C Bodin

Sylvain Takerkart

P Belin

O Coulon

Takerkart

• P Belin

• O Coulon

Anatomo-functional correspondence in the superior temporal sulcus

Keywords: STS, Anatomo-functional, Sulcal depth, Temporal voice areas

Introduction

Given the key role that voice plays in human communication, it is not surprising to find, in our brain, areas preferentially activated when we perceive voices. These temporal voice areas (TVA) are located along the superior temporal gyrus/sulcus (STG/STS) bilaterally [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF]. They show greater response to vocal sounds, whether they carry speech or not, than to other categories of non-vocal sounds. Despite the great reproducibility of these results using the voice localizer paradigm [START_REF] Belin | Human temporal-lobe response to vocal sounds[END_REF][START_REF] Charest | Cerebral processing of voice gender studied using a continuous carryover FMRI design[END_REF][START_REF] Ethofer | Emotional voice areas: anatomic location, functional properties, and structural connections revealed by combined fMRI/DTI[END_REF][START_REF] Kreifelts | Cerebral representation of non-verbal emotional perception: fMRI reveals audiovisual integration area between voice-and face-sensitive regions in the superior temporal sulcus[END_REF][START_REF] Latinus | Norm-based coding of voice identity in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]), it appears from individual maps that the location of voicesensitive peaks varies considerably across subjects. However, recent evidence using cluster analysis suggests that individual functional peaks can be organized, almost symmetrically, in three bilateral ''voice patches'' distributed antero-posteriorly along the STG/STS [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. These patches exhibit distinct levels of activity [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF], and density [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]: bilateral middle patches located near the Heschl's gyrus were those having the highest intensity

The right STS was found to be deeper than the left STS in a portion ventral to Heschl's gyrus, called the ''STAP'' (Superior Temporal Asymmetrical Pit, [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF][START_REF] Glasel | A robust cerebral asymmetry in the infant brain: the rightward superior temporal sulcus[END_REF]). This asymmetry is highly significant and present irrespective of age, handedness and language lateralization. However, it is still not known if this anatomical trait is functionally relevant. A correspondence between sulcal depth and a functional area corresponding to the sensory-motor hand representation has been already found in the central sulcus [START_REF] Boling | Localization of hand motor activation in Broca's Pli de Passage Moyen[END_REF], and observed qualitatively between voice areas and the STS [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. Thus, and this is the third argument, variability of STS depth could be linked to functional variability in voice areas.

The purpose of the present study is to search for correspondences between functional activity elicited by voice perception and STS depth. To this end, we first constructed STS depth profiles, i.e., the depth fluctuations along the STS antero-posterior axis [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF], of 116 subjects from their cortical surface representation generated by the BrainVisa software [START_REF] Mangin | Object-based morphometry of the cerebral cortex[END_REF]). Then, we projected individual functional maps from the voice localizer on these same corresponding cortical surface representations [START_REF] Operto | Projection of fMRI data onto the cortical surface using anatomically-informed convolution kernels[END_REF]. This allowed us to generate individual functional profiles in the same coordinate space than depth profiles for better comparison. We examined anatomo-functional correspondence first qualitatively at the group level, by comparing functional profiles with depth profiles, then quantitatively at the individual level, by estimating the predictive power of the functional signal to infer the value of anatomical features, using a machine learning approach. We hypothesized that the depth within STS would provide landmarks indicating the location of voice-sensitive activity.

Materials and methods

Subjects

One-hundred and sixteen healthy subjects were scanned (n = 116; 53 males, 63 females; mean age 23.7 ± 5.8) as part of published and unpublished experiments of the Voice Neurocognition Laboratory (http://vnl.psy.gla.ac.uk/) of the Institute of Neuroscience and Psychology at University of Glasgow. Participants, drawn mostly from the Glasgow student population, were of various ethnic backgrounds, education and manual lateralization and all provided written informed consent prior to participation. The experiments were approved by the local ethics committee at University of Glasgow. Anatomical analysis was carried out on 95 subjects (51 males, 44 females; mean age 23.9 ± 6.1), whereas functional and anatomo-functional parts were performed on level, which is consistent with initial findings [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF]. In contrast, the right posterior patch showed the highest cluster density, indicating a high concentration of individual peaks in this location that interestingly seems to correspond to the deepest region of the STS [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. However, no study until now has revealed clear anatomical landmarks that could indicate the location of voice-sensitive regions.

At a basic level of description, the STS is described as a long fold divided into an anterior horizontal trunk section (from the temporal pole to the temporo-parietal junction) and posterior ascending branches. This rough subdivision has been found useful by previous authors to investigate anatomo-functional correspondences in the STS. [START_REF] Kreifelts | Cerebral representation of non-verbal emotional perception: fMRI reveals audiovisual integration area between voice-and face-sensitive regions in the superior temporal sulcus[END_REF] observed that the locus of audiovisual integration of emotional stimuli falls at the bifurcation of the posterior branches. In addition to relying on anatomical features, this was the first study to highlight a functional subdivision of the STS. More recently, [START_REF] Sun | Linking morphological and functional variability in hand movement and silent reading[END_REF] explored the individual variability and found that subjects having a longer STS trunk section also had a more widespread silent-reading activation along the antero-posterior axis of the sulcus. Taken together, these findings suggest that STS anatomy can constitute a good indicator to locate or characterize functional activity. However, given the apparent complexity of the functional anatomy of voice areas [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Belin | Human temporal-lobe response to vocal sounds[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF], a finer description of the STS anatomy is required for a better understanding of anatomo-functional correspondences inside these regions.

One potential relevant trait to characterize sulcal anatomy is depth. Indeed, depth fluctuations inside the STS can help to define potential landmarks to segment the sulcus [START_REF] Auzias | Deep sulcal landmarks: algorithmic and conceptual improvements in the definition and extraction of sulcal pits[END_REF][START_REF] Cykowski | The central sulcus: an observer-independent characterization of sulcal landmarks and depth asymmetry[END_REF][START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]. Recent evidence shows that the STS, including its trunk section, contains a relatively invariant number of ''sulcal pits'', which are points of maximum depth within folds being very reproducible across subjects [START_REF] Auzias | Deep sulcal landmarks: algorithmic and conceptual improvements in the definition and extraction of sulcal pits[END_REF][START_REF] Im | Spatial distribution of deep sulcal landmarks and hemispherical asymmetry on the cortical surface[END_REF]. Conversely, the STS would also contain an invariant number of ''plis de passage'' (PP), i.e., annectant gyri buried inside the fold that separate the sulcus into several parts [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]), similar to the ''Pli de passage fronto-parietal moyen'' within the central sulcus [START_REF] Boling | Localization of hand motor activation in Broca's Pli de Passage Moyen[END_REF]. Each PP is thought to be more or less superficial across individuals and hemispheres, causing an important source of variability of both depth and visual external aspect of the sulcus. This variability constitutes the second reason why we should investigate STS depth. In addition to giving a fine description (and consequently a segmentation) of the sulcus, the variability of depth landmarks opens the way to the comparison between individuals, genders or hemispheres. A major finding demonstrating this is the STS depth asymmetry.

74 subjects (34 males, 40 females; mean age 23.4 ± 4.9). Exclusion of subjects was due to algorithmic failure for depth or functional profile extraction (supplementary material figure C). All subjects kept for the study had both lower and upper functional profiles in addition to depth profiles.

MRI acquisition

All scans were acquired on a 3T Siemens (Erlangen, Germany) Tim Trio scanner at the Centre for Cognitive Neuroimaging (http://www.ccni.gla.ac.uk/), University of Glasgow.

High-resolution 3D T1-weighted sagittal scans were acquired for each subject (voxel size 1 mm 3 isotropic; acquisition matrix 256 9 256 9 192).

Functional images were obtained from the voice localizer [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF] constituted of forty 8-s blocks of either vocal (20 blocks) or non-vocal (20 blocks) sounds. A single-shot gradient-echo echo-planar imaging (EPI) sequence was performed with the following parameters: field of view (FOV) = 210 9 210 mm 2 , 32 slices per volume, interleaved slices order, voxel size 3 9 3 9 3.3 mm 3 , acquisition matrix 70 9 70, flip angle = 77°, echo time (TE) = 30 ms, repetition time (TR) = 2 s, acquisition time (TA) = 2 s.

Anatomical analysis

Brainvisa pipeline

All anatomical images were post-processed at Neurospin center (Analysis and information processing unit UNATI, NeuroSpin, CEA Saclay, Gif-sur-Yvette) using the Morphologist pipeline in the BrainVisa software [START_REF] Mangin | Object-based morphometry of the cerebral cortex[END_REF]). The pipeline includes the following steps: correction for spatial inhomogeneities in the T1w signal intensity, splitting of the hemispheres, tissue statistics estimation, white-gray matter classification and white matter reconstruction. As a result, an individual triangular mesh representation of the grey/white matter interface was obtained for both hemispheres of all subjects.

STS identification and drawing

The superior temporal sulcus (STS) is a highly distinguishable fold separating the superior from the middle temporal gyrus in the temporal lobe. It consists of extensive-both in width and length-areas of cortex buried in the upper and lower banks of the sulcus. The left and right STS fundi of each subject were drawn semi-automatically on the inner surface of the cortex (grey-white matter interface, Fig. 1a) using the SurfPaint module of the Anatomist visualization software [START_REF] Troter | An interactive sulcal fundi editor in BrainVisa[END_REF]. We first determined manually the anterior and posterior points of the sulcus based on anatomical landmarks easily identifiable in each subject. The anterior extremity was chosen as the tip of the temporal pole, posterior to polar temporal sulcus. In most cases, this sulcus was perpendicular to the STS and a clearly visible PP separates these two folds [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]). The posterior extremity was chosen as the intersection of the main body of the sulcus with its posterior branches (that were not included).

Finally, the fundus of the STS was drawn automatically between the two extremities, following the deepest path between the two extremities [START_REF] Troter | An interactive sulcal fundi editor in BrainVisa[END_REF]. In short, we selected the ''stem'' of the STS [START_REF] Bonte | Development from childhood to adulthood increases morphological and functional inter-individual variability in the right superior temporal cortex[END_REF][START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]) which we call STS thereafter.

Depth profiles

For both hemispheres of each subject, sulcal depth was computed as the geodesic distance from the sulcal fundus to the outer cortex. More precisely, the algorithm selected the shortest path along the cortical surface from the deepest points of the STS to the gyral crowns [START_REF] Rettmann | Automated sulcal segmentation using watersheds on the cortical surface[END_REF]. The depth was then sampled at 100 equally spaced locations along the antero-posterior axis of the STS fundus. As a result, two standardized individual depth profiles were extracted (left and right), where 0 represented the anterior extremity and 100 the posterior extremity of the STS fundus for each hemisphere, similar to the standardized depth curves used in [START_REF] Cykowski | The central sulcus: an observer-independent characterization of sulcal landmarks and depth asymmetry[END_REF]. An algorithmic failure during this depth profile extraction compelled us to keep only 95 subjects for next anatomical analysis (supplementary material figure C). Once extracted, depth profiles could be easily compared between subjects (Fig. 1a).

An STS depth asymmetry index AI = 2 9 (R -L)/(R ? L), where R and L represent right and left depth, respectively, was computed for each individual at each STS coordinate. Positive AI values reflected a locally deeper right STS. Significant left-right differences were assessed using Mann-Whitney non-parametric tests across individuals, at each position of the STS axis. The resulting p values were corrected for multiple comparisons (Bonferroni).

Functional analysis

Voice localizer

Functional images were preprocessed in SPM8 (Wellcome Department of Cognitive Neurology, University College London). It included corrections for head motion, realignment of scans to the first volume, co-registration of the maps across subjects to get a group-level overview of the voice areas localization on the cortex. The average map is represented in Fig. 3, upper row, on the HipHop138 group template (https://meca-brain.org/software/hiphop138/).

Individual maps were also overlapped with the corresponding STS lines to compute functional profiles as described in the next section.

Functional profiles

The surface-based functional map of each individual was used to generate a functional value for each of the 100 sections of the STS line. More precisely, at each coordinate of the STS fundus, functional values were integrated in the direction perpendicular to the STS fundus, across the upper and lower walls, within a distance of 25 mm. The HipHop process was used to compute the orthogonal direction to the sulcus. We showed a posteriori that this distance allowed integrating the highest level of functional activity (supplementary material figure B). mean EPI image to the corresponding individual anatomical scan and smoothing (Gaussian kernel of 8 mm full width at half maximum), as in [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF].

We then performed first-level (single-subject) analysis using a design matrix that contained separate regressors for vocal and non-vocal sounds [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF], plus realignment parameters to account for residual motion artifacts. Two contrast maps: ''Vocal [ Non-Vocal'' and ''All sounds [ Silence'' were computed per subject.

Surface-based functional maps

Individual 3D functional maps were projected on the corresponding 2D cortical mesh representation (Fig. 1b) using the method presented in [START_REF] Operto | Projection of fMRI data onto the cortical surface using anatomically-informed convolution kernels[END_REF], resulting in individual 2D surface-based functional maps.

Inter-subject cortical surface matching was performed using the Hip-Hop method [START_REF] Auzias | Model-driven harmonic parameterization of the cortical surface: HIP-HOP[END_REF]) available in the Cortical Surface toolbox in BrainVisa (https://mecabrain.org/software). We could then average functional This procedure resulted in upper and lower functional values at each STS coordinate and produced, for each hemisphere, an upper and lower functional profile, i.e., the functional activity located on the upper and lower walls of the sulcus, respectively, as shown in Fig. 1b,c. An algorithmic failure during this functional profile extraction compelled us to keep 74 subjects for which we had both lower and upper functional profiles in addition to depth profiles (supplementary material figure C).

These profiles give information about the functional fluctuations along the STS axis (Fig. 1c). In the end, each subject had four functional profiles: right upper (Rup) and lower (Rlow), and left upper (Lup) and lower (Llow). These functional profiles are defined in the same standardized STS axis than depth profiles, thus making them directly comparable.

Mann-Whitney non-parametric tests at each [0-100] coordinate were performed to investigate functional asymmetry, between hemispheres or between the two banks of the STS. Resultant p values were then corrected for multiple comparisons (Bonferroni).

Anatomo-functional analysis

The principal question of the present study was: can sulcal depth be a landmark of voice-sensitive functional activity? After characterizing the STS anatomy and functional activity, we therefore investigated anatomo-functional relations by comparing depth profiles with functional profiles (Fig. 1c). To this aim, we first examined if there was an anatomo-functional correspondence at the group level by qualitatively comparing fluctuations of depth and functional average profiles along the STS axis.

We then investigated the link between structure and function in a quantitative manner. We proceeded to a correlation analysis between depth and voice-related functional activity along the STS and across subjects. To limit the number of multiple comparisons and avoid an overly conservative Bonferroni correction, we restricted the spatial precision of this analysis. We divided each STS in ten equal sections, by averaging depth and functional values of ten consecutive coordinates into a single value. Correlation scores and corresponding p values were computed for all sections and for the four profiles: right and left upper and lower profiles (see Fig. 1c). We then performed a multiple testing correction (Bonferroni), considering significant scores with p \ 0.05/10.

Finally, to further investigate the anatomo-functional correspondence at the individual level, we used a machine learning approach similar to the one used in [START_REF] Weiner | The mid-fusiform sulcus: a landmark identifying both cytoarchitectonic and functional divisions of human ventral temporal cortex[END_REF], based on supervised classification. According to the results obtained at the group-level (Fig. 3), we asked whether the functional activity estimated at a given point of the STS could predict whether this point was located 'inside' or 'outside' the STAP. We used a simple logistic regression model f(X) = Y, with the input vector X of our model f defined as the concatenation of the values of the upper and lower profiles for the ''Voice minus Nonvoice'' contrast. To define the output Y of the model for each subject, we first located the individual deepest point of the STS within the STAP that had been defined at the group level (see Fig. 2); then, we defined the region centered on this point with the same width as the group STAP (inside: y = ?1), and the points outside of this region were labeled with y = -1. The generalization performances of the model were assessed by measuring the balanced accuracy [START_REF] Brodersen | The balanced accuracy and its posterior distribution[END_REF]) in a tenfold cross-validation (i.e., where we left out one tenth of the subjects for the test set in each fold). Finally, the significance of these results was estimated by repeating this analysis using datasets where the labels y are randomly permuted (N = 10,000 times), which allowed computing the probability that the null hypothesis of no link between structure and function was valid.

Results

STS depth asymmetry

We found a strong asymmetry of the STS, with a deeper sulcus in the right hemisphere (Fig. 2). This trend was highly significant (Mann-Whitney, p \ 0.001) between coordinates 49-93 on the STS axis, and was preserved after Bonferroni correction (Mann-Whitney, p \ 0.05). We considered this region of significant asymmetry as the equivalent of the ''STAP'' (Superior Temporal Asymmetrical Pit) previously described by [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF]. This group-asymmetric segment approximately corresponds to the posterior half of the STS stem. If we consider the whole STS, with its posterior branches, it corresponds to the middle part of the sulcus. In the right hemisphere, the STAP corresponds to the deepest region of the sulcus, whereas in the left hemisphere it corresponds only to a local depth maximum because the deepest region is the intersection of the posterior ascending branches (the posterior extremity in Fig. 2).

At the individual level, we observed a large variability in depth profiles, due to distinct combinations of ''plis de passage'' (PP) for each subject. Nevertheless, 82% of our subjects (n = 78) showed a global rightward asymmetry, i.e., a positive average index of asymmetry (AI), with a mean AI of ?15.4 ± 20%. When we computed the mean AI along the STS axis, we found a maximal value at coordinate 68 (Fig. 2), inside the STAP segment. A second asymmetry peak (AI = 12.7%, coordinate 18) was also

Anatomo-functional correspondence

Our qualitative analysis of average profiles revealed a spatial correspondence between the deepest point of the STS inside the STAP region and the peak of voice activity, for both hemispheres (Fig. 3). More precisely, for both hemispheres and both upper and lower banks of the STS, the maximum of the average functional profile (Voice [ Non-Voice) was very close to that of the average depth profile on our STS axis. Knowing that these depth peaks are not identically located in the two hemispheres (Left: 60; Right: 73), the functional peaks are also shifted as previously described (Lup: 58, Llow: 55; Rup: 77, Rlow: 60), suggesting that inter-hemispheric differences are similar for voice-specific functional activity and sulcal depth. To conclude, an anatomo-functional correspondence seems to exist inside the asymmetric region of the STS (STAP), human voice activity being focused into the deepest region of the sulcus. Note that this was not the case for the global auditory activity (All [ Silence, supplementary material Figure A) where functional peaks are, respectively, at: Lup: 57, Llow: 44; Rup: 58, Rlow: 52 on the STS axis.

Quantitative analysis of the anatomo-functional correspondence is presented Fig. 4. Only one of the ten STS sections survived Bonferroni correction: the sixth section of the left inferior STS. This section corresponded to coordinates [50-59] on the STS axis and was located within the STAP. This is in agreement with previous groupwise qualitative results (Fig. 3). However, this linear correlation seems nevertheless weak, given that only one profile (Llow) showed significant scores after correction. revealed in the anterior part of the sulcus, meaning that there is a trend in our population toward a deeper right STS in this region. The presence of a superficial PP was often observed on the left hemisphere in this part of the sulcus and could explain the peak of asymmetry.

Temporal voice areas (TVA)

As shown by average functional maps (Fig. 3, top), the contrast ''Voice [ Non-Voice'' showed significant BOLD signal differences on both banks of the STS, as well as within the fold, which is the traditional location of the voice areas [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. The contrast ''All [ Silence'' (supplementary material figure A top row) revealed an activity localized around the upper bank of the STS and the primary auditory cortex.

These observations were found again on averaged functional profiles (Fig. 3, bottom row). They revealed a higher activity in the upper bank of the posterior STS bilaterally (p \ 0.05, Bonferroni corrected) for the ''Voice [ Non-Voice contrast''. For ''All [ Silence'' contrast, functional activity is higher on the upper bank than on the lower bank (p \ 0.05, Bonferroni corrected) and more extended along the STS axis bilaterally.

No significant difference between hemispheres was found at any location along the entire STS axis (p [ 0.05 Bonferroni corrected). Regarding the peaks of the groupaverage functional profiles, right voice activity was located at a larger coordinate than on the left side, particularly for the upper bank. Indeed, the functional peaks were, respectively, located at coordinates: Lup: 58, Llow: 55; Rup: 77, Rlow: 60 on the STS axis (Fig. 3). We hypothesize that this small effect was mainly due to the important inter-individual variability (see ''Limitations and future work'' section).

To investigate further anatomo-functional correspondences, we decided to explicitly test the link between high functional activity and STS depth at the individual level. Given the results observed qualitatively at the group-level (Fig. 3), we decided to focus this analysis on the particular region of the STAP, and explicitly test a relationship between functional activity and STAP localization. compare depth and functional profiles in a common referential and to highlight an anatomo-functional correspondence between the voice-sensitivity maxima and depth maxima inside the STAP region of the STS (Fig. 3). Finally, we demonstrated that this correspondence was present irrespective of hemisphere side and subject gender (Fig. 5) and could be found at the individual level too.

STS depth characterization

We found that the STS is highly asymmetric (p \ 0.001) in most of our subjects (82%) with a deeper right STS in a middle portion of the sulcus (Fig. 2). This landmark seems to correspond to the STAP initially described in [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF] and fits with previous reports of a rightward STS asymmetry [START_REF] Bonte | Development from childhood to adulthood increases morphological and functional inter-individual variability in the right superior temporal cortex[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF][START_REF] Li | Mapping longitudinal hemispheric structural asymmetries of the human cerebral cortex from birth to 2 years of age[END_REF][START_REF] Van Essen | A population-average, landmark-and surfacebased (PALS) atlas of human cerebral cortex[END_REF][START_REF] Im | Spatial distribution of deep sulcal landmarks and hemispherical asymmetry on the cortical surface[END_REF][START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]. As in previous investigations [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF][START_REF] Bonte | Development from childhood to adulthood increases morphological and functional inter-individual variability in the right superior temporal cortex[END_REF], the STAP found here represented a large part of the STS trunk section, with a right peak deeper than 20 mm, and was present irrespective of the gender.

The method used here for the STS exploits the 'depth profiles' representation previously validated on the central sulcus (CS) [START_REF] Cykowski | The central sulcus: an observer-independent characterization of sulcal landmarks and depth asymmetry[END_REF][START_REF] Hopkins | Evolution of the central sulcus morphology in primates[END_REF]). However, instead of extracting automatically the sulcus as 3D object, as done in anatomical studies of the CS [START_REF] Cykowski | The central sulcus: an observer-independent characterization of sulcal landmarks and depth asymmetry[END_REF][START_REF] Hopkins | Evolution of the central sulcus morphology in primates[END_REF][START_REF] Sastre-Janer | Three-dimensional reconstruction of the human central sulcus reveals a morphological correlate of the hand area[END_REF]) and the STS [START_REF] Glasel | A robust cerebral asymmetry in the infant brain: the rightward superior temporal sulcus[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF][START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF], we performed a semi-automatic drawing of each individual sulcus. This approach presents several advantages: first, the semi-automatic drawing avoids possible misidentification of sulci: merging can occur between the STS and adjacent sulci, disrupting automatic sulcal identification. Second, defining the anterior and posterior extremities of the STS trunk in a consistent manner is extremely difficult automatically, which is why we favored a manual selection. Third, the two-dimensional sulcal line constructed by our method can go through sulcal interruptions over the PPs, resulting in a single connected line, while automatic representations are segmented in several pieces [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]). Finally and crucially, functional activity can be more easily projected on this 2D sulcal line and analyzed in the same referential.

Because of these advantages and given the fact that our results are in line with those of previous studies, we argue that the semi-automatic method is well adapted to study anatomo-functional correspondence.

Anatomo-functional correspondence between STS pit and voice-sensitivity maxima

The first question in this study sought to determine if STS depth could be related to voice-sensitive functional At the individual level, the generalization performance of our machine learning model, i.e., the supervised classification of functional data into 'inside' or 'outside' the individual STAP, was of 61 and 71% (average balanced accuracies), respectively, in the left and right hemispheres, which is significantly different from chance level (50%) in both hemispheres (p \ 0.0001). We can then examine the behavior of the model at the points of maximal activity, specifically by testing whether it predicts whether these points are located inside the STAP individually: the maximum of left (Linf) and right (Rinf) inferior profiles were located in the individual STAP in 56.8 and 73% of the subjects, respectively, whereas the maximum of left (Lsup) and right (Rsup) superior profiles were both located in the STAP in 100% of the subjects. These results, therefore, quantitatively demonstrate the anatomofunctional correspondence at the individual level.

Gender comparison

We performed additional analysis to examine whether previous findings were modulated by subject gender. There were no significant differences between the respective left and right STS depth profiles of males and females (Fig. 5). Both groups had a significant (p \ 0.05, Bonferroni corrected) asymmetrical STAP region (between 51 and 90 for males, 55-90 for females).

Contrary to depth, functional profiles showed a trend towards an influence of gender. Indeed, Fig. 5 shows several portions of the STS where the functional profile of females differs from that of males (p \ 0.05 uncorrected, plotted in blue for upper profiles, green for lower ones). These portions are more extended for upper profiles, and the differences always reflect higher activity in females than in males. These trends did not survive Bonferroni correction, however.

Noteworthy, the anatomo-functional correspondence between the deepest points of the STS and the peaks of voice activity found in the entire sample is clearly visible in each gender group. The average depth peaks inside the respective STAP regions were located at Left, 61; Right, 73 for females and Left, 55; Right, 76 for males. The average functional peaks were located at: Lup, 72; Llow, 54; Rup, 77; Rlow, 59 for females and Lup, 63; Llow, 57; Rup, 74, Rlow, 53 for males on the STS axis.

Discussion

The present study is the first to demonstrate a link between STS depth and functional activity in the temporal voice areas. We used an innovative method that first allowed us to finely characterize depth anatomy of the STS and to replicate the finding by [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF] of a strong asymmetry in its middle part (the STAP, Fig. 2). Then, to activity, and potentially provide landmarks of this activity. Several reports have shown that the functional activity elicited by human voice perception peaks in the middle region of the STS [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Deen | Functional organization of social perception and cognition in the superior temporal sulcus[END_REF][START_REF] Kreifelts | Cerebral representation of non-verbal emotional perception: fMRI reveals audiovisual integration area between voice-and face-sensitive regions in the superior temporal sulcus[END_REF] or, similarly, at the level of the middle cluster of TVAs [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. However, for the first time, we highlighted a more accurate localization of this functional peak based on anatomical landmarks. We found a bilateral spatial correspondence between the STS depth peak inside the STAP region and the functional peak of voice-sensitive activity (Fig. 3). By looking at surfacebased functional maps and considering the location of the STAP [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF], we can see that this spatial correspondence occurs, on the length axis, at a middle section of the entire sulcus corresponding to the 'posterior sulcal root' [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]) ventral to the planum temporale. To localize this correspondence on the depth axis, we integrated repeatedly the voice-related activity within an increasingly large distance from the STS fundus. Results showed that the peaks were located on the top of right STS banks, around 20 mm from the sulcal fundus. However, on the left hemisphere, the peaks were located beyond the upper bank but also in the fundus of the STS (supplementary material figure B, maps Fig. 3). Hence, the depth peaks of the STS within the STAP constitute landmarks that indicate the position of voice functional peaks along the STS length axis. This landmark has the same localization value than the 'hand-knob' or the 'Pli-depassage fronto-parietal moyen' used to localize the primary hand motor area along the length of the central sulcus [START_REF] Yousry | Localization of the motor hand area to a knob on the precentral gyrus. A new landmark[END_REF][START_REF] Boling | Localization of hand motor activation in Broca's Pli de Passage Moyen[END_REF].

The anatomo-functional correspondence was found in both hemispheres and both sulcal banks, but functional differences could be noticed between these cortical regions. On one hand, we did not find significant differences in functional activity between the two hemispheres (p [ 0.05 Bonferroni corrected), but found a trend towards a higher activity on the right side, confirming previous results [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. On the other hand, we found that activity was higher in the upper bank of the STS the other hand, anatomical studies showed that a deeper right STS is already observable in fetuses [START_REF] Kasprian | The prenatal origin of hemispheric asymmetry: an in utero neuroimaging study[END_REF] and persists throughout the human lifespan [START_REF] Bonte | Development from childhood to adulthood increases morphological and functional inter-individual variability in the right superior temporal cortex[END_REF][START_REF] Glasel | A robust cerebral asymmetry in the infant brain: the rightward superior temporal sulcus[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF]. Taken together, these reports suggest that the establishment of the depth asymmetry of the STS and its sensitization to human voice occur very early during the prenatal life, and that the cortex in this region becomes really selective to voice perception during infancy.

A key region in a deep sulcus

The fact that the most voice-sensitive region is located at the level of an asymmetrical and deep part of the sulcus might be more than mere coincidence. Indeed this STS region, as a sulcal pit, is supposed to gather several key features that would make it particularly interesting.

First of all, sulcal pits have been shown to be highly heritable and reproducible across subjects, with a more pronounced effect in deeper sulci [START_REF] Auzias | Deep sulcal landmarks: algorithmic and conceptual improvements in the definition and extraction of sulcal pits[END_REF][START_REF] Im | Spatial distribution of deep sulcal landmarks and hemispherical asymmetry on the cortical surface[END_REF][START_REF] Lohmann | Sulcal variability of twins[END_REF]. Obviously, this cannot be segregated from the hierarchical emergence of folds during development [START_REF] Chi | Gyral development of the human brain[END_REF][START_REF] Lohmann | Sulcal variability of twins[END_REF]. The STS has been shown to be a primary fold appearing early in the development [START_REF] Chi | Gyral development of the human brain[END_REF][START_REF] Dubois | Mapping the early cortical folding process in the preterm newborn brain[END_REF]) with a rapid growth of depth during the first years of life [START_REF] Meng | Spatial distribution and longitudinal development of deep cortical sulcal landmarks in infants[END_REF]. Crucially, its asymmetry is already present in utero and persist until adulthood [START_REF] Glasel | A robust cerebral asymmetry in the infant brain: the rightward superior temporal sulcus[END_REF][START_REF] Kasprian | The prenatal origin of hemispheric asymmetry: an in utero neuroimaging study[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF]. The importance of these results is twofold: first, the most voicesensitive cortex is located in a region that is stable both inter-and intra-individually. Second, the depth of this region is still evolving during the first years of life, a period which is crucial for vocal social interactions. Conversely, voice perception is early developed in humans [START_REF] Kisilevsky | Effects of experience on fetal voice recognition[END_REF][START_REF] Ockleford | Responses of neonates to parents' and others' voices[END_REF], and is a primitive ability shared with others species including its cerebral substrate [START_REF] Andics | Voicesensitive regions in the dog and human brain are revealed by comparative fMRI[END_REF][START_REF] Petkov | A voice region in the monkey brain[END_REF][START_REF] Sadagopan | High-field functional magnetic resonance imaging of vocalization processing in marmosets[END_REF]. Thus, it can be proposed that the location of the voice activity is in good agreement by the stable and primitive nature of the depth pit.

Another feature that can play a role in the anatomofunctional correspondence observed in this paper is the structural connectivity. In the key study of [START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF], the authors revealed an effect of callosal connectivity on the STS asymmetry, and suggested that the white matter fibers may constrain the folding anatomy. However, given that the asymmetry was reduced but not absent in corpus callosum agenesis and according to new hypothesis on gyrification [START_REF] Ronan | From genes to folds: a review of cortical gyrification theory[END_REF][START_REF] Xu | Axons pull on the brain, but tension does not drive cortical folding[END_REF], we cannot assert that the sulcal pit within the STAP region is only impacted by structural connectivity. Rather, we can hypothesize that white fibers may be a factor among others (Fig. 3) compared to the lower bank. This could be explained by a greater recruitment of the STG rather than MTG during voice perception [START_REF] Belin | Voice-selective areas in human auditory cortex[END_REF][START_REF] Deen | Functional organization of social perception and cognition in the superior temporal sulcus[END_REF][START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. Only differences in the posterior part of the STS survived statistical corrections, meaning that in this region the upper bank of the STS (STG) is much more recruited than the lower bank (MTG). This trend could be driven by the vertical orientation of the posterior sylvian fissure: the activity would follow the STG rather than ventral regions, which seem more implicated in face perception [START_REF] Deen | Functional organization of social perception and cognition in the superior temporal sulcus[END_REF][START_REF] Pelphrey | Functional anatomy of biological motion perception in posterior temporal cortex: an fMRI study of eye, mouth and hand movements[END_REF].

Another important finding was that the anatomo-functional correspondence was found in both genders (Fig. 5). However, one unanticipated result was that females exhibited higher voice activity than males in several portions of the STS (p [ 0.05 uncorrected). Only one study [START_REF] Arhens | Gender differences in the temporal voice areas[END_REF]) describes a gender effect in temporal voice areas using multivariate pattern analysis. They showed a better vocal/non-vocal classification in females than in males in the middle parts of bilateral MTG and the right STG. Here, this trend could be related to the fact that we found gender differences between both lower profiles and upper profiles along the STS axis. This raises interesting questions regarding the possible influence of anatomy on this different functional sensitivity to voice between genders.

Finally, we demonstrated that the anatomo-functional correspondence exists at the individual level. Indeed, machine learning techniques were able to individually predict the depth-relative position of an STS point based on functional information. For the majority of subjects, the maximal functional peak was located inside their individual STAP. Crucially, this correspondence was found in all subjects if we take into account the upper bank of the STS. This finding, while preliminary, supports the idea of an STS depth landmark for voice-sensitive activity previously suggested by the qualitative results (Fig. 3). Nonetheless, there are still many unanswered questions about the link between depth and function.

What link between STS depth and voice-related activity?

The anatomo-functional correspondence found here does not allow us to infer if depth and voice-sensitive activity are causally linked. However, several points can be addressed because they could directly or indirectly link these two features, for example during development.

On one hand, we know from previous studies that voice perception is already functional in infants and even in neonates [START_REF] Kisilevsky | Effects of experience on fetal voice recognition[END_REF][START_REF] Ockleford | Responses of neonates to parents' and others' voices[END_REF]) and that a voice-sensitive cortex is already present in 3-monthold infants [START_REF] Blasi | Early specialization for voice and emotion processing in the infant brain[END_REF][START_REF] Grossmann | The developmental origins of voice processing in the human brain[END_REF]. On that influence the depth of the STS during first years of life [START_REF] Meng | Spatial distribution and longitudinal development of deep cortical sulcal landmarks in infants[END_REF], and that this influence would be heterogeneous along the STS like sensitivity for voice is heterogeneous.

Finally, the voice-sensitive region described in this study could be located in a functional strategic place. At the whole brain scale, STS appears as a hub region for both social perception [START_REF] Allison | Social perception from visual cues: role of the STS region[END_REF][START_REF] Lahnkoski | Naturalistic fMRI mapping reveals superior temporal sulcus as the hub for the distributed brain network for social perception[END_REF]) and language, between the ventral and the dorsal streams [START_REF] Hickok | The cortical organization of speech processing[END_REF]. At a smaller scale, the sulcal pit of interest here is located functionally near the primary auditory cortex on Heschl's gyrus, and anatomically close to the intermediate PP [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]). This sulcal interruption was previously described as asymmetric, with a PP much more superficial in the left STS compared to the right [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]). Nevertheless, the functional significance of these sulcal bridges is interesting and needs to be further explored, in line with the observation that three main PP are crossing the STS similar to the three voice patches described along this sulcus [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF].

Limitations and future work

The main limitation of this study concerns the individual variability that made the anatomo-functional correspondence more arduous to detect. We encountered two sources of variability in this study: the first one was anatomical and concerns essentially the presence of PP inside the sulcus that exhibit variable levels of depth [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF] depending on the subject and the hemisphere. These PP can strongly modify the aspect of two individual depth profiles and make the detection of the depth pit harder. The second source of individual variability was functional and derives from the considerable spatial variability of voice-sensitive peaks [START_REF] Pernet | The human voice areas: spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. Individual functional profiles revealed several voice peaks along the STS and the highest one was not necessarily the one observed at the group level (Fig. 3). These two sources of variability could explain the weakness of the correlation analysis (Fig. 4), despite the existence of a relationship between voice-sensitive functional activity and STAP, as demonstrated by our classification approach.

Another limitation of this study comes from the removal of part of the subjects (see supplementary material, Fig- ure C). Solving the occasional algorithmic failures that led to excluding subjects would certainly improve our statistical analysis.

Other improvements could be made for future investigations. First, it might be possible to use other landmarks on the STS in addition to the extremities (see Fig. 2). This could improve the alignment of both depth and functional profiles between subjects. For example, a strategy would be to label stable gyri around the STS (as in [START_REF] Glasel | A robust cerebral asymmetry in the infant brain: the rightward superior temporal sulcus[END_REF][START_REF] Leroy | New human-specific brain landmark: the depth asymmetry of superior temporal sulcus[END_REF], or the PP, whose number has been shown to be constant across individuals despite variable depth levels [START_REF] Ochiai | Sulcal pattern and morphology of the superior temporal sulcus[END_REF]). Along the same line, other sulcal features than depth could be studied, such as variable morphological traits that could be linked to functional variability, as done by [START_REF] Sun | Linking morphological and functional variability in hand movement and silent reading[END_REF]. In this work, the authors characterized both anatomical and functional individual variability and found correspondences based on correlation analysis.

  Fig. 1 Schematic description of the methods that were used in the paper. a The superior temporal sulcus (STS) was manually drawn on the cortical mesh of each subject. Depth values were computed and standardized on a common coordinate axis to build the depth profiles. b Functional voice [ non-voice maps were individually projected on

Fig. 2

 2 Fig. 2 STS depth asymmetry. Group-average (±SEM) depth profiles of the right (in red) and left (in blue) STS, with depth values (mm) on the left vertical axis. The group-average index of asymmetry (AI %) is shown in black dotted line with values on the right vertical axis. Horizontal dotted line indicates null AI value. The most significant portion of right-left asymmetry (Mann-Whitney, p \ 0.001) is represented by the orange overlay and illustrates the STAP (superior temporal asymmetrical pit, Leroy et al. 2015)

Fig. 3 Fig. 4

 34 Fig. 3 Anatomo-functional correspondence in the STS. Top panel averaged surface-based functional maps of the voice areas projected on the HipHop138 template for the left (L) and right (R) hemisphere. The color bar indicates the mean t value of the functional activity. Lower panel anatomical (black dotted line) and functional (blue line

Fig. 5

 5 Fig. 5 Effect of gender on STS asymmetry and function. Anatomical (black dotted line) and functional averaged profiles (blue line for upper profile and green line for lower profile (±SEM) were represented together with the STAP region (orange overlay) for females (top panel) and males (lower panel). The blue stars line

Acknowledgements We gratefully acknowledge the support of Grant AJE201214 by the Fondation pour la Recherche Me ´dicale (France) to P.B, and of Labex BLRI (Aix-Marseille Universtite ´, France) to CB. Research supported by Grants ANR-16-CONV-0002 (ILCB), ANR-11-LABX-0036 (BLRI) and ANR-11-IDEX-0001-02 (A*MIDEX).


Julien Baste
email: baste@lirmm.fr

Dieter Rautenbach
email: dieter.rautenbach@uni-ulm.de

Degenerate Matchings and Edge Colorings

Keywords: Matching, edge coloring, induced matching, acyclic matching, uniquely restricted matching

A matching M in a graph G is r-degenerate if the subgraph of G induced by the set of vertices incident with an edge in M is r-degenerate. Goddard, Hedetniemi, Hedetniemi, and Laskar (Generalized subgraph-restricted matchings in graphs, Discrete Mathematics 293 (2005) 129-138) introduced the notion of acyclic matchings, which coincide with 1-degenerate matchings. Solving a problem they posed, we describe an efficient algorithm to determining [determine] the maximum size of an r-degenerate matching of a given chordal graph. Furthermore, we study the r-chromatic index of a graph defined as the minimum number of r-degnerate matchings into which its edge set can be partitioned, obtaining upper bounds and discussing extremal graphs.

Introduction

Matchings in graphs are a central topic of graph theory and combinatorial optimization [START_REF] Lovász | Matching Theory[END_REF]. While classical matchings are tractable, several well known types of more restricted matchings, such as induced matchings [START_REF] Cameron | Induced matchings[END_REF][START_REF] Stockmeyer | NP-completeness of some generalizations of the maximum matching problem[END_REF] or uniquely restricted matchings [START_REF] Golumbic | Uniquely restricted matchings[END_REF], lead to hard problems. Goddard, Hedetniemi, Hedetniemi, and Laskar [START_REF] Goddard | Generalized subgraph-restricted matchings in graphs[END_REF] proposed to study so-called subgraph-restricted matchings in general. In particular, they introduce the notion of acyclic matchings. By a simple yet elegant argument (cf. Theorem 4 in [START_REF] Goddard | Generalized subgraph-restricted matchings in graphs[END_REF]) they show that finding a maximum acyclic matching in a given graph is hard in general, and they explicitly pose the problem to describe a fast algorithm for the acyclic matching number in interval graphs. In the present paper, we solve this problem for the more general chordal graphs. Furthermore, we study the edge coloring notion corresponding to acyclic matchings.

Before we give exact definitions and discuss our results as well as related research, we introduce some terminology. We consider finite, simple, and undirected graphs, and use standard notation.

A matching in a graph G is a subset M of the edge set E(G) of G such that no two edges in M are adjacent. Let V (M ) be the set of vertices incident with an edge in M . M is induced [START_REF] Cameron | Induced matchings[END_REF] if the subgraph G[V (M )] of G induced by the set V (M ) is 1-regular, that is, M is the edge set of G[V (M )].

Induced matching are also known as strong matchings. M is uniquely restricted [START_REF] Golumbic | Uniquely restricted matchings[END_REF] if there is no other matching M in G distinct from M that satisfies V (M ) = V (M ). It is easy to see that M is uniquely restricted if and only if there is no M -alternating cycle in G, which is a cycle in G every second edge of which belongs to M [START_REF] Golumbic | Uniquely restricted matchings[END_REF]. Finally, M is acyclic [START_REF] Goddard | Generalized subgraph-restricted matchings in graphs[END_REF]

if G[V (M )] is a forest. Let ν(G), ν s (G), ν ur (G),
and ν 1 (G) be the maximum sizes of a matching, an induced matching, a uniquely restricted matching, and an acyclic matching in G, respectively. Since every induced matching is acyclic, and every acyclic matching is uniquely restricted, we have

ν s (G) ≤ ν 1 (G) ≤ ν ur (G) ≤ ν(G).
We chose the notation "ν 1 (G)" rather than something like "ν ac (G)", because we consider some further natural generalization.

For a non-negative integer r, a graph G is r-degenerate if every subgraph of G of order at least one has a vertex of degree at most r. Note that a graph is a forest if and only if it is 1-degenerate.

An r-degenerate order of a graph G is a linear order u 1 , . . . , u n of its vertices such that, for every i in

[n], the vertex v i has degree at most r in G[{v i , . . . , v n }], where [n] is the set of the positive integers at most n. Clearly, a graph is r-degenerate if and only if it has an r-degenerate order. Now, let a matching M in a graph G be r-degenerate if the induced subgraph G[V (M )] is rdegenerate, and let ν r (G) denote the maximum size of an r-degenerate matching in G.

For every type of matching, there is a corresponding edge coloring notion. An edge coloring of a graph G is a partition of its edge set into matchings. An edge coloring is induced (strong), uniquely restricted, and r-degenerate if each matching in the partition has this property, respectively. Let χ (G), χ s (G), χ ur (G), and χ r (G) be the minimum numbers of colors needed for the corresponding colorings, respectively. Clearly,

χ s (G) ≥ χ 1 (G) ≥ χ ur (G) ≥ χ (G).
In view of the hardness of the restricted matching notions, lower bounds on the matching numbers [START_REF] Henning | Induced matchings in subcubic graphs without short cycles[END_REF][START_REF] Horák | Induced Matchings in Cubic Graphs[END_REF][START_REF] Joos | Induced Matchings in Subcubic Graphs[END_REF][START_REF] Kang | Induced matchings in subcubic planar graphs[END_REF], upper bounds on the chromatic indices [START_REF] Andersen | The strong chromatic index of a cubic graph is at most 10[END_REF][START_REF] Baste | Uniquely restricted matchings and edge colorings[END_REF], efficient algorithms for restricted graph classes [9-11, 13, 25], and approximation algorithms have been studied [START_REF] Baste | Uniquely restricted matchings and edge colorings[END_REF][START_REF] Rautenbach | Two greedy consequences for maximum induced matchings[END_REF]. There is only few research concerning acyclic matchings; Panda and Pradhan [START_REF] Panda | Acyclic matchings in subclasses of bipartite graphs[END_REF] describe efficient algorithms for chain graphs and bipartite permutation graphs.

Vizing's [START_REF] Vizing | On an estimate of the chromatic class of a p-graph[END_REF] famous theorem says that the chromatic index χ (G) of G is either ∆(G) or ∆(G) + 1, where ∆(G) is the maximum degree of G. Induced edge colorings have attracted much attention because of the conjecture χ s (G) ≤ 5 4 ∆(G) 2 posed by Erdős and Nešetřil (cf. [START_REF] Faudree | The strong chromatic index of graphs[END_REF]). Building on earlier work of Molloy and Reed [START_REF] Molloy | A bound on the strong chromatic index of a graph[END_REF], Bruhn and Joos [START_REF] Bruhn | A stronger bound for the strong chromatic index[END_REF] showed χ s (G) ≤ 1.93∆(G) 2 provided that ∆(G) is sufficiently large. In [START_REF] Baste | Uniquely restricted matchings and edge colorings[END_REF] we showed χ ur (G) ≤ ∆(G) 2 with equality if and only if G is the complete bipartite graph K ∆(G),∆(G) .

Our results are upper bounds on χ r (G) with the discussion of extremal graphs, and an efficient algorithm for ν r (G) in chordal graphs, solving the problem posed in [START_REF] Goddard | Generalized subgraph-restricted matchings in graphs[END_REF].

Bounds on the r-degenerate chromatic index

Since, for every two positive integers r and ∆, every r-degenerate matching of the complete bipartite graph K ∆,∆ of order 2∆ has size at most r, we obtain χ r (K ∆,∆ ) ≥ ∆ 2 r .

Our first result gives an upper bound in terms of r and ∆.

{theorem1

Theorem 1 If r is a positive integer and G is a graph of maximum degree at most ∆, then , and E u will be introduced and discussed in the proof of Theorem 2. Note that the set N u ∪N v ∪N u,v is not required to be independent, that is, the sets E w and E w may intersect for distinct vertices w and w in

χ r (G) ≤ 2(∆ -1) 2 r + 1 + 2(∆ -1) + 1. ( 1
N u ∪ N v ∪ N u,v . {fig1}
Let F 1 be the colors that appear on edges in

E u ∪ E v ∪ E u,v .
Clearly, every color in F 1 is forbidden for uv, because each color class must be a matching. Let F 2 be the colors α in [K] that do not belong to F 1 such that

d α u + 2d α u,v + d α v ≥ r + 1,
where d α u is the number of vertices in N u incident with an edge colored α, d α v is the number of vertices in N v incident with an edge colored α, and d α u,v is the number of vertices in N u,v incident with an edge colored α. Note that, since F 1 and F 2 are disjoint, none of the edges contributing to

d α u + 2d α u,v + d α v is incident with u or v.
If there is some α in [K] \ (F 1 ∪ F 2 ), then neither u nor v is incident with an edge of color α, and

d α u +2d α u,v +d α v ≤ r. This implies min{d α v +d α u,v , d α v +d α u,v } ≤ r/2 ≤ r-1 and max{d α v +d α u,v , d α v +d α u,v } ≤ r.
Hence, coloring uv with color α, the edges of G colored α form an r-degenerate matching. As explained above this would complete the proof. Therefore, we may assume that

F 1 ∪ F 2 = [K].
Note that Furthermore,

|F 1 | ≤ |E u ∪ E v ∪ E u,v | (2) {e3a} = (d G (u) -1) + (d G (v) -1) ≤ 2(∆ -1) (3)
(r + 1)|F 2 | ≤ α∈F 2 d α u + 2d α u,v + d α v (4) {e2a} ≤ w∈Nu |E w | + 2 w∈Nu,v |E w | + w∈Nv |E w | (5) {e2b} ≤ (∆ -1)n u + 2(∆ -2)n u,v + (∆ -1)n v (6) {e2c} ≤ (∆ -1)(d G (u) -1) + (∆ -1)(d G (v) -1) (7) {e2d} ≤ 2(∆ -1) 2 . ( 8
) {e2e}
Note that (r + 1)|F 2 | = 2(∆ -1) 2 if and only if equality holds in ( 4) to [START_REF] Cameron | Induced matchings[END_REF], which implies that Altogether, we obtain

(c) d α u + 2d α u,v + d α v = r + 1 for every color α in F 2 (equality in (4)), (d)
|K| = |F 1 ∪ F 2 | = |F 1 | + |F 2 | ≤ 2(∆ -1) + 2(∆ -1) 2 α + 1 ,
contradicting the choice of K. This completes the proof. 2

For r = 1, the bound from Theorem 1 simplifies to ∆ 2 . In view of K ∆,∆ , Theorem 1 is tight in this case, and, as we show next, K ∆,∆ is the only extremal graph.

{theorem2 Theorem 2 If G is a graph of maximum degree at most ∆, then χ 1 (G) = ∆ 2 if and only if G is K ∆,∆ . Proof: By Theorem 1, we have χ 1 (G) ≤ ∆ 2 . Since χ 1 (K ∆,∆ ) = ∆ 2 , it suffices to show that χ 1 (G) = ∆ 2 implies that G is K ∆,∆ . Therefore, we consider a 1-degenerate edge coloring of G using colors in [∆ 2 ]
such that the number of edges colored ∆ 2 is as small as possible. Let uv be an edge colored ∆ 2 .

We use the notation and observations from the proof of Theorem 1. Recall that

|F 1 | ≤ 2(∆ -1) and 2|F 2 | ≤ 2(∆ -1) 2 , which implies |F 1 ∪ F 2 | ≤ ∆ 2 -1. Furthermore, recall that uv can be colored with any color in [∆ 2 -1] \ (F 1 ∪ F 2 )
. By the choice of the coloring, these observations imply that

F 1 ∪ F 2 = [∆ 2 -1], |F 1 | = 2(∆ -1), and 2|F 2 | = 2(∆ -1) 2 .
The latter two equalities imply that the properties (a) to (g) hold. In particular, by (c), for every color α in F 2 , we have

d α u + d α v = 2, that is,
exactly two vertices in N u ∪ N v are incident with an edge colored α.

Let u ∈ N u and let α be the color of the edge uu . We introduce some more notation already illustrated in Figure 1

k β = |E v | + w∈Nv |E w | = ∆(∆ -1).
First, let β in F 1 be such that k β = 1. Since F 1 and F 2 are disjoint by definition, (d) implies that no edge in E u has color β. If k β = 0, that is, no edge in E 2 u has color β, then changing the color of uv to α and the color of uu to β yields a 1-degenerate edge coloring with less edges colored ∆ 2 , which is a contradiction. Hence, k β ≥ 1.

Next, let β be a color in

F 2 with k β = 1. If k β = 0, that is, no edge in E u ∪ E 2 u has color β, then
changing the color of uv to α and the color of uu to β yields a 1-degenerate edge coloring with less edges colored ∆ 2 , which is a contradiction. Hence, k β ≥ 1.

Finally, let β be a color in

F 2 with k β = 2. By (c), no edge in E u has color β. If k β ≤ 1, that
is, there is at most one vertex in N u that is incident with an edge colored β, then changing the color of uv to α and the color of uu to β yields a 1-degenerate edge coloring with less edges colored ∆ 2 , which is a contradiction. Hence, k β ≥ 2.

Altogether, it follows that k β ≥ k β for every β ∈ [∆ 2 -1], and, we obtain

∆(∆ -1) = β∈[∆ 2 -1] k β ≤ β∈[∆ 2 -1] k β ≤ |E u | + w∈N u |E w | ≤ (∆ -1) + w∈N u (∆ -1) ≤ ∆(∆ -1).
Equality throughout this inequality sequence implies that

k β = k β for every β ∈ [∆ 2 -1], all edges from E u ∪ E 2 u have a color from [∆ 2 -1]
, and all vertices in N u have degree ∆. Now, let v ∈ N v . Note that symmetric observations apply to the vertex v as to the vertex u . Let the edge vv have color β. There is exactly one vertex u in N u such that some edge in E u , say u u , has color β. Defining N v and E w for w ∈ N v similarly as above, it follows, by symmetry between u and v , that there is exactly one vertex v in N v such that some edge in E v , say v v , has color α.

If the edge u u is distinct from the edge vv , then changing the color of uv to β and the color of vv to α yields a 1-degenerate edge coloring with less edges colored ∆ 2 , which is a contradiction.

Hence, the edge u u equals vv . Since v is incident with an edge colored ∆ 2 but u is not, we obtain that u equal v , that is, u and v are adjacent.

Since u and v were arbitrary vertices in N u and N v , respectively, it follows, by symmetry, that every

vertex in N u is adjacent to every vertex in N v , that is, G is K ∆,∆ . 2
We believe that, for large values of r, the bound from Theorem 1 is far from being tight. Our next result vaguely supports this.

{proposit Proposition 3 If r is an integer at least 2, then no graph G of maximum degree at most ∆ satisfies

χ r (G) = 2(∆-1) 2 r+1 + 2(∆ -1) + 1.
Proof: For contradiction, suppose that G is a graph of maximum degree ∆ that satisfies χ r (G) = K, where K = 2(∆-1) 2 r+1 + 2(∆ -1) + 1. Similarly as in the proof of Theorem 2, we consider an r-degenerate edge coloring of G using colors in [K] such that the number of edges colored K is as small as possible.

Let uv be an edge colored K. Again using the same notation as in the proof of Theorem 1 and arguing as in the proof of Theorem 2, we obtain that

F 1 ∪F 2 = [K -1], |F 1 | = 2(∆-1), (r +1)|F 2 | = 2(∆-1) 2 ,
and that the properties (a) to (g) hold.

Suppose that there is some color α in F 2 such that d α u and d α v are both positive. In this case, (c) and r ≥ 2 imply that min{d α u , d α v } ≤ r -1 and max{d α u , d α v } ≤ r, and changing the color of uv to α yields an r-degenerate edge coloring with less edges colored K, which is a contradiction. Hence, for every color α in F 2 , we obtain, again using (c), that (d α u , d α v ) ∈ {(0, r + 1), (r + 1, 0)}.

Let u ∈ N u and let uu have color α. Arguing as in the Theorem 2 and using the same notation as there, it follows that every color β in F 1 that appears on some edge in E v appears on at least one edge in E 2 u .

Now, let β be a color in F 2 such that some vertex in N v is incident with an edge colored β. Since

Efficient algorithm for chordal graphs

Let G be a chordal graph. It is well known that G has a tree decomposition (T, (X t ) t∈V (T ) ) such that each bag X t is a clique in G. By applying standard manipulations [START_REF] Bodlaender | Combinatorial Optimization on Graphs of Bounded Treewidth[END_REF], we may furthermore assume that • T is a rooted binary tree,

• if t is the root or a leaf of T , then X t = ∅,

• if some node t of T has two children t and t , then X t = X t = X t (t is a "join node"),

• if some node t of T has only one child t , then either |X t \ X t | = 1 and |X t \ X t | = 0 (t is an "introduce node") or |X t \ X t | = 0 and |X t \ X t | = 1 (t is a "forget node"), and

• given G, the decomposition (T, (X t ) t∈V (T ) ) can be constructed in polynomial time, in particular, n(T ) is polynomially bounded in terms of n(G).

Theorem 5 For a fixed positive integer r, and a given chordal graph G, the maximum size of an r-degenerate matching can be determined in polynomial time.

Proof: By Lemma 4, it follows that the tree decomposition (T, (X t ) t∈V (G) ) as well as the sets R t can all be determined in polynomial time processing T in a bottom-up fashion. Furthermore, (9) allows to extract ν r (G) from the set R t of the root t of T . 2

It is easy to extend the above dynamic programming approach in such a way that it also determines a maximum r-degenerate matching of the given graph. Furthermore, given weights on the edges, also a maximum weight r-degenerate matching can be determined efficiently by replacing the cardinality k within the triples (S, N, k) by the weights of suitable matchings. In order to maintain the important property that the sets R t only contains polynomially many elements, one can prune R t maintaining only those triples (S, N, k) that maximize the weight k for given choices of S and N .

Since no efficient algorithm to determine a maximum r-degenerate induced subgraph, and, in particular for r = 1, a maximum induced forest of a given chordal graph seems to have been published

(cf. comments in [START_REF] Kratsch | Feedback vertex set on AT-free graphs[END_REF]), we want to point out that modifying the above approach easily allows to obtain such an algorithm.

Conclusion

The problem to determine the acyclic matching number of a given graph G is equivalent to the problem to determine an induced forest T of G whose matching number ν(T ) is largest possible. This observation shows that ν 1 (G) is somewhat related to the problem to determine a largest induced forest of a given graph. The latter problem is dual to the feedback vertex set problem, and has received a lot of attention [START_REF] Alon | Large induced degenerate subgraphs[END_REF][START_REF] Alon | Large induced forests in sparse graphs[END_REF][START_REF] Bau | The decycling number of graphs[END_REF][START_REF] Gentner | Feedback vertex sets in cubic multigraphs[END_REF]. In particular, the classes of graphs for which a largest induced forest can be found efficiently [START_REF] Kratsch | Feedback vertex set on AT-free graphs[END_REF][START_REF] Liang | On the feedback vertex set problem in permutation graphs[END_REF][START_REF] Liang | Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs[END_REF][START_REF] Lu | A linear-time algorithm for the weighted feedback vertex problem on interval graphs[END_REF] are good candidates for classes of graphs for which the acyclic matching number might be tractable. Note that, for a connected graph G of maximum degree at most ∆, the value of ν ∆-1 (G) can be determine efficiently. In fact, if G has no perfect matching, then it equals ν(G), otherwise, it equals ν(G) -1.

Further upper bounds on the r-degenerate chromatic index and lower bounds on the r-degenerate matching number for general as well as for restricted graphs seem to deserve additional research.

Figure 1 :

 1 Figure1: Vertices and edges close to uv. The indicated objects N u , u , and E u will be introduced and discussed in the proof of Theorem 2. Note that the set N u ∪N v ∪N u,v is not required to be independent, that is, the sets E w and E w may intersect for distinct vertices w and w in N u ∪ N v ∪ N u,v .

  {e3b} with equality if and only if (a) all edges in E u ∪ E v ∪ E u,v are colored differently (equality in (2)), and (b) u and v have degree ∆ (equality in (3)).

  all edges in w∈Nu∪Nu,v∪Nw E w have a color from F 2 (equality in (5)), (e) all vertices in N u ∪ N u,v ∪ N v have degree ∆ (equality in (6)), (f) n u,v = 0, that is, u and v have no common neighbor (equality in[START_REF] Bruhn | A stronger bound for the strong chromatic index[END_REF]), and (g) u and v have degree ∆ (equality in[START_REF] Cameron | Induced matchings[END_REF]).

  d β v > 0 implies d β u = 0 and d β v = r + 1, there are exactly r + 1 such vertices. If at most r vertices in N u are incident with an edge colored β, then changing the color of uv to α and the color of uu to β yields an r-degenerate edge coloring with less edges colored K, which is a contradiction. Hence, for every such color β, at least r + 1 vertices in N u are incident with an edge colored β, and, since d β u = 0, all these edges belong to E 2 u . Altogether, we obtain the contradiction (∆ -1) 2 ≥ w∈N u |E w | ≥ |E v | + w∈Nv |E w | = ∆(∆ -1), which completes the proof. 2

  . Let N u = N G (u ) \ {u}. For every vertex w in N u , let E w be the set of edges incident with w but not incident with u . Let E 2 u = For every color β in [∆ 2 -1], let k β be the number of vertices w in {v} ∪ N v such that E w contains an edge colored β, and, similarly, let k β be the number of vertices w in {u } ∪ N u such that E w contains an edge colored β. Since the color classes are matchings, for every such color β, each of the sets E w [, w ∈ {u , v} ∪ N v ∪ N u ,] contains at most one edge colored β. By (a), (c), and (d), all edges in E v have a different color from F 1 , all edges in w∈Nv E w have colors from F 2 , k β ∈ {0, 1} for every color β in F 1 , and k β ∈ {0, 1, 2} for every color β in F 2 . By (b), (e), and (f), we have

	E w .
	w∈N u
	β∈[∆ 2 -1]

For every note t of T , let T t denote the subtree of T rooted in t that contains t and all its descendants. Let G t be the subgraph of G induced by s∈V (Tt) X s .

We design a dynamic programming procedure calculating ν r (G) for a fixed positive integer r.

Therefore, for every node t of T , let R t be the set of all triples (S, N, k) such that (i) N ⊆ S ⊆ X t and (ii) there is a matching M ⊆ E(G t ) \ Xt 2 such that Note that the matching M satisfying (a), (b), and (c) may not be uniquely determined by (S, N, k).

We call every such matching suitable for (S, N, k), and denote one (arbitrary yet specific) suitable matching by M t (S, N, k). Intuitively, the vertices in S correspond to those vertices of X t that can be incident with edges e of some r-degenerate matching of the entire graph G containing a suitable matching such that either e has both endpoints in X t or e has one endpoint in X t and the other

The following lemma contains the relevant recursions.

{lemma1} Lemma 4 Let G, (T, (X t ) t∈V (T ) ), and (R t ) t∈V (T ) be as above.

(a) If t is a leaf of T , then R t = {(∅, ∅, 0)}.

(b) If t is an introduce node, t is the child of t, and {x} = X t \ X t , then (S, N, k) ∈ R t if and only if

(d) If t is a join node, and t and t are the children of t, then (S,

Proof: (a) This follows immediately from the definition of R t .

(b) Note that

the definition of R t easily implies that (S, N, k) ∈ R t . Note, in particular, that in the second case, the vertex x has degree |S | ≤ r in the subgraph of G induced by V (M t (S , N, k)) ∪ S ∪ {x}, which ensures the degeneracy conditions.

Conversely, let (S, N, k) ∈ R t . If x ∈ S, then, by the definition of R t , we obtain (S, N, k) ∈ R t . If

x ∈ S, then, since X t is a clique, the set S = S \ {x} has order at most r, and, since all neighbors of

x belong to X t , the vertex x does not belong to N , which implies that (S , N, k) ∈ R t .

(c) Note that G t = G t , and that N G (x) ⊆ V (G t ).

If either (S, N, k) ∈ R t and x ∈ S, or (S, N, k) = (S \{x}, N ∪{y}, k +1) for some (S , N , k ) ∈ R t with x ∈ S \ N and some y ∈ S \ (N ∪ {x}), or (S, N, k) = (S \ {x}, N \ {x}, k ) for some (S , N , k ) ∈ R t with x ∈ N , then the definition of R t easily implies that (S, N, k) ∈ R t . In the first case, this is immediate. In the second case, since x ∈ S \ N has no neighbor in G outside of V (G t ), any suitable matching contains no edge incident with x but x corresponds to a vertex that can eventually be matched to some vertex y in S \ (N ∪ {x}). Since x is adjacent to all vertices in S \ (N ∪ {x}), we add to R t all triples corresponding to the possible choices of y, and increase k by 1 because of the edge xy that lies between V (G t ) \ X t and X t . Similarly, in the third case, the vertex

x is incident with an edge in M t (S , N , k ) whose other endpoint lies in V (G t ) \ X t , and removing x from X t , it has to be removed from S and N as well while the size k of the matching M t (S , N , k )

does not change.

Conversely, let (S,

where M = M t (S, N , k ) and M = M t (S, N , k ). Since N and N are disjoint, M is a matching with

Let u 1 , . . . , u n be a linear order of the vertices in V (M ) ∪ S such that u 1 , . . . , u n -|S| contains the n -|S| vertices in V (M ) \ N in an order of non-increasing depth of the corresponding forget nodes.

More precisely, if 1 ≤ i < j ≤ n -|S|, t i is the forget node of u i , meaning that u i belongs X t i where t i is the child of t i but u i no longer belongs to X t i , and t j is the forget node of u j , then the depth of t i within T is at least the depth of t j . Note that, since (T, (X t ) t∈V (T ) ) is a tree decomposition, the forget nodes of the vertices of G are uniquely determined. Now, if 1 ≤ i ≤ n -|S|, then the neighborhood of u i in the graph G[{u i , . . . , u n }] is completely contained in X t i , because, for all vertices u j of G t i distinct from u i that belong to V (M )∪S, the forget node of u j has strictly larger depth than the forget node t i of u i . Since X t i is a clique, and


M Merai

D Flick

Loïs Guiller

Steven Duret

Onrawee Laguerre

Characterization of airflow inside a refrigerated trailer loaded with carcasses

Keywords: refrigeration, transport, carcasses, airflow, Laser Droplet Velocimetry

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Meat carcasses are highly perishable products and their quality depend on the refrigeration conditions used after slaughtering. To prevent quality loss, since 1964 in the European Union, carcasses should be chilled immediately after the post mortem inspection and kept at a constant core temperature of ≤ 7°C until the consumption. The first step of cooling in slaughterhouses intend to cool down, as fast as possible, the core temperature of carcasses to prevent the growth of undesirable microorganisms [START_REF] Kinsella | Impact of a novel spray-chilling system on surface microflora, water activity and weight loss during beef carcass chilling[END_REF]Savell et al., 2005).

Different techniques, e.G. using air or spray chilling are used depending on slaughterhouses.

The big challenge in this step is to lower the core temperature without affecting the color, tenderness and water loss of carcasses [START_REF] Létang | Brumisation des viandes: économie d'énergie en réfrigération des carcasses de gros animaux grâce à la brumisation[END_REF][START_REF] Kuitche | Modélisation des cinétiques de température et de perte de poids des viandes pendant la réfrigération en régime variable[END_REF]Savell et al., 2005). As a consequence of this regulation, carcasses can be stored in chilled rooms of slaughterhouses for 24-48 hours.

In France, most of the slaughterhouses are located in west part while cutting facilities are dispatched all over the country. In order to provide their clients with acceptable product quality in an acceptable delay, slaughterhouses are asking today for an easing of this regulation. Such a request has been accepted in other European countries such Italy where a national regulation has been created (DL n.286, 1994) that allows carcasses transportation before reaching the European regulatory temperature (≤ 7°C). In this case, only short and fast transportation are allowed [START_REF] Kuffi | CFD Modeling of Air Cooling of Multiple Beef Carcasses Using 3D Geometrical Models[END_REF]. The Belgium Scientific Committee of the FASFC used predictive microbiology to conclude that the transport of pork carcasses at temperature above the regulatory temperature was acceptable for 2 hours of transportation if carcasses are loaded in the trailer with 9°C of surface temperature, or for 3 hours with temperatures of 7°C at the surface and 12°C in the core [START_REF] Anonymous | Regulation (EC) No 853/2004 of the European Parliament and of the Council of 29 April 2004 laying down specific hygiene rules for food of animal origin[END_REF][START_REF] Anonymous | A quantitative microbiological risk assessment on Salmonella in meat 1: source attribution for human salmonellosis from meat[END_REF].

Those studies have been conducted using either empirical studies or predictive microbial tests without taking into account the variability of the transportation step. The refrigerated equipment characteristics, the loading technique, the number of carcasses and the transportation duration have an important impact on the core and surface temperature evolution and as a consequence they can affect the microbial growth. In 1996, James

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc

The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf explained that the main problem in the cold transportation is the non-uniform air distribution inside the trailer. It is thus necessary to understand the airflow pattern in the refrigerated trailer.

The objective of this experimental work is to study the airflow inside a trailer loaded with carcasses at reduced scale. This airflow is characterized by air velocity and flow direction using 2D Laser Doppler Velocimetry (LDV). The identification of zones of low air velocity leads to the information of a potential microbial risk.

MATERIALS AND METHODS

Experimental device

With respect to European regulation, the refrigerated trailers have the internal length of 13.3 m. In order to conduct our experiments on a reasonable dimension, a reduced scale trailer

(1:3.3) has been used. In such a way, Reynolds number is the same order of magnitude of the real trailer (Re~1.3 10 5 ). The side wall of this trailer was made of glass to allow the measurements of internal air velocity using two-dimensional LDV (Dantec Dynamics, Flow explorer-2D). The other walls are made of wood.

As a result of the scale reduction, studied pork carcasses have been also prepared at the same scale (1:3.3). Based on X-ray scans provided by the French Pork Institute (IFIP), 3D carcass geometry has been built up and the scale has been reduced before printing both half-carcasses on polyamide structure (Figure1). The obtained structure has been used to create a silicone mold and to prepare 430 half-carcasses using expanded polyethylene. Once all the reduced scale carcasses are prepared, they have been placed into the reduced scale trailer respecting a real loading configuration observed in real pork carcasses refrigerated transport. This configuration has been approved by both French Pork Institute (IFIP) and French meat companies (Culture Viande) the partners in this work (Figure 2).

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf

Velocity measurements

The measurement of the internal air velocity by LDV requires the use of a wall of glass to allow the penetration of the laser beam inside the trailer. The laser equipment was placed on a displacement system that allowed the automation and the control of the movement according to the three axes x, y and z (Figure 3).

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf The air velocity was determined by measuring the displacement velocity of small particles of paraffin oil (diameter ≈ 1 µm) injected into the blowing device. Those particles are small enough to follow the air displacement inside the trailer. Thus the velocity of this tracer is considered as the air velocity.

The 2D LDV used in this study was able to show the airflow direction (sign +/-) as well as the velocity magnitude. This magnitude is represented by the mean velocity and its fluctuations of two velocity components in the trailer: U (longitudinal in x direction) and V (vertical in y direction). The device is composed of two 70 mW laser diodes emitting two laser beams at 660 nm and 785 nm, a beam-splitter, a Brag cell, a focusing and receiving lens, a pinhole arrangement to collect scattered light and a photomultiplier. Measurements were conducted in a coincidence mode where both laser beams converge in the same measure volume (0.17 mm x 2.8 mm).

In this study, measurements of velocity have been carried out in the symmetry plan (35 cm from the glass wall) of the trailer filled with carcasses over and under the carcasses as shown in Figure 4-A.

A total of 2270 measurement points were undertaken to describe the airflow pattern in the symmetry plan using Matlab (R2015a, Mathworks Inc.). Results were presented using the velocity magnitude defined by the eq. 1.

‖𝑉 ⃗ ‖ = √𝑉 2 + 𝑈 2 (eq. 1)

In addition to this profile, measurements in 8 vertical plans at x= 50, 150, 250 and 350 cm of the reduced scale trailer length over and under carcasses (from z=0 cm to z= 35cm) and just after the inlet section were conducted (Figure 4-B). Measurements were also carried out in

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf two horizontal plans at mid-height at the front (from x=3 cm to x=10 cm and from z=0 cm to z=35 cm) and at the rear of the trailer where no carcasses are present (from x=486 cm to x=497 cm and from z=0 cm to z=35 cm). The airflow rate was then calculated in those plans using the integral of the U velocity component (eq. 2).

𝑄

RESULTS AND DISCUSION

Symmetry plan characterization

In order to illustrate the overall airflow pattern in the symmetry plan, Figure 5 shows the magnitude of the velocity obtained by LDV measurements around carcasses. The zones over and under the carcasses were expanded vertically in this graph to better highlight the velocity in those zones. Over carcasses we can see the evolution of the jet behavior.

The trailer can be divided into two main zones: the first zone from x=0 m to x=10 m and the second one from x=10 m to the end of the trailer. In the front zone (x≤10 m), the air velocity decreases from 11 m.s -1 to 2 m.s -1 . For the rear zone (x>10 m), the air velocity decreases also from the upper part of the trailer (y=0 m) to the upper part of carcasses (y=3.6 m) where the velocity ranges from 0 m.s -1 to 1.5 m.s -1 corresponding to a weak airflow. Similar results have been obtained with the airflow characterization in the empty trailer [START_REF] Moureh | Airflow characteristics within a slot-ventilated enclosure[END_REF].

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf Under carcasses, the airflow rate is different from the over carcasses ones. Those differences can be attributed to a circulation of the air between the carcasses volume.

Simplified model reconstruction

Based on the results obtained in (3.1.) and (3.2.), a simplified schematic airflow can be developed in the filled trailer (Figure 7). It takes into account the hypothesis of vertical and horizontal air circulations between carcasses. These circulations were estimated by the difference between upper and lower airflow rates. Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf

CONCLUSION

Using a reduced scale trailer filled with pork carcasses (1:3.3) with respect of the Reynolds number and a 2D LDV technique, internal flow was successfully characterized in the trailer.

A zone of low airflow has been identified in the rear part of the trailer above carcasses starting from approximately 10 m. Under carcasses the zone of low airflow is larger and extends from the back of the trailer to 6 m of length. In these zones where air is nearly stagnant, the product temperatures maintain is problematic and a potential risk of microorganism's development can be expected. The results of this study on one hand, the understanding of the airflow distribution around carcasses and on the other hand, the development of the simplified model.

Figure 1 .

 1 Figure 1. Steps to produce reduced scale carcasses from the X-ray scan of real pork carcass.

Figure 2 .

 2 Figure 2. Photography of the selected loading configuration (A and B) and the reduced scale trailer filled with pork carcasses (C).

Figure 3 .

 3 Figure 3. Measurement of air velocity above the carcass in a reduced scale trailer by Laser Doppler Velocimetry (LDV).

Figure 4 .

 4 Figure 4. Graphical representation of measurement positions (A) in the symmetry plan of the trailer and (B) in vertical plans over and under carcasses.

Figure 6 .

 6 Figure 6. Dimensionless airflow rate estimation in the filled trailer (blue) over and (red) under carcasses (results obtained from LDV measurements and translated to the real scale of the trailer using Reynolds analogy).

Figure 7 .

 7 Figure 7. Simplified schematic airflow rate distribution in the symmetry plan of the filled trailer. Values represent 𝑄 𝑄 0 ⁄ . Bold values with grey arrow correspond to the calculated airflow; italic values with red arrow correspond to the difference between upper and down airflow; blue arrow correspond to the airflow lateral circulation.

ACKNOWLEDGMENT

This research work was carried out in the framework of DIM ASTREA grant by the Regional Council of Ile-de-France (France). The authors thank "IFIP" and "Culture viande" represented by Brice Minvielle and Franck Bellaca for their contribution in this work.

where the air velocities are around 2-3 m.s -1 and the second zone (6 m < x ≤ 13.3 m) where the air is nearly stagnant.

Airflow rate estimation

Velocity measurements in the vertical plans upper and under carcasses were used to estimate the airflow rate in those regions (Figure 6). Results are shown as dimensionless airflow rate obtained by the expression of the measured airflow divided by the inlet flow (𝑄 𝑄 0 ⁄ ).

In the two first measurements plans (x=1.66 m and x=4.99 m), the airflow rate is higher than the inlet one. This phenomenon is related to the strong blowing airflow in the inlet of the trailer. At 5 m < x< 8 m, the airflow rate decreases rapidly. Those results are similar to those obtained in the empty trailer [START_REF] Zertal-Ménia | Etude numérique et expérimentale de l'aéraulique dans un véhicule frigorifique[END_REF].

Author-produced version of the paper presented at the 3rd International conference on Food and Biosystems Engineering on 1-04 June 2017, Rhodes island, Grc The original publication is available at http://www.fabe.gr/images/fabe_2017/proceedings_3face2017.pdf