Laury Tellouck

Muriel Durieux

Pierrick Coupé

Audrey Cougnard-Grégoire

Joy Tellouck

Thomas Tourdias

Fanny Munsch

Arnaud Garrigues

Catherine Helmer

Florence Malet

Optic Radiations Microstructural Changes in Glaucoma and Association With Severity: A Study Using 3Tesla-Magnetic Resonance Diffusion Tensor Imaging

Keywords: glaucoma, fractional anisotropy, 3T MRI, optic radiations, DTI

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

1

OPTIC RADIATIONS MICROSTRUCTURAL CHANGES IN

GLAUCOMA AND ASSOCIATION WITH SEVERITY: A STUDY USING

3TESLA-MAGNETIC RESONANCE DIFFUSION TENSOR IMAGING

Laury Tellouck [START_REF] Resnikoff | Global data on visual impairment in the year 2002[END_REF][START_REF] Tham | Global prevalence of glaucoma and projections of glaucoma burden through 2040: a systematic review and meta-analysis[END_REF][START_REF] Weinreb | Primary open-angle glaucoma[END_REF] , Muriel Durieux 4 , Pierrick Coupé 5,[START_REF] Yucel | A framework to explore the visual brain in glaucoma with lessons from models and man[END_REF] , Audrey Cougnard-Grégoire 2,3 , Joy Tellouck 1 , Thomas Tourdias 2,4,7 , Fanny Munsch 4 ; Arnaud Garrigues 1 , Catherine Helmer [START_REF] Tham | Global prevalence of glaucoma and projections of glaucoma burden through 2040: a systematic review and meta-analysis[END_REF][START_REF] Weinreb | Primary open-angle glaucoma[END_REF] , Florence Malet 8 , Jean-François Dartigues 2,3 , Vincent Dousset 2,4,7 , Cécile Delcourt 2,3 , Cédric Schweitzer 1,2,3 1: CHU de Bordeaux, Service d'Ophtalmologie, F-33000 Bordeaux, France.

2: Univ. Bordeaux, F-33000 Bordeaux, France. Financial support: This study received financial support from UNADEV (Bordeaux, France). UNADEV did not participate in the design of the study, the collection, management, statistical analysis and interpretation of the data, nor in the preparation, review or approval of the present manuscript. INTRODUCTION Glaucoma affects 64 million people and is the first cause of irreversible blindness, worldwide [START_REF] Resnikoff | Global data on visual impairment in the year 2002[END_REF][START_REF] Tham | Global prevalence of glaucoma and projections of glaucoma burden through 2040: a systematic review and meta-analysis[END_REF] . It encompasses a group of disorders characterized by progressive degeneration of the optic nerve head, loss of retinal ganglion cells and a corresponding pattern of visual field loss [START_REF] Weinreb | Primary open-angle glaucoma[END_REF] . Primary open-angle glaucoma (POAG) is the predominant form of glaucoma in Western countries. Although some risk factors for POAG have been identified (high intraocular pressure, age, high myopia, ethnicity and heredity), several aspects of its pathophysiology remain unclear. As the disease could also affect intra-cerebral visual pathways in addition to optic nerve head degeneration, a neurodegenerative hypothesis raises concerns [START_REF] O'hare | Is primary open-angle glaucoma part of a generalized sensory neurodegeneration? A review of the evidence[END_REF][START_REF] Gupta | Glaucoma as a neurodegenerative disease[END_REF] Central visual pathway degeneration in glaucoma was first suggested in experimental and histological studies, which have evidenced that glaucoma is not strictly limited to the optic nerve [START_REF] Yucel | A framework to explore the visual brain in glaucoma with lessons from models and man[END_REF][START_REF] Yücel | Central nervous system changes in glaucoma[END_REF] . In an animal model of ocular hypertension, brain changes were observed in the lateral geniculate nucleus and superior colliculus, in parallel with retinal ganglion cells loss [START_REF] Zhang | Detection of early neuron degeneration and accompanying glial responses in the visual pathway in a rat model of acute intraocular hypertension[END_REF] . In another study, grey matter of glaucoma patients was reduced compared to healthy subjects, in the approximate retinal lesion projection zones in the visual cortex [START_REF] Boucard | Changes in cortical grey matter density associated with long-standing retinal visual field defects[END_REF] . Moreover, a clinicopathological case in humans highlighted a neural degeneration in intracranial optic nerve, lateral geniculate nucleus and visual cortex [START_REF] Gupta | Human glaucoma and neural degeneration in intracranial optic nerve, lateral geniculate nucleus, and visual cortex[END_REF] . This paradigmatic shift is further supported by several other small-sized clinical studies using brain magnetic resonance imaging, showing reduced volume of all the visual pathways (optic tracts, optic chiasm, lateral geniculate nucleus, optic radiations) measured at 1.5T [START_REF] Gupta | Atrophy of the lateral geniculate nucleus in human glaucoma detected by magnetic resonance imaging[END_REF][START_REF] Zhang | Anterior visual pathway assessment by magnetic resonance imaging in normal-pressure glaucoma[END_REF][START_REF] Zikou | Voxel-based morphometry and diffusion tensor imaging of the optic pathway in primary open-angle glaucoma: a preliminary study[END_REF] or 3T field strength [START_REF] Chen | Structural Brain Abnormalities in Patients with Primary Open-Angle Glaucoma: A Study with 3T MR Imaging[END_REF][START_REF] Dai | Assessment of lateral geniculate nucleus atrophy with 3T MR imaging and correlation with clinical stage of glaucoma[END_REF] . Some other experimental studies may help understand the pathogenesis of the disease. Using in vivo MRI studies is a way to study metabolic and spatiotemporal changes in glaucoma [START_REF] Chan | Evaluation of the retina and optic nerve in a rat model of chronic glaucoma using in vivo manganese-enhanced magnetic resonance imaging[END_REF][START_REF] Chan | Proton magnetic resonance spectroscopy revealed choline reduction in the visual cortex in an experimental model of chronic glaucoma[END_REF][START_REF] Ho | In Vivo Evaluation of White Matter Integrity and Anterograde Transport in Visual System After Excitotoxic Retinal Injury With Multimodal MRI and OCT[END_REF] .

In addition, epidemiological studies have also suggested that glaucoma might be associated with other neurodegenerative disorders, in particular Alzheimer's disease [START_REF] Helmer | Is there a link between open-angle glaucoma and dementia?: The Three-City-Alienor Cohort[END_REF][START_REF] Lin | Glaucoma, Alzheimer's disease, and Parkinson's disease: an 8-year population-based follow-up study[END_REF] , and a few studies have found a reduced volume of brain structures, beyond the visual pathwaysparticularly in the hippocampuswhich is well known to be affected in Alzheimer's disease [START_REF] Chen | Structural Brain Abnormalities in Patients with Primary Open-Angle Glaucoma: A Study with 3T MR Imaging[END_REF][START_REF] Frezzotti | Structural and functional brain changes beyond visual system in patients with advanced glaucoma[END_REF][START_REF] Williams | Evidence for widespread structural brain changes in glaucoma: a preliminary voxel-based MRI study[END_REF] . These data have nevertheless been collected in patients with long standing disease. Whether subtle alterations suggestive of associated neurodegenerative disease can be captured from the early stage of glaucoma, prior to atrophy, is unknown.

Recent improvements in neuroimaging techniques allow more accurate evaluation of brain structure volumes and intra-cerebral microstructural damage. By quantifying microscopic movements of water molecules, Diffusion Tensor Imaging (DTI)a functional MRI techniqueprovides a sensitive evaluation of underlying brain microstructural changes even prior to atrophy [START_REF] Bihan | Diffusion Magnetic Resonance Imaging: What Water Tells Us about Biological Tissues[END_REF] . Therefore, this technique appears particularly promising in the documentation of intra-cerebral damage in glaucoma.

The most commonly assessed DTI parameters include fractional anisotropy (FA, which reflects the degree of cellular structural alignment within fiber tracts and the structural integrity of the fiber tracts) and mean diffusivity (MD, which measures the average motion of water molecules independently of fiber directionality).

Several case-control studies have already shown that FA of the optic radiations is decreased and MD increased in glaucoma patients [START_REF] El-Rafei | A framework for voxel-based morphometric analysis of the optic radiation using diffusion tensor imaging in glaucoma[END_REF][START_REF] Garaci | Optic nerve and optic radiation neurodegeneration in patients with glaucoma: in vivo analysis with 3-T diffusiontensor MR imaging[END_REF] , and some others have suggested that these changes may be progressive with increasing axon loss of the optic nerve [START_REF] Chen | Diffusion tensor magnetic resonance imaging reveals visual pathway damage that correlates with clinical severity in glaucoma[END_REF] . Whereas these studies provide new insights in the understanding of glaucoma disease, they were limited in sample size and mainly included advanced glaucoma patients. Therefore, our study aimed at exploring the potential neurodegenerative hypothesis associated with glaucoma and whether subtle changes could be measurable at the early stage of the disease. Thus we analyzed both the microstructural changes of the visual pathway, in relation with glaucoma severity, as well as changes beyond the visual pathway, in particular in regions affected in neurodegenerative pathologies.

METHODS

Patient Population

This study is an observational case-control study performed at the University Hospital of Bordeaux. Fifty patients with POAG (20 men, 30 women, mean age 61.9 +/-6.9 years) and 50 age-and sex-matched controls (20 men, 30 women, mean age 61.9 +/-7.0 years) were prospectively included. This research followed the tenets of the Declaration of Helsinki. Participants gave written consent for the participation in the study. The design of this study was approved by the Ethical Committee of Bordeaux (Comité de Protection des Personnes Sud-Ouest et Outre-Mer III) in March 2012. This study was registered on the website http://clinicaltrials.gov/ (identifier NCT01621841).

Ophthalmological Examination

All participants underwent a complete ophthalmic examination including measurement of best-corrected visual acuity, intraocular pressure (IOP) using Goldmann aplanation tonometry, gonioscopy, slit-lamp biomicroscopy and optic disc examination by funduscopy. Central corneal thickness and anterior chamber depth were assessed using interferometry (OCT Visante, Carl Zeiss Meditec, Inc., Dublin, CA, USA), and axial length measurement using IOL Master (Carl Zeiss Meditec, Inc., Dublin, CA, USA). All participants underwent a visual field testing (Octopus 101, Haag-Streit, Inc., Bern, Switzerland) and only reliable tests (false-positive errors <15%, false negativeerrors <15%, loss fixations <20%) were included. In addition, visual fields (VF) were reviewed and excluded in the presence of artefacts, such as eyelid or rim artifacts, fatigue effects, inattention, or inappropriate fixation.

A measurement of peripapillary Retinal Nerve Fiber Layer (RNFL) thickness was performed using Spectral-Domain optical coherence tomography (SD-OCT) (Cirrus, Carl Zeiss Meditec, Inc., Dublin, CA, USA). All images were acquired and reviewed by specially trained technicians of the study to control the quality of signal strength and accurate centration and segmentation of the RNFL circle scan acquisition. Signal strength lower than 6 or acquisitions with artifacts were excluded from the analysis.

Glaucoma subjects and controls received a questionnaire requesting for cardiovascular risk factors, familial history of glaucoma, ophthalmological diseases and current medications. Each participant underwent Mini Mental State Examination (MMSE) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF] .

Primary open-angle glaucoma was defined by the following criteria: the presence of glaucomatous optic neuropathy (defined as a loss of neuroretinal rim with a vertical cup-to-disc ratio [VCDR] of >0.7 or an intereye asymmetry of >0.2, with or without notching attributable to glaucoma) associated to compatible VF loss. This VF loss was defined as the presence of at least 3 contiguous non edge test points within the same hemifield on the pattern deviation probability plot at P < 0.05, with at least 1 point P < 0.01, excluding points directly above and below the blind spot, and the presence of glaucomatous hemifield test results outside normal limits. Iridocorneal angle opening was graded 3 or 4 on gonioscopy using Schaeffer classification.

Controls were defined as normal optic disc without notching or abnormal thinning of the neuroretinal rim, no visual field defects, IOP measurement < 21 mmHg and no family history of glaucoma.

Exclusion criteria included any diseases that could affect the visual field, secondary glaucoma including exfoliative and pigmentary glaucoma, diabetes mellitus, any neurological or psychiatric disorders, and a score <26 on the MMSE for global cognition. We also excluded participants according to standard MRI exclusion criteria such as claustrophobia, ferromagnetic implants or pacemakers, and inability to lie still for the MRI acquisition time.

Stage of severity of glaucoma was classified according to the Hodapp-Parrish-Anderson classification [START_REF] Maeda | Automated keratoconus screening with corneal topography analysis[END_REF] . The different stages are:

•

Image Processing

Measurement of DTI metrics along the optic radiations

From the DTI data, the distortions induced by eddy currents were first corrected, then a diffusion tensor model was fitted at each voxel using Olea Medical ® software to generate fractional anisotropy (FA) maps and to investigate the microstructural integrity of the optic radiations. The optic radiations were identified using deterministic tractography between two seed-regions of interest (ROIs) over the proximal and distal optic radiations according to previously published method and landmarks [START_REF] Reich | Damage to the optic radiation in multiple sclerosis is associated with retinal injury and visual disability[END_REF] . The proximal ROI was placed near the lateral geniculate nuclei, while the distal ROI was placed just anterior to its termination in the visual cortex. Fiber tract propagation was terminated for FA<0.2 and angle<35° based on agreed-upon thresholds. ROIs were placed by a specialized neuroradiologist symetrically, based on color-coded FA maps and trace DTI images on the anterior and posterior part of the expected pathway of the optic radiations (green boxes on Figure 1). Fibers whose directions did not correspond to the optic radiations based on anatomic knowledge and DTI-derived atlas were excluded by adding additional ROIs and a logical "not" function [START_REF] Wakana | Fiber tractbased atlas of human white matter anatomy[END_REF] (red boxes on Figure 1). Only fibers that connected the anterior and posterior regions of interest were selected for further analysis. The analysis was independently repeated for a subset of cases (n=25 out of the 100 cases) by a specialized ophthalmologist with an interreader agreement of 0.88.

The median FA and its sub-component (axial and radial diffusivity, AD and RD respectively) as well as the mean diffusivity were measured along the reconstructed optic radiations (green streamlines on Figure 1). Decreased FA and increased MD values is usually considered as a proxy of axonal disruption [START_REF] Wheeler-Kingshott C A | In vivo diffusion tensor imaging of the human optic nerve: pilot study in normal controls[END_REF] .

None of the people participating in FA measurements had any access to the case/control status of the participants, nor to any other clinical data.

MRI volumetric measurements

For volumetric analyses, T1-weighted images were processed using the volBrain system (http://volbrain.upv.es). After denoising [START_REF] Manjón | Adaptive nonlocal means denoising of MR images with spatially varying noise levels[END_REF] , images were affine-registered [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF] into the Montreal Neurological Institute space and the total brain volume was estimated using the Nonlocal Intracranial Cavity Extraction method [START_REF] Manjón | Nonlocal intracranial cavity extraction[END_REF] . Hippocampus was segmented using patch-based multi-template approach [START_REF] Coupé | Patchbased segmentation using expert priors: application to hippocampus and ventricle segmentation[END_REF] following the international consortium from the EADC-ADNI Harmonized Protocol for anatomical definitions of the hippocampus [START_REF] Frisoni | The EADC-ADNI Harmonized Protocol for manual hippocampal segmentation on magnetic resonance: evidence of validity[END_REF] . To control variations in head size between subjects, total brain volumes and hippocampal volumes were scaled using the volumetric scaling factor determined through the affine registration to the MNI brain template.

For DTI analysis within the hippocampus masks, an in-house pipeline (dtiBrain) was used to process diffusion-weighted images. First, diffusion-weighted images were affine-registered to the T1w MRI in the MNI space [START_REF] Manjón | Adaptive nonlocal means denoising of MR images with spatially varying noise levels[END_REF] . Then, to compensate for EPI distortion, a non-rigid registration was performed. Finally, a diffusion tensor model was fitted at each voxel using FSL 5.031 (fmrib.ox.ac.uk/fsl), generating FA and MD maps.

Mean FA and MD were measured within the hippocampal masks previously generated on anatomical T1-weighted MRI.

Statistical analysis

Statistical analysis was performed using SAS 9.3 (SAS Institute Inc, Cary, NC).

Differences of MRI characteristics between glaucoma subjects and healthy controls

were tested using logistic conditional analyses, for parameters both along optic radiations and outside the visual pathway (globally for white and grey matter and in hippocampal and amygdala structures). Additionally, within the group of patients with glaucoma, we used mixed linear regression analyses, adjusted for sex and age (as a continuous variable expressed in years), to test the associations between optic radiation DTI parameters (FA, MD, AD, RD) and the parameters of severity of the disease (VCDR, mean deviation of VF and RNFL). This type of analysis allows taking into account both right and left sides of each patient, while taking into account the intra-individual correlation between sides. In particular, this allowed studying the associations of ocular parameters with homolateral (right optic radiation with right eye and left with left eye) and contralateral (right optic radiation with left eye and vice versa) optic radiations MRI parameters. In these regression analyses, both ocular and brain parameters were entered as z-scores. In addition, for RNFL, we also adjusted for axial length, which is strongly associated with RNFL [START_REF] Sowmya | Effect of Refractive Status and Axial Length on Peripapillary Retinal Nerve Fibre Layer Thickness: An Analysis Using 3D OCT[END_REF] .

RESULTS

Demographic and ophthalmological characteristics

As shown in Table 1, cases and controls were similar for age, gender, history of cardiovascular diseases or risk factors and MMSE. Family history of glaucoma was reported by 58 % of glaucoma patients, and 0 % of controls (since this was an exclusion criterion for controls).

As shown in Table 2, cases and controls did not significantly differ for visual acuity (distance and near), intraocular pressure and axial length. As expected, they were significantly different for central corneal thickness, VCDR, RNFL thickness and visual field parameters. Similar results were observed for the left eye (Table 3).

In our study, 70% of glaucoma patients had an early stage of the disease, 20% a moderate stage and 10% an advanced or severe stage, according to the Hodapp-Parrish-Anderson classification.

Comparison of MRI parameters along optic radiations between glaucoma and control subjects

One patient refused to do MRI examination and 3 MRI examinations were insufficient quality for analysis, leaving 49 glaucoma patients and 47 controls for the comparison of MRI parameters (Table 4). The optic radiations were similarly reconstructed for glaucoma and control subjects (similar length and volume and reconstructed streamlines). Glaucoma patients showed significantly lower FA along the optic radiations than controls (0.57 vs 0.59, p=0.02), which was driven by significant increase in radial diffusivity (52.8 10 -5 mm 2 /s vs 49.7 10 -5 mm 2 /s, p=0.03) while axial diffusivity was unchanged. Mean diffusivity tended to be slightly higher in glaucoma patients, but this did not reach statistical significance (82.4 10 -5 mm 2 /s vs 80.6 10 -5 mm 2 /s p=0.10).

Associations of homo-and contralateral optic radiation parameters with the severity of the disease in glaucoma patients

Table 5 shows the associations of optic radiation parameters (FA, MD, RD and AD) with the ophthalmological parameters of glaucoma severity evaluated by visual field, optic disc cupping and RNFL thickness, only among patients with glaucoma (n=50).

We tested associations of ophthalmological parameters with MRI parameters on the homolateral (right eye -right optic radiation and left eye -left optic radiation) and contralateral (right-left and left-right) sides. For the homolateral side, significant associations were found between optic radiations FA and mean deviation of the visual field (β= -0.22; p= 0.03), VCDR (β= -0.42; p= 0.0003) and RNFL (β= 0.22; p= 0.03).

The direction of the association is opposite for RNFL, since RNFL decreases with higher severity of glaucoma, while other parameters increase with severity. Mean and radial diffusivities increased with the severity of the disease measured by VCDR, (p<0.006 and p<0.0008, respectively), but were not significantly associated with mean deviation of VF or RNFL thickness. By contrast, axial diffusivity, as well as length and volume of optic radiations were not significantly associated with any of the severity parameters.

With regard to the contralateral side, associations of MRI parameters with glaucoma severity parameters were much weaker, and reached statistical significance only for the association of FA and RD with VCDR (p=0.01 and p=0.02, respectively).

Brain volumes analyses between cases and controls

Finally, we did not evidence any statistically significant difference between glaucoma subjects and controls for volumes and DTI parameters of cerebrum white and grey matters, hippocampus and amygdala (Table 6).

DISCUSSION

Our study demonstrates microstructural changes of the optic radiations in glaucoma, as evaluated by lower FA driven by higher RD, and a correlation between the level of structural modifications and disease severity.

Using MRI at 1.5T [START_REF] Murai | Positive correlation between the degree of visual field defect and optic radiation damage in glaucoma patients[END_REF] or 3T [START_REF] Garaci | Optic nerve and optic radiation neurodegeneration in patients with glaucoma: in vivo analysis with 3-T diffusiontensor MR imaging[END_REF][START_REF] Chen | Diffusion tensor magnetic resonance imaging reveals visual pathway damage that correlates with clinical severity in glaucoma[END_REF][START_REF] Engelhorn | A new approach to assess intracranial white matter abnormalities in glaucoma patients: changes of fractional anisotropy detected by 3T diffusion tensor imaging[END_REF] a few case-control studies have also reported such modifications of diffusion parameters in optic radiations of glaucoma patients. All these studies found significantly lower FA in glaucoma patients compared with control patients. In the present study, which included 70 % of early stages of glaucoma, FA differences between cases and controls are numerically small (about 0.02 for a mean of about 0.60, i.e about 3.3 %). However, the standard deviation is also small (about 0.04), showing low inter-individual variability in this parameter, and the difference is substantial when related to the standard deviation (about 0.5 SD), suggesting a major effect of glaucoma on this highly conserved parameter. In other studies, the differences in FA of optic radiations observed between glaucoma patients and controls were larger, but these studies generally included more severe cases. The study by Engelhorn et al included 22 severe glaucoma cases, and observed a difference in FA of optic radiations ranging from 17 % to 30 % according to the localization (anterior, central, posterior) [START_REF] Engelhorn | A new approach to assess intracranial white matter abnormalities in glaucoma patients: changes of fractional anisotropy detected by 3T diffusion tensor imaging[END_REF] . The study by Murai et al included 18 severe glaucoma cases, 9 moderate and only 2 mild, and observed a 14 % difference in FA of the optic radiations [START_REF] Murai | Positive correlation between the degree of visual field defect and optic radiation damage in glaucoma patients[END_REF] . The study by Garaci et al included 4 pre-perimetric glaucoma cases, 4 early, 4 moderate and 4 severe cases and observed a 36 % differences in FA of the optic radiations [START_REF] Garaci | Optic nerve and optic radiation neurodegeneration in patients with glaucoma: in vivo analysis with 3-T diffusiontensor MR imaging[END_REF] . Finally, the study by Chen et al included a majority of severe cases (36 eyes with MD>9.5 dB out of 50) but did not report numerically the averages of optic radiations FA [START_REF] Chen | Diffusion tensor magnetic resonance imaging reveals visual pathway damage that correlates with clinical severity in glaucoma[END_REF] . Furthermore, we observed higher RD value in glaucoma patients and its correlation with disease severity whereas AD was not significantly different between glaucoma and control patients. Although, AD and RD are the two components of FA, these parameters have been scarcely analyzed in the literature and some studies have already reported increasing RD in glaucoma subjects compared to controls [START_REF] El-Rafei | A framework for voxel-based morphometric analysis of the optic radiation using diffusion tensor imaging in glaucoma[END_REF] .

Even though the underlying pathological alterations are not specifically known, animal studies have suggested that higher RD could mainly represent myelin loss while lower AD could be a more specific marker of neuronal loss. However these considerations were based on simplistic models and whether alterations of optic radiations truly predominate on myelin or axon component cannot be formally ascertained for glaucoma patients, for whom other modifications such as microglia activation may confound the data. Additionally, we observed a trend towards higher mean diffusivity value in the glaucoma group without reaching statistical significance. However, we observed a significant positive correlation between mean diffusivity and disease severity measured with VCDR. Two studies also showed higher mean diffusivity in glaucoma patients [START_REF] Garaci | Optic nerve and optic radiation neurodegeneration in patients with glaucoma: in vivo analysis with 3-T diffusiontensor MR imaging[END_REF][START_REF] Chen | Diffusion tensor magnetic resonance imaging reveals visual pathway damage that correlates with clinical severity in glaucoma[END_REF] . While FA measures the degree of cellular structural alignment within fiber tracts and their structural integrity, mean diffusivity measures the average motion of water molecules independently of fiber directionality and is considered as an additional marker of axonal disruption. As these studies included patients with advanced glaucoma, our findings might be explained by a lack of statistical power and a lower grade of disease severity in our glaucoma group. Such converging evidence of loss of fiber integrity in optic radiations in glaucoma cannot be measured in terms of length and volume of the optic radiations, which were similar in both groups probably because our measurements were made prior to fiber loss or major disorganization.

We also observed an association of diffusivity parameters (mainly FA and RD), with the severity of glaucoma (assessed by mean deviation of the visual field, VCDR and RNFL measured with SD-OCT), suggesting that microstructural changes to the optic radiations is one of the components of the severity that could participate in the clinical status of the patients and the alteration of the visual field. Although we mainly included early and moderate glaucoma as defined by the Hodapp-Parrish-Anderson classification, our findings are consistent with some previous studies, which included more advanced cases [START_REF] Chen | Diffusion tensor magnetic resonance imaging reveals visual pathway damage that correlates with clinical severity in glaucoma[END_REF][START_REF] Murai | Positive correlation between the degree of visual field defect and optic radiation damage in glaucoma patients[END_REF][START_REF] Engelhorn | A new approach to assess intracranial white matter abnormalities in glaucoma patients: changes of fractional anisotropy detected by 3T diffusion tensor imaging[END_REF][START_REF] Michelson | DTI parameters of axonal integrity and demyelination of the optic radiation correlate with glaucoma indices[END_REF] . All these studies also evidenced significant associations of optic radiations FA with structural parameters of optic nerve head degeneration evaluated with VCDR or time-domain RNFL thickness, as well as functional visual field alterations. For example, Michelson et al. found a correlation between FA and visual field [START_REF] Michelson | DTI parameters of axonal integrity and demyelination of the optic radiation correlate with glaucoma indices[END_REF] . Thus, all these results illustrate the fact that FA could be a strong biomarker of glaucoma severity.

Our study also assessed the associations of glaucoma severity according to FA of homolateral and contralateral optic radiations. Interestingly, glaucoma severity parametersin particular VCDRwere more strongly associated with homolateral optic radiations diffusion parameters than with contralateral parameters. As chiasmatic decussation of optic pathways results in approximately 50% crossing of axons on the contralateral side [START_REF] Pietrasanta | The Corpus Callosum and the Visual Cortex: Plasticity Is a Game for Two[END_REF] , we would expect similar associations of glaucoma severity parameters with homolateral and contralateral diffusion parameters. However, our findings might also be related to an increased vulnerability of some specific retinal nerve fiber bundles of the optic nerve head resulting in an atrophy of optic radiations more predominant on the homolateral side of the decussation than on the controlateral. Indeed, several studies have demonstrated a specific vulnerability of the temporal and temporal-inferior sides of the optic nerve head to glaucoma damage [START_REF] Strouthidis | Longitudinal change detected by spectral domain optical coherence tomography in the optic nerve head and peripapillary retina in experimental glaucoma[END_REF][START_REF] Tan | Detection of macular ganglion cell loss in glaucoma by Fourier-domain optical coherence tomography[END_REF][START_REF] Yang | Comparison of optic nerve morphology in eyes with glaucoma and eyes with non-arteritic anterior ischemic optic neuropathy by Fourier domain optical coherence tomography[END_REF] . Thus, we could expect an increased atrophy of the corresponding optic radiation predominant on the homolateral side that could explain our findings. However, even if temporal and temporal-inferior nerve fiber layers are more vulnerable to glaucomatous damage, the meaning of our findings should be interpreted with caution and would need further exploration to be confirmed and to identify the exact underlying mechanism. Indeed, distribution of RNFL is not homogeneous around the optic nerve head with superior and inferior sectorial RNFL thicker than nasal or temporal RNFL sectors. Furthermore the mean optic disc-fovea angle deliminating superior and inferior nerve fiber layers, is around 8°4 [START_REF] Gupta | Glaucoma as a neurodegenerative disease[END_REF] . Thus the exact distribution of nerve fiber layers of the retina that decussates to the contralateral optic tract or remains on the ipsilateral optic tract and finally leads to a vertical delimitation through the fovea on the hemivisual field test remains unclear. Hence, in our study, the corresponding optic radiations in the homo or contralateral side could not be accurately matched to specific sectors of the retina or the optic nerve head.

Although high intraocular pressure is the main risk factor of glaucoma, this disease is increasingly considered as a neuro-ophthalmological and neurodegenerative disease [START_REF] Chang | Glaucoma 2.0: Neuroprotection, Neuroregeneration, Neuroenhancement[END_REF] . Furthermore, there are still controversies on the association between glaucoma and some other neurodegenerative diseases as Alzheimer's disease. In particular, in a cohort of elderly subjects followed every 2 years, we observed an association of POAG with incident dementia [START_REF] Helmer | Is there a link between open-angle glaucoma and dementia?: The Three-City-Alienor Cohort[END_REF] . Volume changes beyond the visual system in glaucoma patients have also been reported in several studies but with inconsistent results. For instance, Frezzotti et al. reported that POAG patients had brain atrophy in some grey matter regions and the visual cortex [START_REF] Frezzotti | Structural and functional brain changes beyond visual system in patients with advanced glaucoma[END_REF] . By contrast, Williams et al. found five cerebral structures larger in the glaucoma group than in the control group [START_REF] Williams | Evidence for widespread structural brain changes in glaucoma: a preliminary voxel-based MRI study[END_REF] . Chen et al. revealed both a decreasing grey matter volume in some regions and an increasing grey matter volume in some others [START_REF] Chen | Structural Brain Abnormalities in Patients with Primary Open-Angle Glaucoma: A Study with 3T MR Imaging[END_REF] . In the present study, we analyzed brain globally and focused on brain regions that are well known to be affected in the course of Alzheimer's diseaseparticularly hippocampusand did not evidence any significant difference for any of the studied regions of interest, neither in volume nor in parameters of diffusivity. However, as we included subjects with MMSE ≥26 at baseline, the risk of brain structures atrophy was probably limited. Regarding the hippocampus, results have been particularly inconsistent, since Frezzotti et al.

reported decreased hippocampus volume in glaucoma patients [START_REF] Frezzotti | Structural and functional brain changes beyond visual system in patients with advanced glaucoma[END_REF] , while Williams et al. reported no significant difference of hippocampus volume between glaucoma and controls, but an increase in hippocampus volume with disease severity in patients with glaucoma [START_REF] Williams | Evidence for widespread structural brain changes in glaucoma: a preliminary voxel-based MRI study[END_REF] .

Such differences between study results may be explained by differences in study methodology, in particular regarding the selection of subjects and severity of the disease, MRI sequences used or definition of regions of interest. For instance, in a recent study by Frezzotti et al, only severe cases of glaucoma (but not early) showed grey matter atrophy of the visual cortex and hippocampus [START_REF] Murphy | Retinal Structures and Visual Cortex Activity are Impaired Prior to Clinical Vision Loss in Glaucoma[END_REF] . The evolution of brain volume in the course of glaucoma and its association with other neurodegenerative diseases would need further investigation and prospective follow-up of subjects.

Finally, functional MRI may offer new insights into the brain modifications associated with glaucoma, as suggested by two recent studies, showing functional modifications of the visual cortex at the earliest stages of the disease [START_REF] Murphy | Retinal Structures and Visual Cortex Activity are Impaired Prior to Clinical Vision Loss in Glaucoma[END_REF][START_REF] Frezzotti | Early changes of brain connectivity in primary open angle glaucoma[END_REF] .

In conclusion, we confirmed microstructural changes of optic radiations in glaucoma and its association with glaucoma severity. In accordance with several other studies, DTI appears as an objective measurement for evaluating alterations of the visual pathways in glaucoma and provides new insight in the pathophysiological process of glaucoma. A prospective evaluation of our cohort of patients would be of interest to observe the evolution of these microstructural modifications of optic radiations and to analyze the evolution of brain volume in association with the evolution of glaucoma disease. DTI could represent a future way to explore central nervous system of glaucomatous subjects, leading to a better understanding of the pathophysiology and, potentially, to help clinical trials evaluate new therapeutic strategies based on neuroprotection or brain repair. Green boxes are the markers placed manually on anterior and posterior parts of the optic radiations. Green lines are optic radiations automatically reconstructed by the Olea Medical ® software. Red boxes are markers manually placed to manually deleted fibers outside the expected area.

3 : 5 :

 35 INSERM, U1219 -Bordeaux Population Health Research Center, F-33000 Bordeaux, France. 4: CHU de Bordeaux, Service de Neuro-Imagerie, F-33000 Bordeaux, France. Université de Bordeaux, LaBRI, UMR 5800, PICTURA, F-33400 Talence, France. 6: CNRS, LaBRI, UMR 5800, PICTURA, F-33400 Talence, France. 7 : INSERM U1215, Neurocentre Magendie, F-33000 Bordeaux, France. 8: Centre d'Ophtalmologie Point Vision, F-33000 Bordeaux, France. Corresponding author/reprint request: Cécile Delcourt, Inserm U1219, ISPED, 146 rue Léo Saignat, 33076 Bordeaux Cedex. Tel: + 33 557 57 11 91. Email: cecile.delcourt@isped.fr Word count: 4089

  Stage 0: no or minimal defect • Stage 1: MD ≥ -6.0 dB (early defect) • Stage 2: -12.0 ≥ MD ≥ -6.0 dB (moderate defect) • Stage 3: -20 ≥ MD ≥ -12.0 dB (advanced defect) • Stage 4: MD ≥ -20.0 dB (severe defect) • Stage 5: End-stage disease MRI Data Acquisition MRI examinations were performed on a 3-T Discovery MR750w scanner (GE Medical Systems, Milwaukee, WI, USA) using a 32-channel phased array head coil within 30 days following the ophthalmologic examinations. The protocol included a DTI sequence to look for microstructural alterations along and beyond the optic radiations, a 3D-T1-wi sequence to look for global or focal atrophy. The parameters of acquisitions were as follows. The DTI sequence consisted in dual echo-planar imaging: 40 axial slices; repetition time, 12000 ms; echo time, 100.9 ms; slice thickness, 3.5 mm; matrix, 160x160; field of view, 24 cmx24 cm; b values, 0 and 1000 s/mm² applied in 32 non-collinear directions. The 3D-T1 was an inversion recovery gradient echo sequence: 288 slices; repetition time, 11.4 ms; echo time, 4.3 ms; inversion time, 400 ms; flip angle, 15°; slice thickness, 0.8 mm; matrix, 384x384; field of view, 25 cmx25 cm.

Figure 1 .

 1 Figure 1. Building of optic radiations and measurement of fractional anisotropy, using the Olea Medical® software (image: CHU de Bordeaux, department of neuroimaging).

Table 2 : Ophthalmological characteristics; right eye (mean (standard deviation))

 2

				Glaucoma	Control
	Characteristics			(n=50)	(n=50)	p-value*
	Best-Corrected	Visual	Acuity
	(logMAR)
	Distance			0.03 (0.09)	0.006 (0.02)	0.09
	Near			0.19 (0.07)	0.18 (0.00)	0.2
	Central corneal thickness (μm)	525.9 (30.9) 544.0 1 (31.9) 0.005
	Axial Length (mm)			24.1 (1.1)	23.7 (1.2)	0.15
	IOP (mmHg)			14.6 (3.7)	14.7 (2.5)	0.82
	Vertical Cup : Disc Ratio		0.8 (0.2) 1	0.3 (0.2) 1	<0.0001
	RNFL Thickness (μm)		76.9 (17.7)	93.2 (9.9)	<0.0001
	Visual field (dB)
	Mean Sensitivity			21.9 (6.1)	26.6 (1.1)	<0.0001
	Mean Deviation			5.0 (6.0)	0.3 (1.0)	<0.0001
	Loss Variance			22.6 (26.3)	5.0 (1.8)	<0.0001
	RNFL: Retinal Nerve Fiber Layer
	IOP: Intraocular Pressure
	* Chi-square for categorical variables and Student t-test for continuous variables 1 One missing data

Table 3 : Ophthalmological characteristics; left eye (mean (standard deviation))

 3

				Glaucoma	Control
	Characteristics			(n=50)	(n=50)	p-value*
	Best-Corrected	Visual	Acuity
	(logMAR)
	Far			0.03 (0.08)	0.006 (0.02)	0.08
	Near			0.2 (0.1)	0.2 (0)	0.1
	Central corneal thickness (μm)	524.6 (30.0) 1 542.9 (28.4) 2 0.003
	Axial Length (mm)			24.0 (1.0)	23.5 (0.8)	0.003
	IOP (mmHg)			14.2 (3.2)	15.2 (2.4)	0.09
	Vertical Cup : Disc Ratio
	Funduscopy			0.8 (0.2)	0.3 (0.2)	<0.0001
	RNFL Thickness (μm)		70.1 (18.7)	93.1 (9.7)	<0.0001
	Visual field (dB)
	Mean Sensitivity: MS		21.8 (4.6)	26.6 (1.2)	<0.0001
	Mean Deviation: MD		5.1 (4.5)	0.3 (1.1)	<0.0001
	Loss Variance: LV			28.1 (28.0)	4.7 (2.2)	<0.0001
	RNFL: Retinal Nerve Fiber Layer
	IOP: Intraocular Pressure
	* Chi-square for categorical variables and Student t-test for continuous variables 1 One missing data 2 Two missing data

Table 4 : Comparison of MRI parameters along optic radiations between glaucoma and control subjects (mean (standard deviation))

 4

		Glaucoma	Control	P-value*
		(n=49)	(n=47)
	Fractional anisotropy	0 . 5 7 ( 0 . 0 4 )	0 . 5 9 ( 0 . 0 3 )	0 . 0 2
	Mean diffusivity (10 -5 mm 2 /s)	82.38 (6.43)	80.62 (4.87)	0.10
	Axial diffusivity (10 -5 mm 2 /s)	141.59 (7.14) 142.37 (6.74)	0.53
	Radial diffusivity (10 -5 mm 2 /s)	52.78 (6.74)	49.74 (5.04)	0.03
	Length (mm)	63.91 (6.80)	63.72 (7.15)	0.74
	Volume (cm 3 )	20.79 (9.13) 24.96 (12.33)	0.11

Table 5 : Associations of homo-and contra-lateral optic radiation parameters with the severity of the disease in glaucoma patients

 5

		Visual field		Funduscopy		SD-OCT examination
		M e a n D e v i a t i o n		V e r t i c a l C u p / D i s c R a t i o	R N F L t h i c k n e s s
		β[95%CI]	P	β [95%CI]	P	β [95%CI]	P
	Homolateral
	Fractional anisotropy	-0.22 [-0.41;-0.02]	0.03	-0.42 [-0.64;-0.21]	0.0003 0.22 [0.03;0.41]	0.03
	Mean diffusivity	0.02 [-0.17;0.21]	0.80	0.29 [0.08;0.50]	0.008 -0.15 [-0.34;0.04]	0.12
	Axial diffusivity	-0.11 [-0.31;0.08]	0.26	0.09 [-0.14;0.31]	0.45	-0.06 [-0.26;0.14]	0.56
	Radial diffusivity	0.09 [-0.10;0.28]	0.33	0.38 [0.17;0.59]	0.0006 -0.18 [-0.37;-0.007] 0.06
	Controlateral
	Fractional anisotropy	-0.17 [-0.37;0.02]	0.09	-0.28 [-0.55;-0.06]	0.01	0.10 [-0.09;0.30]	0.30
	Mean diffusivity	0.13 [-0.06;0.31]	0.19	0.20 [-0.02;0.41]	0.07	0.01 [-0.18;0.21]	0.91
	Axial diffusivity	0.04 [-0.15;0.24]	0.67	0.07 [-0.16;0.30]	0.55	0.09 [-0.11;0.30]	0.34
	Radial diffusivity	0.15 [-0.04;0.34]	0.11	0.25 [0.04;0.46]	0.02	-0.03 [-0.23;0.16]	0.72

CI: Confidence Interval; RNFL: Retinal Nerve Fiber Layer P-value adjusted for age and gender, with additional adjustment for axial length for RNFL

Table 6 : Comparison of DTI parameters and MRI-based volume between glaucoma and control subjects (mean (standard deviation))

 6

			C e r e b r u m w h i t e m a t t e r	C e r e b r u m g r e y m a t t e r	Hippocampus			Amygdala
					P-			P-			P-			P-
			G l a u c o m a	C o n t r o l		Glaucoma	Control		Glaucoma Control		Glaucoma Control
					value*			value			value			value
			( n = 4 8 )	(n=48)		(n=48)	(n=48)		(n=48)	(n=48)		(n=48)	(n=48)
	Volume (cm 3 )	275.28	238.25	0.06	270.19	283.09	0.18	3.95	3.84	0.16	0.60	0.55	0.22
			(115.61)	(59.03)		(59.03)	(38.17)		(0.53)	(0.46)		(0.23)	(0.18)
	DTI parameters	(n=41)	(n=41)		(n=41)	(n=41)		(n=41)	(n=41)		(n=41)	(n=41)
			0.38	0.37		0.17	0.17		0.19	0.19		0.15	0.15
	FA				0.39			0.45			0.44			0.09
			(0.02)	(0.02)		(0.01)	(0.01)		(0.01)	(0.02)		(0.02)	(0.01)
	MD (10 -5 mm	2 /s)	78.34	78.19	0.79	121.70	120.01	0.25	98.83	97.72	0.42	80.45	80.56	0.87
			(3.58)	(3.21)		(9.63)	(8.81)		(8.82)	(5.79)		(4.03)	(3.40)
	AD (10 -5 mm 2 /s)	110.91	110.35	0.26	138.84	136.96	0.24	116.86	115.46	0.39	92.64	92.21	0.57
			(2.80)	(2.53)		(10.50)	(9.67)		(9.78)	(6.17)		(4.43)	(3.31)
	RD (10 -5 mm 2 /s)	62.24	62.27	0.96	113.20	111.60	0.26	90.02	88.90	0.44	74.35	74.73	0.57
			(4.17)	(3.78)		(9.23)	(8.44)		(8.40)	(5.74)		(3.99)	(3.53)
	Abreviations: AD: Axial diffusivity; FA: Fractional anisotropy; MD: Mean diffusivity; RD: Radial diffusivity
	*P-value : logistic conditional analyses

*P-value: logistic conditional analyses


Rubén Hernández
email: rhernandeza6@gmail.com

Jesús De León

Vincent Léchappé
email: vincentlechappe@gmail.com

Franck Plestan

A Decentralized Second Order Sliding-Mode Control of Multi-Agent System with Communication Delay

This paper addresses the formation control problem of a multi-agent system (MAS) under the leader-follower scheme. Thus, the followers must to keep a desired relative distance with respect to the leader trajectory against known and constant communication delay. Thus, in order to ensured the coordinated behavior and formation control against parametric uncertainties and external disturbances, a decentralized control approach based on a state predictor and a Second Order Sliding Mode control is proposed. Sufficient conditions are provided to guarantee the stability of the closed-loop system. Finally, simulation results illustrate the effectiveness of the proposed method.

I. INTRODUCTION

Motion control in MAS has been trendy topic in the last years, due that there are tasks that can be performed more efficiently by a MAS instead of by a single agent. A MAS is a set of autonomous units that interact with each other in a common environment under cooperative or coordinated schemes. In coordinated schemes there are only interconnections from the leader or dominant system to rest, while in cooperative schemes there are interconnections among all the agents, such that they have influence on the combined dynamics [START_REF] Nijmeijer | Synchronization of Mechanical Systems[END_REF]. Thus, an example of MAS is a group of mobile robots, that interact in order to perform tasks such as surveillance, exploration, or to move large objects, where it is required to keep a desired formation with respect to global 1 reference given by the leader robot. A formation is a set of coordinates that defines a geometric structure in a coordinate space. Thus, leader-follower formation objective is such that followers form a desired formation given by its own desired relative distances with respect to the leader trajectory, where the leader is independent of their followers. This objective can be addressed by a decentralized approach where only local information from the neighbors is required [START_REF] De | Dynamic model based formation control and obstacle avoidance of multi-robot systems[END_REF]. Several approaches based on decentralized control have 1 "global" is used in this paper to refer to variables that involve to the MAS and are denoted by typeface in bold, whereas "local" is used to refer to variables related to each particular agent. been proposed in order to tackle the formation control problem. For example in [START_REF] Brandao | Decentralized control of Leader-Follower formations mobile robots with obstacle avoidance[END_REF] a leader-followers based formation controller is presented for a group of non disturbed unicycle robots. In this control approach it is assumed that the linearization is possible, this being a disadvantage against parametric uncertainties. On the other hand, traditional sliding modes control is used in many applications including formation and consensus control, as it enables accuracy tracking and robustness to disturbances and plant parameter variations. In [START_REF] Bae | Adaptive controller design for spacecraft formation flying using sliding mode controller and neural networks[END_REF] an adaptive sliding mode formation controller based on leader-follower approach is considered for a six degree of freedom (DOF) nonlinear systems. Under this approach, the upper bounds of uncertainties are estimated via neural networks to compensate the effects of unknown disturbances and modeling errors. Nevertheless, the main disadvantage in these works is chattering phenomenon. Approaches based on high-order sliding modes technique represent an interesting alternative, because chattering phenomenon is attenuated. In [START_REF] Yu | Finite-time consensus for second-order multi-agent systems with disturbances by integral sliding mode[END_REF], the finite time consensus problem of disturbed secondorder MAS is addressed through Super Twisting Algorithm (STA) for the disturbance-rejection. However, it should be pointed out that in practical multi-agent system network time delay is unavoidable and is a factor that causes instability in MAS, and should be taken into account in consensus controller design. The communication delay is related to the information flow between agents [START_REF] Liu | Consensus Problem of second-Order Multi-Agent Systems with Input Delay[END_REF]. In [START_REF] Wang | Consensus Seeking for Discrete-time Multi-agent Systems with Communication Delay[END_REF] a consensus protocol based on Smith Predictor, finite spectrum assignment method and the Arstein reduction is proposed with the aim to avoid the effect of large time delay. It should be mentioned that this control approach is not robust to model uncertainties and external disturbances.

In this work, the leader-follower formation control problem for a group of unicycle mobile robots with communication delays is addressed. In order to solve this problem, a decentralized formation control based on recent predictive method [START_REF] Léchappé | New Predictive Scheme for the control of LTI Systems With Input Delay and Unknown Disturbances[END_REF] and STA is proposed. Thus, the delay is compensated and robustness against parametric uncertainties and external disturbances is ensured. This paper is organized as follows: in Section II, a system description and mathematical model of a mobile robot are provided. Thus, a MAS model based on leader-follower approach is introduced. The synthesis of control scheme is derived in Section III, where a state predictor is presented in order to compensate delay effects. Then, a Super-Twisting Control design for driving trajectories of the mobile robots is addressed. Simulation results are given in Section IV with the aim to illustrate the feasibility of the proposed scheme. Finally, conclusions of this work are drawn.

II. SYSTEM DESCRIPTION

In this section, a system description is presented. In the first paragraph of this section, the mathematical model for a single agent is described. Thus, a global representation of MAS under leader-follower scheme is derived in the second paragraph.

A. Unicycle mobile robot model

The unicycle mobile robot represent an attractive combination of a simple wheel configuration with a high traction through pneumatic tires (see Fig. 1). The mathematical model without external disturbances [START_REF] Lawton | A decentralized approach to formation maneuvers[END_REF] of the i-th robot for i = 1, 2, . . . , N ; is defined as

Ẋi = f i (X i ) + g(X i )u i , Y i = h i (X i ), (1)
where

X i = [x i , y i , v i , ω i , θ i ]
T ∈ R 5 denotes the state vector.

The input vector is denoted by

u i = [ F i τ i ] T ∈ R 2
and the terms F i , τ i correspond to the translation force and to the torque applied to guide the robot in the cartesian space, respectively. The position of the i-th robot in the cartesian space is a measurable function denoted by the hand position [START_REF] Lawton | A decentralized approach to formation maneuvers[END_REF] and corresponds to system output. The functions f i (X i ), g i (X i ) are smooth vector fields defined as

h i (X i ) = [x i y i ] T ∈ R 2
f i (X i ) =       v i c θi -l i ω i s θi v i s θi + l i ω i c θi 0 0 ω i       , g i (X i ) =       0 0 0 0 1 mi 0 0 1 Ii 0 0       , ( 2
) where the terms θ i denoted the orientation, v i the linear speed, and ω i the angular speed; s (•) and c (•) correspond to sine and cosine functions. Moreover, m i correspond to the mass and I i to the moment of inertia. In addition, it is considered that the center of mass coincide with the geometric center. On the other hand, system (1) has relative degree vector ρ = (2, 2). Then, directed relationship between the output Y i and input u i is generated after the 2 nd derivative of the output as Ÿi = L 2 fi h i + L gi L fi h i u i , where L fi h i is the Lie derivative of h i with respect to f i for i = 1, . . . , N ; Thus, there exists a diffeomorphism Φ : R 5 → R 5 defined as

Φ(X i ) =       h 1i (X i ) h 2i (X i ) L f h 1i (X i ) L f h 2i (X i ) η i (t)       =       x i y i v i c θi -l i ω i s θi v i s θi + l i ω i c θi θ i       , (3) so that X i = Φ -1 ([ζ i (t), η i (t)]), for ζ i (t) = [ζ 1i (t), ζ 2i (t), ζ 3i (t), ζ 4i (t)] T ∈ R 4 and η i (t) ∈ R. Note that, from (3) the vector [ζ 1i (t), ζ 2i (t)] T = [x i , y i ] T is the hand position and its derivative as [ζ 3i (t), ζ 4i (t)] T = [ ẋi , ẏi ] T .
Thus, if the trajectories of ( 1) are perturbed by a bounded external input disturbance d i (t), the system can be written in transformated coordinates as

ζi (t) = Aζ i (t) + B [α i (ζ i , η i ) + β i (ζ i , η i )(u i (t) + d i (t))] , (4) ηi (t) = φ i (ζ i , η i ), (5)
Y i (t) = Cζ i (t), i = 1, 2, ..., N ; (6)
where, the matrix A, B, and C are given by

A =     0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0     , B =     0 0 0 0 1 0 0 1     , C = [ 1 0 0 0 0 1 0 0 ] , (7)
from (3), the matrix

α i (ζ i , η i ) ∈ R 2 , β i (ζ i , η i ) ∈ R 2×2 and φ i (ζ i , η i ) ∈ R for i = 1, 2
, . . . , N ; are continuous functions where "(t)" it is omitted of its argument, and are defined as

α i (Φ(X i )) = [ -v i ω i s θi -l i ω 2 i c θi v i ω i c θi -l i ω 2 i s θi ] , β i (Φ(X i )) = [ 1 mi c θi -li Ii s θi 1 mi s θi li Ii c θi ] , (8)
φ i (Φ(X i )) = -1 Li ẋi s θi + 1 Li ẏi c θi . ( 9
)
The unknown bounded function

β i (ζ i , η i ) = βi (ζ i , η i ) + ∆β i (ζ i , η i ) is invertible and positive definite for all ζ i (t) and t ∈ [0, ∞). The term βi (ζ i , η i ) = L Gi L Fi h i is the nominal part known of β i (ζ i , η i ),
where the functions F i ∈ R 5 , G i ∈ R 5×2 are smooth vector fields (similar to (2)) with the nominal value of mass mi and moment of inertia Îi . The function 8), and has unknown variations on mass ∆m i and moment of inertia ∆I i . The equation ( 9) represent the internal dynamics which are rendered unobservable and uncontrollable by the transformation (3) [START_REF] Lawton | A decentralized approach to formation maneuvers[END_REF]. The zero dynamic is stable, due to ηi = 0 for ζ 3i , ζ 4i = 0 [START_REF] Lawton | A decentralized approach to formation maneuvers[END_REF]. On the other hand, in order to provide adequate structure for the design of the proposed scheme, the control input u i is implemented by auxiliary control

∆β i (ζ, η i ) ∈ R 2×2 is a bounded perturbation with the same structure of β i (ζ i , η i ) in (
Ω i (t) ∈ R 2 as u i (t) = β-1 i (ζ i , η i )Ω i (t), (10)
Now, the system (4) with the feedback (10) can be expressed as perturbed linear system:

ζi (t) = Aζ i (t) + BΩ i (t) + Bp i (t), (11)
where the perturbation term p i (t) ∈ R 2 is, In the next section, the MAS under leader-follower scheme will be formulated.

p i (t) = α i (Φ(X i )) + β i (ζ i , η i )d i (t) +(β i (ζ i , η i ) β-1 i (ζ i , η i ) -I 2 )Ω i (t). (12)

B. MAS model based on Leader-Follower approach

Let us define a set of N linear systems described by ( 11) and ( 6), given by

ζ ζ ζ(t) = (I N ⊗ A)ζ ζ ζ(t) + (I N ⊗ B)Ω Ω Ω(t) + (I N ⊗ B)p p p(t), Y Y Y (t) = (I N ⊗ C)ζ ζ ζ(t), ( 13
)
where operator ⊗ denotes the Kronecker product (or tensor product) [START_REF] Horn | Matrix analysis[END_REF];

ζ ζ ζ(t) = [ ζ T 1 (t), . . . , ζ T N (t)
] T represents the global state vector, and its elements are the state vectors of the followers. The p p p(t) = [p 1 (t), . . . , p N (t)] T denotes the vector perturbation term and

Ω Ω Ω(t) = [ Ω T 1 (t), . . . , Ω T N (t)
] T is the formation control vector. The multi-agent system follower (13) must track the leader trajectory, that is described in global form as

ζ ζ ζo (t) = (I N ⊗ A)ζ ζ ζ o (t) + (I N ⊗ B)Ω Ω Ω o (t) + Bp p p o (t), (14)
Ω Ω Ω o (t) = 1 1 1 N ⊗ Ω o (t)
, where the vector Ω 0 (t) is the leader input.

Assumption 1. The control signal Ω o (t) ∈ R 2 is a locally integrable function at unbounded interval [-r, +∞[. Assumption 2. The unknown disturbance signal p o (t) ∈ R 2 is a locally integrable function at unbounded interval R ≥ 0 for all t ≥ 0, with a p max ≥ 0 as upper bound, i.e., ∥p o (t)∥ ≤ p max < +∞. A desired formation is a set of references that defines a geometric structure in a coordinate space. Thus, in order to keep a desired formation, each subsystems in (13) cooperate to keep a desired relative distance with respect to the leader trajectory [START_REF] Andrzej | Finite Spectrum Assignment Problem for Systems with Delays[END_REF]. Desired relative distance for i-th robot is denoted by L 1i (t) and L 2i (t) on x-axis and y-axis, respectively, and satisfies the follow equation:

Li (t) = AL i (t) + B Li (t), (15)
where

L i (t) = [L 1i (t), L 2i (t), L1i (t), L2i (t)] T ; the expression B Li (t) = B[ L1i (t) L2i (t) ] T , where B = I 4 -C T C.
Thus, the expression (15) can be written in global form as

L L L(t) = (I N ⊗ A)L L L(t) + (I N ⊗ B) L L L(t), (16)
that will be used in the sliding surface design (presented in the next section), where L L L(t) = [L T 1 (t), . . . , L T N (t)] T . Now, the interconnection topology is given by network star configuration, where the leader trajectory is the centroid of the desired formation. However, only delayed leader information ζ o (t -r) is available and the consensus objective is not ensured. Therefore, the prediction ζo (t) ≃ ζ o (t + r) will be transmitted to provide the approximation of actual value ζ o (t) for each robot (see Figure 2). The prediction of leader state is estimated thank to a state predictor, that will be introduced in the next section.

III. CONTROL SCHEME

In this section, a control scheme based on STA with a Robust Predictor (RP) is presented. In order to compensate time delay, the prediction of leader state is transmitted through communication channel. Thus, the approximate actual value arrive to the followers (see Figure 2). Therefore, the RP will be introduced in the first part of this section, subsequently the sliding surface and the STA are presented at the end of this section.

A. State Predictor [13]

Since the leader trajectory is delayed in communication channel, a RP will be implemented to calculate an estimation of the prediction of leader state. The prediction state of ( 14) without disturbance is denoted by X X X o (t) = 1 1 1 N ⊗ X o (t) and reads as

X o (t) = e Ar ζ o (t) + ∫ t t-r e A(t-Θ) BΩ o (Θ)dΘ, (17)
Now, with the aim to improve the approximated prediction X X X o (t), disturbances effect can be estimated from available state in [START_REF] Andrzej | Finite Spectrum Assignment Problem for Systems with Delays[END_REF] and previous information from (17) as

ζ o (t) -X o (t -r) = ∫ t t-r e A(t-Θ) Bp o (Θ)dΘ, (18)
that is well defined for all t ≥ 0. Finally, the RP is defined as

ζ ζ ζo (t) = X X X o (t) + ζ ζ ζ o (t) -X X X o (t -r), ( 19
)
where the term

ζ ζ ζ o (t) -X X X o (t -r) = 1 1 1 N ⊗ [ζ o (t) -X o (t -r)],
and

ζ ζ ζo (t) ≃ ζ ζ ζ o (t + r).
Remark 1. The disturbance compensation in (18) depends on time delay, because only previous leader information is available. Then, the compensation disturbance in the prediction is given at time (t -r).

B. Super-Twisting Algorithm

In this paragraph, a controller described in [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] is extended to a formation control of MAS with communication delay. In order to ensure the formation control based on consensus, the formation error ε ε ε(t) is defined as 20) corresponds to the relative position of follower with respect to the leader. Thus, if formation error ε ε ε(t) → 0 0 0 2N , the relative distances between leader and followers converge to desired relative distances, i.e.,

ε ε ε(t) = ζ ζ ζ(t) -ζ ζ ζo (t -r) -L L L(t). ( 20
) Note that term ζ ζ ζ(t) -ζ ζ ζo (t -r) of (
ζ ζ ζ(t)-ζ ζ ζo (t-r) → L L L(t). The vector ζ ζ ζo (t-r) = 1 1 1 N ⊗ ζo (t-r) ≃ 1 1 1 N ⊗ ζ o (t)
. Now, considering that the speed is available, the chosen sliding variable is written as a function of the states predicted as

S S S(t) = λ λ λ(I N ⊗ C)ε ε ε(t) + (I N ⊗ CA)ε ε ε(t), ( 21
)
where 21) is linked to the relative position error and the second term to its derivative. λ λ λ = diag{λ i } ∈ R 2N ×2N is a diagonal matrix gain, and λ i ∈ R 2×2 as local diagonal matrix gain definite positive for i = 1, . . . , N ; and is chosen such that all elements of S S S(t), Ṡ Ṡ Ṡ(t) converges to 0 in finite time.

S S S(t) = [s T 1 (t), . . . , s T N (t)] T ∈ R 2N is a global vector composed of local sliding variables of each agent s i (t) = [s i1 (t), s i2 (t)] T ∈ R 2 for i = 1, . . . , N ; the first term in (
Remark 2. Taken 0 0 0 2N ∈ R 2N as a null vector of dimension 2N . The sliding variable (21) is designed taking into account that the desired compensated dynamics of the system (13) are achieved in the sliding mode S S S(t) → 0 0 0 2N . From (21), ( 13) and ( 19), the input-output dynamics of (21) result as

Ṡ Ṡ Ṡ(t) = a a a(ζ ζ ζ(t), ζ ζ ζ o (t -r), t) + Ω Ω Ω(t), (22)
where, for simplicity, a a a

(ζ ζ ζ(t), ζ ζ ζ o (t -r), t) = a a a and, a a a = λ λ λ(I N ⊗ CA)(ζ ζ ζ(t) -ζ ζ ζ o (t -r) -L L L(t)) -Ω Ω Ω o (t) -(1 + r)λ λ λp p p o (t -r) + (I N ⊗ B T ) L L L(t) + p

p p(t).

Assumption 3. There exists unknown positive constant δ 1 such that the derivative of the function

a a a(ζ ζ ζ(t), ζ ζ ζ o (t -r), t) is bounded as ∥ȧ ȧ ȧ(ζ ζ ζ(t), ζ ζ ζ o (t -r), t)∥ ≤ δ 1 .
The problem is to drive the global sliding variable S S S(t) and Ṡ Ṡ Ṡ(t) to 0 0 0 2N in finite time in presence of the bounded perturbation with the boundary δ 1 > 0, where the argument "(t)" from sliding variable (21) and ( 22) will be omit. The robust control Ω Ω Ω(t) = Ω Ω Ω is obtained using the following STA algorithm given by

Ω Ω Ω = -K K K 1 D D D 1/2 s sign(S S S) + ν, ν = - 1 2 K 2 K 2 K 2 sign(S S S) (23)
where the function sign(S S S) returns a vector with the signs of the corresponding elements of S S S ∈ R 2N ; the matrix

D D D s = diag{D si }. The submatrices D si = diag{|s i ι|} for ι = 1, 2,
where s i1 , s i2 are elements of the local sliding surface of each agent s i = [s i1 , s i2 ] T for i = 1, . . . , N , i.e., s iι ∈ s i for ι = 1, 2; so that s i ∈ S S S. The control gains are given by

K K K 1 = diag{k 1i } and K K K 2 = diag{k 2i }
, where all submatrices k 1i , k 2i ∈ R 2×2 are diagonal matrices, for i = 1, . . . , N . From the system ( 22) and ( 23), the closed loop system is given by

Ṡ Ṡ Ṡ = a a a -K K K 1 ς 1 + ν, ν = - 1 2 K K K 2 sign(S S S). ( 24
)
In that case, stability criteria are given in the following theorem.

Theorem 1: Consider the system ( 13) and ( 21) in closedloop with control (23) and references given by state predictor (19). Then, from the assumptions 1, 2 and 3, finite time convergence of S S S and Ṡ Ṡ Ṡ is provided under STA to 2sliding mode for a unknown bound δ 1 > 0 and given initial conditions ς(0) and S S S(0), if the control gains satisfies the matrix inequalities ||K K K 1 || > 0 and λ j (K K K 2 ) > 2δ 1 , where the unknown bounded function 0

< ∥ϱ ϱ ϱ(ζ ζ ζ(t), ζ ζ ζ o (t -r), t)∥ ≤ δ 1 ,
and λ j (K K K 2 ) denotes the real part of the eigenvalues of the matrix K K K 2 for j = 1, . . . , 2N .

Proof: Consider the following vector [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] ς

= [ ς T 1 , ς T 2 ] T = [ sign(S S S) T D D D 1/2 s , a a a T + ν T ] T . ( 25
)
Then, system (24) can be written as

ς = 1 2 [ -D D D -1/2 s K K K 1 D D D -1/2 s -D D D -1/2 s K K K 2 0 0 0 2N×2N ] ς + [ 0 1 ] ⊗ ȧ ȧ ȧ, (26)
Furthermore, in similar way than [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] and due to the Assumption 5, the boundary of the uncertain function ȧ ȧ ȧ =

1 2 D D D -1/2 s ϱ ϱ ϱ(ζ ζ ζ(t), ζ ζ ζ o (t -r), t)ς 1 exists.
In that case, the system (26) can be rewritten as follows

ς = 1 2 (I 2 ⊗ D D D s ) -1/2 A A Aς, ( 27
) taken A A A = [ -K K K 1 I 2N -(K K K 2 -2ϱ ϱ ϱ(ζ ζ ζ(t), ζ ζ ζ o (t -r), t)) 0 0 0 2N×2N ] ,
where

∥ς 1 ∥ ≥ ∥(I 2 ⊗ D D D s )∥ 1/2 = ∥D D D s ∥
V = ς T P P P ς, ( 28
)
where P P P = P P P T > 0 is a constant, symmetric and positive matrix. Now, the derivative of strict Lyapunov function result

V ≤ ∥D D D s ∥ -1/2 ς T [ A A A T P P P + P P PA A A ] ς. ( 29
)
Thus, since the matrix A is Hurwitz, the matrix P P P = P P P T > 0 is an unique solution that satisfies the Algebraic Lyapunov Equation [ A A A T P P P + P P PA A A ] = -Q and V can be written as

V ≤ -∥D D D s ∥ -1/2 λ M in ( Q)∥ς∥ 2 , ( 30
)
taken λ M in ( Q) as the minimum eigenvalue of the matrix Q.

On the other hand, since ∥ς∥ ≥ ∥D D D s ∥ 1/2 it is true that -∥ς∥ ≥ -∥D D D s ∥ -1/2 ∥ς∥ 2 , then one gets that V ≤ -λ M in ( Q)∥ς∥. Now, since that V ≤ λ M ax (P P P )∥ς∥ 2 , taken λ M ax (P P P ) as the minimum eigenvalue of the matrix P P P , this leads to show that

V ≤ - λ M in ( Q) λ 1/2 M ax (P P P ) V 1/2 . ( 31
)
Therefore, since ς 1 and ς 2 convergence in finite time, convergence of S S S and Ṡ Ṡ Ṡ in finite time is ensured.

IV. SIMULATIONS

In this section, simulation results are provided to illustrate the feasibility and the performance of the proposed methodology. In order to compare the performance of the proposed scheme, standard predictor [START_REF] Andrzej | Finite Spectrum Assignment Problem for Systems with Delays[END_REF] with STA has been also considered. System and control scheme have been developed with MATLAB/Simulink environment, using Runge-Kutta solver with an integration step of 0.001s. The MAS is composed by three unicycle robots (one leader and two followers), where the system parameters are given in the Table I. Note that simulation are made with parametric uncertainties in order to evaluated the robustness of proposed approach (for example, mass of the system m i = 0.2kg [see Table I], whereas nominal mass mi = 0.16kg in the controller [see Table II]). The simultaneous time-varying external disturbances applied to each robot are displayed in Figure 3. On the other hand, According to leader-follower formation control approach, the follower robots must keep a desired relative distance with respect to the leader (see Figure 4), where the control of the leader and followers Ω i (t) are based on (23) for i = o, 1, 2. In order to evaluate the performance of the STA with RP, STA using standard predictor (SP) is tested. A time delay r = 0.4 seconds in flow leader information is considered. The norm of prediction error for SP and RP are displayed on Figure 5-Top; it is clear to see that RP has a higher accuracy than SP. The simulation results are shown in Figure 5-Bottom, where the norms of formation error for each approach (STA+SP and STA+RP) are displayed. The trajectories of robots are shown on Figures 7-6. From the graphs, it appears that formation control is ensured with high accuracy with the STA+RP scheme versus external disturbances and parametric uncertainties. Thus, the tracking error is smaller with for STA+RP than with STA+SP.

V. CONCLUSION

In this work, the leader follower formation control problem with time delay communication has been considered. Thus, a control strategy combining a robust state predictor and super twisting algorithm has been proposed. On the other hand, since a good prediction plays an important role in consensus, the predicted leader trajectory is estimated by a robust state predictor. Therefore, dynamics of state predictor is treated as disturbance term in the input-output dynamics of sliding surface. Thus, in order to ensure finite time convergence, sufficient conditions of stability have been presented. Finally, simulation results demonstrated the feasibility and robustness of the proposed scheme.

VI. FUTURE WORKS

This work present some opportunities for extending, for example: evaluate the proposed scheme in experimental way; extend the result to adaptive sliding mode control; include the

Figure 1 .

 1 Figure 1. Differential-drive mobile robot.

  where the perturbation term p p p o (t) = 1 1 1 N ⊗ p o (t) has the same structure than (12), and 1 1 1 N is a N dimensional vector of ones. The state vector ζ ζ ζ o (t) = 1 1 1 N ⊗ ζ o (t) and ζ o (t) = [ζ 1o (t), ζ 2o (t), ζ 3o (t), ζ 4o (t)] T respect to position and speed in coordinated space; finally, the control input

Figure 2 .

 2 Figure 2. Transmission of state predicted trough channel communication.

Figure 3 .

 3 Figure 3. Input disturbances for all agents, where force d F i ∈ d i (t) and torque dτ i ∈ d i (t) concerns to 41.5% of the input F i and 92.91% of the input τ i , respectively, for i ∈ {o, 1, 2}.

Figure 4 .

 4 Figure 4. General scheme of desired formations with respect leader trajectory at different interval times, where j = 1, 2; denote x and y axis, respectively.

Figure 5 .

 5 Figure 5. Comparison of error norm of the leader state prediction (Top) for ε * (t) = ζ ζ ζo(t) -ζ ζ ζo(t -r) and r = 0.4 seconds. Comparison of the norm of formation error (Bottom) for delay r=0.4 seconds, with ε ε ε(t) = ζ ζ ζ(t)-ζ ζ ζo(t)-L L L(t).

Figure 6 .

 6 Figure 6. MAS trajectories with STA + SP, where switching of formation is denoted by vertical dashed line at 15 seconds

Figure 7 .

 7 Figure 7. MAS trajectories with STA + RP, where switching of formation is denoted by vertical dashed line at 15 seconds

Table I MODEL

 I PARAMETERS FOR i = o, 1, 2.

	Parameter	value	Description
	m i	0.2kg	Mass of the system
	I i	6.89 × 10 -5 kgm 2	Moment of Inertia
	l i	0.045m	Distance from center of mass
		Table II
	PARAMETERS OF CONTROL SCHEME FOR i = o, 1, 2.
	Parameter	value	Description
	r	0.4sec	Communication time delay
	mi	0.16kg	Nominal Mass of the system
	Îi	1.2 × 10 -4 kgm 2	Nominal Moment of Inertia
	K K Kι	100 I 4	Control gain for ι = 1, 2
	λ λ λ	I 4	Sliding gain