We report the case of a 58-year-old man (Figure ) in a profound coma after a 30-min-long cardiac arrest who benefited of a brain MRI 6 weeks after the accident (1.5-T MRI scanner, SIEMENS Avanto MR, Erlangen, Germany) with DTI acquisition (30 directions, b = 1000 sec/mm 2, TR = 6800 msec, TE = 99 msec, FOV=230×230 mm 2 , matrix 128×128, 3.5 mm slice thickness). FA, parallel diffusion (D // ) and perpendicular diffusion (D ﬩ ) maps were computed and compared voxelwise to a probabilistic voxel-based atlas of fractional anisotropy as well as parallel and perpendicular diffusion based on 19 healthy subjects. Zscore maps were computed. The patient died from sepsis. An autopsy was performed. Brain sections were stained with Luxol fast blue cresyl violet to analyse myelin and immunostained for neurofilaments to detect white matter axons. DTI images analysis demonstrated extensively reduction of white matter FA, whereas D // and D ﬩ were elevated. D // and D ﬩ modifications reflect axonal and myelin lesions, respectively, whereas FA reflects white matter global disorganization. Histological analysis was in accordance with DTI data demonstrating extensive demyelination and widespread axonal loss. Global cerebral anoxia after cardiac arrest resuscitation is one of the most common causes, with traumatic cerebral injury, of chronic disorders of consciousness [START_REF] Higashi | Epidemiological studies on patients with a persistent vegetative state[END_REF].

The distribution and severity of brain damage following global ischemia is proportional to the duration and severity of ischemia and is modified by the selective vulnerability of the different cell types and brain regions [START_REF] Petito | The neuropathology of global brain ischemia[END_REF]. Neurons are the most vulnerable cells, in particular in the CA1 region of the hippocampus, followed by those in the basal ganglia, cerebral cortex and the Purkinje cells of the cerebellum [START_REF] Petito | The neuropathology of global brain ischemia[END_REF]. Moreover, border zones, between two vascular territories, are more vulnerable because of the poor blood supply (2). Among glial cells, oligodendrocytes are the most vulnerable cells (2). The grey matter lesions observed are not uniform in the different brain regions and can extend from selective neuronal necrosis to tissue necrosis [START_REF] Petito | The neuropathology of global brain ischemia[END_REF][START_REF] Sevestre | Anoxic encephalopathy after cardiocirculatory insufficiency. Neuropathological study apropos of 16 cases[END_REF]. These lesions are associated with white matter leukoencephalopathy, combining demyelination, axonal loss and focal regions of necrosis [START_REF] Petito | The neuropathology of global brain ischemia[END_REF].

MRI is able to assess these lesions, on T2-weighted images demonstrating signal abnormalities in the hippocampus, the cortexpreferentially in the border zonesand in the parietal, occipital and frontal regions, the basal ganglia and the cerebellum (4-7). Moreover, the MRI appearance of the lesions depends on the delay between the cardiopulmonary arrest and the date the MRI was performed, extending in the gray matter from an edematous appearance at the early phase to atrophy sometimes associated with cortical laminar necrosis at the late phase [START_REF] Arbelaez | Diffusion-weighted MR imaging of global cerebral anoxia[END_REF][START_REF] Wijman | Prognostic value of brain diffusionweighted imaging after cardiac arrest[END_REF]. White matter abnormalities can also be detected on T2-weighted MR images, but are usually delayed, observed only after the late subacute period (14-20 days) (4).

Diffusion-weighted imaging (DWI) seems to be more accurate for the detection of lesions and the determination of their extension in the early acute period than other conventional MR sequences [START_REF] Arbelaez | Diffusion-weighted MR imaging of global cerebral anoxia[END_REF][START_REF] Mckinney | Diffusion-weighted imaging in the setting of diffuse cortical laminar necrosis and hypoxic-ischemic encephalopathy[END_REF]. DWI could help in determining the prognosis at the early phase after cardiopulmonary resuscitation, as the extension of the lesions and low ADC values within the lesions seem to be related to poor outcome [START_REF] Mckinney | Diffusion-weighted imaging in the setting of diffuse cortical laminar necrosis and hypoxic-ischemic encephalopathy[END_REF][START_REF] Wijdicks | MR imaging in comatose survivors of cardiac resuscitation[END_REF][START_REF] Wijman | Prognostic value of brain diffusionweighted imaging after cardiac arrest[END_REF][START_REF] Barrett | Brain injury after cardiopulmonary arrest and its assessment with diffusion-weighted magnetic resonance imaging[END_REF][START_REF] Wu | Comatose Patients with Cardiac Arrest: Predicting Clinical Outcome with Diffusion-weighted MR Imaging[END_REF]. Diffusion tensor imaging (DTI) is a recent MRI technique that can characterize the neuronal architecture of the brain white matter in vivo by probing the diffusion of water molecules in tissues. The degree of directionality of water in the tissue is described by the fractional anisotropy (FA). The diffusion process is often modeled in each voxel as a 3×3 symmetric definite positive matrix. The largest eigenvalue of this matrix is related to the main diffusion direction along the fiber bundle, namely the parallel diffusion (D // ), and the mean of the two other eigenvalues is related to the diffusion direction perpendicular to the fiber bundle, namely the perpendicular diffusion (D ﬩ ).

Recent studies have shown that D // and D ﬩ provide additional information on white matter structures that is more specific to underlying histological processes, as compared to FA. D // seems to reflect diffusivity along the axon in relation to axonal integrity, whereas D ﬩ seems to reflect diffusivity perpendicular to the axon, in relation to degree of myelination [START_REF] Harsan | New insights on neuronal alterations in jimpy mutant brain[END_REF][START_REF] Harsan | Brain dysmyelination and recovery assessment by noninvasive in vivo diffusion tensor magnetic resonance imaging[END_REF].

Diffusion tensor imaging could be of particular interest in the evaluation of white matter injuries in patients with global cerebral anoxia after cardiac arrest resuscitation, as demonstrated in a hypoxic-ischemic neonatal rat model that combined FA and trace : increased D ﬩ with no significant change in D // appears to characterize noncystic white matter injury with reduced myelination, whereas reduction in both D ﬩ and D // characterize severe damage with loss of structural integrity and necrosis [START_REF] Wang | Characterization of white matter injury in a hypoxic-ischemic neonatal rat model by diffusion tensor MRI[END_REF].

In conclusion, DTI modifications, particularly D // and D ﬩ , seem to be in accordance with histological data. DTI acquisition and postprocessing are easy to perform and could contribute additional information on biological processes in white matter injuries, as compared to morphological MRI. However, these preliminary data need to be confirmed on a larger cohort of patients. -Point 0 : we modified the presentation of the manuscript.

-Point 1 : We completed the description of the lesions and their pathophysiology -Point 2 : We agree with the reviewers comment about grey matter lesions after cardiac arrest resuscitation. But we focused our case report only on white matter abnormalities studied with DTI. -Point 3 : The DTI examination has been performed 6 weeks after the cardiac arrest. We think that it could be more interesting to perform the MRI examination earlier. Diffusion-weighted imaging seems to be more accurate for the detection of lesions and the determination of their extension in the early acute period than other conventional MR sequences. Moreover Diffusion-weighted imaging could help determine the prognosis at the early phase after cardiopulmonary resuscitation, as the extension of the lesions on diffusion-weighted magnetic resonances images and the low value of ADC in the lesion seem to be related to poor outcome. But in this case the delay between DTI examination and autopsy was very short (8 days) and allows a direct comparison of both techniques.

We are looking forward to receive your editorial decision,

Sincerely yours

Stéphane Kremer *Detailed Response to Reviewers / Reponse aux lecteurs

  DIFFUSION TENSOR IMAGING IN HUMAN GLOBAL CEREBRAL ANOXIA: CORRELATION WITH HISTOLOGY IN A CASE WITH AUTOPSY. IMAGERIE DU TENSEUR DE DIFFUSION DANS LE COMA POST-ANOXIQUE : CORRELATION AVEC L'HISTOLOGIE Stéphane Kremer (MD, PhD) (1), Félix Renard (PhD) (2), Vincent Noblet (PhD) (2), Roxana Mialin (MD) (1), Renée Wolfram-Gabel (MD,PhD) (3), Chantal Delon-Martin (PhD) (4), Sophie Achard (PhD) (5), Maleka Schenck (MD) (6), Michel Mohr (7), Jean L Dietemann (MD) (1), Francis Schneider (MD, PhD) (6) 1. Service de Radiologie 2 -Hôpital de Hautepierre -CHU de Strasbourg; LINC; Université de Strasbourg, France 2. Laboratoire des Sciences de l'Images, de l'Informatique et de la Télédétection -Université de Strasbourg, France 3. Grenoble Institut des Neurosciences, Institut national de la santé et de la recherche médicale, Grenoble, France 4. Laboratoire d'Anatomie Normale, Faculté de Médecine de Strasbourg; Université de Strasbourg, France 5. Grenoble Image Parole Signal Automatique, Centre National de la Recherche Scientifique, Grenoble, France 6. Service de Réanimation Médicale -Hôpital de Hautepierre -CHU de Strasbourg; Université de Strasbourg, France 7. Service d'Anatomopathologie -CHU de Strasbourg -Université de Strasbourg, France Corresponding author: Dr. S. Kremer -Service de Radiologie 2 -Hôpital de Hautepierre -Avenue Molière -CHU de Strasbourg -67098 Strasbourg cedex -France

Figure :

 : Figure : Axial FLAIR-weighted MR images at the occipital lobe level (a) demonstrating

Figure

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Acknowledgements : The authors would like to thank Ms. Sandrine Decocq for her help in histological preparations.

No conflict of interest


Marc Fournier

Benoît Combès

Neil Roberts

José Braga

Sylvain Prima

Mapping the distance between the brain and the inner surface of the skull and their global asymmetries

Keywords: Brain-skull MRI segmentation, Brain-skull relations, Surface-based brain hull, Brain-endocast distance, Brain asymmetry, Endocranial asymmetry, Average brain-endocast shapes, Brain-endocast asymmetry t-test

The primary goal of this paper is to describe i) the pattern of pointwise distances between the human brain (pial surface) and the inner surface of the skull (endocast) and ii) the pattern of pointwise bilateral asymmetries of these two structures. We use a database of MR images to segment meshes representing the outer surface of the brain and the endocast. We propose automated computational techniques to assess the endocast-to-brain distances and endocast-and-brain asymmetries, based on a simplified yet accurate representation of the brain surface, that we call the brain hull. We compute two meshes representing the mean endocast and the mean brain hull to assess the two patterns in a population of normal controls. The results show i) a pattern of endocast-to-brain distances which are symmetrically distributed with respect to the mid-sagittal plane and ii) a pattern of global endocast and brain hull asymmetries which are consistent with the well-known Yakovlevian torque. Our study is a first step to validate the endocranial surface as a surrogate for the brain in fossil studies, where a key question is to elucidate the evolutionary origins of the brain torque. It also offers some insights into the normal configuration of the brain/skull interface, which could be useful in medical imaging studies (e.g. understanding atrophy in neurodegenerative diseases or modeling the brain shift in neurosurgery).

INTRODUCTION

The human brain has been often reported to exhibit a global, counterclockwise, so-called Yakovlevian, torque 1 about a vertical axis (i.e. parallel to the long axis of the human body). The magnitude of this torque has been quantified variously in terms of the i) greater widths or volumes, or ii) anterior and posterior protrusion of the brain in the right as compared to the left frontal lobe and in the left compared to the right occipital lobe or iii) leftward (resp. rightward) deviation of the inter-hemispheric fissure between the frontal (resp. occipital) lobes. Measuring the torque is of the utmost importance, as it has often been hypothesized to be the neural substrate for cerebral hemisphere dominance for language in humans. As of today, it is not clear i) when this pattern appeared during the course of human evolution and ii) whether this pattern is shared by other species, and especially by our closest extant relatives (bonobos and chimpanzees). The first question is typically addressed by physical anthropologists, using CT imaging of fossil skulls and segmentation of the inner surface of the skull to obtain a so-called virtual endocast [START_REF] Holloway | Brain endocast asymmetry in pongids and hominids: some preliminary findings on the paleontology of cerebral dominance[END_REF] . The second question can be addressed using MR imaging of the brain and segmentation of the outer cortical surface of the brain [START_REF] Lyttelton | Positional and surface area asymmetry of the human cerebral cortex[END_REF] . However, both approaches led to controversial results. First, it is not clear whether the endocast is a faithful surrogate representation of the brain as it is separated from it by the dura mater, the pia mater and the arachnoid (blood vessels, CSF); these layers are of varying thickness between anatomical regions. Second, local intra-subject (i.e. differences between hemispheres) and intersubject sulcogyral variability makes it difficult to assess the torque, which is a global pattern.

In this paper, we propose to address these two questions using MRI by i) measuring the local distances between the brain and the endocast and ii) measuring the asymmetries of the brain and of the endocast. To alleviate the problems due to cortical variability, we base our measures on a simplified yet accurate representation of the brain (that we call the brain hull) which closely follows the gyri without entering the sulci. Both the brain hull and the endocast are represented by 3D meshes. In the first study, using a dataset of 37 healthy control subjects, we compute individual pointwise endocastto-brain distances, and we map them onto a mean endocast computed from the population. In the second study, using a subset of 14 right-handed males, we compute individual pointwise asymmetry maps of endocast and brain hull separately, and we map them onto, respectively, a mean endocast and mean brain hull computed from the subpopulation. Student's t-test is used to assess statistically significant brain and endocranial asymmetries. The results show i) a pattern of endocast-to-brain distances which are symmetrically distributed with respect to the mid-sagittal plane and ii) a pattern of global endocast and brain hull asymmetries which are consistent with the well-known Yakovlevian torque. The first pattern of distances gives an insight into the normal spatial brain-skull configuration which can be useful in medical imaging applications such as brain shift modeling for neurosurgery [START_REF] Miller | Modelling brain deformations for computer-integrated neurosurgery[END_REF] or the study of brain atrophy which occurs in pathologies such as Alzheimer's disease [START_REF] Fritzsche | Automated MRI-based quantification of the cerebral atrophy providing diagnostic information on mild cognitive impairment and Alzheimer's disease[END_REF] and multiple sclerosis [START_REF] Losseff | Progressive cerebral atrophy in multiple sclerosis A serial MRI study[END_REF] . Section 2 of this paper describes the material and methods while in Section 3 we provide and discuss some results, before concluding in Section 4.

MATERIAL AND METHODS

Material

In this paper we use a database of MR images of 37 healthy human subjects (25 to 45 years old) including 19 males and 18 females. The data were acquired at the MARIARC, University of Liverpool (http://www.liv.ac.uk/mariarc), using a MPRAGE T1-weighted sequence with a 3T Siemens Trio, CP Head Coil MRI scanner. The in-plane resolution is 0.625x0.625mm with a data matrix size of 320x320. The whole scan is composed of 192 slices with a thickness of 1.0mm per slice.

Brain and endocast segmentation

The first step of our method is to extract the brain and endocranial surfaces from the MR images. We use BrainMask [START_REF] Mikheev | Fully automatic segmentation of the brain from T1-weighted MRI using bridge burner algorithm[END_REF] to process the MR images in order to correct the intensity non-uniformity, find the white matter seed and output the binary mask of the whole brain without hemisphere split. This software was chosen for its fully automatic pipeline process and for the quality of its segmentation results. We use FSL [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF] to process the MR images in order to find and output the binary mask of the skull. We found that only a few computer programs can extract the skull from MR images, and FSL provides satisfying results for our application. Then the marching cube [START_REF] Lorensen | Marching Cubes: a high resolution 3D surface reconstruction algorithm[END_REF] isosurface extraction algorithm is applied to both binary volume masks in order to generate mesh surfaces of the brain and endocast. Figure 1(a) shows an example of this volume segmentation preprocessing step.

Brain hull computation

The second step of our method consists in computing the brain hull from the detailed brain mesh. We first compute the 3D convex hull of the brain mesh vertices (http://www.qhull.org) [START_REF] Barber | The quickhull algorithm for convex hulls[END_REF] . This algorithm generates a mesh with few large triangles corresponding to a coarse representation of the brain which is not sufficiently accurate for our purpose. We subdivide this mesh to obtain a uniform triangulation with enough vertices to accurately describe the initial brain mesh gyri positions. Then for each vertex of the subdivided convex hull, we compute the minimal sphere centered on this vertex which intersects the boundary of the initial detailed brain mesh. Finally we move the vertex in its opposite normal direction (towards the brain surface) on the minimal sphere surface. This procedure illustrated in Figure 1(b) attracts vertices to the gyri positions and prevents them to penetrate inside the sulci, between neighboring gyri. It produces an accurate brain hull shown in Figure 1(c) and 1(d) with respectively the brain surface and the endocranial mesh. The brain hull represents the brain global shape, which closely follows the gyri. superposed on the brain surface mesh obtained from the volume mask. (d) Endocranial mesh (from the volume mask) superposed on the brain hull surface.

Brain to endocast distance

In order to evaluate the distance pattern, we compute the local distance between the brain hull and endocast of each subject. For each vertex of the endocast, the distance to the brain hull surface is computed. Then we evaluate the average local distance for all subjects by computing an average endocranial shape. Endocasts of all subjects are registered using the ICP algorithm [START_REF] Besl | A method for registration of 3-D shapes[END_REF] and the average endocranial shape is computed using a shape-based [START_REF] Flores | Shape-based averaging for craniofacial anthropometry[END_REF] method which consists of a weighted average in the implicit distance transform domain of the meshes. Finally the same local weights used to evaluate the average shape are used to compute the corresponding local average distances.

Brain and endocranial asymmetries

In order to evaluate both asymmetry patterns, we compute local surface-based asymmetries separately for the brain hull and endocast of each subject. For this study we use a subsample of the database composed of the 14 right-handed male subjects among the 37 subjects. Both average asymmetry patterns are evaluated for all subjects by computing separately the average shapes of the brain hull and the endocast using the same procedure as for the average local distance pattern evaluation. Both brain and endocranial asymmetry measurements are achieved by flipping the brain surface over the mid-sagittal plane. The latter is defined as the median plane of the scanned MRI volume, whose position and orientation are defined by an automated atlas-based procedure [START_REF] Van Der Kouwe | On-line automatic slice positioning for brain MR imaging[END_REF] . The asymmetries are defined as the local distances between the original brain surface and the flipped one. These distances are signed according to the surface normal in order to quantify the asymmetry orientation. We also perform the Student's t-test in order to locate significant asymmetries on the brain hull and the endocast. In order to better interpret this distance map pattern, we cut the underneath part (Figure 2(b) view) of the average endocranial mesh to create an open boundary mesh. We unfold this mesh onto a plane using mesh parameterization and we consider the distance pattern as a height function. We resample this distance pattern on a regular grid for display purpose and show the result in Figure 3(a) using the same color map as in Figure 2. Figure 3(a) shows all views of Figure 2 (except Figure 2(b)) embedded on the same chart using an alternative representation of the brain to endocast distance pattern. This representation displays the distance pattern in a synthetic way and suggests it is approximately symmetrical about to the mid-sagittal plane. The peaks, ridges and flatter regions are quite similar on the left and right sides of Figure 3(a). To quantify the left-right differences of this distance pattern, we compute its mirror image about the mid-sagittal plane shown in Figure 3(b) and we evaluate the local differences between the two. This symmetry difference map shown in Figure 3(b) illustrates that the three major differences (red and blue colors) are located on the peak (and its symmetrical counterpart) identified as #1 and on the two ridges (and their symmetrical counterparts) identified as #2 and #3 in Figure 3(b). Besides these three highlighted local components ranging from 0.4mm to 0.5mm, most of the other absolute differences are below 0.3mm. This observation shows that most of the symmetrical differences (< 0.3mm) are less than 5% of the maximum distance (6.7mm) between the brain and the endocast.

RESULTS AND DISCUSSION

Brain to endocast distance

Brain and endocranial asymmetries

Figure 4 shows the local asymmetry patterns on the average brain hull and endocast for the data subset composed of the 14 right-handed males. Green color means there is no local asymmetry on average. Red color (versus blue color) means the original surface is "above" (resp. "below") the surface obtained when mirroring it about the mid-sagittal plane. This can be interpreted in different ways: as a left-right difference of area, volume, width, protrusion, etc. What is noteworthy is that, for both the brain hull and the endocast, the right frontal pole is "above" the left one on average, while the opposite is observed in the occipital poles (Figures 4(a and (h). In short, it suggests that both the brain hull and the endocast exhibit the same global asymmetry pattern as the brain, i.e. the Yakovlevian torque [START_REF] Toga | Mapping brain asymmetry[END_REF] . The only difference is the asymmetrical regions of the endocast seem more spread out compared to the brain hull ones according to the t-test results. The measured range of local asymmetries is the same for the brain hull and the endocast. The maximum asymmetry magnitude is also the same in both directions. In order to quantify the asymmetry differences between the brain hull and the endocast, we compute the local differences between both asymmetry maps. The obtained asymmetry differences map is shown for different views in Figure 5(a), 5(b) and 5(c). The front and rear views (Figure 5(b), (c)) mostly have differences near 0mm (blue color). That explains the similar Yakovlevian torque asymmetry pattern retrieved on both the brain hull and endocranial mean shapes in Figure 4. However Figure 5(a) shows approximately symmetrical regions (with respect to the mid-sagittal plane) at the top view with mid-range (green color) to larger (red color) differences. These differences are the result of larger asymmetries observed on the endocast compared to the brain hull. These differences are from 1mm (green) to 2mm (red) at the most, which is around 30% of the maximum asymmetry range. This result suggests that asymmetries in these regions measured on fossil endocasts for example should not necessarily be considered as asymmetries also belonging to the brain of extinct species. This is an interesting observation even if that region of interest is outside of the region where the typical torque is usually observed.

Figure 5 also shows t-test results (H 0 : "the difference between pointwise brain hull and endocranial asymmetry is null", p-value = 0.05, not corrected for multiple comparisons) that support the previous observations. The green color (no statistically significant difference) is observed on most of the shape including the frontal (Figure 5(e)) and occipital (Figure 5(f)) lobes where the Yakovlevian torque is observed on both the brain hull and the endocast. However the two regions with larger differences (red color on Figure 5(a)) do have more significant differences according to the t-test (light blue color on Figure 5(d)) in these two regions.

CONCLUSION

In this paper we presented an asymmetry and relative distance patterns study of the human brain (pial surface) and the inner table of the skull (endocast). We described preprocessing of MR images in order to extract the surfaces of interest from the volume data. We proposed a mesh processing method to obtain an accurate brain hull surface which closely follows the gyri positions without entering the neighboring sulci. This coarse representation is useful for efficient registration and average shape computation. It prevents results from being influenced by inter-subject variability in the patterns of gyri and sulci. It enables a global brain shape asymmetry mapping and comparison with the endocast. It also provides a more accurate distance mapping between the brain and endocast.

We evaluated the local distances between the endocast and brain hull and the bilateral asymmetry of both surfaces with respect to the mid-sagittal plane. Average shape meshes were computed and the distance and asymmetry patterns were projected onto these average shapes using color maps for visualization purposes. We found that our brain hull representation allows recovering the well-known global brain asymmetry (Yakovlevian torque). We also found that local distances between the endocast and brain hull follow an approximately regular and bilaterally symmetrical pattern. This explains why we also observed that the endocranial asymmetry is related to that of the brain hull.

We performed t-tests on the brain hull and endocranial asymmetries to support these findings. These results suggest that the inner surface of the skull is a relevant surrogate of brain shape, and thus that virtual endocasts can be used in paleoneurological studies to infer the evolutionary history of global brain asymmetry with respect to the Yakovlevian torque. However we also observed that a region at the top of the endocast, outside the typical torque regions, is asymmetrical in a significantly different way when compared to the brain. This suggests that this specific asymmetry, which could be observed on virtual endocasts, may not be representative of the global brain asymmetry pattern.

Future work will include other processing pipelines using different tools to validate the findings in this paper. In particular, new state-of-the-art methods to compute the mid-sagittal plane [START_REF] Combès | New algorithms to map asymmetries of 3D surfaces[END_REF] , the asymmetries [START_REF] Combès | New algorithms to map asymmetries of 3D surfaces[END_REF] and the average shapes [START_REF] Combès | An efficient EM-ICP algorithm for symmetric consistent non-linear registration of point sets[END_REF] for both the brain hull and the endocast will be investigated. This will also involve a more extensive study over other subgroups within humans, and over other species too, such as chimpanzees that are our closest extant relatives. We will also use an available database of CT images and MR images of the same subjects to compare MR and CT-based endocasts segmentation accuracy. We will provide further quantification of the patterns of endocast-to-brain distance and endocast-and-brain asymmetries. This will involve spherical parameterization and vector field computation and

Figure 1 .

 1 Figure 1. MRI volume segmentation and mesh processing. (a) Brain and skull segmentation masks superposed on MRI image. (b) Illustration of the brain hull computation with: V i (resp. V f ) the initial (resp. final) vertex position of the subdivided convex (resp. brain) hull; -n the opposite normal vertex displacement direction. (c) Brain hull mesh(a) (b) (c) (d)

Figure 2 Figure 2 .

 22 Figure2shows the local distance pattern between the endocast and brain hull for the dataset of 37 subjects. The average local distances are converted into a color map that is mapped onto the average endocranial shape. The distance is globally greater at the top of the brain compared to the bottom (Figures2(a) and 2(b)). It seems that at the top of the brain the distance follows a mid (green) to high (red) values pattern in the left to right axis as illustrated by the curve in Figure2(a). The distance is also globally greater at the front compared to the back of the brain (Figures2(c) and 2(e)). However this observation is influenced by gravity since the subjects were scanned in supine position. Indeed relative brain-skull displacements[START_REF] Asiminei | An image based approach for in vivo evaluation of the brain-skull relative displacement and brain deformation in quasi-static conditions[END_REF] are monitored under external forces such as gravity attraction. On both left and right sides of the brain, homogenous minimal distances are observed (Figures2(d) and 2(f)).

Figure 3 .

 3 Figure 3. Height function of the brain to endocast distance pattern. (a) Height function of the distance pattern projected onto a plane displayed using the same color map as in Figure (2). (b) Symmetry difference of the distance pattern about the mid-sagittal plane shown using an equivalent color map scaled to the symmetrical difference data.

  Figure4shows the local asymmetry patterns on the average brain hull and endocast for the data subset composed of the 14 right-handed males. Green color means there is no local asymmetry on average. Red color (versus blue color) means the original surface is "above" (resp. "below") the surface obtained when mirroring it about the mid-sagittal plane. This can be interpreted in different ways: as a left-right difference of area, volume, width, protrusion, etc. What is noteworthy is that, for both the brain hull and the endocast, the right frontal pole is "above" the left one on average, while the opposite is observed in the occipital poles (Figures4(a), (c), (e) and (g)). These four results are statistically significant, as shown by two t-tests on the brain hull and endocast (H 0 : "the pointwise asymmetry is null", p-value = 0.05, not corrected for multiple comparisons). The significantly asymmetrical vertices are shown in blue in Figures4(b), (d), (f)

Figure 4 .Figure 5 .

 45 Figure 4. Asymmetry color map patterns of brain hull and endocast and their t-test shown on average shapes. (a) Brain hull front view asymmetry. (b) Brain hull front view t-test. (c) Brain hull rear view asymmetry. (d) Brain hull rear view t-test. (e) Endocast front view asymmetry. (f) Endocast front view t-test. (g) Endocast rear view asymmetry. (h) Endocast rear view t-test. The linear color map scale is in millimeters and the asymmetry values are shown on the color bar. The t-test results show significantly asymmetrical vertices (p-values lower than 0.05); green color corresponds to regions with no asymmetry and blue color to regions with asymmetry.

decomposition of a more specific analysis of the endocast-to-brain distance pattern. We will also design an automated comparison method between the endocast and the brain hull asymmetries including a correlation on specific anatomical regions for easier interpretation of the results.


Benaych-Georges Florent

Nathana And

Ël Enriquez

Nathanaël Enriquez
email: nenriquez@u-paris10.fr

PERTURBATIONS OF DIAGONAL MATRICES BY BAND RANDOM MATRICES

Keywords: . 2000 Mathematics Subject Classification. 15A52, 46L54 Random matrices, band matrices, Hilbert transform, spectral density

We exhibit an explicit formula for the spectral density of a (large) random matrix which is a diagonal matrix whose spectral density converges, perturbated by the addition of a symmetric matrix with Gaussian entries and a given (small) limiting variance profile.

Perturbation of the spectral density of a large diagonal matrix

In this paper, we consider the spectral measure of a random matrix D ε n defined by D ε n = D n + ε n X n , for D n a deterministic diagonal matrix whose spectral measure converges and X n an Hermitian or real symmetric matrix whose entries are Gaussian independent variables, with a limiting variance profile (such matrices are called band matrices). We give a first order Taylor expansion, as ε → 0, of the limit spectral density, as n → ∞, of D ε n . The proof is elementary and based on a formula given in [START_REF] Shlyakhtenko | Random Gaussian band matrices and freeness with amalgamation[END_REF] for the Cauchy transform of the limit spectral distribution of D ε n as n → ∞.

For each n, we consider an Hermitian or real symmetric random matrix X n = [x n i,j ] n i,j=1 and a real diagonal matrix D n = diag(a n (1), . . . , a n (n)). We suppose that:

(a) the entries x n i,j of X n are independent (up to symmetry), centered, Gaussian with variance denoted by σ 2 n (i, j), (b) for a certain bounded function σ defined on [0, 1] × [0, 1] and a certain bounded real function f defined on [0, 1], we have, in the L ∞ topology, For ε ≥ 0, let us define, for all n,

σ 2 n (⌊nx⌋, ⌊ny⌋) -→ n→∞ σ 2 (x, y) and a n (⌊nx⌋) -→ n→∞ f (x), (
D ε n = D n + ε n X n .
It is known, from Shlyakhtenko in [START_REF] Shlyakhtenko | Random Gaussian band matrices and freeness with amalgamation[END_REF]Th. 4.3] (see also [START_REF] Anderson | A CLT for a band matrix model[END_REF], which also provides a fluctuation result), that as n tends to infinity, the spectral distribution of D ε n tends to a limit µ ε with Cauchy transform

C ε (z) = 1 x=0 C ε (x, z)dx,
where C ε (x, • ) is defined by the fact that it is analytic, maps the upper half-plane C + into the lower one C -, and satisfies the relation

(1) C ε (x, z) = 1 z -f (x) -ε 1 y=0 σ 2 (x, y)C ε (y, z)dy .
Our goal here is to understand µ εµ for small values of ε. Let us introduce the set T of test functions we shall use here. We define

T = t -→ 1 z -t ; z ∈ C + .
Let us now define the Hilbert transform, denoted by H[u], of a function u:

H[u](s) := p. v. t∈R u(t) s -t dt = y∈R u(s -y) -u(s) y dy.
Before stating our main result, let us make some assumptions on the functions σ and f :

(d) the push-forward µ of the uniform measure on [0, 1] by the function f has a density ρ with respect to the Lebesgue measure on R, (e) there exists a symmetric function τ ( • , • ) such that for all x, y, σ 2 (x, y) = τ (f (x), f (y)), (f) there exist η 0 > 0, α > 0 and C < ∞ such that for almost all s ∈ R, for all t ∈ [sη 0 , s

+ η 0 ], |τ (s, t)ρ(t) -τ (s, s)ρ(s)| ≤ C|t -s| α .
Note that by hypothesis (f) and by the boundedness of the function f , the function

s -→ ρ(s)H[τ (s, • )ρ( • )](s)
is well defined and compactly supported.

Theorem 1. Under the hypotheses (a) to (f ), as ε → 0, for all g ∈ T ,

g(s)dµ ε (s) = g(s)dµ(s) -ε g ′ (s)F (s)ds + o(ε), with F (s) := -ρ(s)H[τ (s, • )ρ( • )](s).
As a consequence, if the function F ( • ) has bounded variations, then

µ ε = µ + εdF + o(ε).
Remark 1. Roughly speaking, this theorem states that

lim ε→0 lim n→∞ spectral law(D ε n ) -spectral law(D n ) ε = dF.
It would be interesting to let ε and n tend to 0 and ∞ together, and to find out the adequate rate of convergence to get a deterministic limit or non degenerated fluctuations. We are working on this question.

Remark 2. This result provides an analogue, for our random matrix model, of the following formula about real random variables (valid when Y is centered and independent of X):

density X+ √ εY (s) = density X (s) + ε E[Y 2 ] 2 density ′′ X (s) + o(ε).
Remark 3. In the case where X n is a GUE or GOE matrix, the limiting spectral distribution of D ε n as n → ∞ is the free convolution of the limiting spectral distribution of D n with a semi-circle distribution. Several papers are devoted to the study of qualitative properties (like regularity) of the free convolution (see [START_REF] Biane | On the Free convolution by a semi-circular distribution[END_REF][START_REF] Bercovici | Regularity questions for free convolution[END_REF][START_REF] Belinschi | The Lebesgue decomposition of the free additive convolution of two probability distributions[END_REF][START_REF] Belinschi | A note on regularity for free convolutions[END_REF][START_REF] Belinschi | Regularization by Free Additive Convolution, Square and Rectangular Cases[END_REF]). Besides, it has recently been proved that type-B free probability theory allows to give Taylor expansions, for small values of t, of the moments of µ t ⊞ ν t for two time-depending probability measures µ t and ν t (see [START_REF] Belinschi | Free probability of type B: analytic aspects and applications[END_REF][START_REF] Février | Infinitesimal non-crossing cumulants and free probability of type B[END_REF][START_REF] Février | Higher order infinitesimal freeness[END_REF]). Our work differs from the ones mentioned above by the fact that we allow to perturb D n by any band matrix, but also by the fact that it is focused on the density and not on the moments, giving an explicit formula rather than qualitative properties.

Proof. For all z ∈ C + , we have

(2) |C ε (x, z)| ≤ 1 ℑz .
Indeed, for all y, z such that z ∈ C + , C ε (y, z) ∈ C -. As a consequence, the imaginary part of the denominator of the right hand term of ( 1) is larger than ℑ(z).

Hence by ( 1) and ( 2), as

ε → 0, C ε (x, z) -→ 1 z-f (x) uniformly in x. From what precedes, C ε (x, z) - 1 z -f (x) = ε 1 y=0 σ 2 (x, y)C ε (y, z)dy (z -f (x) -ε 1 y=0 σ 2 (x, y)C ε (y, z)dy)(z -f (x)) = ε 1 (z -f (x)) 2 1 y=0 σ 2 (x, y)C ε (y, z)dy + o(ε) = ε 1 (z -f (x)) 2 1 y=0 σ 2 (x, y) z -f (y) dy + o(ε) where each o(ε) is uniform in x ∈ [0, 1].
But for all a = b,

1 (z-a) 2 (z-b) = 1 (a-b) 2 1 z-b -1 z-a -b-a (z-a) 2 , hence since the Lebesgue measure of the set {y ∈ [0, 1] ; f (y) = f (x)} is null, we have 1 (z -f (x)) 2 1 y=0 σ 2 (x, y) z -f (y) dy = 1 y=0 σ 2 (x, y) (f (x) -f (y)) 2 1 z -f (y) - 1 z -f (x) - f (y) -f (x) (z -f (x)) 2 dy.
As a consequence, it follows by an integration in x ∈ [0, 1] that

C ε (z) -C(z) = ε 1 x=0 1 y=0 σ 2 (x, y) (f (x) -f (y)) 2 1 z -f (y) - 1 z -f (x) - f (y) -f (x) (z -f (x)) 2 dydx + o(ε),
where C(•) is the Cauchy transform of µ.

Let us now recall that the push-forward of the uniform law on [0, 1] by f is the measure ρ(x)dx and that σ 2 (x, y) can be rewritten σ 2 (x, y) = τ (f (x), f (y)). Hence

C ε (z) -C(z) = ε s∈R t∈R { 1 z -t - 1 z -s - 1 (z -s) 2 (t -s)} τ (s, t) (s -t) 2 ρ(s)ρ(t)dtds + o(ε).
This allows us to write that for any test function g ∈ T ,

lim ε→0 µ ε (g) -µ(g) ε = Λ(g),
where

Λ(g) = (s,t)∈R 2 {g(t) -g(s) -g ′ (s)(t -s)} τ (s, t) (t -s) 2 ρ(s)ρ(t)dtds.
Note that by the Taylor-Lagrange formula, for all s, t,

{g(t) -g(s) -g ′ (s)(t -s)} τ (s, t) (t -s) 2 ρ(s)ρ(t) ≤ ρ(s)ρ(t) × τ ( • , • ) L ∞ g ′′ L ∞ 2 ,
so that, since ρ is a density, by dominated convergence,

Λ(g) = lim η→0 (s,t)∈R 2 |s-t|>η {g(t) -g(s) -g ′ (s)(t -s)} τ (s, t) (t -s) 2 ρ(s)ρ(t)dsdt.
But by symmetry, for all η > 0, Set also M := f L ∞ . Then the support of the function ρ is contained in [-M, M ], and so does the support of the function θ η , for any η > 0. For almost all s ∈ [-M, M ], lim η→0 θ η (s) exists by the formula

(
θ η (s) = t∈[s-2M,s-η]∪[s+η,s+2M ] τ (s, t)ρ(s)ρ(t) -τ (s, s)ρ(s)ρ(s) s -t dt
and by Hypothesis (f). Moreover, for η 0 as in Hypothesis (f),

|θ η (s)| ≤ 2Cρ(s) s+η 0 t=s+η (s -t) α-1 dt + t∈[s-2M,s-η 0 ]∪[s+η 0 ,s+2M ] τ (s, t)ρ(s)ρ(t) s -t dt ≤ 2Cρ(s) α (η 0 ) α + 1 η 0 t∈R τ (s, t)ρ(s)ρ(t)dsdt ≤ 2Cρ(s) α (η 0 ) α + τ ( • , • ) L ∞ η 0 ρ(s).
Hence by dominated convergence, s∈R g ′ (s) lim η→0 θ η (s)ds = lim η→0 s∈R g ′ (s)θ η (s)ds, i.e.

Λ(g) = s∈R g ′ (s) lim η→0 t∈R |s-t|>η τ (s, t)ρ(s)ρ(t)
st dt ds.

Examples

2.1. Perturbation of a uniform distribution by a standard band matrix. Let us consider the case where f (x) = x (so that µ is the uniform distribution on [0, 1]) and

σ 2 (x, y) = ✶ |y-x|≤ℓ ,
where ℓ is a fixed parameter in [0, 1] (the width of the band). In this case, τ (

• , • ) = σ 2 ( • , • ) and
F (s) = ✶ (0,1) (s) log ℓ ∧ (1 -s) ℓ ∧ s .
For small values of ε and large values of n, the density ρ ε of the eigenvalue distribution µ ε of

D ε n is approximately ρ ε (s) = ρ(s) + ε ∂ ∂s F (s) + o(ε) = ✶ (0,1) (s) -ε ✶ (0,ℓ) (s) s + ✶ (1-ℓ,1) (s) 1 -s + o(ε),
which means that the additive perturbation ε n X n alters the spectrum of D n essentially by decreasing the amount of extreme eigenvalues. This phenomenon is illustrated by Figure 1 (where we ploted the cumulative distribution functions rather than the densities for visual reasons).

(0.2 0.0 0.2 0.4 0.6 0.8 1.0

F D ε n ( • )-F Dn ( • ) ε (with F D ε n ( • ) and F Dn ( • ) the cumulative eigenvalue distribution functions of D ε n and D n ) for different val- ues of ℓ.
2.2. Perturbation of the triangular pulse distribution by a GOE matrix. Let us consider the case where ρ(x) = (1 -|x|)✶ [-1,1] (x) and σ 2 ≡ 1 (what follows can be adapted to the case σ 2 (x, y) = ✶ |y-x|≤ℓ , but the formulas are a bit heavy). In this case, thanks to the formula (9.6) of H[ρ( • )] given p. 509 of [START_REF] King | Hilbert transforms[END_REF], we get

F (s) = (1 -|s|)✶ [-1,1] (s) {(1 -s) log(1 -s) -(1 + s) log(1 + s) + 2s log |s|} .
For small values of ε and large values of n, the density ρ ε of the eigenvalue distribution µ ε of

D ε n is approximately ρ ε (s) = ρ(s) + ε ∂ ∂s F (s) + o(ε),
which implies that the additive perturbation ε n X n alters the spectrum of D n by increasing the amount of eigenvalues in [-1, -0.5] ∪ [0.5, 1] and decreasing the amount of eigenvalues around zero. This phenomenon is illustrated by Figure 2.

F D ε n ( • )-F Dn ( • ) ε (with F D ε n (

2.3.

Free convolution with a semi-circular distribution and complex Burger's equation. Let us consider the case where σ 2 ≡ 1, which happens for example if the matrix X n is taken in the Gaussian Orthogonal Ensemble. In this case, by the theory of free probability developped by Dan Voiculescu (see e.g. [START_REF] Voiculescu | Limit laws for random matrices and free products[END_REF] or [START_REF] Anderson | An Introduction to Random Matrices[END_REF]Cor 5.4.11 (ii)]), for all t ≥ 0,

µ t = µ ⊞ λ t ,
where λ t is the semi-circular distribution with variance t, i.e. the distribution with support [-2 √ t, 2 √ t] and density 1 2πt √ 4tx 2 . In this case, we know by the work of Biane [8, Cor. 2] that for all t > 0, µ t admits a density ρ t . By the implicit function theorem, and the formula given in [8, Cor. 2], one easily sees that the function (s, t) -→ ρ t (s) is regular. Then, by Theorem 1 and the fact that the linear span of T is dense in the set of continuous functions on the real line with null limit at infinity, one easily recovers the following PDE, which is a kind of projection on the real axis of the imaginary part of complex Burger's equation given in [START_REF] Biane | On the Free convolution by a semi-circular distribution[END_REF]Intro.] (3) ∂ ∂t ρ t (s) + ∂ ∂s {ρ t (s)H[ρ t ( • )](s)} = 0, ρ 0 (s) = ρ(s).

For example, if µ = λ c for a certain c > 0, then by the semi-group property of the semi-circle distribution [START_REF] Anderson | An Introduction to Random Matrices[END_REF]Ex. 5.3.26], for all t ≥ 0, µ t = λ c+t and ρ t (s) = 1 2π(c+t) 4(c + t)s 2 . One can

  c) the set of discontinuities of the function σ is closed and intersects a finite number of times any vertical line of the square [0, 1] 2 .

2 ( 9 Figure 1 .

 291 Figure 1. Perturbation of a uniform distribution by a standard band matrix: plot of the functions F ( • ) and

Figure 2 .

 2 Figure 2. Perturbation of the triangular pulse distribution by a GOE matrix: Left: plot of the functions F ( • ) and

  • ) and F Dn ( • ) the cumulative eigenvalue distribution functions of D ε n and D n ). Right: plot of the eigenvalues histogram of D ε n and of the spectral density ρ of D n . On the right figure, the (infinitesimal) increase of eigenvalues with respect to ρ on [-1, -0.5] ∪ [0.5, 1] and the (infinitesimal) decrease around zero can be observed, in agreement with the fact that, as the left figure shows, F ′ ≫ 0 on (approximately) [-1, -0.5] ∪ [0.5, 1] and F ′ ≪ 0 around zero. Both figures were made with the same simulation (n = 6.10 3 and ε = 10 -2 ).

Acknowledgements. It is a pleasure to thank Guy David for his useful advices about the Hilbert transform.

This work was partially supported by the Agence Nationale de la Recherche grant ANR-08-BLAN-0311-03.

then verify (3), using the formula (9.21) of H[ρ t ( • )] given p. 511 of [START_REF] King | Hilbert transforms[END_REF].