The use of a SUVmax threshold [START_REF] Rosario | Calsolari Fluorine-18-fluorodeoxyglucose positron emission tomography in thyroid nodules with indeterminate cytology: a prospective study[END_REF] or SUVmax ratio (SUVmax nodule/SUVmax contralateral thyroid lobe) [START_REF] Piccardo | Indeterminate thyroid nodules. The role of 18F-FDG PET/CT in the "era" of ultrasonography risk stratification systems and new thyroid cytology classifications[END_REF] has been proposed to improve specificity and positive predictive value (PPV), but without consensus or optimal threshold recognized in the literature. Several studies investigated the contribution of ultrasound and FNAC subgroups and the additional or combined value of 18 F-FDG-PET/CT in this context (Table 3), without finding any added value of 18 F-FDG-PET/CT over and above ultrasound characterization with TIRADS or EU-TIRADS score [START_REF] Deandreis | Is (18)F-fluorodeoxyglucose-PET/CT useful for the presurgical characterization of thyroid nodules with indeterminate fine needle aspiration cytology[END_REF], [START_REF] Piccardo | Indeterminate thyroid nodules. The role of 18F-FDG PET/CT in the "era" of ultrasonography risk stratification systems and new thyroid cytology classifications[END_REF].

In conclusion, the most recent data on 18 F-FDG-PET/CT showed lower sensitivity and NPV than previously reported. Semi-quantitative analysis (i.e., SUV) increases specificity and PPV but at the expense of sensitivity and NPV, and there is no consensual threshold reported in the literature. Data combining ultrasound/cytological classifications and visual analysis of 18 F-FDG-PET/CT do not appear to show any added value of 18 F-FDG-PET/CT in predicting malignancy in all thyroid nodules with indeterminate FNAC. Finally, a recent prospective study investigated the value of another PET tracer, 18 F-Fluorocholine in the characterization of ≥ 15 mm thyroid nodules, indeterminate on FNAC (Bethesda III-IV). The results showed a high NPV of 96% but a low PPV of 29%. Further studies are needed to better assess the contribution of this radiotracer in this indication [START_REF] Ciappuccini | F-Fluorocholine Positron Emission Tomography/Computed tomography is a highly sensitive but poorly specific tool for identifying malignancy in thyroid nodules with indeterminate cytology: the Chocolate study[END_REF]. Recommendation 5.7:

18 F-FDG-PET/CT is not recommended for FNAC-indeterminate thyroid nodules (Bethesda III-IV), due to suboptimal NPV in recent studies and the lack of added value over and above combined ultrasound/cytology classification approach. Grade A ++

Radioactive iodine therapy for autonomous thyroid nodules

Iodine-131 is a high-energy emitter with a sufficiently long half-life (8.04 days) to be used in therapy. In autonomous nodules, treatment is given as a single dose, in the form of a capsule, usually in an outpatient setting. In exceptional cases, treatment can be delivered in hospital, and/or in liquid form (subject to a specific temporary authorization).

Autonomous nodules are treated radically, surgically or with radioactive iodine (RAI) when they are the cause of frank hyperthyroidism, whatever the patient's age, and in sub-clinical hyperthyroidism, depending on the context.

Thus, in overt hyperthyroidism with TSH < 0.1mIU/L, radical treatment is recommended for all subjects over 65 years of age, and in symptomatic or asymptomatic patients under 65 years of age but with cardiac comorbidity, osteoporosis or menopause without hormone replacement. Treatment may be discussed for asymptomatic patients under 65 years of age.

In subclinical hyperthyroidism with TSH between 0.1 and 0.4 mIU/L, treatment is discussed on a case-by-case basis, depending on age, symptomatic features of the hyperthyroidism, and comorbidities [START_REF] Ross | American Thyroid Association Guidelines for diagnosis and management of hyperthyroidism and other causes of thyrotoxicosis[END_REF].

The choice of RAI therapy rather than surgery depends on the patient's age and comorbidities, contraindications to surgery, tolerance of the hyperthyroidism, compressive nature of the nodule and the patient's wishes. RAI therapy is the treatment of choice for non-compressive autonomous nodules, not suspicious on ultrasound, and for hyperthyroidism not requiring an urgent therapeutic solution. It is, of course, contraindicated in case of pregnancy and breastfeeding.

Preparing the patient for therapeutic administration of iodine 131 involves several stages:

• ultrasound evaluation of the volume of the autonomous nodule, possibly weighted by functional volume on scintigraphy [START_REF] Clerc | Radioiodine therapy of thyroid autonomy Q[END_REF]; • informing the patient about the procedure for this type of metabolic radiotherapy and the associated radiation protection measures [START_REF] Silberstein | The SNMMI practice guideline for therapy of thyroid disease with 131I 3.0[END_REF]; • checking possible iodine overload;

• if synthetic anti-thyroid drugs have been used previously, it is strongly recommended that they should be discontinued 5-7 days before treatment, to avoid increasing the risk of permanent secondary hypothyroidism [START_REF] Bernard | Radioiodine therapy in benign thyroid disorders[END_REF]. Any synthetic anti-thyroids may be resumed at a minimum 7 days after RAI therapy. If the TSH is low but not suppressed and there is still iodine uptake in the non-autonomized thyroid parenchyma, a short course of LT3-based hormone therapy may be considered in preparation for RAI therapy, after assessment of the clinical context and depending on tolerance. A short course of LT3 is generally well tolerated in the absence of cardiac comorbidity and, by extinguishing iodine uptake in the non-autonomized thyroid parenchyma, may reduce the risk of secondary hypothyroidism, although this was not demonstrated in case-control cohort studies [START_REF] Greilsamer | Management of thyroid nodules incidentally discovered on MIBI scanning for primary hyperparathyroidism[END_REF], [START_REF] Zakavi | Davachi Comparison of four different protocols of I-131 therapy for treating single toxic thyroid nodule[END_REF], [START_REF] Ceccarelli | Pinchera Outcome of radioiodine-131 therapy in hyperfunctioning thyroid nodules: a 20 years' retrospective study[END_REF]. The therapeutic goal in hyperthyroidism due to an autonomous nodule is ideally the restoration of euthyroidism, avoiding both too low a level of radioactivity, leading to failure, and too high a level, leading to hypothyroidism in the long term.

There is no international consensus to determine radioactivity level 131. The data on therapeutic efficacy and tolerance come from studies of heterogeneous patient populations, all causes of hyperthyroidism combined. Similarly, practices in France are heterogeneous [START_REF] Bernard | Radioiodine therapy in benign thyroid disorders[END_REF]. There are several approaches to metabolic radiotherapy in hyperthyroidism:

• fixed radioactivity, depending on the cause of hyperthyroidism; however, practices are non-standardized and heterogeneous, varying in France from 185MBq to 740 MBq [START_REF] Bernard | Radioiodine therapy in benign thyroid disorders[END_REF]; • calculated radioactivity level:

∘ according to a semi-quantitative method based on the mass of tissue to be treated (depending on the publications from 3.3 MBq to 8 MBq/g of tissue), ∘ or according to dosimetric methods taking account not only of mass but also of the dose (in Grays) to be delivered, iodine uptake, and the effective half-life of the iodine in the tissue to be treated [START_REF] Gharib | American Association of Clinical Endocrinologists, American College of Endocrinology, and Associazione Medici Endocrinologi Medical Guidelines for Clinical Practice for the Diagnosis and Management of Thyroid Nodules--2016[END_REF]. This method uses Marinelli's approach based on iodine uptake observed several days after administration of low-activity iodine-131. Dosimetric measurements based on iodine-123 uptake at 2 h, 4 h or 24 h are also feasible, by applying matching factors.

Few studies compared efficacy and safety between these methods in this indication. Only one randomized study in patients with autonomous nodules showed equivalent efficacy with lower risk of hypothyroidism for calculated than fixed radioactivity methods [START_REF] Zakavi | Davachi Comparison of four different protocols of I-131 therapy for treating single toxic thyroid nodule[END_REF], [START_REF] Leitha | Dosimetry-guided radioiodine therapy of hyperthyroidism: long-term experience and implications for guidelines[END_REF], [START_REF] Reiners | Radioiodine therapy of thyroid autonomy[END_REF], [START_REF] Rokni | Efficacy of different protocols of radioiodine therapy for treatment of toxic nodular goiter: systematic review and meta-analysis of the literature[END_REF]. However, although more complex to implement, dosimetric methods should ideally be preferred, as internal vectored radiotherapy should be performed with "the lowest effective activity possible" (ALARA principle: As Low As Reasonably Achievable).

The rate of successful restoration of euthyroidism after a single course of iodine-131 in this indication ranges from 75% to 95% [START_REF] Ceccarelli | Pinchera Outcome of radioiodine-131 therapy in hyperfunctioning thyroid nodules: a 20 years' retrospective study[END_REF], [START_REF] Reiners | Radioiodine therapy of thyroid autonomy[END_REF], [START_REF] Nygaard | Long-term effect of radioactive iodine on thyroid function and size in patients with solitary autonomously functioning toxic thyroid nodules[END_REF], including the case of large nodules weighing more than 50 g [32]. The speed of regression of hyperthyroidism depends on factors influencing iodine uptake, and correlates with radioactivity level [START_REF] Reiners | Radioiodine therapy of thyroid autonomy[END_REF]. The risk of secondary hypothyroidism persists for up to 20 years after RAI therapy, with cumulative rates of approximately 7% at 6 months to 30% at 20 years [START_REF] Leitha | Dosimetry-guided radioiodine therapy of hyperthyroidism: long-term experience and implications for guidelines[END_REF], [32]. The risk of hypothyroidism correlates with patient age, radioactivity level, TSH value at the time of therapy, and pre-existing underlying thyroid disease (presence of antithyroid antibodies before RAI therapy) [START_REF] Ceccarelli | Pinchera Outcome of radioiodine-131 therapy in hyperfunctioning thyroid nodules: a 20 years' retrospective study[END_REF], [START_REF] Leitha | Dosimetry-guided radioiodine therapy of hyperthyroidism: long-term experience and implications for guidelines[END_REF], [START_REF] Nygaard | Long-term effect of radioactive iodine on thyroid function and size in patients with solitary autonomously functioning toxic thyroid nodules[END_REF]. This risk of long-term hypothyroidism justifies regular longterm biological monitoring, with annual TSH assay. The rate of recurrence or persistence of hyperthyroidism due to incomplete treatment is about 3% to 5.5% [START_REF] Nygaard | Long-term effect of radioactive iodine on thyroid function and size in patients with solitary autonomously functioning toxic thyroid nodules[END_REF], favored by rapid turnover of iodine in the thyroid tissue, as shown by the 24 h/4 h iodine fixation ratio, by too low a dose, or underestimation of the mass to be treated. Repeat RAI treatment can be proposed after 6 months, allowing a return to euthyroidism in nearly 80% of cases [32]. Adverse effects consist in increased thyrotoxicosis by hormone release in case of large nodules, and, exceptionally, induction of thyroid autoimmunogenicity after RAI therapy [START_REF] Reiners | Radioiodine therapy of thyroid autonomy[END_REF]. Recommendation 5.8: RAI therapy should be performed in first line patients with overt hyperthyroidism (or subclinical hyperthyroidism requiring radical treatment) related to an autonomous nodule. Surgical treatment should be preferred in case of compressive nodule, a suspicious nodule on ultrasound, if pregnancy is planned within 6 months, or according to patient preference. Grade A ++ Recommendation 5.9: Preparation with beta-blockers (propranolol) and synthetic anti-thyroid drugs is not systematic before RAI therapy, but may be considered in cases of cardiovascular risk and high T3L, preferably using imidazoles, with two precautions: • maintaining TSH suppression to limit the risk of permanent hypothyroidism secondary to RAI therapy;

• interrupting treatment 5 to 7 days beforehand to preserve the effectiveness of the RAI therapy.

Resumption of synthetic anti-thyroid drugs should be discussed with the endocrinologist.

Grade B ++. Recommendation 5.10:

The administred activity should be as low as possible, according to the ALARA principle, possibly based on dosimetric methods. Grade B +++ Recommendations 5.11:

After treatment with iodine-131, biological monitoring of TSH is necessary at 1, 3, 6 and 12 months in order to assess therapeutic efficacy. Grade A ++.

After the first year, annual TSH measurements are justified in the long term to detect late hypothyroidism. Grade A ++.

Table 1 .

 1 Comparison of 99m Tc and 123 I.

	Radio Advantages	Disadvantages
	99m

Table 2 .

 2 Performance of 99mTc-MIBI scintigraphy in predicting malignancy of thyroid nodules. Results obtained with 99mTc-MIBI washout analysis.

	Authors	Number and types of nodules	Number of cancers	Sensitivity Specificity PPV NPV
	Campenni [10]	≥ 15 mm 40 Bethesda III 65 Bethesda IV	11/40 (27.5%) 17/65 (26.1%)	100%	91%	80% 100%
		10-42 mm (median: 22 mm)
	Giovanella [11]	61 Bethesda IV + ultrasonographic	16/61 (26.2%) 100%	96%	88% 100%
		criteria: EU-TIRADS 5

Table 3 .

 3 Performance of 18F-FDG-PET/CT in the prediction of malignancy of thyroid nodules according to recent meta-analyses and studies.

	Authors	Type of study	Number and type of nodules	Number of cancers	Sensitivity Specificity PPV NPV
	Vriens et al., 2011 [12]	Meta-analysis (6 studies)	225 patients Bethesda III/IV	58/225 (25.8%)	95%	48%	39% 96%
	Castellana et al., 2019 [13]	Meta-analysis (8 studies)	331 nodules Bethesda III/IV	104/331 (24.1%)	74%	58%	34% 74%
	Deandreis et al., 2012 [16]	Prospective study	56 nodules Bethesda III/IV Median: 21 mm	10/56 (17.8%)	77% 80% a	62% 52% a	57% 81% 27% a 92% a
	Piccardo et al., 2016 [15]	Prospective study	87 nodules TIR3 b > 10 mm Median: 20 mm	18/87 (20.7%)	94%	58%	37% 98%

Acknowledgements

Thanks to the reviewers: Dr C. Bournaud, Pr M. Klein, Pr B. Goichot, Pr M.C. Vantyghem, and Pr J.L. Wemeau.

Disclosure of interest

The authors declare that they have no competing interest.