Ph.D Julie Coloigner
email: julie.coloigner@inria.fr

Jean-Marie Batail

Olivier Commowick

Isabelle Corouge

Christian Barillot

Dominique Drapier

White matter abnormalities in depression: a categorical and phenotypic diffusion MRI study

Keywords: diffusion-weighted imaging, voxel-based analysis, fractional anisotropy value, depression, categorical and phenotypic approach

Mood depressive disorder is one of the most disabling chronic disease with a high rate of everyday life disability that affects 350 million people around the world. Recent advances in neuroimaging have reported widespread structural abnormalities, suggesting a dysfunctional frontal-limbic circuit involved in the pathophysiological mechanisms of depression. However, a variety of different white matter regions has been implicated. These inconsistent results might be attributed to various factors: symptoms and duration of the disease as well as clinical phenotype. In this study, we examined WM integrity in a large cohort (114 patients) compared to a healthy control group. Consistent with previous studies, reduced white matter integrity was observed in the genu of the corpus callosum extending to the inferior fasciculus and posterior thalamic radiation, confirming a frontal-limbic circuit abnormality. Our analysis also reported other patterns of increased fractional anisotropy and axial diffusivity as well as decreased apparent diffusion coefficient and radial diffusivity in the splenium of the corpus callosum and posterior limb of the internal capsule. Moreover, we sought to identify specific WM alterations in relation to specific depressive phenotypes such as anhedonia (i.e. lack of pleasure), anxiety and psychomotor retardation -three core symptoms involved in depression. A positive correlation between FA and anhedonia was found in superior longitudinal fasciculus as well as a negative correlation in the cingulum. Then, the analysis of the anxiety and diffusion metric revealed that increased anxiety was associated with greater FA values in genu and splenium of corpus callosum, anterior corona radiata and posterior thalamic radiation. Finally, the motor retardation analysis showed a correlation between increased Widlöcher depressive retardation scale scores and reduced FA in the body and genu of the corpus callosum, fornix, and superior striatum. Through this twofold approach (categorical and phenotypic), this study has underlined the need to move forward to a research area of biomarkers, which help to understand the pathophysiology of mood depressive disorder and to stratify precise phenotypes of depression with targeted therapeutic strategies.

Introduction

Mood Depressive Disorder (MDD) is one of the most disabling chronic diseases with a high rate of everyday life incapacity that affects 350 million people around the world [START_REF] Smith | Mental health: a world of depression[END_REF]. This pathology is considered underdiagnosed, without adequate therapeutic resources, which emphasizes the need to be a public health priority [START_REF] Ferrari | Burden of depressive disorders by country, sex, age, and year: findings from the global burden of disease study 2010[END_REF]. One of the most important causes of misdiagnosis is the poor interrater reliability of actual classifications [START_REF] Freedman | The initial field trials of DSM-5: new blooms and old thorns[END_REF]. Mood Depressive Episode criterion of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) [START_REF] Edition | Diagnostic and statistical manual of mental disorders[END_REF] indeed encompass a wide range of clinical phenotypes of depression "from gravely disabled melancholic patients to many individuals in the general population who do not seek treatment" [START_REF] Freedman | The initial field trials of DSM-5: new blooms and old thorns[END_REF]. This suggests that DSM-5 does not provide a valid tool for helpful and reliable diagnosis and therapeutic strategies. In this context, despite the extensive therapy options available for depression, up to 80% of patients will suffer from a relapse [START_REF] Gotlib | Neuroimaging and depression: current status and unresolved issues[END_REF]. On the neurobiological side, this disease is characterized by a profound and persistent dysregulation of affect and mood [START_REF] Fitzgerald | Gray colored glasses: is major depression partially a sensory perceptual disorder[END_REF], which coexist with additional disturbance including cognitive dysfunction, insomnia, fatigue and appetite disturbance [START_REF] Drevets | Neuroimaging and neuropathological studies of depression: implications for the cognitive-emotional features of mood disorders[END_REF].

Consequently, understanding the neural correlates underlying depression, and its phenotype variability, will optimize the diagnosis and treatment of individual depressed patients.

Over the last decades, recent advances in neuroimaging have greatly increased our knowledge of MDD, particularly its neural bases. Widespread structural abnormalities have been reported including regional tissue loss in the hippocampus, amygdala, basal ganglia, prefrontal cortex and anterior cingulate cortex [START_REF] Arnone | State-dependent changes in hippocampal grey matter in depression[END_REF][START_REF] Zou | Changes of brain morphometry in first-episode, drug-naïve, non-late-life adult patients with major depression: an optimized voxel-based morphometry study[END_REF]. These results suggested that a dysfunctional cortical-subcortical neural circuit is involved in the pathophysiology and psychopathology of the depression [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF][START_REF] Sacchet | Structural abnormality of the corticospinal tract in major depressive disorder[END_REF]. More particularly, a frontal-limbic circuit abnormality is considered as the locus of the dysfunction underlying mood-regulation in MDD [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF][START_REF] Sacchet | Structural abnormality of the corticospinal tract in major depressive disorder[END_REF]. Structural connectivity analysis provides important information allowing for the characterization of cortical and subcortical abnormalities. White matter (WM) investigations have become a rapidly growing area of research in psychotic disorders including schizophrenia [START_REF] Buchsbaum | MRI white matter diffusion anisotropy and PET metabolic rate in schizophrenia[END_REF], bipolar disorder [START_REF] Benedetti | White matter microstructure in bipolar disorder is influenced by the serotonin transporter gene polymorphism 5-HTTLPR[END_REF][START_REF] Magioncalda | Patterns of microstructural white matter abnormalities and their impact on cognitive dysfunction in the various phases of type I bipolar disorder[END_REF] and depression [START_REF] Chen | Intrinsic disruption of white matter microarchitecture in first-episode, drugnaive major depressive disorder: A voxel-based meta-analysis of diffusion tensor imaging[END_REF]. In this context, in the brain, diffusion-weighted imaging (DWI), a noninvasive magnetic resonance imaging (MRI) technique based on the extent of water diffusion, enables the quantification of the fiber orientation and integrity of WM pathways within neural networks [START_REF] Assaf | Diffusion tensor imaging (DTI)-based white matter mapping in brain research: a review[END_REF][START_REF] Taylor | Diffusion tensor imaging: background, potential, and utility in psychiatric research[END_REF]. The most common DWI measure is the fractional anisotropy (FA), an invariant property of DWI that reflects a nonspherical diffusion tensor with a preferential orientation.

Several MDD studies [START_REF] Chen | Intrinsic disruption of white matter microarchitecture in first-episode, drugnaive major depressive disorder: A voxel-based meta-analysis of diffusion tensor imaging[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF] have revealed significant FA reductions in the body of the corpus callosum (CC) [START_REF] Han | Cortical thickness, cortical and subcortical volume, and white matter integrity in patients with their first episode of major depression[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF], bilateral anterior limb of the internal capsule (ALIC) [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF][START_REF] Jia | High-field magnetic resonance imaging of suicidality in patients with major depressive disorder[END_REF][START_REF] Zou | Alterations of white matter integrity in adults with major depressive disorder: a magnetic resonance imaging study[END_REF] and superior longitudinal fasciculus (SLF) [START_REF] Ota | White matter abnormalities in major depressive disorder with melancholic and atypical features: A diffusion tensor imaging study[END_REF] in MDD patients relative to healthy controls. However, existing knowledge about WM integrity is limited due to the small number of participants in most studies and the variety of different WM regions implicated [START_REF] Liao | Is depression a disconnection syndrome? Meta-analysis of diffusion tensor imaging studies in patients with MDD[END_REF][START_REF] Murphy | Meta-analysis of diffusion tensor imaging studies shows altered fractional anisotropy occurring in distinct brain areas in association with depression[END_REF][START_REF] Sexton | A systematic review of diffusion tensor imaging studies in affective disorders[END_REF]. These inconsistent results might be attributed to various factors. The high heterogeneity of MDD individuals in terms of disease characteristics (i.e. symptoms and duration of the disease), could be considered a main cause of these inconsistencies [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF]. Previous studies have indeed demonstrated the influence of episodes or illness duration (de [START_REF] De Diego-Adelino | Microstructural white-matter abnormalities associated with treatment resistance, severity and duration of illness in major depression[END_REF], illness severity [START_REF] Henderson | A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability[END_REF] and antidepressant drugs [START_REF] Dusi | Brain structural effects of antidepressant treatment in major depression[END_REF][START_REF] Taylor | Cingulum bundle white matter lesions influence antidepressant response in late-life depression: a pilot study[END_REF] on WM. The FA reduction was associated with depression severity and illness duration in MDD patients in the genu of corpus callosum, anterior thalamic radiation, anterior cingulum, and sagittal striatum, which indicates that DWI may be of clinical value in measuring and tracking disability in MDD [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF][START_REF] Henderson | A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability[END_REF]. One additional source of heterogeneity could be the clinical phenotype [START_REF] Fried | Problematic assumptions have slowed down depression research: why symptoms, not syndromes are the way forward[END_REF], such as anhedonia (i.e. lack of pleasure), anxiety and psychomotor retardation -three core symptoms involved in depression. As mentioned above, DSM-5 diagnosis criteria include a wide panel of emotional, motivational and cognitive symptoms [START_REF] Edition | Diagnostic and statistical manual of mental disorders[END_REF]. Some of these dimensions can be negatively correlated such as motivation and anhedonia severity [START_REF] Batail | Apathy and depression: Which clinical specificities? Personalized Medicine in Psychiatry[END_REF]. Then, the high variability of diagnosis criteria could lead to poor diagnosis interrater reliability [START_REF] Freedman | The initial field trials of DSM-5: new blooms and old thorns[END_REF]. For the last two decades, neuroimaging studies in MDD have identified many putative biomarkers, based on actual disease classification, which remain inconsistent tools. Therefore, there is a need to find new imaging biomarkers which could potentially improve our understanding the pathophysiology of MDD [START_REF] Young | Is there progress? An overview of selecting biomarker candidates for major depressive disorder[END_REF]. Then, the investigation of core dimensions of depression such as anxiety, anhedonia and psychomotor retardation, is of great importance for identifying the fundamental and specific changes in the microarchitecture of WM, elucidating the essential pathophysiology of MDD, and help to stratify precise phenotypes of depression with targeted therapeutic strategies.

Therefore, the first aim of this study was to examine WM integrity in a larger cohort of MDD patients to address more consistent and replicable WM microarchitecture abnormalities than previously found in previous studies. The second aim of our study was to identify specific WM alterations in relation to specific dimensions of depression such as anhedonia, anxiety and psychomotor retardation.

Materials and Methods

Participants

One hundred and fourteen depressed patients were recruited from routine care units in the psychiatric university hospital of Rennes between November 2014 and January 2017 and were enrolled in a naturalistic prospective cohort study. The study was approved by an ethic committee and is registered in www.clinicaltrial.gov (NCT02286024), written informed consents were obtained from all subjects.

The study was proposed to patients suffering from a Mood Depressive Episode (MDE) under DSM-5 criteria with or without personal history of Mood Depressive Disorder (unipolar or bipolar subtype). Exclusion criteria included other Axis-I disorders (except for Obsessive-Compulsive Disorder and four anxious comorbidities such as posttraumatic stress disorder, social phobia, generalized anxiety disorder or panic disorder), which were explored using the Mini-International Neuropsychiatric Interview [START_REF] Lecrubier | The Mini International Neuropsychiatric Interview (MINI). A short diagnostic structured interview: reliability and validity according to the CIDI[END_REF]. Patients with severe chronic physical illness were not included. Other exclusion criteria were potential safety contraindications for MRI (pacemakers, metal implants, pregnancy, and lactation), diagnosed neurodegenerative disorders (e.g. Parkinson's disease, Alzheimer's disease, Huntington's disease), a history of significant head injury, or diagnosed dementia (according to DSM-5 criteria).

Depressed patients underwent clinical interview and examination, including routine neuropsychological testing and MRI. For each subject, demographic data, comorbidities and medication, as well as clinical variables were collected (see Table 1). A composite measure of medication load for each patient was assessed using a previously established method [START_REF] Sackeim | The definition and meaning of treatment-resistant depression[END_REF].

The control group was composed of 65 healthy volunteers, who previously participated in neuroimaging studies.

The two groups were matched in terms of age (CTL: 48.6 ±11.8, MDD: 48.3±15.4; Student t-test p-value=0.5) and gender (CTL: 17M/48F, MDD: 43M/71F; Chi-square p=0.5).

Clinical assessment

Patients were assessed by a single structured clinical interview by a trained psychiatrist. Depression severity was assessed using Montgomery and Åsberg Depression Rating Scale (MADRS) [START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF]. Additionally, the Widlöcher Depressive Retardation Scale (WDRS) [START_REF] Widlöcher | Psychomotor retardation: clinical, theoretical, and psychometric aspects[END_REF] was used to estimate psychomotor retardation. State anxiety was measured using State Trait Anxiety Inventory A (STAI-YA) [START_REF] Spielberger | Manual for the state-trait anxiety inventory[END_REF].

Then, the anhedonia scores were assessed by Snaith Hamilton Pleasure Scale (SHAPS) [START_REF] Snaith | A scale for the assessment of hedonic tone the Snaith-Hamilton Pleasure Scale[END_REF].

MRI acquisition

Patients and healthy subjects were scanned on a 3T whole body Siemens MR scanner (Magnetom Verio, Siemens Healthcare, Erlangen, Germany) with a 32-channel head coil. The 3D T1-weighted image was acquired covering the whole brain (176 sagittal slices) with TR = 1.9 s, TE = 2.26 ms, flip angle =9°, in-plane resolution = 2 mm × 2 mm, FOV = 256 mm × 256 mm and thickness/gap =1.0/0 mm). DWI data were gathered on 60 slices using an interleaved slice acquisition, slice thickness of 2 mm, no gap, in-plane resolution = 1 mm × 1 mm and in a 256 mm × 256 mm field of view. The acquisition matrix was 128 × 128, the reconstruction matrix was 128 × 128, using 30 directions and a b-value of 1000 s/mm 2 . TR/TE = 11 000/99 ms, flip angle was 90°, pixel bandwidth was 1698 Hz, and the imaging frequency was 123.2 MHz.

Image processing

Preprocessing of diffusion images was mainly performed using the open source medical image processing toolbox Anima (https://github.com/Inria-Visages/Anima-Public/wiki). Diffusion images were corrected for eddy currentinduced image distortion using Block-Matching Distortion Correction method ensuring an opposite symmetric transformation [START_REF] Hedouin | Block-Matching Distortion Correction of Echo-Planar Images With Opposite Phase Encoding Directions[END_REF]. Then, rigid realignment was then performed to compensate for subject motion and ensure voxel-to-voxel correspondence across gradients. Then, denoising step using blockwise non-local means filtering was applied [START_REF] Coupé | An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images[END_REF]. Skull stripping was performed using an atlas-registration-based method. The structural image of each patient was transformed to the atlas image, using the linear and non-linear block-matching algorithms (Commowick et al., 2012a, b;[START_REF] Ourselin | Block matching: A general framework to improve robustness of rigid registration of medical images[END_REF]. Then, the atlas intracranial brain mask was inversely transformed by applying the inverse of two calculated transformation matrix and multiplied with the original structural image of the patient. Diffusion tensor images were then estimated using a log-Euclidean estimation method proposed by Fillard et al. [START_REF] Fillard | Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics[END_REF], from which diffusion scalar maps were calculated.

Then, a two-step co-registration method was used to transform the diffusion data into the Montreal Neurological Institute (MNI) template space. First, the FA image was co-registered with the structural image; then the structural image was non-linearly transformed to the MNI template space, using the block-matching algorithms (Commowick et al., 2012a, b;[START_REF] Ourselin | Block matching: A general framework to improve robustness of rigid registration of medical images[END_REF]. Then, we applied these parameters to the other diffusion metrics to normalize them to the MNI space. Next, an average FA image was generated and thresholded to FA >0.3 to confine the statistical analysis within the WM regions [START_REF] Du | Diffusion tensor imaging of the structural integrity of white matter correlates with impulsivity in adolescents with internet gaming disorder[END_REF]. Then, this map was used to mask the FA, apparent diffusion coefficient (ADC), axial diffusivity (AD) and radial diffusivity (RD) maps, to include the major WM tracts and avoid small peripheral tracts that would cause excessive inter-participant variability.

Statistical analysis

Statistical tests of the diffusion metrics (FA, ADC, AD and RD) in the MNI space between MDD and CTL were performed using voxel-based two-sample t-test with age, gender, duration of disease and medication load as nuisance covariates. The result was corrected for multiple comparison using False Discovery Rate (FDR) correction with (p<0.05) [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF].

To investigate the relationship between diffusion metrics and clinical variables (WDRS, STAI-YA and SHAPS) in the MDD group, Pearson correlation was performed in a voxelwise manner. This analysis was restricted to FA values, because the metric is more likely to capture microstructure-induced diffusion abnormalities that ADC, AD and RD [START_REF] Westin | Processing and visualization for diffusion tensor MRI[END_REF]. Thresholds were set at a corrected p < 0.05, with multiple comparison correction using FDR correction (p<0.05) [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF].

Results

Demographics and clinical measures

Women, middle-aged predominantly represented the patient population. They have severe characteristics of disease with a long mean duration of illness (range: 25%<3 years and 17%>21 years) and episode (range: 50%≥ 14 episodes). A large majority of patients suffered from recurrent depressive episodes (50% of the patients had more that 3 episodes). Actual depressive episode was moderately intense (MADRS total score = 27.4±5.9). Indeed, 78% of this cohort suffered from high anxiety (36≤AES≤65) and 27% from very high anxiety higher than 65. All patients were under treatment (Table 1).

Whole brain structural alteration Data: Differences between Groups

Compared with CTL, the MDD group had lower FA in the cingulum, the inferior longitudinal fasciculus (ILF), posterior thalamic radiation (PTR), cingulum and the genu of CC (see Figure 1). However, the MDD group had greater FA in the posterior limb of the internal capsule (PLIC) and splenium of CC. The maximum FA differences in genu of CC and splenium of CC were -33% and +11%, respectively. The identified WM regions and maximum FA values for these clusters (corrected for age, gender, disease duration and medication load) are summarized in Table 2. For these significant regions, the results of two-sample t-test were also reported in the Table for the other diffusion metrics (ADC, AD and RD). Of the 6 clusters examined, all of them showed significant RD differences after FDR correction, whereas only 3 and 5 regions showed significant differences for ADC and AD, respectively.

Correlation of structural connectivity and clinical scores

Analyses revealed eight clusters, as displayed in

Discussion

Anatomic connectivity abnormalities in depression

This is the first study analyzing diffusion metrics in such a large depression cohort. Analyses revealed widespread white matter abnormalities in the MDD group compared to CTL. As previously reported in many studies [START_REF] Chen | Intrinsic disruption of white matter microarchitecture in first-episode, drugnaive major depressive disorder: A voxel-based meta-analysis of diffusion tensor imaging[END_REF][START_REF] Duffy | Cognitive impairment with and without depression history: an analysis of white matter microstructure[END_REF][START_REF] Henderson | A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF][START_REF] Olvet | A comprehensive examination of white matter tracts and connectometry in major depressive disorder[END_REF], reduced WM integrity (i.e. decreased of FA and AD as well as increased ADC and RD) was observed in the genu of the CC extending to the ILF and PTR in MDD patients relative to healthy controls. Decreased FA and AD were observed in the cingulum combined with a normal or slightly increased RD and ADC. This pattern may represent decreased myelination [START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF] ; however, it could also result from a decrease in the number of normally myelinated fibers or an altered WM microstructure in the affected WM tracts (e.g., axonal degeneration [START_REF] Bennett | Age-related differences in multiple measures of white matter integrity: A diffusion tensor imaging study of healthy aging[END_REF]).

The CC is the largest WM pathway providing a physical pathway for information transfer between the two hemispheres. The genu is the most rostral region of the corpus callosum connecting between prefrontal brain regions. The most caudal region, the splenium, connects association areas of the parietal and temporal lobes (anterior splenium) and occipital lobes (posterior splenium). In addition, the body is the midsection between the genu and the splenium associating the motor areas. In general, the corpus callosum changes structurally throughout life, but most dramatically during childhood and adolescence. Moreover, patients with disruption in axonal myelination exhibit executive deficits as well as cognitive dysfunction. Therefore, this tract constitutes key components of the frontotemporal and frontolimbic connections whose dysfunction is thought to underlie many of the emotional, cognitive and behavioral deficits associated with depression. However, the splenium is less vulnerable to damage than the genu and body of the CC, which are both late-myelinating. Moreover, reduced callosal thickness was found in this area only in late-onset depression compared to control, reflecting a more widespread compromise in corticocortical connectivity [START_REF] Ballmaier | Mapping callosal morphology in early-and late-onset elderly depression: an index of distinct changes in cortical connectivity[END_REF].

The cingulum bundle lies within the cingulate gyrus and is an important association pathway linking prefrontal and parahippocampal regions. To date there is no conclusive evidence for cingulum bundle microstructural alterations in MDD groups. No changes in FA were demonstrated in adults [START_REF] Carballedo | Reduced fractional anisotropy in the uncinate fasciculus in patients with major depression carrying the met-allele of the Val66Met brain-derived neurotrophic factor genotype[END_REF][START_REF] Zhang | Quantitative tract-specific measures of uncinate and cingulum in major depression using diffusion tensor imaging[END_REF]. However, reductions of FA in this region were reported in treatment-resistant MDD (de [START_REF] De Diego-Adelino | Microstructural white-matter abnormalities associated with treatment resistance, severity and duration of illness in major depression[END_REF]. The structural brain tissue markers FA and MD of the middle-anterior and middle-posterior cingulum bundle was associated with executive functioning and divided attention [START_REF] Schermuly | Association between cingulum bundle structure and cognitive performance: an observational study in major depression[END_REF]. The PTR joins the occipital and posterior parietal cortex with the posterior thalamus, including the pulvinar, and includes the optic radiation from the lateral geniculate body. A previous meta-analytic study that compared 231 MDD patients with 261 healthy participants found reduced FA in posterior thalamic radiation [START_REF] Liao | Is depression a disconnection syndrome? Meta-analysis of diffusion tensor imaging studies in patients with MDD[END_REF]. This reduction of WM integrity in thalamus may be related to the role of the thalamus in motivation circuit, contributing to the motivational deficits associated with depression [START_REF] Henderson | A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability[END_REF] or to dysfunctional facial recognition in patients with MDD, as shown by other functional MRI studies [START_REF] Liao | Is depression a disconnection syndrome? Meta-analysis of diffusion tensor imaging studies in patients with MDD[END_REF].

The ILF connects the anterior part of temporal and occipital lobes, and could be related to emotion and visual processing information deficits of depression (de [START_REF] De Diego-Adelino | Microstructural white-matter abnormalities associated with treatment resistance, severity and duration of illness in major depression[END_REF].

From a clinical perspective, our findings provide robust evidence that the reduction of white-matter integrity in the interhemispheric connections and fronto-limbic neuronal circuits may lead to a better understanding of the pathophysiology of MDD. These results are consistent with an overall hypothesis that depression involves a disconnection of prefrontal, striatal, and limbic emotional areas.

However, this analysis also reported other patterns with an increase of FA and AD as well as decreased ADC and RD in the PLIC and sCC in patients with depression. These findings in this large MDD cohort contrast with the previous results obtained with small and heterogeneous samples of patients suffering from depression. Previous studies reported inconsistent results concerning the sCC, varying from reduced WM integrity [START_REF] Ota | White matter abnormalities in major depressive disorder with melancholic and atypical features: A diffusion tensor imaging study[END_REF][START_REF] Reppermund | White matter integrity and late-life depression in community-dwelling individuals: diffusion tensor imaging study using tract-based spatial statistics[END_REF] to no effect. The white matter abnormality in the sCC and PLIC could potentially be associated with anxiety [START_REF] Kim | White matter correlates of anxiety sensitivity in panic disorder[END_REF]. As described in the section 4.2, patients of this cohort indeed suffered from high anxiety.

Phenotypic approach

This study builds on the research domain criteria (RDoC) approach. This initiative was proposed by the American National Institute of Mental Health to develop "new ways of classifying mental disorder based on behavioral dimensions and neurobiological measures" [START_REF] Nimh | National Institute of Mental Health. The National Institute of Mental Health strategic plan[END_REF]. This approach promises to increase our understanding of basic network-level abnormalities and their relation to psychopathology [START_REF] Insel | Research domain criteria (RDoC): toward a new classification framework for research on mental disorders[END_REF]. Depression, as currently defined, spans two of the RDoC domains: the Loss construct within the Negative Valence Systems domain and various Reward constructs within the Positive Valence Systems domain. This study focuses specifically on these two dimensions by analyzing the relation between diffusion metrics and two clinical measures, called SHAPS and STAI-YA measures [START_REF] Cuthbert | Toward the future of psychiatric diagnosis: the seven pillars of RDoC[END_REF][START_REF] Insel | Research domain criteria (RDoC): toward a new classification framework for research on mental disorders[END_REF] Then, although RDoC program does not propose a behavioral domain, the relationship with psychomotor retardation using WRDS was investigated because of its central role in depression and discriminative potential for different phenotypes (melancolic subtype, bipolar/unipolar and consequences on treatment) [START_REF] Bennabi | Psychomotor retardation in depression: a systematic review of diagnostic, pathophysiologic, and therapeutic implications[END_REF]. Through these three core dimensions of depression, this study highlights that depressive phenotypes can be underlined by wide differences according to the level of anxiety, anhedonia or psychomotor retardation. As developed below, phenotypic approach leads to a more specific, consistent and closer to physiopathology biomarkers.

The relationship between WM integrity and anhedonia

Consistent with previous studies, we found decreased FA as anhedonia increased in the cingulum. Indeed, Bracht et al. suggested that this tract could be a biomarker of vulnerability to depression [START_REF] Bracht | A review of white matter microstructure alterations of pathways of the reward circuit in depression[END_REF]. Thus, the FA value in the cingulum bundle was also described negatively correlated with trait anhedonia [START_REF] Keedwell | Cingulum white matter in young women at risk of depression: the effect of family history and anhedonia[END_REF].

In addition, the same abnormality was described in women with a family history of depression [START_REF] Keedwell | Cingulum white matter in young women at risk of depression: the effect of family history and anhedonia[END_REF], or in first depressed un-medicated patients [START_REF] Yang | White matter microstructural abnormalities and their association with anticipatory anhedonia in depression[END_REF]. Alltogether, these results suggested that cingulum WM alteration is involved in the pathophysiological mechanism of depression and is probably linked with reward sensitivity dimension.

Increased FA related to increased anhedonia was also found in SLF. Some studies reported the role of this tract in linking aspects of default mode network and cognitive control and its consequence on executive functioning [START_REF] Jenkins | Shared white matter alterations across emotional disorders: A voxel-based meta-analysis of fractional anisotropy[END_REF]. Moreover, patients suffering from depression could exhibit a deficit in executive functioning and related with a deficit in programing actions relative to hedonic valence. However, based on these arguments, the opposite correlation between FA and SHAPS was found in SLF. This result may represent a compensatory neuroplasticity or selective neurodegeneration mechanism [START_REF] Jenkins | Shared white matter alterations across emotional disorders: A voxel-based meta-analysis of fractional anisotropy[END_REF][START_REF] Mole | Increased fractional anisotropy in the motor tracts of Parkinson's disease suggests compensatory neuroplasticity or selective neurodegeneration[END_REF].

Other significant decreased FA were found associated with decreased anhedonia in PTR. This result echoes the literature that pointed out the involvement of thalamic cortical projections in MDD [START_REF] Hermesdorf | Reduced fractional anisotropy in patients with major depressive disorder and associations with vascular stiffness[END_REF][START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF]. Previous studies have found that such WM alterations were particularly significant in melancholic MDD [START_REF] Korgaonkar | Loss of white matter integrity in major depressive disorder: Evidence using tract-based spatial statistical analysis of diffusion tensor imaging[END_REF]. Our study is the first that links anhedonia and PTR. Moreover, decreased FA was related with increased anhedonia in gCC. WM abnormalities of gCC were linked with the severity of depression (Chen et al., Sci reports, 2016). This tract is involved in decision making, reward processing (i.e.

anhedonia) and emotional regulation [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF]. WM alteration in this area leads to disturbances in cognitive functioning such as memory, executive functions and emotions [START_REF] Chen | Disorganization of white matter architecture in major depressive disorder: a meta-analysis of diffusion tensor imaging with tract-based spatial statistics[END_REF]. Taken together, our result can lead to the hypothesis that WM alteration in gCC and PTR participates to the severity of depression through one specific phenotype involved in emotion regulation, executive functioning, reward, and decision-making such as anhedonia.

Anxiety and WM integrity

To our knowledge, no study has correlated patient anxiety with white matter integrity in depressed patients. Most of the studies have chosen a categorical approach in comparing MDD with or without anxious comorbidity [START_REF] Canu | Brain structural abnormalities in patients with major depression with or without generalized anxiety disorder comorbidity[END_REF][START_REF] Delaparte | A comparison of structural connectivity in anxious depression versus nonanxious depression[END_REF]. Despite a large sample, no significant connectivity abnormality has been linked

with anxiety [START_REF] Delaparte | A comparison of structural connectivity in anxious depression versus nonanxious depression[END_REF].

In this study, the phenotypic analysis of anxiety revealed a relationship between anxiety and FA values in genu and splenium of CC, ACR and PTR. Kim et.al. also reported a positive correlation between anxiety and FA values in these three tracts [START_REF] Kim | White matter correlates of anxiety sensitivity in panic disorder[END_REF]. Moreover, a meta-analysis has identified that patients suffering from emotional disorders, such as post-traumatic stress disorders had decreased FA in superior corona radiata and anterior thalamic radiation compared with healthy controls [START_REF] Jenkins | Shared white matter alterations across emotional disorders: A voxel-based meta-analysis of fractional anisotropy[END_REF].

Psychomotor retardation and WM integrity

Our phenotypic analysis with motor retardation revealed a correlation between increased WRDS scores and reduced FA in body and genu of CC, superior corona radiata and superior striatum. To date, only two studies had explored the relationship between white matter integrity and motor retardation in depression, using a behavioral metric called actimetry. First, a positive association was found between activity level and structural alterations of cortico-cortical white matter in motor pathways, such the connection between the rostral anterior cingulate cortex and pre-Supplementary Motor Area (SMA) and between the dorsolateral prefrontal cortex and pre-SMA in MDD [START_REF] Bracht | Cortico-cortical white matter motor pathway microstructure is related to psychomotor retardation in major depressive disorder[END_REF]. The authors attributed this result to a relation between these circuits and movement initiation.

Then, the second study has pointed out a negative link between FA and left parahippocampal gyrus [START_REF] Walther | Frontal white matter integrity is related to psychomotor retardation in major depression[END_REF], suggesting a plausible link between cerebral alteration in this region and impaired motor planning.

Consistent with these results, we exhibited the implication of parahippocampal gyrus and the cingulate.

Furthermore, we found that striatal pathway could also play a role in psychomotor retardation. This circuit was also reported in [START_REF] Bennabi | Psychomotor retardation in depression: a systematic review of diagnostic, pathophysiologic, and therapeutic implications[END_REF], showing a link between this behavior impairment and dopamine as well as noradrenergic transmission in basal ganglia such as striatum and caudate nucleus.

To sum up, this phenotypic approach pointed out abnormal structural connectivity that have never been demonstrated in depression. One critical findings concern anxiety which have linked with higher connectivity in areas such as corpus callosum, anterior corona radiata and posterior thalamic radiation whereas categorical for depression have shown strictly the opposite findings for posterior thalamic radiation and genu of corpus callosum. These results support the stratification of patients suffering from depression according to these core dimensions. This study allowed a better understanding of critical mechanism involved in pathophysiology of depression and to more reliable imaging biomarkers.

Limitations

The depression cohort is heterogeneous with regard to demographic and clinical parameters such as age and gender of subjects, age of onset of depression, potentially increasing variability. However, the patient heterogeneity is also a strength of this study because it disrupts coincidental associations.

Another investigation will be needed to determine whether diffusion changes are associated with poorer neurocognitive performance. Further studies should include neurocognitive tests in order to refine clinical phenotype such as executive functioning, memory, attention, information treatment speed, and emotional patterns, and therefore permit a better accuracy of biomarkers.

Another limitation is the use of self-completed questionnaire to assess anhedonia (SHAPS) and psychomotor retardation (WRDS). Furthermore, these latter scales despite their good psychometric properties are not considered as valid scales for dimensional approach such as RDoC.

All patients were under psychotropic medication. There is some evidence that psychotropic treatment could lead to decreased FA in schizophrenic patients after medication [START_REF] Wang | White-matter microstructure in previously drug-naive patients with schizophrenia after 6 weeks of treatment[END_REF]. Another study have pointed out that such confounding factor could open to type II errors [START_REF] Hafeman | Effects of medication on neuroimaging findings in bipolar disorder: an updated review[END_REF]. We assumed that this is a limitation but have chosen to control this issue using a medication load score as described above.

Finally, it is important to note that this study cannot be included in a RDoC program. In fact, it does not strictly emphasize the recommendations such as translational perspective (normal to pathologic, from basic science to behavioral science), or the use of reliable and valid measures of fundamental components [START_REF] Cuthbert | Toward the future of psychiatric diagnosis: the seven pillars of RDoC[END_REF].

Nevertheless, this study addresses the utility of a deconstruction of actual phenotype of depression from a simple disease-based biomarker to a more specific, symptom-based, severity-related, biomarker.

Conclusions

Our study aimed to conduct both categorical and dimensional approach in the study of anatomical connectivity biomarkers in MDD.

First, our results highlight the well-known WM abnormalities in a large cohort of currently depressed patients and compared to controls. The frontolimbic disconnection hypothesis has been replicated as well as large decreased connectivity in widespread brain areas (corpus callosum extending to the inferior fasciculus and posterior thalamic radiation). In addition to previous studies with smaller samples, our results reveal other patterns of increased FA in posterior limb of internal capsule and splenium of corpus callosum.

On the other side, the phenotypic approach has identified some specific patterns related to anxiety, anhedonia and psychomotor retardation -three core symptoms of depression. This latter approach highlighted that within a depressive population, there is a wide variability of anatomic connectivity depending on the severity of each core dimension. This study pointed out that such an approach probably allows more specific and accurate biomarkers than categorical ones, which emphasized widespread abnormalities.

In the future, a dimensional approach could help stratifying different depressive phenotypes, better understanding the pathophysiology characterizing core dimensions, and consequently propose a robust framework for developing targeted therapeutic strategies.

Figure 1 :

 1 Figure 1: Axial, coronal and sagittal brain slices showing significant differences of the FA values between MDD and control groups, overlaid on the FA skeleton of Johns Hopkins University diffusion white matter atlas (green). Voxels with negative t-values (MDD<Control) are shown in blue and positive values (MDD>Control) with red. Arrows indicate the five clusters: genu (1) and splenium (3) of corpus callosum, posterior thalamic radiation/Inferior longitudinal fasciculus (2), bilateral posterior limb of internal capsule (4) and cingulum (5).

Figure 2 .

 2 As the psychomotor retardation increased (i.e. WRDS values increased), the FA value decreased in the body and the genu of the CC, the superior corona radiata, the superior striatum (see Figure2.A.). Additionally, as displayed in Figure2.B., a positive correlation was found between the FA values and anxiety, measured by STAI-YA, in the genu and splenium of the CC, anterior corona radiata and posterior thalamic radiation. However, anxiety was negatively correlated with FA values in the body of CC. Similarly, a negatively correlation was found between the FA values and anhedonia with cingulum, genu of CC and posterior thalamic radiation (see Figure2.C.). Furthermore, increased anhedonia (i.e. SHAPS values increased) was associated with greater FA values in superior longitudinal fasciculus.

Figure 2 :

 2 Figure 2: Axial, coronal and sagittal brain slices showing significant correlation between FA values and clinical variables: (A) WDRS, (B) STAI-YA and (C) SHAPS within the MDD group, overlaid on the FA skeleton of Johns Hopkins University diffusion white matter atlas (green). Voxels with negative correlation are shown in blue and positive correlation with red. Eight clusters are indicated by arrows: genu (1), body (2) and splenium (5) of corpus callosum, anterior corona radiata (ACR) (3), posterior thalamic radiation/Inferior longitudinal fasciculus (4), superior longitudinal fasciculus (SLF) (6) and cingulum (7).

Table 1 :

 1 Demographic and clinical characteristics of the MDD group.

		MDD group (n=114)
		Mean	SD	Range
	Sociodemagraphic
	variables
	Age (years)	48.2	15.3	18 -77
	Gender (M/F)	43M/71F	-	-
	Education (years)	12.2	3.6	45078.0
	Duration of illness (years)	15.1	13.9	0 -60
	Number of episodes	4.4	4.5	0-30
	Duration of episode (weeks)	30.0	37.8	0-170
	Number of suicidal attempts	1.1	2.0	0-10
	Medication
	Medication load	3.1	1,2	0-7
	Antidepressant	74.6%	-	-
	Mood stabilizer	35.1%	-	-
	Antipsychotic	14.0%	-	-
	Benzodiazepine	49.1%	-	-
	Clinical variables
	WDRS	21.4	9.0	2 -43
	MADRS	27.1	5.9	15 -43
	STAI-YA
	SHAPS	5.5	4.0	0 -14
	AES	40.3	8.9	24 -69

All results except gender and medications are given as mean, standard deviation (sd) and the range. The percentage of patients having used the prescribed medicines: antidepressant, mood stabilizer, antipsychotic and benzodiazepine have been reported. WDRS: Widlöcher Depressive Retardation Scale; MADRS: Montgomery-Åsberg Depression Rating Scale; STAI: State-Trait Anxiety Inventory A; SHAPS: Snaith Hamilton Pleasure Scale.

Table 2 :

 2 Brain areas with significant FA differences between the MDD and CTL groups.

	Inter-group comparison			MDD vs.CTL
	Cerebral regions	Tracts	Right or Left	MNI coordinates	t-value	cluster size
	Anterior cingulate	gCC	R/L	[±11,35,5]	-7.41	6959
	Cingulum gyrus	Cingulum	R/L	[±11,-25,33] -9.36	36859
	Insula	ILF	R/L	[±6,-16,31]	-3.39	4338
	Thalamus	PTR	R/L	[±27, -28,7]	-10.15	4109
	Parahippocampal gyrus	PLIC	R/L	[±15,18,-16] 7.01	451
	Posterior cingulate	sCC	R/L	[-8,38,7]	3.91	398

The t-value corresponds to the maximum value of the cluster. The cluster size is given in mm 3 . gCC and sCC: genu and splenium of the corpus callosum, ILF: Inferior Longitudinal Fasciculus, PTR: Posterior Thalamic Radiation (ATR), PLIC:

Posterior Limb of the Internal Capsule.

Table 3 :

 3 Statistic results of diffusion metric (FA, ADC, AD and RD) between the MDD and control groups.

	Inter-group comparison			MDD vs.CTL
	Cerebral regions	Tracts	FA	ADC	AD	RD
	Anterior cingulate	gCC	↘	↗	↘	↗
	Cingulum gyrus	Cingulum	↘	↗	NS	↗
	Insula	ILF	↘	↗	↘	↗
	Thalamus	PTR	↘	↗	↘	↗
	Parahippocampal gyrus	PLIC	↗	↘	↘	↘
	Posterior cingulate	sCC	↗	↘	↘	↘

Up and down arrows indicate increased and decreased of the diffusion metric (FA, ADC, AD and RD) in MDD group compared to healthy control group. The arrows in bold front correspond to changes, which survived after FDR correction (p≤0.05). NS means no significant difference was found. gCC and sCC: genu and splenium of the Corpus Callosum, ILF: Inferior Longitudinal Fasciculus, PTR: Posterior Thalamic Radiation, PLIC: Posterior Limb of the Internal Capsule.

Table 4 :

 4 Voxel-wise correlations between FA and clinical variables.

	Cerebral regions	Tracts	Right or	MNI	t-value	cluster
			Left	coordinates		size
	Correlation with WDRS
			R	[0,-13,20]	-3,84	3638
	Anterior cingulate	bCC
			L	[-14,-3,27]	-4,67	1877
	Cingulate gyrus	SCR	L	[-19,-22,35]	-3,39	576
	Parahippocampal gyrus	SS/PLIC	R/L	[±33, -45,-6]	-3,81	384
	Superior frontal gyrus	gCC	R	[21,45,5]	-3,24	394
	Correlation with STAI-YA
	Anterior cingulate	gCC	R/L	[0, 26, 8]	4,08	995
	Insula	ACR	L	[-24, 29, 10]	3,53	1786
	Precuneus	PTR	R/L	[±26,-69, 19]	4,01	938
	Posterior cingulate	sCC	R	[-4,38,20]	3,88	306
	Cingulate gyrus	bCC	R	[-23,25,38]	-3,02	562
	Correlation with SHAPS
	Cingulate gyrus	Cingulum	L	[5,12,29]	-3,36	855
	Anterior cingulate	gCC	R	[9,33,14]	-3,53	450
	Cuneus	PTR	L	[-22,-72,11]	-3,49	342
	Superior temporal gyrus	SLF	L	[-57,-29,4]	3,44	322

The t-value corresponds to the maximum correlation of the cluster. The cluster size is given in mm 3 . WDRS: Widlöcher Depressive Retardation Scale; MADRS: Montgomery-Åsberg Depression Rating Scale; STAI-YA: State-Trait Anxiety Inventory; SHAPS: Snaith Hamilton Pleasure Scale, gCC, bCC and sCC: genu, body and splenium of the Corpus Callosum, ACR and SCR: anterior and superior corona radiata, SS: sagittal stratum, PLIC: posterior limb of the internal capsule, PTR: posterior thalamic radiation, SLF: superior longitudinal fasciculus, PTR: posterior thalamic radiation..

Acknowledgments

MRI data acquisition was supported by the Neurinfo MRI research facility from the University of Rennes I. Neurinfo is granted by the European Union (FEDER), the French State, the Brittany Council, Rennes Metropole, Inria, Inserm and the University Hospital of Rennes. This work has been funded by Institut des Neurosciences Cliniques de Rennes (INCR). The authors thank Mr Steṕhane Brousse and Mr Jacques Soulabaille for their involvement in the conduct of the study.


domain: spi phys

text:
Jean-Baptiste B Layly
email: jean-baptiste.layly@onera.fr

Fabien Tholin
email: fabien.tholin@onera.fr

Anne Bourdon

Anne Bourdon Modelling

Modelling of an electric contact under lightning current

Keywords: Simulation, contact resistance, out-gassing, sparking

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The contact resistance between two conducting surfaces is a consequence of the constriction of the current lines as they cross the spots, so-called "a-spots" in the literature, where roughness micro-peaks of one surface meet those of the mating surface, as shown on Figure 1 (partly taken from [START_REF] Slade | Principles and Applications[END_REF]).

Figure 1 -Schematics of an electric contact between two surfaces on a microscopic scale [START_REF] Slade | Principles and Applications[END_REF]: three a-spots represented.

An electric contact then consists in a distribution of many aspots in parallel, each one characterized by a constriction radius a of a few micrometres. To be able to study a realistic distribution of a-spots, a model for a single a-spot is then needed. In the first part of this study, the current distribution through a single a-spot is modelled via a 2D axisymmetric finite volume method. The electrical resistance of a single spot is then compared with the values obtained in the literature by analytical approaches. Then, the distribution of the current density and Joule heating in the spot is emphasized as well as thermal diffusion and phase transitions as the solid metal is heated up at temperatures higher than melting and boiling points. In the second part, the electrostatic interaction between different spots is addressed by means of 3D finite volume simulations. Based on these results, a simple pseudo-analytic model is finally derived in the third part of this work. It makes it possible to study the current distribution and contact resistance evolution of a cluster of many a-spots in parallel, more representative of a real contact. It is able to simulate the complete vaporization of the smallest a-spots, and the redistribution of the current from the destroyed spots to the largest ones. A parametric study is conducted with this contact model and a strong emphasis is placed on the effect of the initial spot distribution for a given contact resistance.

Single a-spot model

The resistance of a single a-spot can be seen as the resistance Rc (Ω) of a constriction of radius a between two cylinders of radius b (see Figure 1). Rc can be approximated by the widely used Holm's formula [START_REF] Fustin | Fuel tank safety conditioning aspect for sparking phenomenology on metallic assemblies[END_REF], where σ is the conductivity of the material (S m -1 ) [START_REF] Holm | Electric contact theory and application[END_REF].

𝑅 c = 1 2𝜎𝑎 (1)
This simple formula relies on the hypothesis that the constriction is axisymmetric with a zero thickness (l = 0 on Figure 1) and with perfectly spherical isopotential lines at infinity, which means that the radius b must fulfil b ≫ a.

According to (1), the resistance of many such constrictions in parallel corresponds to the resistance of a single equivalent constriction with a radius equal to ∑a, the sum of all the radii of the constrictions. In Holm's Theory, a distribution of many a-spots in an electric contact is then modelled by a single equivalent a-spot with radius ∑a. On Figure 2, the black curve shows the evolution of this constriction resistance with radius according to Holm's formula in the range 20 -200 µm for an Al-Al contact (σ = 10 7 S m -1 ). Moreover, for a single constriction, Holm's theory gives the norm j of the current density vector j (A m -2 ) in the constriction (2):

𝑗(𝑟) = 𝐼 2𝜋𝑎 1 √𝑎 2 -𝑟 2
(2)

I (A)
is the total current flowing in the constriction and r the distance from the symmetry axis. Assuming a constant temperature in the constriction and isotherms parallel to the isopotential lines, stationary solutions of the Fourier's law with Joule effect have been obtained that make it possible to compute the maximum temperature in the contact as a function of the potential difference, as for example the socalled φ-θ relation [START_REF] Holm | Electric contact theory and application[END_REF]. However, equation ( 2) gives an infinite current density at the periphery of the a-spot (r = a), which is a direct consequence of the idealized zero thickness hypothesis for the constriction, and reveals the limits of Holm's model. It is then not possible to compute a realistic unsteady current, Joule heating, and temperature distribution within Holm's theory. For this purpose, 2D axisymmetric finite volume simulations have been performed. The geometry consists in a single 2D axisymmetric constriction in cylindrical coordinates (r , z), with a thickness l = 80 µm, and a radius a varying in the range 20-200 µm. Figure 3 is a closed view of the a-spot geometry for a radius a = 200 µm, and b = 2 mm. The electrostatic current conservation equation ( 3) is solved thanks to the SuperLU solver [START_REF] Demmel | A supernodal approach to sparse partial pivoting[END_REF], where φ is the electric potential (V):

𝜵 • 𝒋 = -𝜵 • 𝜎 𝜵𝜑 = 0 (3)
Dirichlet boundary conditions are imposed on the top and bottom boundaries in z = +/-2 mm respectively to ensure a current setpoint, and a Neumann boundary is imposed at r = 2 mm. On Figure 2, the simulation results (orange curve R s ) are compared with Holm's formula (black curve R c ). In both cases the resistance decreases as the radius of the spot increases but an important difference remains. This difference is mainly due to the non-zero thickness l of the simulated spot, which resistance R a is the association in series of two different resistances (4): A constriction resistance Rc, given by (1), and the resistance Rcyl of a cylinder of radius a and thickness l, given by ( 5): Because R cyl is proportional to a -2 , while R c is proportional to a -1 , its contribution to the total resistance R a is dominant for small radii, as shown on Figure 2 (red curve). Then, a very good agreement is found between the numerical results and equation ( 4). Figure 3 shows the current streamlines and the 2D current density distribution inside the spot of radius a = 200 µm subject to a D-wave current (I = 100 kA at t = 3 µs) at the very beginning of the wave (t = 0.1 µs, I = 13 kA). Qualitatively, a good agreement is obtained with Holm's equation ( 2) with a current density in the constriction increasing with the radial position r, and a maximum with a sharp gradient on the rim of the spot. It appears from the different simulations performed in this study that this is a quite general geometric effect that barely depends on the exact geometry of the spot. Then the Joule effect -𝒋 • 𝜵𝜑 (W m -3 ) may be computed, and used as a source term in the energy conservation equation ( 6), where e is the volume internal energy (J m -3 ), λ the thermal conductivity (W m -1 K -1 ), and T the temperature ( K ).

𝑅 𝑎 = 𝑅 c + 𝑅 cyl (4)
𝑅 𝑐𝑦𝑙 = 𝑙 𝜎 π 𝑎 2 (5)
𝜕 𝑡 𝑒 = -𝒋 • 𝜵𝜑 + 𝜵 • 𝜆𝜵𝑇 (6) 𝑒 = ∫ 𝜌𝑐 v (𝑇)𝑇𝑑𝑇 + 𝑇 𝑇 0 𝜌( (𝑌 l + 𝑌 g )𝐿 f + 𝑌 g 𝐿 v ) (7)
The volume internal energy is given by equation ( 7), where T 0 is the room temperature, ρ is the density (kg m -3 ), c v the specific thermal capacity at constant volume (J kg -1 K -1 ), Y l (resp. Y g ) the mass fraction of liquid (resp. gaseous) metal and L f (resp. L v ) the fusion (resp. vaporization) latent specific heat (J kg -1 ). Equation ( 7) has been tabulated, and equations ( 3) and ( 6) are coupled by an explicit temporal scheme. This system of equations makes it possible to compute the evolution of the temperature as well as phase transitions. In order to compute the physical properties in the bi-phase regions, an assumption of homogeneous phase distribution is assumed, that results in equation ( 8) for the electric conductivity between phase 1 and phase 2 for example:

𝜎 12 = 𝜌 1 𝑌 1 𝜎 1 + 𝜌 2 𝑌 2 𝜎 2 (8)
The temperature dependent thermal capacity and thermal and electric conductivities of the solid metal up to the melting point have been obtained from references [START_REF] Haynes | CRC handbook of chemistry and physics: A ready-reference book of chemical and physical data[END_REF] and [6]. However, the conductivity of the metallic vapour state is difficult to address since in the hypothesis of isochoric heating the pressure may reach thousands of bars, which would imply a fast expansion, followed by a density decrease and a metal-insulator transition in the vapour, far beyond the scope of this study [START_REF] Sheftman | Evaluation of electrical conductivity and equations of state of non-ideal plasma through microsecond timescale underwater electrical wire explosion[END_REF]. As a first step, it has been assumed that the metallic vapour density and conductivity decrease fast enough so that it does not modify significantly the current distribution. Then, a conductivity of 10 4 S m -1 , representative of an aluminium plasma at atmospheric pressure has been considered for the gaseous phase [START_REF] Cressault | Properties of airaluminum thermal plasma[END_REF]. This conductivity being much smaller than the conductivity of the solid phase (~10 7 S m -1 ), no current flows through the gaseous phase and the vaporization of the metal of the a-spot increases its electric resistance. Because of the imposed total current flowing through the a-spot, the resistance increase due to vaporization results in a very fast increase of the Joule heating, the temperature, and then the vaporization rate itself. This thermo-electrical instability first takes place where current density and Joule heating are the most important, which means on the rim of the a-spot. As a consequence, the vaporization dynamics results in a decrease with time of the conducting radius of the spot, defined as the radius separating the liquid phase from the gaseous phase. This important phenomenon can be observed on Figure 4: the current density is shown for the same spot and current waveform as on Figure 3, but at t = 0.7 µs, just after the beginning of the vaporization. The current density maximum is always offaxis, but not attached to the rim of the spot and closer to the centre, due to the radius decrease.

Multi-spot interaction

The response of a single a-spot to high current levels is closely related to the spatial distribution of the current density and the Joule effect. Then the question arises if the distribution of current density inside a-spots can be significantly modified by the presence of other spots in its neighbourhood. If the a-spots are very distant from each other relatively to their size, they can be considered as well separated from each other, and the Holm's theory is valid. On the contrary, if they are close enough, the current lines going through them will interfere. According to the Greenwood formula (9) [START_REF] Greenwood | Constriction resistance and the real area[END_REF], this interaction results in an additional term in the Holm's resistance formula (1) for an electric contact with n spots:

𝑅 𝑐 = 1 2𝜎 ∑ 𝑎 𝑖 𝑛 𝑖=1 + 1 𝜋𝑛 2 ∑ ∑ 1 𝑠 𝑖𝑗 𝑛 𝑗,𝑗≠𝑖 𝑛 𝑖=1 (9)
In this formula, sij (m) is the distance between a-spots i and j and a i the radius of a-spot i. To study into more details this purely electrostatic interaction between several a-spots, 3D numerical simulations have been performed with Code_Saturne [10], the EDF's open source CFD code, solving the current conservation equation ( 3) with a finite volume method. Figure 5 shows a sliced view of the current density and the current lines going through two a-spots in parallel, each spot having the same radius of 200 µm. It can be observed that the current lines between the two a-spots are influencing each other, resulting in the additional interaction resistance of Greenwood. Figure 6 shows the evolution of the resistance as a function of the separating distance. It seems clear that as soon as the distance between the spots is higher than about 20 times their radii, the resistance remains constant, meaning that the interaction becomes negligible. Same conclusion arises looking at the Joule effect distribution: when the distance between the spots is small, the current and the Joule effect distributions are modified compared to the axisymmetric distribution of an isolated spot, with reinforcement on the outer edge, and a screening effect on the inner edge. This result has also been observed on 4-spots simulations, and it could have a significant influence on the thermo-electrical response of the contact. However, since this effect disappears when the distance is about two times the radius, it seems very unlikely to occur in a real contact, or would concern only a very small number of a-spots.

Pseudo analytic model for real contacts

Realistic electric contacts may consist in a large number of spots. It does not seem reasonable to perform detailed numerical simulations on such complex geometries with many different spatial scales. Moreover, according to previous results it is probably not necessary to get a realistic description of contacts under high current levels. The distribution in number, size and location of the a-spots, depend mainly on the mechanical load applied to the contact, and on the mechanical properties of the materials [START_REF] Slade | Principles and Applications[END_REF][START_REF] Holm | Electric contact theory and application[END_REF]. A mechanical model could be used to determine a realistic aspot distribution, as performed for example by the author of [START_REF] Schoft | Joint resistance depending on jointforce of high current aluminium joints[END_REF]. This kind of method will be part of further developments, but this study is focused on the thermoelectrical constraints, and the initial a-spot distribution for a given contact is considered as an input for the simulations in the following. According to previous simulation results on multi-spot interaction, seems reasonable to assume that all the a-spots are well isolated from each other. Then, if a current I is flowing through an electric contact, the current in each a-spot is known by solving a simple system of resistances in parallel: The current I i going through the a-spot i is given by equation ( 10), with 𝑅 𝑖 the resistance of a-spot i, given by equation [START_REF] Demmel | A supernodal approach to sparse partial pivoting[END_REF].

𝐼 𝑖 = 1 𝑅 𝑖 ( 1 ∑ 1/𝑅 𝑗 𝑛 𝑗=1 ) 𝐼 (10)
To be able to take into account the complex redistribution of the current from vaporized a-spots to the remaining ones, it is important to model the fast temperature increase and phase transitions in the spots due to Joule heating. As shown previously, heating and phase transitions start on the rim of the spots due to geometric current concentration effects. It is then reasonable to consider that as soon as an a-spot reaches the boiling point, all the energy dissipated by Joule effect generates a vaporization front going from the a-spot periphery to its centre. The evolution of the radius a i of the a-spot i is then given by the following non-linear differential equation:

2π 𝑎 𝑖 𝜕 𝑡 𝑎 𝑖 = - 𝑅 𝑖 (𝑎 𝑖 ) 𝐼 𝑖 2 𝐿 v (11)
A simple pseudo analytical model has been derived that solves the coupled equations ( 4), ( 10) and ( 11) with an explicit scheme, and considers a uniform temperature increase up to the boiling point with temperature dependent material properties (c v , σ and λ). Figure 7 shows the evolution of the radius of a 200 µm a-spot subject to a D-wave current. The results from the pseudo-analytic model are compared to the 2D axisymmetric simulation results obtained in part 2. The definition of the radius of the a-spot in the 2D simulation is not straightforward. Contrary to the pseudo-analytic model, there is a bi-phase region in the volume of the a-spot in the 2D simulations and the vaporization front is not precisely located. Then, 2 radii have been defined: the first radius is the minimum distance from the symmetry axis where the metal is fully vaporized, which means that the volume fraction V g of the gaseous phase is equal to 1. The second radius corresponds to the minimum distance from axis where a biphase region with V g = 0.5 is found (half of the liquid metal is vaporized). Both models predict similar dynamics: the spot vaporization occurs very fast, in less than 0.2 µs. This is due to the increase of the resistance as the spot is vaporized, that results in an increase of the Joule effect and the vaporization rate. The two models seem to predict the starting of the vaporization at around 0.65 µs, which means at the very beginning of the D-wave. This is a strong indication that when a lightning current is flowing through an assembly with many electric contacts, current redistribution phenomena may take place on very short timescales. The differences in the radius evolution between the two models may be due to the hypothesis of vaporization front in the analytical model, but also to the fact that thermal conduction in 2D simulations act as a dissipation process that may delays the rapid collapse of the spot. To study the behaviour of a cluster of many spots in parallel, several distributions have been considered that always correspond to a contact resistance of 0.1 mΩ. The different distributions differ by the number of spots N s , and their size: Two kinds of distributions have been studied for the a-spots' radius: constant-radius distributions, where all the spots are identical and uniform distributions, where radii are evenly distributed between two values a min and a max , computed in order to obtain the desired contact resistance. Figure 8 shows the evolution of the resistance (a) and the vaporized volume (b) of the 0.1 mΩ contact for the different a-spots distributions of Table 1, and assuming that a D-wave current is flowing through the contact. The complexity of the multi-spot interaction results in very different contact behaviours even if the initial contact resistance is always the same. It appears clearly that for all the distributions the vaporization takes place on very short timescales, and lead to a fast increase of the contact resistance. More surprisingly, the distributions a, b, c exhibit almost the same dynamics, while d, e and f differ strongly. It seems that for a given contact resistance, the radius range of the distribution does not play a significant role, while the number of spots have a significant influence only in the range 1 < N s < 10. Contacts with small N s seem to be able to support higher current levels, but they may lead to much more energetic outgassing phenomena at breaking point. 1.

Conclusion

Electric contacts in aeronautic assemblies consist on a microscopic scale in many a-spots in parallel, where the current density and the Joule effect may become very important under lightning stroke conditions, leading to intense heating and metal vaporization. 2D numerical simulations on single a-spots under high currents have revealed that as soon as the vaporization of the metal takes place, the conducting radius of the a-spot decreases very fast due to a thermoelectrical instability. On the other hand, 3D numerical simulations strongly suggest that electrostatic interactions between the a-spots of a given contact have a negligible influence, and that it can be neglected for practical applications. Then, a simple pseudo-analytic model has been derived that successfully mimic the behaviour of individual aspots, and makes it possible to consider realistic contacts with many a-spots in parallel. The evolution of the total contact resistance has been studied against different a-spots distributions for a given contact initial resistance. The number of a-spots N s seems to have an influence when small (N s <10), but no significant influence has been observed for higher values. This is a first prediction of the model that could be compared with experimental studies in the future. This model also allows us to predict the redistribution of the current in complex assemblies taking into account the non-linear effects occurring at high current levels. Moreover, it is able to compute macroscopic parameters, such as the electric-field, the energy dissipated in the contact, or the amount of metallic vapour produced, that could be of great interest regarding sparking and outgassing phenomena in aeronautic assemblies subject to lightning stroke.

Figure 2 -

 2 Figure 2 -Resistance of an a-spot as a function of its radius a: 2D-axisymetric simulations (R s ) and Holm's theory (R c )

Figure 3 :

 3 Figure 3: Current density distribution in a a-spot at the beginning of a D-wave current (t = 0.1 µs). a = 200 µm, r = 2 mm, l = 80 µm.

Figure 4 :

 4 Figure 4: Current density distribution in a 200 µm a-spot at the beginning of vaporization (t = 0.7 µs) for a D-wave.

Figure 5 :

 5 Figure 5: Sliced view of the current density flowing in two a-spots in parallel

Figure 6 :

 6 Figure 6: Total resistance of a two-spots contact and Joule power distribution as a function of the separating distance.

Figure 7 :

 7 Figure 7 : Evolution of the radius of a 200 µm a-spot subject to a D-wave current: pseudo-analytic model and 2D simulations.

Figure 8 :

 8 Figure 8: Evolution of the resistance (a) and volume of vaporized metal (b) in a 0.1 mΩ contact under a D-wave current for the different a-spot distributions of Table1.

Table 1

 1 summarizes the different distributions simulated with the pseudo-analytic model.

	Id	N s	a min [µm]	a max [µm]
	a	1000	0.0252	5.0497
	b	100	9.736	9.736
	c	100	0.08599	17.194
	d	10	37.083	37.083
	e	2	74.853	149.706
	f	1	196.705	196.705

Table 1 :

 1

a-spot distributions considered that correspond to a contact resistance of 0.1 mΩ with l=80 µm.

Acknowledgements

We gratefully acknowledge the funding received toward J. B. Layly's PhD from the Plas@par Labex.