R El Jurdi

Caroline Petitjean

Paul Honeine

Fahed Abdallah

Rosana El Jurdi

CoordConv-Unet: Investigating CoordConv for Organ Segmentation

Keywords: Medical Image Segmentation, Fully Convolutional Networks, Prior-based Losses, CoordConv, MRI, CT

Introduction

Medical image segmentation is the process of classifying each pixel within an image into an instance corresponding to an anatomical object of interest. Generally, medical images are largely versatile in nature, depending on the acquisition process and the type of object to be segmented. They can be acquired with magnetic resonance imaging (MRI), computed tomography (CT), nuclear medicine functional imaging, ultrasound imaging, fundus photography, to name a few. Hence, they vary in characteristics and nature and are broad with regards to the anatomical object of interest. As such, guaranteeing high performance within medical image segmentation can be considered very challenging when compared to other types of images or segmentation tasks. Regardless, segmentation within the medical domain is considered a key step in performing non-invasive diagnostic procedures, assisting early disease detection or surgical planning.

Early attempts to automate or semi-automate the process of medical image segmentation started with optimization-based approaches. Such methods generally involve optimization of an energy functional, where the image can be considered continuous [START_REF] Xu | Image segmentation using deformable models[END_REF] or discrete (eg. a graph [START_REF] Shi | Normalized cuts and image segmentation[END_REF][START_REF] Boykov | Graph cuts and efficient nd image segmentation[END_REF]). In order to counter-react noise and low contrast, works have aimed to perform optimization processes in such a way that the automated predictions conform with particular rules relative to the anatomical object characteristics [START_REF] Nosrati | Incorporating prior knowledge in medical image segmentation: a survey[END_REF]. This information may include the object appearance, size, smoothness or compactness [START_REF] Vicente | Graph cut based image segmentation with connectivity priors[END_REF][START_REF] Ayed | Area prior constrained level set evolution for medical image segmentation[END_REF][START_REF] Foulonneau | Multi-reference shape priors for active contours[END_REF]. These rules and characteristics are known as prior knowledge of the datasets. Anatomical priors refer to medical knowledge and domain expertise that capture spatial as well as topological guidelines and character-istics with respect to the understudied anatomical objects.

In the recent era, deep learning has registered a pivotal milestone in many fields including pattern recognition, object detection, natural language processing, with medical image segmentation being no exception to the rule. Convolutional neural networks (CNNs), a class of deep learning models, have been known to register considerable results due to their generalization ability and powerful predictive notions. Due to the fact that not only what is inside the image is to be specified, but also where, semantic segmentation through CNNs must consider a trade-off between contextual and spatial understanding.

First CNN architectures known for their success in image segmentation are the fully convolutional neural networks (FCNs) [START_REF] Long | Fully convolutional networks for semantic segmentation[END_REF]. FCNs are structures derived from typical deep classification models such as VGG16, AlexNet or GoogLeNet by removing the corresponding classification layers, replacing their fully connected layers with convolutional ones and adding an upsampling layer that is dedicated to transforming coarse outputs into dense predictions. FCNs are very well-known in the medical [START_REF] Zhou | Three-dimensional ct image segmentation by combining 2d fully convolutional network with 3d majority voting[END_REF][START_REF] Li | Adaptive seeded region growing for image segmentation based on edge detection, texture extraction and cloud model[END_REF] and non-medical [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF] fields as they have paved the way for encoder-decoder networks for segmentation problems.

Many works within the field advocate to go deeper with FCN layers in order to increase the depth and precision of the learnt contextual features [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF]. However, increasing model's prediction ability by adding additional layers is not an easy task. Thus, as one goes deeper within the layers, insight on location features are hence lost. As a result, deep FCNs often fail to consider global and spatial information and are prone to producing fuzzy coarse-grained predictions [START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF]. Moreover, deepening the convolutional network will often increase the model's complexity, thus subjecting the training to additional challenges such as vanishing gradients. As a result, deep FCN may suffer from performance saturation or degradation while training.

One powerful architecture known for the ability to preserve semantic information while achieving promising segmentation performance is the wellknown U-Net [START_REF] Ronneberger | Unet: Convolutional networks for biomedical image segmentation[END_REF]. U-Net is a symmetrical encoder/decoder structure composed of a contracting path of stacked convolutional and max pooling layers representing the encoder branch and a corresponding expanding path composed of deconvolutional layers representing the decoder. U-Net is able to achieve a trade-off between extracting contextual features from the encoder convolutions on one hand and semantic features from the decoder convolutions on the other hand by concatenating their respective feature maps from different levels of the corresponding symmetric encoder and decoder layers via skip-connections. In this way, U-Net combines low-level detail and con-textual information with high-level semantic and location attributes, thus achieving a trade-off between the two. Several variants of U-Net consist in changing the backbone model used for encoding, e.g. VGG and DenseNet, and/or replacing deconvolution layers with super-resolution ones for more concise localization ability [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] Zhang | Road extraction by deep residual u-net[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF].

There is no doubt that, thanks to the complex and powerful architectures such as U-Net and its variants, the segmentation performance has reached a serious breakthrough. Even so, multiple challenges still remain within medical imaging. Thus, these deep networks often require large amounts of annotated training data, which is not easy to obtain given the medical field. Rather, unannotated or partially labeled data are more easily available or less computationally expensive. Moreover, even with sufficient data, automated systems generally and CNNs particularly still lack the anatomical plausibility that a medical expert has.

Prior to deep learning, prior information such as shape and the topology of organs have often been investigated within variational approaches in order to increase the anatomical correctness of automated segmentation. Recent advances in the domain have attempted integrating these prior onto CNN training in order to overcome the problem of lack of data and to evade production of anatomically aberrant errors. For example, the method of [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] constrains segmentations produced by a regular U-Net to conform with particular upper and lower bound sizes of the heart within a cardiac dataset. Another example is in [START_REF] Jurdi | Bb-unet: U-net with bounding box prior[END_REF] where connected component numbers were preserved by introducing bounding box prior at the level of the skip connections. These bounding boxes allow the network to learn focused features concerning small components of the heart that are dissolved within the deep layers of a normal U-Net baseline network. Nevertheless, prior constraint neural networks still face difficulties regarding which information to model, how they are modeled and integrated into the deep neural networks.

Integration of the prior onto CNN learning can be conducted at the level of the network structure, hereby the name structural constraints [START_REF] Jurdi | Bb-unet: U-net with bounding box prior[END_REF][START_REF] Oktay | Attention u-net: Learning where to look for the pancreas[END_REF] or at the level of the loss function [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF][START_REF] Clough | A topological loss function for deep-learning based image segmentation using persistent homology[END_REF] or a combination of both [START_REF] Oda | Besnet: Boundary-enhanced segmentation of cells in histopathological images[END_REF][START_REF] Oktay | Anatomically Constrained Neural Networks (ACNNs): Application to Cardiac Image Enhancement and Segmentation[END_REF]. Structure prior consists in designing parts of the network in order to take into consideration these external specifications [START_REF] Jurdi | Bb-unet: U-net with bounding box prior[END_REF][START_REF] Jurdi | Towards semi-supervised segmentation of organs at risk using deep convolutional neural networks[END_REF][START_REF] Jurdi | Investigating coordconv for fully and weakly supervised medical image segmentation[END_REF][START_REF] Oktay | Attention u-net: Learning where to look for the pancreas[END_REF][START_REF] Oktay | Anatomically Constrained Neural Networks (ACNNs): Application to Cardiac Image Enhancement and Segmentation[END_REF]. Imposing constraint via loss functions, on the other hand, consists of formalizing the prior as an additional term in the loss function. The loss term computes the error demonstrating the degree of violation of the prior constraint. Given the state of the art, prior-based losses have been widely adopted as means of enhancing consistency and plausibility of segments produced by powerful neural network architectures within fully supervised learning [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF][START_REF] Trullo | Joint segmentation of multiple thoracic organs in CT images with two collaborative deep architectures[END_REF][START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF].

On the other hand, they could be used to counteract the problem of lack of complete data annotations [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF].

The types of prior knowledge cover a breadth of notions in the literature [START_REF] Nosrati | Incorporating prior knowledge in medical image segmentation: a survey[END_REF]. Aside from the lowlevel prior which includes ground-truth transformation such as distance maps [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF][START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF] or Laplacian filters [START_REF] Arif | Shape-aware deep convolutional neural network for vertebrae segmentation[END_REF], high-level prior integrate actual expert knowledge into the learning system. This type of prior can include the size of the organ [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF], its compactness [START_REF] Dolz | Unbiased shape compactness for segmentation[END_REF], its topological structure [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF] or its convexity [START_REF] Mirikharaji | Star shape prior in fully convolutional networks for skin lesion segmentation[END_REF] among others. High-level prior losses is currently a growing trend that has captured the attention of many researchers within the field since they offer a versatile way to integrate external knowledge in a generic manner and can be plugged into any backbone.

The CoordConv layers, recently introduced in [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF], are extensions of convolutions that allow convolution filter to take into account the spatial coordinates of the pixels. The goal of CoordConv is to learn a mapping between coordinates in the Cartesian space and coordinates in the one-hot pixel space. CoordConv has shown promising potential for object localization [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF][START_REF] Levine | Endto-end Training of Deep Visuomotor Policies[END_REF], and has rightfully raised interest for image segmentation [START_REF] Qi | UPI-Net: Semantic Contour Detection in Placental Ultrasound[END_REF][START_REF] Yao | Land Use Classification of the Deep Convolutional Neural Network Method Reducing the Loss of Spatial Features[END_REF]. However, the CoordConv's added value has not been yet assessed in image segmentation generally and in prior guided segmentation particularly.

This paper investigates CoordConv as a proficient substitute to convolutional layers in U-Net models for medical image segmentation. We explore the effect of CoordConv on model performance and rate of convergence when learning is conducted under anatomical prior-based losses, particularly the size loss proposed by [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and the skeleton loss proposed by [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF]. We propose a new U-Net variant based on the CoordConv layer, the CoordConv-Unet, and demonstrate its role in enhancing stability and performance of the above losses. Finally, we expose the dual role of CoordConv-Unet as a regularizer with the ability to stabilize network training under prior-based losses on one hand, and the ability to increase system performance significantly by evading local solutions on the other.

The use of CoordConv with UNet has been used in the papers of [START_REF] Wang | Automated segmentation of pulmonary lobes using coordination-guided deep neural networks[END_REF] and [START_REF] Wang | Segmentation-based method combined with dynamic programming for brain midline delineation[END_REF]. However, in this work, we investigate the significance of U-Net/CoordConv combinations when trained under prior-based losses which according to our knowledge has not been investigated prior to this paper. The significance of CoordConv is investigated on two datasets: a cardiac dataset which consists of MR images covering the entire atrium. The understudied organ within this dataset is characterized with multi-connected components and large size variability. The spleen dataset is a CT image dataset where the spleen is characterized by a largely varying size and shape convexity issues at boundary level. The contributions of the paper are summarized as follows:

• We investigate the significance of Coord-Conv solution given organ segmentation under prior-based loss training.

• We propose a novel architecture, the CoordConv-Unet as a proficient substitute to U-Net given prior constrained problems.

• We shed light on the dual role that CoordConv-Unet has in increasing and stabilizing system performance.

The rest of the paper is organized as follows. Section 2 provides a brief overview of the state of the art regarding incorporating constraints onto deep learning networks. Section 3 elaborates on the proposed CoordConv-Unet model as well as the multiple frameworks and paradigms explored. Section 4 presents the datasets and experimental settings. Section 5 evaluates the significance of the proposed CoordConv-Unet relative to the proposed datasets. Finally, Section 6 concludes with future works and perspectives.

Related works on Constrained Convolutional Neural Networks

Prior knowledge can be integrated into the CNN learning in the form of structural constraints or at the level of the loss function.

Structural Prior Constraint

Among structural constraint methods, integration of prior can be done either externally in conjunction with the segmentation network [START_REF] Trullo | Joint segmentation of multiple thoracic organs in CT images with two collaborative deep architectures[END_REF][START_REF] Ghafoorian | Location sensitive deep convolutional neural networks for segmentation of white matter hyperintensities[END_REF][START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF][START_REF] Khoreva | Simple does it: Weakly supervised in-stance and semantic segmentation[END_REF][START_REF] Jurdi | Organ segmentation in ct images with weak annotations: A preliminary study[END_REF] or at the intermediate level [START_REF] Jurdi | Bb-unet: U-net with bounding box prior[END_REF][START_REF] Jurdi | Investigating coordconv for fully and weakly supervised medical image segmentation[END_REF][START_REF] Trullo | Joint segmentation of multiple thoracic organs in CT images with two collaborative deep architectures[END_REF].

In [START_REF] Ghafoorian | Location sensitive deep convolutional neural networks for segmentation of white matter hyperintensities[END_REF], the authors propose a cascade of several deep CNN architectures that consider multiscale patches in order to incorporate anatomical location in their decision making process. Thus, spatial location of patches extracted from the image into a CNN model is injected posterior to the convolutional layers. Similar to [START_REF] Ghafoorian | Location sensitive deep convolutional neural networks for segmentation of white matter hyperintensities[END_REF], authors of [START_REF] Trullo | Joint segmentation of multiple thoracic organs in CT images with two collaborative deep architectures[END_REF] demonstrate two collaborative architectures in order to iteratively refine the posterior probability and provide information about neighboring organs. In this work, anatomical constraints are obtained from an auxiliary network that are later used by the segmentation network (U-Net) in order to refine ill-defined organ boundaries. Similarly, the SR-UNet in [START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF] jointly adds an external network to the segmentation one in order to take into consideration the incomplete, overor under-segmented shape masks provided by the U-Net. However, unlike [START_REF] Trullo | Joint segmentation of multiple thoracic organs in CT images with two collaborative deep architectures[END_REF] that fine-tunes illdefined segmentations by U-Net using the constraints obtained from the external network, [START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF] maps the segmentations to conform to a manifold of permissible training shapes.

Instead of integrating prior via external networks, BB-UNet as proposed by [START_REF] Jurdi | Bb-unet: U-net with bounding box prior[END_REF] aims at integrating location prior represented by bounding boxes onto a U-Net at the level of the skip connections in order to capture focused features and preserve connected components properties that are lost in deep layers given normal U-Net functioning. Whereas BB-UNet imposes external constraints at the level of the skip connection, Attention-UNet [START_REF] Oktay | Attention u-net: Learning where to look for the pancreas[END_REF] do so by extracting these constraints from the bottle-neck layers of the baseline U-Net.

Loss prior constraints

Incorporating prior at the level of the loss function offers a versatile way to constraint neural network predictions while preserving computational complexity and generality. Prior integrated can be low-level, which resembles reformulated ground-truth representation such as: distance maps [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF][START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF], Laplacian filters [START_REF] Arif | Shape-aware deep convolutional neural network for vertebrae segmentation[END_REF] or internal layer feature maps [START_REF] Mosinska | Beyond the pixel-wise loss for topology-aware delineation[END_REF]. Prior could also be high-level representing actual external medical information such as the shape of the organ, compactness [START_REF] Dolz | Unbiased shape compactness for segmentation[END_REF] or size [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and are optimized directly based on ground-truth prior tags.

Low-level Prior knowledge. Both works of [START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF] and [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF] exploit distance maps to improve boundary consistency. Whereas [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF] do so by the extraction of shape bio-markers and allow the network to differentiate between hard-to-segment boundaries of different anatomical classes, [START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF] aims to fine-tune probability outputs by these distance maps in order to overcome the problem of class imbalance between empty/full images with high boundary precision. In the same context, [START_REF] Arif | Shape-aware deep convolutional neural network for vertebrae segmentation[END_REF] demonstrates a shape aware loss function that constraints predictions to conform to permissible manifold in vertebrae segmentation. In order to do so, the method takes into consideration the average point to curve Euclidean distance factor between predicted and the ground-truth contours. Moreover, authors of [START_REF] Yang | Major vessel segmentation on x-ray coronary angiography using deep networks with a novel penalty loss function[END_REF] exploit Laplacian filters in order to develop a boundary enhanced loss term that invokes the network to generate strong responses around boundary areas while producing a zero response in pixels that are at the periphery. Afar from boundary criteria, [START_REF] Mosinska | Beyond the pixel-wise loss for topology-aware delineation[END_REF] is able to close small gaps in neuronal membranes and alleviate topology mistakes by leveraging the topological information or shape descriptors present within the internal layers of VGG16 networks when introduced to both label and predicted segments. All the above-mentioned approaches enhance segmentation consistency by reformulating and refining transformations of posterior probabilities. However, they do not optimize the prior attribute directly between ground-truth and predicted segments. As a result, there is no guarantee that the ground-truth prior specification will be met.

High-Level Prior Loss. In [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and [START_REF] Peng | Discretely-constrained deep network for weakly supervised segmentation[END_REF], organ size is taken into consideration where the size prior is directly optimized relative to known ground-truth size bounds. Whereas [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] integrates the prior via an additional penalty loss term and computes the errors relative to approximate upper and lower bounds, [START_REF] Peng | Discretely-constrained deep network for weakly supervised segmentation[END_REF] relates to the discrete nature of size prior and optimizes the network via discrete based optimization techniques. Another type of prior is topology which is concerned with the properties of spatial objects by abstracting their connectivity, while ignoring their detailed form [START_REF] Ségonne | Integration of Topological Constraints in Medical Image Segmentation[END_REF]. Both works of [START_REF] Clough | A topological loss function for deep-learning based image segmentation using persistent homology[END_REF] and [START_REF] Hu | Topology-preserving deep image segmentation[END_REF] use notions of persistent homologies, which is a method for capturing topological structures via a series of thresholding on prediction maps in order to evade prohibited broken vessels and connections. Skeletonization is yet another way to represent topological properties of objects and is exploited in [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF] in order to conduct vessels and neuron segmentations. Other properties, such as compactness [START_REF] Dolz | Unbiased shape compactness for segmentation[END_REF], star-shape [START_REF] Mirikharaji | Star shape prior in fully convolutional networks for skin lesion segmentation[END_REF], inter-region relations [START_REF] Ganaye | Removing segmentation inconsistencies with semi-supervised non-adjacency constraint[END_REF], are also investigated as means of improving segmentation consistency via prior-based losses.

Combined Structural and Loss Constraints

In many works, authors introduce interchangeably both structural and loss constraints. In [START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF], a non-linear shape regularization model is trained jointly along U-Net. The main function of their adjoint network is to learn projections of arbitrary shapes onto a manifold space. The method then incorporates a loss function that updates the segmentation network (U-Net) parameters based on the regularized predicted segments, the rough predicted segments as well as the ground-truth labels. The authors of [START_REF] Oktay | Anatomically constrained neural networks : Application to cardiac image enhancement and segmentation[END_REF] adopt a similar regularization approach to that in [START_REF] Ravishankar | Learning and incorporating shape models for semantic segmentation[END_REF]. However, they target the decoder layer with their U-Net-like structure and train the up-sampling layers through super resolution ground-truth maps. On the other hand, authors of [START_REF] Oda | Besnet: Boundary-enhanced segmentation of cells in histopathological images[END_REF] introduce a boundary enhanced loss function similar to that of [START_REF] Caliva | Distance map loss penalty term for semantic segmentation[END_REF] and [START_REF] Arif | Shape-aware deep convolutional neural network for vertebrae segmentation[END_REF]. Instead of weighting by the errors through distance maps, [START_REF] Oda | Besnet: Boundary-enhanced segmentation of cells in histopathological images[END_REF] adds an extra decoder branch to the U-Net network in order to predict hard to segment boundaries. In the same manner, the method in [START_REF] Zotti | Gridnet with automatic shape prior registration for automatic MRI cardiac segmentation[END_REF] integrates center of mass and contour prior into the loss function which was obtained from an encoder/decoder structure trained end-to-end along the segmentation network.

Proposed CoordConv-Unet Architecture

In this section, we present the CoordConv layer and its implementation within the segmentation framework. We further elaborate on the proposed CoordConv-Unet model, the corresponding building blocks, as well as the different integration strategies.

CoordConv Component

The CoordConv layer is a simple extension of the standard convolutional layer wherein convolution is condition by spatial coordinates. The goal is to establish mappings between the Cartesian space and the pixel space, by enabling the filters to know where pixels are located. In a general sense, convolutions are mainly characterized by three specific characteristics: few training parameters, fast optimization via modern GPUs, and translational invariance. However, given many tasks, there is a controversy with regards to whether translational invariance will truly help model performance or not. CoordConv allows the network to keep or drop the property of translational invariance according to what is needed in the task at hand. In doing so, CoordConv ensures the best of both convolutional and spatial features. The implementation of CoordConv is done by concatenating two additional and channels to the input channel as shown in Figure 1(top figure). In such a way, CoordConv allows the learning of a function characterized by a certain degree of translational dependence, if the weights connecting the coordinate layers of the CoordConv with the convolutional are non-zero or could mimic a regular convolutional layer if they were set to zero. In the proposed experiments, CoordConv is implemented via a PyTorch library1 where a linear scaling is applied in order to bound the values of the coordinate layers between -1 and 1.

CoordConv-Unet

In the proposed architecture, we extend upon U-Net by replacing convolutional blocks with the CoordConv ones. In such a way, we allow the network to take into consideration spatial and geometric aspects while training. As previously stated, U-Net is a symmetric encoder/decoder structure with equivalent distribution of convolutional and de-convolutional blocks connected via skip connections. Each convolutional block is composed of two consecutive ensembles of convolutional layers and batch normalization, whereas the decoder block adds a bilinear upsampling layer to the previous ensemble. Our main contribution targets the first convolutional layer of the convolutional blocks consisting the U-Net model. Thus, instead of directly convoluting the input of the convolution layer with that of the one before it, rather coordinates for each feature are taken into consideration. The proposed network is represented in Figure 1(bottom figure).

Computational complexity

Given a U-Net, parameters generally involve the number of learnable quantities or weights connecting a convolutional layer with the corresponding precedent or following layer. In order to quantify the computational overhead induced by the CoordConv component, we will adopt the following mathematical notations. Let represent the width of convolutional filter and its height. Let be the number of filters in the preceding layer and the number of filters in the current one. Then, the number of parameters involving the conventional convolutional layer is hence computed according to ( + 1) , where 1 represents the bias term for each filter. Given CoordConv component, since 2 additional channels are added at the level precedent to the convolutional layer, the number of filters in the preceding layer is hence +2, thus resulting in a new number of parameters of ( ( + 2) + 1) . In such a way, each convolutional layer adds to the computational complexity 2 operations.

Integration Strategies

We have investigated various integration strategies of the CoordConv block onto the U-Net architecture. The first setting is one that mimics the state of the art, where the and channels are added only to the first convolutional layer. We call this model CoordConv(+1). The proposed CoordConv-Unet consists of replacing the first convolutional layer of each convolutional block within the encoding path with the CoordConv layer. We call the proposed method CoordConv-Unet.

Loss functions

The CoordConv-Unet is trained with two priorbased losses: the size loss [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and the skeleton loss [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF].

The size loss is a penalty loss function that integrates size information by computing the mean squared error between the grouping of pixel probabilities indicating predicted organ size and a predefined upper or lower bound indicating groundtruth size. The significance of the loss lies in its ability to impose some constraint on the size of the predicted segments.

The clDice loss exploits the topological notion of skeletonization in order to reveal subtle

Comparison to the state of the art

The CoordConv concept in essence is not a new topic proposed in the paper. Introduced by [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF], the CoordConv layer was initially designed to investigated supervised Coordinate classification, object detection, supervised coordinate regression, and generative adversarial modeling. The method in [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF] raises questions with regards to Co-ordConv's ability to make the training more stable given the coordinate classification task. However, implementations in [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF] did not address the segmentation problem, which is quite important in current research. To add to this, the method includes adding the CoordConv layer solely at the primary convolution of the entire architecture Since segmentation is a classification problem done at pixel-level, investigating the efficiency of CoordConv in aiding prior-constrained medical segmentation is an interesting research direction to be explored. In this context, works within the state-of-the-art most relevant to our method are [START_REF] Wang | Automated segmentation of pulmonary lobes using coordination-guided deep neural networks[END_REF] and [START_REF] Wang | Segmentation-based method combined with dynamic programming for brain midline delineation[END_REF]. In [START_REF] Wang | Automated segmentation of pulmonary lobes using coordination-guided deep neural networks[END_REF], the authors investigate the role of CoordConv in conducting 3D segmentation of pulmonary lobes via a V-Net. Instead of replacing the encoder layers of the network as our proposed CoordConv-Unet model, authors of [START_REF] Wang | Automated segmentation of pulmonary lobes using coordination-guided deep neural networks[END_REF], perform this interchange at the level of the up-sampling layers. Thus, the method first exploits a 2D automated lung segmentation model followed by the CoordConv embedded V-Net architecture. On the other hand, authors of [START_REF] Wang | Segmentation-based method combined with dynamic programming for brain midline delineation[END_REF] propose a 3-stage framework in order conduct brain mid-line delineation. Within the segmentation step, they introduce CoordConv component at the input level of the network at the intermediate segmentation step, midway between alignment (first step) and delineation (third step).

Whereas [START_REF] Wang | Segmentation-based method combined with dynamic programming for brain midline delineation[END_REF] integrates CoordConv solely at the input level (as is originally implemented in [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF]), our proposed CoordConv-Unet in this paper interchanges the first convolution of each stage of the U-Net architecture with that of the Coord-Conv layer. Moreover, we do not conduct preor post processing steps as in [START_REF] Wang | Segmentation-based method combined with dynamic programming for brain midline delineation[END_REF]. Unlike [START_REF] Wang | Automated segmentation of pulmonary lobes using coordination-guided deep neural networks[END_REF], where CoordConv component is incorporated at the level of the upsampling layers, the proposed method do so at the encoder convolution level.

Experimental Setting

Datasets

Experiments are conducted on two well-known public datasets from the Decathlon challenge for medical image segmentation2 . Namely, a cardiac dataset and a spleen dataset. In Table 1, a brief summary of data characteristics is specified.

Atrium dataset is a mono-modal MRI cardiac dataset dedicated to segmentation of the left atrium. The heart as shown in Figure 2 is a multiconnected component object with up to 4 elements of varying sizes and lying in close proximity to each other. The dataset is also characterized by a huge class imbalance with respect to background and foreground pixel distribution.

Spleen dataset is a CT dataset as presented in Figure 3. The objective is to segment a single organ (the spleen), characterized with largely varying size and mild convexity issues at boundary levels.

For pre-processing, we have resized the images to a size of 256 × 256 and normalized them to a pixel value between 0 and 1. Deploying the framework presented by [START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF], we have kept negative samples for training. Negative samples are empty images, meaning that the organ of interest is not present. The datasets were split into train and validation based on an 80 % , 20 % partition respectively. Cross-validation was done on three folds of the data based on three Monte-Carlo simulations [START_REF] Arlot | A survey of cross-validation procedures for model selection[END_REF].

Model architecture and training

To insure reproducibility, we deploy a wellknown experimental framework presented by [START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF]. The U-Net [START_REF] Ronneberger | Unet: Convolutional networks for biomedical image segmentation[END_REF], of which we integrate the Coord-Conv layers onto, is a 3-stage U-Net composed of convolutional and de-convolutional blocks, bottleneck and skip connections. Each stage within the encoder is composed of convolutional blocks containing an ensemble of convolutional and batch normalization layers. On the other hand, each stage within the decoder path is composed of 2 consecutive convolutional blocks followed by an upsampling layer. The bottleneck is constituted of 2 convolutional blocks separated by a residual block [START_REF] Zhang | Road extraction by deep residual u-net[END_REF].

Since prior losses essentially suffer from training instability and local solutions [START_REF] Kervadec | Boundary loss for highly unbalanced segmentation[END_REF], we have trained both size loss [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and clDice loss [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF] in conjunction with the Dice baseline loss weighted by a hyper-parameter , where is dynamically updated through training according to the following equation

= (1 -) + .
Thus, starting from a value of = 0.01, is increased by a value of 0.01 at each training epoch. Models were evaluated using the Dice index and Hausdorff distance. Training was conducted via the Adam optimizer with a batch size of 8 over 200 epochs. The learning rate was set to 5 × 10 -4 and halved each 20 epochs if the validation performance did not improve.

Investigation of CoordConv-Unet performance relative to the Datasets

In this section, we present results for the CoordConv-Unet model when trained on the atrium and the spleen datasets via just the Dice loss, the Size loss [START_REF] Kervadec | Constrained-cnn losses for weakly supervised segmentation[END_REF] and the clDice loss [START_REF] Shit | cldice -a topology-preserving loss function for tubular structure segmentation[END_REF]. We note that the Atrium is characterized by multiconnected small components that are very close to each other, whereas the spleen is a complexshape organ of non-convex curves and edges. Results obtained on Dice accuracy and Hausdorff distance are compared in reference to the regular U-Net baseline trained under the considered losses as well as the CoordConv(+1), which represents the addition of the CoordConv layer solely at the input stage as originally designed by [START_REF] Liu | An intriguing failing of convolutional neural networks and the coordconv solution[END_REF].

Results on the Spleen dataset

Results on Dice accuracy and Hausdorff metric for the spleen as shown in Table 2 and Table 3 indicate the significance of CoordConv layers This mainly indicates the ability of CoordConv-Unet to learn curvature features relative to the inter-distance position of the pixels relative to the spleen.

Results on the Atrium dataset

For the Atrium, we benchmark the results in Table 4 and Table 5. A closer look at the tables, one can realize the significance of the clDice loss generally against the size as well as the Dice baseline loss. We predict that, since the clDice is generally based on the skeleton concept, the clDice was hence able to distinguish between the different boundaries of the connected components relative to the heart. Moreover, training CoordConv-Unet under the clDice loss has increased model performance by about 3% on atrium dataset from baseline training. This indicates the proper role of CoordConv-Unet in increasing system performance under topological losses. This is also verified by the Hausdorff distance metric where CoordConv-Unet under clDice scored second best relative to the other frameworks.

Comparing relative to the size loss, CoordConv-Unet maintains system performance considerably over all its paradigms. From here, we can realize the dual role that the CoordConv plays in enabling learning of spatial dependent attributes when needed (case of the clDice) or mimicking typical convolutional functioning. Given the latter, one would anticipate that the behavior of the CoordConv would be typical to that of a regular convolution. However, a closer look at the evolution of the Dice accuracy over the number of training epochs given folds where CoordConv solution equates that of a regular U-Net from Figure 4, one would realize yet another significant advantage of using CoordConv-Unet in place of regular U-Net. Thus, CoordConv-Unet insures model stability and convergence by evading the undershoot evident when training with regular U-Net.

Analysis

Based on the results presented on both the spleen and atrium datasets, one can gather that CoordConv-Unet plays a dual role given prior constraint neural networks. Thus, CoordConv can either maintain segmentation performance while regularizing training and stabilizing evolution, or it increases system performance by evading the local solutions that prior losses suffer from.

Despite the significance of CoordConv-Unet as shown by the above study, however, limitations still persist with regards to CoordConv-Unet performance. Thus, when tested against the Dice loss baseline alone (first row of each table), the addition of the CoordConv components degrades the system performance considerably. One could cast the clarification of the phenomena to the complexity/regularizing trade-off relative to CoordConv-Unet. We have clarified earlier 2 main roles of CoordConv:1) the role of stabilize the undershoot evident when training against prior losses. 2) the role of enhancing segmentation performance by evading local solutions. Addressing the two attributes relative the Dice baseline training, we can gather the following: The undershoot revealed in the plotted diagrams are a result of the interchange between the main pixel-wise Dice loss on one hand and the anatomical prior loss on the other in prior-constrained training problem. In the Dice baseline, the undershoot is rather nonexistent; hence CoordConv cannot play the role of the regularizer that evades training in stability and ends up decreasing model performance by adding complexity to the system. The second role of CoordConv is that it increases model performance under prior-based losses by evading local solutions. As we have previously said, prior-based losses are interestingly used because they integrate expert knowledge onto the automatic training. However, designing these loss functions is often tedious and subjected to various differentiability and stability challenges. One of the reasons for these issues is the discrete nature of these prior information vs the continuous soft probability output of the network. With CoordConv-Unet, the network can act as a regular U-Net if needed, i.e., if no stability problems persist or the network could integrate spatial knowledge thus fulfilling the true CoordConv concept. In regular Dice baseline training, prior does not exist, hence, addressing stability is not an issue. From all of the above, we can hence realize the insignificance of using CoordConv component if there is no prior constraint problem involved.

Conclusion

In this paper, we have proposed a new model, the CoordConv-Unet model as a proficient substitute to U-Net given prior-constraint tasks. We have exposed the dual role of CoordConv-Unet given these constrained tasks. Thus, CoordConv-Unet can either improve performance by evading stability problems, or can mimic a regular U-Net if the translation invariance attribute is required. In the later role, CoordConv-Unet still poses a further significance that can help stabilize the network performance under the prior-based losses.

Future work includes designing certain frameworks that can impose whether the weights connecting the coordinate layers with the convolutional ones to be trained or fixed so as to resolve the problem faced when training CoordConv-Unet under a regular unconstrained Dice loss. Moreover, efficiency of CoordConv-Unet given multiorgan and lesion segmentation is also to be explored.

grant ANR-18-CE23-0014). This work is part of the DAISI project, co-financed by the European Union with the European Regional Development Fund (ERDF) and by the Normandy Region.

Figure 1 :

 1 Figure 1: Proposed CoordConv-Unet model. In the top panel, the CoordConv layer consists in concatenating the x-layer and y-layer to the convolutional layer. CoordConv-Unet consists of replacing the first convolutional layer of each stage with the CoordConv layer.

Figure 2 :

 2 Figure 2: Atrium dataset from Decathlon challenge with multiple components of variable size that are in close proximity of each other

Figure 3 :

 3 Figure 3: Spleen dataset from Decathlon challenge showing organ or large size variability and convexity issues at boundary level

Figure 4 :

 4 Figure 4: Evolution of the Dice accuracy in validation under Dice +clDice (LEFT), Dice +Size (RIGHT) for the atrium dataset.

Table 1 :

 1 Dataset Description: Patient Dist.: Patient distribution. Org. Size: percentage of pixels occupied by the anatomical object relative to the entire image. Mod.: number of dataset modality. Con. Comp: number of connected components.

		Patient Dist. Org. Size	Class Mod.	Con.
		Train Test (% of image)			Comp
	Atrium	16	4	≤ 1.42	1	1	0 ∼ 4
	Spleen	32	9	≤ 4.36	1	1	0 ∼ 1

Table 2 :

 2 Dice accuracy results on the spleen dataset

		U-Net	CoordConv(+1) CoordConv-Unet
	Dice Loss	78.58±5.46	64.65±5.7	65.48±8.48
	Size Loss	86.44±15.87	94.86±1.72	94.96±1.59
	clDice loss 87.15±13.61	87.04±9.98	94.54±1.06

Table 3 :

 3 Hausdorff distance results on the spleen dataset

		U-Net	CoordConv(+1) CoordConv-Unet
	Dice Loss	1.30 ± 0.24	1.89±0.26	1.71±0.32
	Size Loss	1.02 ± 0.56	0.76±0.13	0.76±0.12
	clDice loss 1.07 ± 0.53	1.11±0.31	0.85±0.07
	when training via prior-based losses. Thus, Co-
	ordConv(+1) and CoordConv-Unet increase (case
	of size loss) or maintain (case of CoordConv(+1)
	with clDice loss) system performance relative to
	the regular U-Net baseline. CoordConv-Unet in-
	creases the Dice accuracy by over 8% from the reg-
	ular U-Net baseline under the prior losses. This
	added value of CoordConv is further verified by
	the error computed on Hausdorff distance. Thus,
	the CoordConv-Unet results in a 28% decrease
	in the Hausdorff metric (from 1.02 0.76) under
	size loss and a 20% decrease in Hausdorff dis-
	tance (from 1.07 0.85) under clDice loss relative
	to the U-Net and CoordConv-Unet respectively.

Table 4 :

 4 Dice accuracy on the atrium dataset

		U-Net	CoordConv(+1) CoordConv-Unet
	Dice Loss	83.67 ± 3.66	82.10±2.51	82.52 ± 2.33
	Size Loss	84.59±2.62	84.63±1.67	84.48±1.5
	clDice loss 83.85 ± 2.56	85.35 ± 1.65	86.15±1.39

Table 5 :

 5 Hausdorff distance results on the atrium dataset

		U-Net	CoordConv(+1) CoordConv-Unet
	Dice Loss	1.62 ± 0.16	1.64±0.11	1.64±0.11
	Size Loss	1.59±0.17	1.57±0.08	1.57±0.10
	clDice loss 1.64 ± 0.16	1.60±0.14	1.59±0.14

https://github.com/walsvid/CoordConv

http://medicaldecathlon.com/

Acknowledgments

The authors would like to acknowledge the National Council for Scientific Research of Lebanon (CNRS-L) and the Agence Française de la Francophonie (AUF) for granting a doctoral fellowship to Rosana El Jurdi, as well as the ANR (project APi,


Matthieu Malléjac

Cédric Payan

Lilian D'hondt

Serge Mensah

Aroune Duclos

Matthieu Cavaro

Non-linear Bubbly Helmholtz Resonator

Keywords: Nonlinear Resonant Acoustic Spectroscopy, Two-phase, Helmholtz Resonator, Bubble cloud, Void fraction, Microbubble

Microbubble clouds greatly affect the acoustic behaviour of systems such as liquid-filled Helmholtz resonators. Gas microbubbles change the resonator's behaviour from quasi-linear to mostly nonlinear, together with the appearance of hysteretic phenomena and desymmetrisation of the temporal response or a softening effect with a drop in the resonance frequencies with increasing excitation amplitudes. The aim of this study is to model the non-linear behaviour of a diphasic Helmholtz resonator filled with water containing air microbubbles. The microbubble cloud is therefore modelled with a "damped-mass-spring" second-order equation. The impact of the two phases is accounted for by considering both an equivalent stiffness and an equivalent damping value. The model is then compared to experimental data, which showed its ability to reproduce both linear and non-linear behaviour.

Introduction

In a context where the global electricity demand continues to increase, the French Atomic Energy and Alternative Energies Commission (CEA) conducts research on the development of sodium-cooled fast nuclear reactors (SFR) within the scope of the Generation IV International Forum (GIF).

As liquid sodium is opaque, acoustic non-destructive testing (NDT) and sodium viewing methods have to be developed, which requires accurate knowledge of liquid sodium's physical properties. The presence of argon gaseous microbubbles in the liquid sodium -mainly due to nucleation and gas entrainment -largely modifies its compressibility and celerity. This has led to the development of bubbly liquid characterisation techniques to evaluate both the void fraction , which represents the ratio of the gas volume to the total volume, and the cloud histogram, which refers to the distribution of bubble radii.

Different bubble cloud characterisation methods exist [START_REF] Leighton | The detection of tethered and rising bubbles using multiple acoustic techniques[END_REF]. The CEA has focused its research on acoustic techniques, such as mixing frequency techniques [START_REF] Cavaro | Microbubble cloud characterization by nonlinear frequency mixing[END_REF], attenuation and velocity spectroscopy [START_REF] Cavaro | A void fraction characterisation by low frequency acoustic velocity measurements in microbubble clouds[END_REF], and low-frequency acoustic velocity measurements (LF-AVM) [START_REF] D'hondt | Mensah -Acoustical characterisation and monitoring of microbubble clouds[END_REF].

Another way of characterising a bubbly liquid is to confine the micro-bubble cloud in a resonator and to study the impact of the two-phase medium on the resonance. Both the linear and non-linear resonant behaviour of the system depends on the properties of the cloud. First, an estimate of the void fraction and the mean bubble radius can be made in the linear regime (small excitation amplitudes). The resonance frequency of the system can be compared with that predicted either by Wood's model, which couples the speed of sound (and thus the system's resonance frequency) and the void fraction [START_REF] Wood | A textbook of sound[END_REF], or by Prosperetti's model, which includes the impact of radii [START_REF] Prosperetti | The speed of sound in a gas-vapour bubbly liquid[END_REF]. Second, a softening effect, i.e., a decrease in resonance frequency with increasing excitation amplitude, can be observed in the non-linear regime [START_REF] Coste | Acoustic resonances in a liquid with vapor bubbles: Effect of liquid-vapor transition on sound velocity and attenuation[END_REF].

2

Characterisation techniques based on this softening effect, such as the Non-linear Resonant Acoustic Spectroscopy (NRAS), have been developed to assess the fatigue of rocks and concrete [START_REF] Payan | Caracterisation non destructive du beton : etude du potentiel de l'acoustique non lineaire[END_REF][START_REF] Johnson | Nonlinear resonant ultrasound spectrocopy[END_REF][START_REF] Abeele | On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation[END_REF] and might be transposed to the characterisation of bubble clouds. This method consists in representing the relative variation of the resonance frequency as a function of the excitation amplitude. Three zones can generally be observed: a linear zone with an almost zero variation of the resonance frequency, followed by a quadratic evolution and finally a linear variation of the resonance frequency, whose slope is quantified according to the properties of the media (fatigue state and microcracks in the case of concrete [START_REF] Payan | Caracterisation non destructive du beton : etude du potentiel de l'acoustique non lineaire[END_REF][START_REF] Johnson | Nonlinear resonant ultrasound spectrocopy[END_REF][START_REF] Abeele | On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation[END_REF]). However, to the best of our knowledge, the physical parameters cannot yet be determined on the basis of this information. Some experimental NRAS tests on bubbly liquids [START_REF] Coste | Acoustic resonances in a liquid with vapor bubbles: Effect of liquid-vapor transition on sound velocity and attenuation[END_REF][START_REF] Cavaro | Linear and nonlinear resonant acoustic spectroscopy of micro bubble clouds[END_REF] have been undertaken without further characterisation.

The absence of an analytical modelling of such two-phase resonators prevents a thorough investigation of the influence of each of the bubble cloud parameters on the NRAS technique. To fill this gap, we propose to model in this study a two-phase Helmholtz resonator, capable of reproducing both linear and non-linear resonant behaviour observed experimentally.

The paper is organized as follows. We first expose the proposed model derivation, before providing analytical results to validate its linear resonant response with Wood's and Prosperetti's well-known effective medium models. The third part of the study compares the model with the experimental results. Future prospects are then discussed in the conclusion.

Theoretical background

As two-phase medium compressibility is highly sensitive to void fractions, this property is used as the basis for our model. Moreover, since the behaviour of both microbubbles and Helmholtz resonators can be modelled by an analogue damped-mass-spring system [START_REF] Leighton | The Acoustic Bubble[END_REF][START_REF] Alster | Improved calculation of resonant frequencies of helmholtz resonators[END_REF][START_REF] Howe | On the helmholtz resonator[END_REF], the two-phase Helmholtz resonator is modelled likewise.

We adopt here a macro-micro description of the two-phase Helmholtz resonator. The entire system is described as an effective medium (macroscopic view), the properties of which are directly derived from the microbubble reaction to an external excitation (microscopic view).

We then adopt first a macroscopic description to model the response of the effective two-phase Helmholtz resonator.

The system's behaviour can be described as the displacement of a fluid mass in the neck of length resulting from an excitation pressure , applied to the opening of the Helmholtz resonator with a cross-section = and a cavity volume as shown in Fig. 1. The fluid mass displacement in the resonator neck is governed by a second-order differential equation such as

+ + = , (1)
with the excitation pulsation. The effective mass of moving fluid in the neck of length , = , accounts for the radiation at the openings by considering an effective length . Cummings [START_REF] Cummings | Acoustics of a cider bottle[END_REF] and to that of an infinite baffled piston [19] for the opening to the resonator cavity " , i.e.,

= + ! + " = + # $% + ! ! &' (2)
where ( = 2 is the internal opening diameter, ( &' is the external diameter, and is the cavity radius. The equivalent stiffness and damping are defined in sub-sections 2.1 and 2.2 respectively.

Based on the elementary Helmholtz resonator model, the stiffness of the two-phase resonator creates a restoring force at the mouth's cross-section . The bubble population, which has its own compressibility, also acts on this restoring force. We then adopt a microscopic view to derive these equivalent properties from the microbubble cloud as schemed in Fig. 2. The polydisperse bubble population can be represented by a histogram (number of microbubbles versus bubble radii), which can be divided into several radius classes, *. Each population class gathers every single bubble whose radius falls in the range of the class. (

) 3
The equivalent stiffness of the gaseous phase, depicted in Fig. 2, results from a series combination of the stiffness of each radius range B and of the C B identical microbubbles in the * D class, B , leading to

3 = E∑ 6 G B H 16 , (4)
with

B = 5∑ 6 G7 I G A 16 = G7 I G . ( 5
)
We assumed here that each bubble in the * D class shares the same stiffness since the variation in volume is the same.

An elementary stiffness, modelled similarly to that of a spring [START_REF] Bernardot | Un archetype d'oscillateur : le resonateur acoustique de helmholtz[END_REF], is assumed and assigned to each microbubble of the cloud

= ? ! @ =< J , (6)
with the subscript . B omitted for readability.

The compressibility of each individual bubble of equilibrium radius K from the population is determined by solving the Laplace equation, L = MNOPQ, or

R S + T U J V K L = R S + + T U V L , (7)
so as to estimate the variation in the volume

W = -K = 2% $ 9 $ -K $ ;, (8)
and the variation in the internal pressure

W = -K = S + 2X R 6 U - 6 U J V, (9)
where S is the ambient pressure (atmospheric pressure), X is the surface tension, . K refers to steady state, and Y to the polytropic coefficient. An isothermal behaviour is assumed considering that the average thermal penetration depth

, Z = [ \ 8
~160 /m (with D a = 2.185.10 c m 2 .s -1 , the gaseous thermal diffusivity) is much larger than the expected bubble radius values in the reactor (ranging from some micrometres to a few tens of micrometres) within the resonance frequency range of the bubbly water filling the resonator (see Table 1). Nevertheless, to remain general, the polytropic coefficient Y is estimated at each step of the resolution, i.e., for each radius range, using Devin's model [START_REF] Devin | Survey of thermal, radiation, and viscous damping of pulsating air bubbles in water[END_REF].

Knowing the volume and pressure variations in each bubble, its compressibility can then be determined as such

d = - 6 < e< ef g ? ≈ - 6 < J i< if . (10)
By considering a series combination, the total displacement can be assumed to be divided into elementary displacements absorbed by a volume change in each phase and microbubble.

Equivalent damping

The damping of individual bubbles has been already widely discussed [START_REF] Commander | Linear pressure waves in bubbly liquids: Comparison between theory and experiments[END_REF][START_REF] Prosperetti | Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids[END_REF][START_REF] Medwin | Counting bubbles acoustically: a review[END_REF] and is known to be due to three mechanisms: thermal exchange, radiation and viscosity. However, neither the cloud nor bubbly liquid damping coefficient are well defined in literature. Only Commander and Prosperetti [START_REF] Commander | Linear pressure waves in bubbly liquids: Comparison between theory and experiments[END_REF] recommend defining a complex wave number to determine an attenuation coefficient.

In this damping-mass-spring analogy, we propose to define the equivalent damping as a parallel combination of dampers, corresponding to the damping of each bubble of the same class, in series with the equivalent damper of each class, themselves in series with a damper corresponding to the damping of the Helmholtz resonator.

This choice is motivated by the fact that the damping is directly proportional to the vibration velocity and therefore to the bubble radius. Bubbles of the same class have the same velocity, which leads to a parallel combination of dampers. On the other hand, the velocity is different for bubbles of different radius classes, leading to a series combination of equivalent dampers of each class.

The total system damping is therefore given by , 1 1 1 = = ,r eq eq eff eq eq m b δ δ δ + [START_REF] Abeele | On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation[END_REF] with r eq , δ the equivalent damping constant of the resonator defined in [START_REF] Howe | On the helmholtz resonator[END_REF] by the three same components, that is

j ,k = 2 &ll m @ #%< " + 2 n J √ U ! pnq 4 + nr 4 9s -1;t, (12)
where

e m L S / = 2 Λ
is the neck's acoustic conductivity, l ν is the kinematic viscosity and r 4 is the thermal diffusivity of the liquid. M is the sound velocity of the effective medium (eq. 18), and s is the adiabatic index.

As the study focuses on the Helmholtz resonance, i.e., much lower than the bubble resonance frequency, the low-frequency approximation of bubble damping [START_REF] Commander | Linear pressure waves in bubbly liquids: Comparison between theory and experiments[END_REF] is used

Model validation

Methods

To check the accuracy of our microscopic modelling of the two-phase Helmholtz resonator, multiple analytical simulations of the model are performed to test its response in the linear and non-linear domains. Equation ( 3) is solved with a fourth-order Runge-Kutta algorithm for 5 seconds of sweep excitation in the [20 -500] Hz range.

On the one hand, linear accuracy is examined by comparing the dependence of the resonance behaviours predicted by our model with that given by the classical homogenisation model. On the other hand, we check whether the non-linear phenomena (time amplitude desymmetrization, hysteresis, harmonic generation and softening effect) observed experimentally on a similar system in Ref. [START_REF] Coste | Acoustic resonances in a liquid with vapor bubbles: Effect of liquid-vapor transition on sound velocity and attenuation[END_REF] are well captured by the proposed analytical model.

Linear regime

To validate the linear behaviour predicted by the proposed model based on a microscopic description of the bubble cloud, we compared the bottle's variation in linear (small excitation amplitude) resonance with the variation in the sound speed (resp. frequency) predicted by Wood's model [START_REF] Wood | A textbook of sound[END_REF]. The latter is based on a homogenisation scheme applied to the bubbly liquid (macroscopic view), which is seen as an effective fluid or mixture characterised by effective compressibility d and density d = 91 -;d 4 + d 3 ,

= 91 -; 4 + 3 .

(17) Subscripts . 4 and . 3 correspond to the liquid and the gaseous phases respectively. Using eq. ( 16) and eq. ( 17) and noting that gas compressibility is much higher than liquid compressibility, the effective sound velocity of the mixture M can be written as [START_REF] Lamarre | Instrumentation for the measurement of sound speed near the ocean surface[END_REF] 6 ! @ = . d =

: @ L 8 @ + 961:; @ > @ + 91 -; R u > f J + 3 d 4 V. ( 18
)
The effective velocity is then used in the Helmholtz resonance formula [START_REF] Mechel | Formulas of acoustics[END_REF]

v k = ! % [ ? ! w & < " , (19)
to predict Wood's resonance frequency for the two-phase Helmholtz resonator.

We then compare in Table 1 the linear resonance frequency given by the proposed cloud's microdescription model to that obtained from the Wood's approximation for several two-phase populations, with different void fractions and bubble radii. The good agreement reached validates our equivalent stiffness and damping modelling of the two-phase Helmholtz resonator response for small excitation amplitudes. The slight over-estimation with our model can be attributed to a stiffening issue (as observed with discretisation in the finite-element method for example). Moreover, our modelling also reproduces well the limit case of a water single-phase ( = 0) Helmholtz resonator with a resonance at 383 Hz (382.3 Hz expected with the Helmholtz resonance formula).

Non-linear regime

Due to the lack of a non-linear model for this type of system, the non-linear response predicted by our analytical modelling cannot be compared with existing models. We therefore compare our analytical prediction with measurements and check whether the proposed model reproduces the non-linear response observed experimentally.

Experimental set-up

A glass bottle filled with water, acting as a Helmholtz resonator, the resonance of which is computed at 382.3 Hz (eq. 19), is used for this experimental study. The continuum cross-sectional variation in a champagne bottle is designed to avoid bubble accumulation at discontinuities and to maintain a stable void ratio. The neck cavity limit of the champagne bottle is defined by the cross-section for which a tangent to the bottle shape intersects the revolution line of the bottle's mouth [START_REF] Cummings | Acoustics of a cider bottle[END_REF]. The cavity radius at this point is defined by as shown in Fig. 3.

The 75 cL champagne bottle is instrumented with a B&K 8103 hydrophone and a laminar microfluidic bubble generator. A B&K 4809 shaker mounted above the bottle's mouth is used to produce sweep excitation, as shown in Fig. 3.

By adjusting the pressure in the microfluidic generator, one can control the properties of the generated bubble clouds, both in terms of void fractions and radius ranges. Microfluidic generators can be optically characterised to create histogram abacus evaluating, for a given pressure, the radius histogram of the bubble population composing the cloud.

Experimental observation and qualitative non-linear validation

Figures 4 (a,c) show respectively the measured temporal responses and their power spectral density estimated with a Welch periodogram, for a population of radii included in the [50, 70] µm range (measured by optical camera) and an excitation amplitude ranging in [14 -494] m.s -1 .

As expected, when the excitation amplitude is increased, a softening effect [START_REF] Cavaro | Linear and nonlinear resonant acoustic spectroscopy of micro bubble clouds[END_REF] (coupled to harmonic generation, and amplitude desymmetrisation) is observed. The sweep time is now reduced to 1 s to ensure constant cloud characteristics during the experiments. In addition, to guarantee that the void fraction remains constant during the experiment, the phase of increasing the excitation amplitude is followed by a phase of decreasing the amplitude to ensure that the same linear resonance frequency is returned. Finally, between each set of measurement, the bottle is degassed.

Thanks to Wood's model, the void fraction can be estimated by measuring the linear resonance frequency decrease due to the presence of the bubble cloud, i.e., from 382.3 Hz without bubbles to 357.1 Hz, which corresponds to a void fraction of = 8.10 1x .

The analytical model captures well this response as evidenced by Figs. 4 (b,d). It is worth noting here that the analytical model predicts either a softening or a hardening behaviour depending on excitation amplitude sign, as remarked by [START_REF] Doc | Nonlinear acoustic propagation in bubbly liquids: Multiple scattering, softening and hardening phenomena[END_REF]. As the bubble radius, volume, internal pressure, and thus compressibility are solved in quasi-steady-state conditions (Laplace equation solved for a static excitation pressure), the model solution depends on the amplitude sign. Since it has been shown experimentally that bubbly liquid under sinusoidal sweep excitation presents a softening effect, with the resonance frequency progressively decreasing with increasing amplitude, only the softening solution was retained.

Toward a Non-linear Resonant Acoustic Spectroscopy characterization

We now investigate the possibility of characterizing the cloud properties using the non-linear softening effect captured by our analytical modelling. We then test different bubble population characteristics (different void ratio, different bubble size) to see their influence on the non-linear response of the two-phase Helmholtz resonance. The softening effect predicted can be quantified by applying a non-linear resonant acoustic spectroscopy (NRAS) method [START_REF] Van Damme | Nonlinear Resonant Acoustic Spectroscopy[END_REF], i.e., by representing the relative frequency variation with the excitation amplitude and then interpolating it, as shown in Fig. 6. It is clear that the non-linear softening effect depends on the void ratio. The higher the void ratio, the larger the gaseous phase volume, the stronger the non-linear effect and the lower the resonance frequency.

Figure 7 shows the effect of the bubble radii on the NRAS for a monodispersed bubble cloud for a void fraction of 10 1c . The bubble size also affects the resonant behaviour of a microbubble cloud, especially for very small radii (below 10 µm). The NRAS method is much less sensitive in the case of larger bubbles. The balance between the Laplace pressure and the atmospheric pressure (eq. 7) can explain this. For wide bubbles, the atmospheric pressure becomes predominant compared with the ratio of the surface tension to the bubble radius. In a water-air bubbly liquid, radii smaller than a few micrometres are required to achieve a balance (surface tension of water at 298.25 K is X = 0.073), whereas the resonance behaviour of a bubbly liquid is sensitive up to 70 µm in a sodium-argon bubbly liquid (surface tension of liquid sodium at 773.25 K is X = 0.156).

Conclusion and future prospects

A damped-mass-spring two-phase Helmholtz resonator model was developed to reproduce both the linear and non-linear behaviour of a bubbly resonator with the goal to study the influence of the bubble cloud parameters (radius histogram, void fraction). This model has been validated in the linear domain by comparing its resonant behaviour with that predicted by Wood's analytical model. The model is also able to predict the dependence of the non-linear effects according to the cloud histogram and the void fraction.

The results of the model have been compared with experimental measurements performed on a 75 cL champagne bottle. The bubble generation system does not make it possible to control the void fraction and population histogram separately, or to generate sufficiently small bubbles to study the relative influence of each parameter in depth. However, our experimental validation was successful. In the case of linear behaviour, good quantitative agreement was found for the resonant frequencies. In the case of non-linear behaviour, the model qualitatively estimated the overall behaviour, e.g. the amplitude-dependant frequency shift (softening), as well as the time and frequency response shapes.

The future prospects for this work include finding a new bubble generation system to multiply the experimental measurements (notably with small bubbles, with radii lower than 15 µm) so the experimental data can be better compared with the model's results. Other prospects include more in-depth investigation of the bubble cloud's equivalent damping, even if its impact remains small on non-linear features, as well as of the accuracy of the model to describe the other nonlinear phenomena observed experimentally: the desymmetrisation of the temporal response, the harmonic energy transfer, and hysteresis.

Figure 1 :

 1 Figure 1: Two-phase Helmholtz resonator modelled as a damped mass spring system

Figure 2 :

 2 Figure 2: Microbubble population ( = 1 -15 /m and = 10 12 ) modelled as a damped mass spring system

  is the gaseous thermal diffusivity and l µ the liquid dynamic viscosity.As explained previously, to estimate the equivalent cloud damping, dampers corresponding to each single bubble of the th w class of the population are organised in a parallel combination dampers of the various classes of bubbles are organised in series .

Figure 3 :

 3 Figure 3: Experimental set-up

Figure 4 :

 4 Figure 4: Measured temporal response (a) and PSD (c) of the bottle filled with bubbly water. Analytical temporal response (b) and PSD obtained with the proposed model with the microbubble cloud parameters estimated using optical characterisation and Wood's model: R0 = 50-70 µm and τ = 8.10 -6

Figures 5 (

 5 Figures 5 (a-b) shows respectively the temporal and frequency responses for a bubble cloud of radii K = 1-15 µm and a void fraction of = 10 12 .

Figure 5 :

 5 Figure 5: Analytical results of the model's non-linear behaviour for R0 = 1-15 µm, = 10 12 :

Figure 6 :

 6 Figure 6: Analytical NRAS for a bubble population of radii R0 = 1-15 µm and a void ratio τ = [10 -2 , 10 -3 , 10 -4 , 10 -5 , 10 -6 , 10 -7 , 10 -8 ]: (a) resonance magnitude versus frequency, (b) relative resonance frequency versus amplitude of excitation for 16 pressure levels [500, 80000] Pa

Figure 7 :

 7 Figure 7: Analytical NRAS for τ = 10 -5 and R0 = [1, 4, 10, 15, 50, 100] µm: (a) resonance magnitude versus frequency, (b) relative resonance frequency versus excitation amplitude for 16 amplitudes [500, 80000] Pa

Table 1 :

 1 Linear resonance frequency predicted by the two-phase Helmholtz resonator model and with Wood's model

	Void fraction	10 1#	10 1x	10 12	10 1	1
	Our model for K = y1 -15z μ Our model for K = y1 -79z μ	383.0 Hz 383.0 Hz	379.3 Hz 379.5 Hz	226.1 Hz 224.3 Hz	28.1 Hz 27.6 Hz	87.8 Hz 87.8 Hz
	Wood model	382.3 Hz	378.2 Hz	215.2 Hz	26.1 Hz	87.6 Hz

This work has been funded by the French Atomic Energy and Alternative Energies Commission (CEA) at Cadarache. The authors would like to acknowledge Dr. J-B. Doc for his helpful advice and patience in explaining the dual softening/hardening behaviour, as well as Verralia France for sharing the industrial drawings of their champagne bottle.


Emilie Feisthauer
email: emiliefeisthauer@gmail.com

Audrey Farrugia

Alice Ameline

Laurie Gheddar

Nadia Arbouche

Jean-Sébastien Raul

Pascal Kintz

milieux biologiques comme marqueur d'une exposition au fentanyl ou à un analogue structural [Interest in testing for 4-ANPP in biological matrices as a precursor of exposure to fentanyl or to a fentanyl derivative]

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

(Article court)

Intérêt de rechercher la 4-ANPP dans les milieux biologiques comme marqueur d'une exposition au fentanyl ou à un analogue structural

Interest in testing for 4-ANPP in biological matrices as a precursor of exposure to fentanyl or to a fentanyl derivative E.Feisthauer 1, *, A. Farrugia 1 , A.Ameline 1 , L.Gheddar 1 , N. Arbouche 1 , J-S. Raul 1 , P.Kintz 1,2 1 Institut de médecine légale, 11 rue Humann, 67000 Strasbourg, France 2 X-Pertise Consulting, Mittelhausbergen, France Une méthode analytique par UHPLC-MS/MS (Xevo TQD) permettant de doser le fentanyl et ses métabolites (norfentanyl et 4-ANPP) a été utilisée. Le fentanyl a été dosé à 44 ng/mL dans le sang cardiaque et à 30 ng/mL dans le sang fémoral, ce qui tend à démontrer une possible redistribution post mortem avec un ratio de 1,5. Le norfentanyl est à 2 ng/ml et la 4-ANPP à 0.6 ng/mL dans les 2 échantillons de sang.

On retrouve également le fentanyl, le norfentanyl ainsi que la 4-ANPP dans les autres échantillons : urine (20, 2 et 0,2 ng/mL), bile (112, 10 et 1,6 ng/mL), contenu gastrique (101, 1 et 0.3 ng/mL) et enfin cheveux (652, 92 et 9,5 pg/mg).

Les concentrations de fentanyl sont supérieures aux concentrations thérapeutiques usuelles et sont potentiellement létales. Les faibles concentrations de norfentanyl sont en faveur d'un décès rapide. La présence de 4-ANPP dans tous les prélèvements confirme son statut de marqueur d'exposition aux dérivés du fentanyl.

Mots clés :

Fentanyl, Norfentanyl, 4-ANPP, marqueur, intoxication, nouveaux opiacés de synthèse (NOS).

Summary :

In the context of fatal fentanyl poisoning, the authors aimed to verify that 4-ANPP (4anilino N-phenethyl-piperidine) is found in several biological matrices of forensic interest. In France, fentanyl is used in hospitals and its misuse is mainly limited to health personal. To escape legal regulations, fentanyl is one of the most copied molecules, in the family of new psychoactive substances (NPS), more than fifty analogues do exist. To overcome the impossibility of maintaining all standards in the laboratory, the targeted research of 4-ANPP seems to be a suitable approach for screening for this class of narcotics. A 36-year-old man was found dead at his home, and after investigation, the individual could have used a fentanyl patch by applying it under his tongue. This man was known as user of narcotics. No traumatic cause of death was found during the autopsy, and the Public Prosecutor requested a reference toxicological expertise. An analytical method by UHPLC-MS/MS (Xevo TQD) to test for fentanyl and its metabolites (norfentanyl and 4-ANPP) was used.

Fentanyl concentration was 44 ng/mL in cardiac blood and 30 ng/mL in femoral blood, which tends to demonstrate a possible postmortem redistribution, with a ratio of 1.5. Norfentanyl concentration was 2 ng/ml and 4-ANPP concentration was 0.6 ng/mL in all 2 blood samples. Fentanyl, norfentanyl and 4-ANPP were also found in the other samples, inluding urine (20, 2 and 0.2 ng / mL), bile (112, 10 and 1.6 ng / mL), gastric contents (101, 1 and 0.3 ng / mL) and hair (652, 92 and 9.5 pg / mg). These fentanyl concentrations are higher than usual therapeutic concentrations and are potentially lethal. Low concentrations of norfentanyl are in favor of rapid death.

The presence of 4-ANPP confirms its status as a marker of exposure to fentanyl derivatives.

Key-words: Fentanyl, Norfentanyl, 4-ANPP, precursor, intoxication, New synthetic opioids.

Introduction :

En 2019, Salomone et al publiaient que la 4-ANPP (4-anilino N-phenethyl-piperidine), précurseur et métabolite de nombreux dérivés du fentanyl (figure 1), pouvait servir de marqueur d'exposition dans les cheveux à cette classe de molécules ultra-actives.

Présentation du cas

Un homme de 36 ans est retrouvé décédé à son domicile. D'après des témoins, l'individu aurait consommé un patch de fentanyl en l'appliquant sous la langue. Cette victime était connue comme consommateur de produits stupéfiants. Lors de l'autopsie, aucune cause traumatique n'a été constatée. Le procureur a ainsi requis une expertise toxicologique de référence. Les matrices classiques ont été prélevés (sang fémorale, sang cardiaque, urine, bile, contenu gastrique, cheveux noirs de 4-5 centimètres, en vrac). À ce jour, la provenance du patch de fentanyl consommé par la victime n'est pas encore déterminée. Le patch étant dégradé, nous n'avons pas pu déterminer le dosage de fentanyl.

Extraction des liquides biologiques

Les différents liquides biologiques (sang, urines, bile et contenu gastrique) sont analysés dans les mêmes conditions et avec une méthode validée. [START_REF] Chetraru | Décès impliquant un surdosage de fentanyl par diversion d'un dispositif transdermique. A propos d'un cas original avec mastication[END_REF] Le fentanyl et ses métabolites sont extraits dans 1 mL de matrice en présence de 20 ng de standards internes (fentanyl-D5 et norfentanyl-D5), dans 1 mL de tampon borate saturé à pH = 9.5 et 5 mL mélange de diethyl éther/dichlorométhane/n-hexane/alcool isoamilyque (50:30:20:0.5). Après agitation (15 minutes), centrifugation (15 minutes, 3000 rpm) et évaporation à sec, l'extrait est reconstitué avec 30 µL de tampon ammonium formate ajusté pH 3.0.

Préparation des cheveux et méthode d'extraction

Le fentanyl et ses métabolites ont également été analysés dans les cheveux. Lors de l'autopsie, les cheveux du défunt ont été prélevé en vrac ; ils sont de couleur noire et de longueur 4-5 cm, environ.

Le fentanyl, le norfentanyl et la 4-ANPP, sont extraits dans 20 mg de cheveux décontaminés (2 x 5 mL de dichlorométhane). Après décontamination, les cheveux ont été découpés en segments fins (1mm) avec des ciseaux. Les cheveux sont incubés toute une nuit dans 1 mL de tampon borate pH 9,4 en présence de 200 pg de standard interne (fentanyl D5 + norfentanyl D5). Après incubation, on ajoute dans la préparation 5 mL de mélange de diethyl éther/dichlorométhane/n-hexane/alcool isoamilyque (50:30:20:0.5). Après agitation (15 minutes), centrifugation (15 minutes, 3000 rpm) et évaporation à sec, l'extrait est reconstitué avec 30 µL de tampon ammonium formate ajusté pH 3.0.

Analyses en UPLC-MS/MS

Résultats et discussion :

La demande du Procureur de la République mène les auteurs à analyser toutes les matrices disponibles afin de rechercher une cause de décès. Les analyses ont mis en évidence une prise récente de cannabis, de morphine et de fentanyl. La méthode « fentanyl » étant dédiée a plus d'une vingtaine d'analogues, leur absence confirme que seul le fentanyl a été utilisé. L'alcoolémie était nulle. La morphine (11 ng/mL) et le cannabis (THC : 2,4 ng/mL ; 11-OH-THC : 3,4 ng/mL ; THC-COOH : 24,5 ng/mL) présentent des concentrations non létales. Les concentrations de fentanyl se sont révélées être potentiellement létales par dépression respiratoire, même pour un consommateur régulier. (Tableau 1) [START_REF] Randall | Disposition of toxic drugs and chemicals in man[END_REF] Dans le sang fémoral, la concentration obtenue de fentanyl est de 30 ng/mL. Elle est de 44 ng/mL pour le sang cardiaque. Ainsi, une possible redistribution post mortem est à envisager avec un ratio de 1,5. Un ratio de 1,6 a été rapporté dans la littérature.

[3] Les patchs de fentanyl sont commercialisés aux doses suivantes 12, 25, 50, 75 et 100 µg/H. Les concentrations thérapeutiques de fentanyl rapportées dans la littérature, varient entre 0,3 µg/L à 3,8 µg/L. [START_REF] Randall | Disposition of toxic drugs and chemicals in man[END_REF] Dans ces mêmes prélèvements, les concentrations de norfentanyl sont de 1,7 ng/mL pour le sang fémoral et 1,5 ng/mL pour le sang cardiaque. La transformation du fentanyl en norfentanyl s'effectue par l'intermédiaire du cytochrome P450 (CYP) 3A4 et est donc dépendant d'un éventuel polymorphisme génétique. Les concentrations de fentanyl et de norfentanyl ont été rapportés par plusieurs auteurs. Une première étude présente les concentrations thérapeutiques retrouvées chez 11 patients sous traitement par patch. La concentration moyenne est de 0,785 ng/mL pour le fentanyl et de 0,531 ng/mL pour le norfentanyl soit des valeurs très superposables. [START_REF] Bista | Development, validation and application of an HPLC-MS/MS method for the determination of fentanyl and nor-fentanyl in human plasma and saliva[END_REF] En cas de décès, si la concentration de fentanyl est peu élevée, celle du norfentanyl est proche : 5,6 et 5,9 ng/mL, respectivement. [START_REF] Teske | Fatal outcome in a child after ingestion of a transdermal fentanyl patch[END_REF] Au contraire, si la concentration de fentanyl est très élevée, celle de norfentanyl est plus faible, le décès survenant rapidement. C'est ce qui a été observé dans notre cas, mais aussi dans le travail de Coopman et al [START_REF] Coopman | LC-MS/MS analysis of fentanyl and norfentanyl in a fatality due to application of multiple Durogesic transdermal therapeutic systems[END_REF], avec des concentrations de 28,6 et 3,0 ng/mL. Les concentrations retrouvées pour

[ 1 , 2 , 3 ]

 123 Dans le cadre d'une intoxication fatale au fentanyl sur un homme âgé de 36 ans, les auteurs ont voulu vérifier que la 4-ANPP, pouvait également être retrouvée dans d'autres matrices biologiques d'intérêt médico-judiciaire. En France et plus généralement en Europe, le fentanyl et ses analogues structuraux sont moins fréquemment responsable de décès qu'aux Etats Unis. [4,5] Dans les pays européens, le fentanyl est principalement utilisé comme analgésique chez les patients cancéreux. Son usage détourné est surtout limité au personnel hospitalier, mais peut s'observer par diversion de patchs (Durogésic). [6] Pour échapper aux régulations légales, le fentanyl fait partie des molécules les plus copiées et l'on retrouve, dans la grande famille des nouvelles substances psychoactives (NPS), plus d'une cinquantaine d'analogues, dont le carfentanil, le furanylfentanil [7,8] ou encore l'ocfentanil. [9,10] Face à cette prolifération et à la quasi-impossibilité pour un laboratoire de disposer de tous les standards de référence, la recherche ciblée des métabolites et plus spécifiquement de la 4-ANPP semble être un excellent moyen de dépistage pour cette classe de narcotiques. [11,12,13] Matériels et Méthodes : Matériels L'Ammonium formate (99%), et le di-sodium tetraborate decahydraté proviennent de chez VWR (Louvain, Belgique). L'acide formique (99-100%), de qualité LC-MS, l'acétonitrile (ACN), le dichlorométhane, le méthanol, et l'hexane proviennent de chez VWR (Louvain, Belgique). L'alcool isoamylique SUPELCO provient de chez Merck (Molsheim, France). Les standards de fentanyl, norfentanyl, fentanyl-D5 et norfentanyl-D5 ont été achetés chez Lipomed (Arlesheim, Suisse). La 4-ANPP a été obtenue chez LGC (Molsheim, France)

  Le fentanyl, ses métabolites et les standards internes ont été analysés par chromatographie en phase liquide ultra haute performance couplée à un spectromètre de masse en tandem (Waters XEVO TQD). La séparation est effectuée sur une colonne HSS C18 Acquity UPLC Waters (1,8 µm, 2,1 x 150 mm) maintenue à 50°C. Un gradient d'élution a été réalisé à l'aide d'un tampon ammonium formate ajusté à un pH 3.0 (Phase mobile A1) et d'acétonitrile + 0.1% acide formique (Phase mobile B1). Le débit est à 0.4 mL/min. Le gradient initial est de 13% de phase organique B et le gradient final, à 12.25 min, est de 95.0% de phase organique B. Le volume d'injection est de 2 µL. Un Xevo TQD triple quadripôle est utilisé pour la détection en opérant en mode electrospray positif (ESI+), Le capillaire est optimisé à 1.50 kV et la température de la source est de 150°C. Le gaz de désolvatation est chauffé à 600°C et est délivré à un débit de 1L/h. L'énergie de collision est ajustée pour obtenir un signal optimisé sur les deux meilleures transitions. Les transitions retenues sont les suivantes : pour le fentanyl : m/z 337.1 > 188.0 (48 V ; 38 eV) et m/z 337.1 > 105.0 (48 V ; 38 eV). Le norfentanyl et la 4-ANPP ont respectivement les transitions suivantes : m/z 233.0 > 83.9 (30 V ; 18 eV) et m/z 233.0 > 56.2 (30 V ; 18 eV) ainsi que m/z 280.9 > 104.9 (34 V ; 30 eV) et m/z 280.9 > 188.0 (34 V ; 30 eV). Pour quantifier le fentanyl et la 4-ANPP, le standard interne utilisé est le fentanyl-D5, ses transitions sont les suivantes : m/z 342.1 > 188.1 (26 V ; 20 eV) et m/z 342.1 > 105.0 (26 V ; 48 eV). Enfin pour quantifier le norfentanyl, le standard interne utilisé est le norfentanyl-D5, sa transition est la suivante m/z 238.0 > 83.9 (50 V ; 32 eV). Le logiciel utilisé est MassLynx 4.1.

la 4 -

 4 ANPP sont les suivantes : 0,6 ng/mL pour le sang fémoral et 0,7 ng/mL pour le sang cardiaque. Les trois molécules d'intérêt (fentanyl, norfentanyl et 4-ANPP) ont également été retrouvées dans les autres liquides biologiques (urines, contenu gastrique et bile). Dans les urines, le fentanyl est retrouvé à une concentration de 20 ng/mL, le norfentanyl à 2 ng/mL et enfin la 4-ANPP à 0.2 ng/mL. Pour le contenu gastrique et la bile, les concentrations sont respectivement les suivantes : fentanyl (101 ng/mL, 113 ng/mL), norfentanyl (1 ng/mL, 10 ng/mL) et 4-ANPP (0.3 ng/mL, 1.6 ng/mL). Un dosage dans les cheveux a également été effectuée. Avec une moyenne de pousse d'un centimètre par mois environ et une longueur de 4-5 cm, l'analyse des cheveux permettra d'avoir une visibilité de la consommation de fentanyl sur les 5 derniers mois. Le fentanyl, le norfentanyl et la 4-ANPP ont été retrouvés dans les cheveux. Le fentanyl est présent à une concentration de 652 pg/mg, le norfentanyl a une concentration de 92 pg/mg e la 4-ANPP est présente à une concentration de 9,5 pg/mg. Selon Salomone et al, la moyenne des concentrations de fentanyl retrouvées dans les cheveux pour au moins une exposition est de 3 pg/mg. Dans le cas de la 4-ANPP, la moyenne des concentrations était de 3 pg/mg également. [1] Au vu de la littérature et des résultats obtenus pour le fentanyl et la 4-ANPP, nous pouvons émettre l'hypothèse que le défunt était un consommateur régulier de fentanyl sur les 5 derniers mois avant son décès. Au total, la 4-ANPP a été retrouvée dans l'ensemble des prélèvements biologiques. Les résultats obtenus pour la 4-ANPP dans notre cas confirment les écrits de la littérature mettent en avant l'intérêt de cette molécule dans l'identification future des nouveaux opioïdes de synthèse. Dans la littérature, des cas de décès au furanyl fentanyl sans fentanyl ont également été rapportés, la 4-ANPP a été identifiée confirmant son statut de marqueur de groupe. [1,7,8] Conclusion : Dans cette étude, les concentrations de fentanyl retrouvées sont supérieures aux concentrations thérapeutiques usuelles et sont potentiellement létales par dépression respiratoire, chez un sujet consommateur régulier de cette molécule. Les faibles concentrations de norfentanyl tendent en faveur d'un décès rapide de l'individu après consommation. Retrouvée dans toutes les matrices biologiques, la 4-ANPP confirme son statut de marqueur d'exposition aux dérivés du fentanyl. Quelques observations complémentaires au laboratoire sur d'autres dérivés du fentanyl ont confirmé cette approche, le concept étant qu'un criblage positif de la 4-ANPP donne lieu à des investigations très poussées pour identifier la source. Déclaration d'intérêts : Les auteurs déclarent ne pas avoir de conflit d'intérêt en relation avec cet article.

Figure 1 .

 1 Figure 1.

  Tableau 1. Tableau des résultats du fentanyl, norfentanyl et 4-ANPP dans les différentes matrices

	Matrices	Concentration en Fentanyl	Concentration en Norfentanyl	Concentration en 4-ANPP
	Sang fémoral (ng/mL)	29,6	1,7	0,6
	Sang cardiaque (ng/mL)	44,4	1,5	0,7
	Urines (ng/mL)	20,1	2,1	0,2
	Bile (ng/mL)	112,5	10,3	1,6
	Liquide gastrique (ng/mL)	101,1	1,1	0,3
	Cheveux (pg/mg)	651,9	91,7	9,5