Karim Aderghal
email: aderghal.karim@gmail.com

Karim Afdel
email: k.afdel@uiz.ac.ma

Jenny Benois-Pineau
email: jenny.benois-pineau@u-bordeaux.fr

Gwénaëlle Catheline
email: gwenaelle.catheline@u-bordeaux.fr

Improving Alzheimer's stage categorization with Convolutional Neural Network using transfer learning and different magnetic resonance imaging modalities

Keywords: Alzheimer's Disease, Magnetic Resonance Imaging (MRI), Diffusion Tensor Imaging (DTI), Multi-modality, Image classification, Convolutional Neural Network (CNN), Transfer Learning

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

$ Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/ how_to_apply/ADNI_Acknowledgement_List.pdf.

Introduction

Alzheimer's disease (AD) is a major worldwide public health priority [START_REF] Association | Alzheimer's disease facts and figures[END_REF], it is a progressive degeneration that devastates cells in the human brain, and causes dementia for elderly individuals, mostly for people aged 65 or older. Alzheimer's patients suffer from shortterm memory loss, deterioration of mood, behavior, and ability to continue daily activities. Globally, there are around 44 million people who have been diagnosed with AD in the world [START_REF] Lane | Alzheimer's disease[END_REF], and this number may reach to 131.5 million people in 2050 [START_REF]Alzheimer's Disease International[END_REF]. In USA 5.7 million individ-uals are living with AD in 2018, and this number will approximately triplicate by 2050 [START_REF] Association | Alzheimer's disease facts and figures[END_REF]. Recent statistical studies estimate 900.000 persons diagnosed with AD in France [START_REF] Lane | Alzheimer's disease[END_REF]. Mild Cognitive Impairment (MCI) is considered as the transitional stage where subjects have the risk to convert to AD [4]. There is no known cure for AD, however specific treatments including occupational individual therapy can make it easier for people to live with the disease [5], and that is why it is very important to diagnose and detect affected patients in the early stage of MCI before progression of the disease. MCI patients may remain stable, regress to a normal condition (s-MCI), or may develop dementia due to AD considered converters (c-MCI).

Monitoring and observing brain structural changes in AD subjects has been well studied, for a long time the structural magnetic resonance imaging (sMRI) has been the most used imagery technique for diagnostics and analysis of progressive brain deterioration. It permits tracking structural changes in the brain and measuring the inevitable atrophy caused by the neurodegenerative aspect of the AD pathology [5]. Most studies use the structural MRI to explore the deterioration process of some Regions-of-Interest (ROIs) in the brain, known to be affected by AD such as the hippocampus. The latter is very sensitive to be affected first [START_REF] Deweer | Memory disorders in probable Alzheimer's disease: the role of hippocampal atrophy as shown with MRI[END_REF][START_REF] Dickerson | MRI-derived entorhinal and hippocampal atrophy in incipient and very mild Alzheimer's disease[END_REF]. Temporal and cingulate gyri, and precuneus are amongst other affected structures [START_REF] Braak | Neuropathological stageing of Alzheimerrelated changes[END_REF]9].

However, sMRI scans have insufficient contrast to delineate micro-structural changes due to AD. For an accurate measuring of the white matter integrity, diffusion Tensor Imaging (DTI) is an MRI technique based on the motion of water molecules in brain tissues. It quantifies the random water diffusion in cerebral gray and white matter tracts. The interpretation of diffusion tractography of water infers the tissue architecture continuity of the fiber bundles in brain from voxel to voxel [10,11]. For diagnostic tasks, Mean Diffusivity (MD) and Fractional Anisotropy (FA) are the main density maps derived from DTI data. MD describes the magnitude of self-diffusion of water regardless of direction, while FA is used to characterize the degree of anisotropy of the diffusion [11]. In the AD patients studies, the computed DTI maps have shown an abnormal increase of MD values and decrease of FA values both in gray and white matter, and in some regions including Hippocampus. Quantifying the neuro-degeneration in the Hippocampus area leads to measure the pathological atrophy changes and discriminate AD subjects from those with Mild Cognitive Impairment (MCI) or those in Normal Control (NC) [START_REF] Mielke | Regionally-specific diffusion tensor imaging in mild cognitive impairment and Alzheimer's disease[END_REF].

Deep learning approaches, and specifically Convolu-tional Neural Networks (CNN) have recently emerged as a powerful analysis approach for Computer-Aided Diagnosis (CAD) systems. Various CNN-based frameworks have been used in medical image analysis for disease classification, segmentation, detection and shape modeling problems [START_REF] Dolz | 3D fully convolutional networks for subcortical segmentation in MRI: A large-scale study[END_REF]. Due to their remarkable performances, there exists a wide range of applications for AD classification using multi-modal imaging [14]. However, they require a large number of data samples to well train a network. If the training dataset is not sufficiently large, then the over-fitting phenomena occurs [START_REF] Srivastava | Dropout: A simple way to prevent neural networks from overfitting[END_REF]. This happen because the depth of the dataset is not representative of the variability of brain structures, and consequently the classification results on train set are better while on the test set are not, whereas they should be the same when the over-fitting is absent. Specific data augmentation and regularization methods such as Dropout can solve the problem of insufficiency of samples [START_REF] Srivastava | Dropout: A simple way to prevent neural networks from overfitting[END_REF]16]. An alternative approach called "Transfer Learning" can battle the overfitting phenomena, which consists in reusing the weights of a trained model implemented for different application domain. This approach achieves good results compared to those trained only with an arbitrary initialization of network parameters ("from scratch") [START_REF] Yosinski | How transferable are features in deep neural networks?[END_REF]. Hence, numerous works use models pretrained on large publicly available datasets, such as LeNet [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], VGGNet [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF], GoogleNet [START_REF] Szegedy | Going deeper with convolutions[END_REF], and ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] which were built for various image classification tasks. For example, cross-domain transfer learning using the AlexNet architecture has been reported in various medical applications, resulting an improved performance [START_REF] Sivaranjini | Deep learning based diagnosis of Parkinson's disease using convolutional neural network[END_REF][START_REF] Tajbakhsh | Convolutional neural networks for medical image analysis: Full training or fine tuning?[END_REF]. Nevertheless, they remain limited in case of considerably small datasets as it is the case in AD studies: merely few scans from both sMRI and DTI modalities are available for numerous cohorts in a public ADNI dataset [START_REF] Frisoni | The clinical use of structural MRI in Alzheimer disease[END_REF] or in particular cohorts such as "Bordeaux-3City cohort"

[26]. This paper is an extension of previous works [START_REF] Aderghal | Classification of Alzheimer's Disease on Imaging Modalities with Deep CNNs Using Cross-Modal Transfer Learning[END_REF]16,[START_REF] Aderghal | FuseMe: Classification of sMRI images by fusion of Deep CNNs in 2D+ε projections[END_REF] where we proposed a method that combines sMRI and DTI-MD imaging modalities focusing only on the hippocampal region. Moreover, we presented a fusion framework based on the concept of using multiple sources, including cross transfer learning approach. The method was much more promising, not only in improving the accuracy of classification, but also reducing the problem of overfitting. The originality of the approach allows to measure improvements of various techniques on small datasets. Moreover, the dataset is even smaller since we use just few 2D slices from a reduced region of the brain -the hippocampus. These double constraints lead us to deal with this challenge by com-bining various techniques, cross-modality being the one in focus in this paper.

In the present work we propose an adapted transfer learning scheme based on two processes: first, we implement an inter-modal approach using naturally similar dataset (sMRI and DTI-MD). Second, we carry out the knowledge transfer between two different datasets with non medical data, and use a pretrained model which is very similar to ours. As result, we show that such a transfer learning between close domains, even with a limited number of samples in source domain and shallow network, is more efficient than a plain cross-domain learning when the domains are too different.

The remainder of this paper is organized as follows: In section 2, we review related work, next in section 3, we describe the data preprocessing and our proposed approach in details. Then in section 4, we present the experimental results, and finally we conclude our work and outline its perspectives.

Related work

An improved classification performance of famous CNN models such as AlexNet, VGGNet, GoogleNet, and ResNet with transfer learning has been reported in various applications in medical domain. There are some recent related works relevant to our methodologies using the 2D-based CNN classification instead of working with the 3D or 4D imaging. In the follow-up, we focus on the works which use these known networks and adapt them to the medical image classification, and on the application of transfer learning method as a solution for the limited volume of datasets.

Works based on popular CNN networks

In [29,30,31] the authors presented an adaptive pipeline approach for Alzheimer's Disease binary classification. They used the structural and functional magnetic resonance imaging (sMRI, fMRI) modalities from the ADNI dataset project. They have designed and adopted architectures based on the famous networks LeNet [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] and GoogleNet [START_REF] Szegedy | Going deeper with convolutions[END_REF] and adjusted them to binary classification problem for medical imaging data. In the case of sMRI modality, they converted the images from 3D data to a stack of 2D slices and resized them to 28 × 28 pixels to match the input to LeNet network. The average accuracy for 2-Way (AD/NC) classification was 96.85%. In the second setup, they used the adopted GoogleNet model for binary classification, and the highest overall accuracy rate achieved was 98.84%. With the same concept of reusing an efficient architecture, authors in [START_REF] Billones | DemNet: A Convolutional Neural Network for the detection of Alzheimer's Disease and Mild Cognitive Impairment[END_REF] used a modified version of VGGNet [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] network called DemNet that takes 2D images as input. It was composed of 13 convolutional layers and of three fully connected layers with dropout after each pooling layer, to reduce the over-fitting. The neural network took as an input a 2D slice from MRI data with 224 × 224 resolution for both 2-way and 3-way classification problems. The authors selected 20 slices for each brain, classified each of them and measured the accuracy. The results showed that the first and the last two slices (111, 129, and 130) had significantly lower accuracy than the average accuracy per slice. They have achieved an overall accuracy of 98.33% for AD/NC classification. In [START_REF] Valliani | Deep Residual Nets for Improved Alzheimer's Diagnosis[END_REF], the authors used two networks, a baseline single-layer CNN and a pretrained ResNet network, they used a single 2D axial slice per subject (median slice from the 3D volume) as an input, the baseline CNN network was composed of only one convolutional layer and two FC layers. They studied the impact of transfer learning from RestNet trained on ImageNet, and the data augmentation in a real time, at training phase, they conclude that the ResNet architecture successfully fits to the MRI domain, and pretraining with data augmentation improves the prediction. In a recent study, Bumshi et al. [START_REF] Lee | Using Deep CNN with Data Permutation Scheme for Classification of Alzheimer's Disease in Structural Magnetic Resonance Imaging (sMRI)[END_REF] used a modified AlexNet network which is known as a high-performance pretrained model. The architecture is composed of five layers for convolution computation and a last FC layer with two outputs for binary classification AD/NC, or three outputs for 3-way classification (AD/MCI/NC). Due to the presence of some noisy in the dataset, they proposed a data permutation scheme with outlier rejection, and slice selection methods by removing pixels to eliminate interfering data. All 2D slices (obtained by permutation from axial, sagittal, and coronal planes) were used for training the network, and finally the network was finetuned using OASIS and ADNI datasets. They achieved 98.74%, and 95.35% accuracy in AD/NC classification task, respectively on the OASIS and ADNI datasets. For 3-way classification their method achieved 98.06% on ADNI 3T dataset.

Works based on a transfer learning approach

Overall, the more the architecture is deep the more it needs huge data, and consequently a considerable time for training. The crucial issue here is that, in general, data in medical area are not sufficiently available. For this reason, numerous research works have used transfer learning approach, whether based on popular networks or on novel methods to address the lack of data.

In [START_REF] Glozman | Hidden Cues: Deep Learning for Alzheimer's Disease Classification CS331B project final report[END_REF] authors used AlexNet architecture pretrained on a general purpose large-scale ImageNet dataset to fine-tune last layers in the model on target sMRI and PET modalities. In the conclusion of their work, the authors state that as neuroimaging data differs significantly from the source domain data, such a transfer method is not optimal. Indeed, the accuracy values achieved for classification of AD/NC on sMRI modality are rather low (around 66%) and even worse when AD/MCI/NC classification problem is addressed. Another study [START_REF] Wang | Automatic Recognition of Mild Cognitive Impairment from MRI Images Using Expedited Convolutional Neural Networks[END_REF] employs the transfer learning on the same kind of images, such as brain scans from the same modality (sMRI) but with the source database different from target data to deal with the limited target data to recognize MCI on MRI images. Here two different datasets OASIS1 and LIDC2 have been used for the pretraining stage. They achieved best performance with accuracy of 90.6% for MCI/NC. [START_REF] Cheng | Multi-domain transfer learning for early diagnosis of alzheimer's disease[END_REF] integrates a method called Multi-Domain Transfer Feature Selection (MDTFS) to select discriminant features for classification of AD/NC. In their case, an auxiliary domain corresponds to classification problem on the same data but for different target classes. When classifying AD/NC they consider s-MCI/c-MCI and MCI/NC. Their experiments have been conducted on ADNI-1 sMRI and classification accuracies achieved were of 95.2% and 82.1% for AD/NC, and MCI/NC respectively. The DTI image maps are often seen as a good modality for the detection of Alzheimer's disease. Thus the authors in [38] have compared the NC, AD and MC using MD and FA maps. Their results showed that MD was a better indicator of brain atrophy than FA. In a recent work, Ahsan et al.

[39] proposed multiple deep 2D neural networks for binary AD/NC classification. They introduced two architectures that use the transfer learning approach based on the InceptionV3 and Xcepetion models whose weights are pretrained on Imagenet LSVRC. In addition, a custom CNN network is built with the help of separable convolutional layers. They used 96 central 2D-slices from each subject's brain by ignoring the first and the last 40 outer slices. They used three datasets with different settings from the OASIS project, which are composed of 416 T1-weighted MRI scans. The two datasets are respectively (i) a balanced with 180 and (ii) unbalanced sets with 114 subjects. The third dataset is the one used in the work of [40]. The authors used different configurations of fold cross-validation over the three datasets for the experiments. They achieved an average accuracy of 64% on the first dataset, 82.79% on the second dataset, and 99.45% on the third dataset for AD/NC classification. The latter is doubtful as they used the dataset from [40], and it was mentioned that there could be a problem of leakage in this dataset [START_REF] Wen | Convolutional neural networks for classification of alzheimer's disease: Overview and reproducible evaluation[END_REF]. They concluded that the transfer learning approaches outperform non-transfer learning-based approaches. This demonstrates the effectiveness of these approaches for the binary AD classification task.

Contrarily to all these approaches we propose a transfer learning scheme from one modality to another, on the same dataset. Indeed the sMRI modality shows good discrimination performance for AD diagnosis in brain atrophy analysis, it is our source modality. The supplementary modality is of the same nature, it is the DTI-MD modality which represent our target domain. Otherwise, in our previous work [START_REF] Aderghal | FuseMe: Classification of sMRI images by fusion of Deep CNNs in 2D+ε projections[END_REF] we have shown that multimodal approach increases performances, hence it is interesting to explore if the fusion framework can be more efficient with a transfer learning. We remain using a shallow CNN-based architecture with only a small number of convolutional/pooling layers since the input region definition is rather low (28×28×28).

The contributions of the paper are as follows:

• we transfer knowledge between sMRI and DTI modalities using a shallow architecture specifically designed for our "2-D+ε" approach.

• we use a similar architecture, LeNet trained on a large set (60K) of images from MNIST 3 which has (28×28) resolution as input, i.e. the same size as of our Hippocampal region-of-interest, and then finetune this model on ADNI dataset.

Hence in both cases we perform the "cross" transfer, therefore in the first case, it is a cross-modality transfer. The modalities are similar as the target pathology is expressed by the same image deformations yet in the opposite luminance. In the second case, it is a crossdomain transfer. The domains are different; character images (MNIST) have nothing to do with MRI scans except the fact that they are not coloured. In the next section we present our approaches.

Methodology and approach

Acquisition method and Data selection

We work with two brain imaging modalities: sMRI and DTI. We briefly describe these modalities, and we also present our selected ADNI dataset that we use in the preparation of this work.

structural MRI and DTI modalities

For a long time, the structural MRI remains a popular technique for examining the anatomy and the pathology of the brain, it is a widely used imaging technique in research as well as in clinical practice. This modality provides information describing the shape, size, enhancement patterns and integrity of gray and white matter. In AD diagnosis, High-resolution T1-weighted sequences are used to distinct the anatomical boundaries and to detect structural changes in the brain [5]. Diffusion Tensor Imaging (DTI) is another modality used in AD studies, it is a recent imaging technique that is able to track and quantify water diffusion along fiber bundles, to detect and describe the anisotropy on surrounding tissue micro-structure [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF]. Three eigenvalues (λ 1 , λ 2 and λ 3 ) and eigenvectors are calculated from water's molecular motion tensor in a three-dimensional space to represent the main diffusion directions [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF]. Fractional anisotropy (FA) and mean diffusivity (MD) are the two most common measures (scalar maps) derived from the diffusion tensor imaging. In case of the brain degeneration related to AD, the cerebro-spinal fluid fills-in the cavities and these effects are perceived both in FA and MD maps as the motion of water molecule becomes chaotic.

• Mean diffusivity: Mean diffusivity is an inverse measure of membrane density. It represents the average magnitude of molecular displacement by diffusion, as it informs on the microstructure of (WM) being sensitive to cell density, axon size, and quantities of water. MD is the average of the three eigenvalues (λ 1 + λ 2 + λ 3 )/3. A higher MD value reflects more isotropy of the tissue.

MD = λ = λ 1 + λ 2 + λ 3 3 (1)
• Fractional anisotropy: Fractional anisotropy is a measure of the degree of diffusion anisotropy. It reflects the directions of molecular motion in a certain voxel. FA is calculated from the standard formula:

FA = 3 2 (λ 1 -λ ) 2 + (λ 2 -λ ) 2 + (λ 3 -λ ) 2 λ 2 1 + λ 2 2 + λ 2 3 ( 2
)
Where λ is the mean diffusivity (MD). The values of FA vary between 0, which means the voxel space is totally isotropic, and 1, which means infinite anisotropic diffusion. In CSF, the value of FA is zero due to the equality of the diffusion in all directions.

Brain imaging data: ADNI Dataset

Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer's disease (AD).

Since we work with multimodal imaging, we consider two subsets of the whole data from ADNI. The first one is the data that were selected from the ADNI-1 screening baseline with only anatomical MRI T1weighted sequences, in this set all subjects underwent whole-brain MRI scanning on 1.5 Tesla at 14 acquisition sites. It is the same dataset as used in [16]. With the same demographic information for each of the diagnosis groups (NC, AD and MCI), The data sample consists of 815 structural MRIs including 188 Alzheimer's Disease (AD) patients, 228 cognitively normal (NC) and 399 subjects with Mild Cognitive Impairment (MCI). The second subset includes images of subjects screened with both structural MRI and DTI modalities. It is a union of data from the ADNI-2&Go, and ADNI-3. The table 1 presents demographic characteristics of subjects, including age, gender, and the Mini Mental State Examination (MMSE) score. The age of different groups ranges between 54 and 95 years old, and the proportions of male and female are close in AD/NC groups while the proportions of male are higher than female in MCI groups. We carried out a visual checking step of all sMRI images and DTI-MD maps to ensure the data quality, and exclude scans with excessive motion and/or artifacts. The Mini-Mental State Examination (MMSE) was used to provide a global measure of cognitive status [43].

Image preprocessing

Different imaging modalities of the same subject have to refer to the same physical structures in the brain. For this we need to align and normalize them, but an important preliminary step is noise removal which has not been fulfilled on all subsets of ADNI data we use.

Noise correction

Depending on the equipment used in the data acquisition, the multimodal imaging data are recorded as raw, unprocessed, unthresholded, and noisy. Resulting images usually have artifacts caused by the magnetic susceptibility in biological tissues. According to the ADNI project, in the ADNI-3 phase two main preprocessing steps are to be completed for each subject, the eddy current and N4 Bias Field correction, while in ADNI-GO and ADNI-2, this correction has been already performed. We shortly describe these steps. The DTI scans were first corrected from the proprietary scanner formats, by using the eddy current (Foucault currents) correction tool of FSL (Version 5.0, FMRIB, Oxford, UK, 4 ). This correction consists in compensating nonlinear susceptibility [44] and eddy-current distortions induced by head motion [START_REF] Andersson | An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging[END_REF]. Then the mean of the nondiffusion-weighted (b0) volumes was used to create the DTI brain mask via the FSL "BET" utility [46]. Fitting of diffusion tensors to the data with "DTIfit" module of the Software Library FSL was the last step to generate MD maps [47].

Regarding MRI scans, bias field correction was performed to minimize the influence of the bias field signal obscuring gray/white matter boundaries [48]. Here N4ITK correction methodology has been used from the Advanced Normalization Tools (ANTs) [START_REF] Avants | Advanced normalization tools (ANTS)[END_REF][START_REF] Tustison | N4ITK: improved N3 bias correction[END_REF]. Note that in ADNI-3, and due to the modern MRI scanners, there is no need to provide MPRAGE / IRFSPGR prepossessing 5 .

Alignment and Normalization

We work on AD bio-marker, i.e. the hippocampal region. To select it in the brain scans, we need to superimpose on the anatomical atlas AAL [51]. Thus we need to align sMRI modality on the common template 4 http://www.fmrib.ox.ac.uk/fsl/ 5 http://adni.loni.usc.edu/ MNI [START_REF] Frisoni | Structural correlates of early and late onset Alzheimer's disease: voxel based morphometric study[END_REF] constructed at Montreal Neurological Institute (MNI) which corresponds to AAL. The MNI template is built by averaging 152 scans of normal subjects. Thus each T1-weighted anatomical (sMRI) scan was spatially aligned with MNI as illustrated in Fig. 1 The process is based on the search for 12 affine deformation parameters (m 1 to m 12 ) of the matrix M which performs the homogeneous transformation. The goal is to estimate these parameters for a given image (f) to fit a template image (g), including translation, rotation, scaling and shearing deformations [54]. In order to guide finding appropriate parameters, the algorithm rely on using objective function to minimize transformation error, e.g. (MSE) mean square error criterion.

The matrix M is decomposed as product of four matrices, translation, rotation, scaling and shearing.

M = M Translation × M Rotation × M Scaling × M Shearing (3) M =     m 1 m 4 m 7 m 10 m 2 m 5 m 8 m 11 m 3 m 6 m 9 m 12 0 0 0 1     (4)
We have used the sum of squared differences (SSD) as objective function to minimize the error between the subject ( f ) and the template images (g) which is the (MNI). The optimization method is the Gauss-Newton algorithm [55]. An additional parameter w is added to the function to correct the difference scale that can be produced in images. The function to minimize is then:

SSD( f , g) = I ∑ i=1 ( f (M * x i y i ) -wg(x i )) 2
(5) where x i is the position in the image (f), and y i is the target in (g), we refer this mapping equation as y = M × x, where M is the mapping matrix.

Finally, as the scanners provide images in different scale of the intensity, we performed intensity normalization along the whole dataset. We used the algorithm developed by "Nyul and Udupam" [56], which is based on the use of the image histogram. In order to have similar intensities for similar structures, the method trains a standard histogram through the whole dataset, and makes deformation for the histogram of each image to obtain a correspondence of intensities of this image with the standard histogram. The process was done using the software SPM8 (Wellcome Trust Centre for Neuroimaging at UCL, London, UK) 6 to fulfill the registration and the normalization [57].

Co-registration for DTI-MD

We aim to extract patches related to the same ROI as in sMRI from the DTI-MD map. Therefore, both modalities have to be co-registered [58]. This means sMRI and DTI-MD images for each subject must be comparable. The co-registration between sMRI and MD consists in estimating transformation parameters using mutual information criterion in some specific areas to fit the standard space (MNI). Thus we affinely co-register MD image to the coordinate system of its corresponding sMRI scan. The co-registration algorithm uses brightness of voxels. In order to avoid distortions which may be induced by bright skull voxels, skull stripping task has to be performed to pull out only the brain from the sMRI modality (see Fig 2) [59]. The process of skull removal was performed with SPM8 software. The brain scans were segmented into grey matter (GM), white matter (WM) and Cerebro-Spinal Fluid (CSF). Merging these three maps we can subtract skull region from original sMRI scans. Thus, for each patient there is an sMRI and MD-DTI aligned images of the same definition of 121 × 145 × 121 and each voxels have a spatial resolution of 1.5 × 1.5 × 1.5 mm 3 .

Selection of the region of interest (ROI)

The next step is the selection of the hippocampal region which is convinced to be the first region in the brain affected by AD [START_REF] Deweer | Memory disorders in probable Alzheimer's disease: the role of hippocampal atrophy as shown with MRI[END_REF][START_REF] Dickerson | MRI-derived entorhinal and hippocampal atrophy in incipient and very mild Alzheimer's disease[END_REF]. So, after alignment of sMRI modality on MNI template and co-registration of MD-DTI to same space, the ROI can be selected on the automated anatomical labeling Brain atlas (AAL) [51]. By selecting voxels labeled as Hippocampal region in AAL and computing their 3D bounding boxes we get a sub-volume of the whole 3D scan which encircles Hippocampus in both modalities sMRI and DTI as presented in Fig. 1 (step 2). Hippocampus is a symmet- rical anatomical structure in the brain consisting of two regions. To get only one ROI for a given patient we calculate the mean of the two regions when flipping the left hippocampal region F l on the right F r . Note that the resolution of normalized sMRI and MD volumes is quite low (121×145×121, see above), thus the hippocampal ROI occupies a small amount of voxels (28×28×28). Finally, we extract 2D-slice of 28 × 28 resolution from the computed 3D-volume to feed the CNN classifier (Fig. 1 step 3 & 4).

Domain-specific data augmentation strategy

Basically, deep networks need a sufficient dataset for training the model by optimizing network parameters, such as synaptic weights. However, we designed a specific domain data augmentation (DA) approach to increase the number of images introduced in [60, 16]. The method consists in applying a number of geometric transformations to the original image including random translation and Gaussian blur. For the Gaussian blur filter (Formula. 6), the scale sigma-value σ of the kernel varied randomly from 0.1 to 1.2 with a step 0.1. Then for each sigma-value, random translation for each dimension was performed with voxel step size in the interval from -2 to 2 in each dimension.

G(x, y, σ ) = 1 2πσ 2 e -(x 2 -y 2 ) 2σ 2 (6)
The final hippocampal region is then computed as:

F(x, y, z) = 1 2 (F r (x, y, z) +F (F l (x -δ x, y -δ y, z -δ z)) (7)
Where F is the flipping operator, and δ x, δ y and δ z are the shifts between left and right regions, F l and F r are left and right hippocampal regions respectively.

Convolutional Neural Network approach

The 2-D+ε Network Architecture

In our previous work we have proposed the so-called "2-D+ε" approach for ROI classification [16]. We use a 2D convolution in a CNN architecture feeding it with three neighbouring slices for each projection Sagittal, Axial, and Coronal. The median slice of Hippocampal ROI and its two neighbours have been selected (see Fig. 3). Then the classification results of all three projections were fused. The choice of the depth of the CNN architecture depends on the resolution of input data. The CNN we proposed in [16] is relatively shallow. Indeed, due to the low resolution (28×28) of the ROI in each projection, the optimal architecture consisted of two convolutional layers followed by a max pooling layer for each one, and a fully connected layer. We get three networks associated with each modality, for the Sagittal, Axial and Coronal projections. Different tested fusion schemes resulted in application of the (best) late fusion with majority vote fusion operator [START_REF] Aderghal | FuseMe: Classification of sMRI images by fusion of Deep CNNs in 2D+ε projections[END_REF] on the six binary classification tasks AD/NC, NC/MCI, and MCI/AD on three projections. We use the same fusion scheme in the present work.

Transfer learning for brain image

Transfer learning [START_REF] Yosinski | How transferable are features in deep neural networks?[END_REF] is a popular way of dealing with limited volumes of training datasets. Actually, the CNN models can be either learned from scratch with random initialization of parameters or with the finetuning approach from pretrained models. From the domain knowledge of medical research, we retain that the shrinkage of hippocampal ROI which accompanies the development of AD is observable on both modalities sMRI and MD. Fig. 4 illustrates this phenomenon, it presents two examples of subjects: for the left a normal control (NC) subject, and the right an (AD) subject, with both modalities; the (A) is the MD map, and (B) is sMRI scan. It shows the hippocampal region from different projection views. From the top to the bottom, also the Axial, Sagittal, and Coronal planes respectively.

As we can observe from the illustrative figure, the atrophy of the hippocampus can be recognized from both modalities by conserving the same shape, but in inverted representation. This means the signal spawned from CSF flows surrounding the Hippocampus portion can be interpreted by a dark area in sMRI scans, while it is bright in MD maps. For this reason, we could adopt a transfer learning strategy between these two types of data, from designing trained models in the source domain of sMRI towards the target domain MD called cross-modal transfer learning.

In the framework of learning CNN parameters, we can formally define the transfer learning strategy from the source modality (sMRI) to target MD as follows:

W 0 ← W φ W i+1 ← F(W i ) (8)
Where W φ is the best trained model on the large sMRI dataset, we initialize the training with the parameters of W φ , and fine-tune all or some layers of the used architecture. F is the optimization scheme.

In this work, we fine-tune only FC layer of our proposed architecture. We use Stochastic Gradient Descent with Nesterov momentum as in [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], and as a cost function J(W i ) to minimize we used cross-entropy loss as in [16]. The weights update formula are defined as follows:

V i+1 ← µV i -α∇J(W i + µV i ) (9) W i+1 ← W i +V i+1
where W i are the parameters of each layer at iteration i, α is the learning rate, µ is the momentum and V i is the velocity. The transfer of the parameters in our CNN architecture is illustrated in Fig. 5. The arrows depict initialization of optimization process for each convolutional and fully connected layer.

Adapted cross transfer learning schemes

It is believed that transfer learning improves classification performance, especially when the source and the target domains are awfully close. In order to validate our proposed scheme of intra-domain, we compare the classification efficiency across two different domain by using both known pretrained model and dataset. We have selected the LeNet network owing to similarity at the design level [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. It takes the same input definition 28×28 as ours, and almost the same depth of layer except the fully connected layers. In this view, we take this model which it is already pretrained on MNIST dataset, and apply it to our brain image data DTI-MD and sMRI. Nevertheless, the model has been modified in FC layers and adapted to 2-way classification problem instead of ten. We freeze the two first convolutional layers which are already captured the universal features, and then we fine-tune it on the Alzheimer's disease dataset by optimizing weights only in the two FC layers of the model.

We evaluate the approach through two mechanisms as follows: One-level transfer scheme: In this first approach, we realize the transfer using our LeNet-like model for both modalities from MNIST to sMRI and from MNIST to DTI-MD followed by our fusion scheme (Majority vote). Two-level transfer scheme: In this second approach, we multiply the transfers; it is applied as MNIST-DTI-MD across sMRI. We first use the LeNet-like model trained on MNIST dataset. This model is used as the basis for training of sMRI classification model. Next from sMRI model we transfer to the DTI-MD images of the same domain. After that, we built our fusion framework from this model combined with the model we obtained from the cross-modal transfer (sMRI to DTI).

Experiments and Results

Learning setup parameters and metrics

Data Augmentation parameters: In order to sufficiently increase our dataset size, we applied the data augmentation strategy as presented in (see Section 3.3). Thus, we have set an augmentation factor F upon which the calculations are based. Indeed, we have proposed an approach to increase the data in an equitable manner. The method consists in setting the factor (a multiplication coefficient) for the most represented class, and by multiplying this factor to the cardinal of this class we obtain large enough class (this one). The next step of the approach is increasing all other classes to reach the same size of the first one. In this way we obtain a balanced dataset.

Hence, we defined the factor F to 100 for the both datasets: The subset "1" and subset "2" as introduced in Section 3.1.2. However, the factor was set for MCI class in subset "1" since it is the most represented class, in the same way, NC class was selected for subset "2" as well. The max shift was set to 2 slices (note that two slices of sagittal axis for example represent about 7.4% of the Hippocampus 3D Bounding Box) and the maximum scale parameter of smoothing Gaussian Blur was set to 1.2 (See Algorithm 1). Indeed, the original sig-nal on both modalities is blurred and a stronger blurring would destroy the structure of the ROI. The parameters were generated randomly and selected to avoid similar augmentation for the same brain scan. Metrics: To evaluate single and fusion binary classification tasks, we considered the metrics widely used in medical statistics. We denote tp, tn, fp, and fn respectively True positives, True negatives, False positives, and False negatives. The metrics used are as follows:

Algorithm 1 DA pseudo algorithm Input: D dataset, F for augmentation factor, S for Max-Shift, and σ for Max-Blur.

Output: augmented and balanced dataset.

N ← N -1 Accuracy (Acc) = t p + tn t p + tn + f n + f p (10) Sensitivity (Sen) = t p t p + f n (11) Specificity (Spe) = tn tn + f p (12)
Finally, balanced accuracy (BAcc) measure, which is the average of sensitivity and specificity is defined as:

BAcc = 1 2 (Sen + Spe) = 1 2 t p t p + f n + tn tn + f p (13)
Hardware configuration: The experiments were conducted on a GPU-based high-performance computing platform featuring an Intel(R) Xeon(R) CPU E5-2680 v2 @2.80GHz processor, 187 Gb of RAM, equipped with two Nvidia TESLA P-100 graphic cards with 16GB dedicated memory. The computational time for one epoch at the training step with batch-size of 64 samples takes 2.03 seconds in average.

Optimization settings: To pick the best learning parameters, we have trained our basic 2-D+ε network on the sMRI dataset. The choice of appropriate hyperparameters was achieved using various training setups. Thus we finally use the exponential learning rate decay policy, α = α 0 • γ i , where α 0 is the initial value of learning rate, and γ ∈ [0, 1]. In our case we set γ to 0.95, and α 0 = 0.0001, and we use the Gradient Descent with Nesterov Momentum method (See Formula. 9). The batch-size set to 64 as mentioned above. After several iterations, the stabilization of the training is observed around the 30th epoch. The network was implemented with "Caffe" Deep learning Framework [START_REF] Jia | Caffe: Convolutional Architecture for Fast Feature Embedding[END_REF].

2-D+ε single and fusion architecture

We use the "2-D+ε" network (see Section 3.4.1) we proposed in our previous work [16,[START_REF] Aderghal | FuseMe: Classification of sMRI images by fusion of Deep CNNs in 2D+ε projections[END_REF] on sMRI data, It was shown that further improvements can be achieved through the fusion approach in particular the majority vote. Likewise, in this section of this work we use only sMRI data but with a larger number of subjects compared to the previous work. In addition, the best fusion method the "Majority Vote" as used in [START_REF] Aderghal | FuseMe: Classification of sMRI images by fusion of Deep CNNs in 2D+ε projections[END_REF] is adopted here. We base our classification on two different models: single network for each projection (Axial, Coronal, and Sagittal), and a late fusion which is designed to improve and enhance classification performances.

We have designed two architectures to perform the classification: (i) 3-way classification (AD/MCI/NC) and (ii) 2-way classification (AD/MCI, AD/NC, and MCI/NC) as the most works in the literature. We built a 3-way classification baseline model, and accuracies of 60.23%, 58.71%, 56.84%, and 66.49% for Sagittal, Coronal, Axial, and fusion, respectively, were obtained. We found that the 3-way model performs somewhat faintly. However, we considered only the 2-way classification in our work since the application domain requires only to test positive or negative for the AD diagnosis. Besides, most of the related works provide only binary classification results with which we can make the comparison. Hence, we performed our method as presented above, and Table 3 presents an overview of the 2-Way classification results.

On average, the single network on the sagittal projection shows better results than other projections, but the fusion method with majority vote achieves the best results for each classification tasks in terms of accuracy. For example, for the AD/NC classification we obtain 82.92% of accuracy for the single sagittal network, while with the fusion we get 91.86%. For the classification tasks involving MCI class (e.g., AD/MCI, and MCI/NC), we can notice that the accuracy is lower than AD/NC, we have 69.65% for AD/MCI, and 68.52% for MCI/NC. This class is a special class, as it includes two subclasses: The early MCI (e-MCI), and the late MCI (l-MCI). We completely grasp that (l-MCI) have more similarity of the atrophy with AD in our ROI, making it difficult to distinguish between subjects in MCI and AD classes. The same conclusion can be derived from the MCI/NC classification.

Evaluation of transfer Learning.

In order to evaluate the efficiency of our method, different adaptive transfer learning schemes were adopted, in this section we provide experimental results on the two proposed approaches: the cross-modal and crossdomain.

4.3.1. Transfer learning from sMRI to DTI-MD with the 2-D+ε approach With the similarity between the structural MRI and DTI-MD, we proposed a cross-modal transfer learning from sMRI dataset as a source to the DTI-MD dataset, which is considered the target dataset. The model was first trained on the sMRI dataset and then fine-tuned with the DTI-MD dataset. Obviously, with the experiments, the cross-modal method yields slightly better results than the training from scratch (by random initialization of network parameters). We can see the difference of the behavior in training (loss) and validation (loss and accuracy) in Figures 6,7 for AD/NC and AD/MCI respectively. We get improved accuracy at the final 30th epoch with transfer learning, and the loss is lower on both sets along training epochs. Figure 8 illustrates that with the transfer from sMRI to DTI-MD the overfitting is slightly reduced (a) compared to training from scratch (b).

From MNIST knowledge to sMRT and DTI crossdomain learning

In this part of the experiment, we apply our second proposed method for cross-domain transfer learning as presented. We construct the LeNet-like model, and perform the experiences as follows: One-level transfer scheme: The transfer is realized for both modalities from MNIST to sMRI and from MNIST to DTI-MD followed by our fusion scheme. Table 5 presents an overview of results for both experiments and also results for the fusion. Analyzing the table, we obtain accuracies around 3% in average lower than the previous results for each classification task. For AD vs. NC as example, we passed from 82.92% to 80.02% for sMRI, and from 84.93 % to 81.85% for DTI-MD on the sagittal projection. For the other classification tasks, the situation is pretty much the same. The results are lower than those for cross-modal transfer in the same domain, see table 4. Two-level transfer scheme: In this setting we perform the experiments using the scheme as explained in 3.4.3. Table 6 presents the results. We can notice that the use of the Two-level transfer scheme, may clearly give better results which we will analyze in the following section. Hence, we experienced three knowledge transfer types: cross-modal with LeNet-like designed architecture and cross-domain one-level and two-level transfer using LeNet Architecture. Comparing the results presented in Tables 4, 5, and 6, we can conclude the following. The cross-domain transfer, which is a very popular transfer learning scheme, performs the worst even on very distinct classes such as AD and NC. Hence definitely, the cross-modal transfer in the same domain (sMRI and DTI in our case) is a better solution. When multiplying transfers such as in the two-level crossdomain transfer scheme, we manage to get slightly better results for the most difficult classification tasks.

Indeed with the transfer from MNIST to sMRI and then DTI, we get a nearly 5% accuracy increase in the classification AD/MCI and MCI/NC. We note that in other metrics, such as Specificity, Sensitivity, and BAcc, the methods perform similarly. Thus, the two-level transfer increases the metrics by more than 5% for the most difficult classification tasks, which is an interesting result. Indeed, the transfer from a pre-trained model does not cost too much; hence the first step of it can be done from a different domain using publicly available trained models such as LeNet on MNIST. Although further transfer in the same domain is needed to improve the result. Now we will compare our best results with methods from literature, see table 7. We have to note that an exact comparison in the medical image domain is not possible, as different ADNI databases are used in each work. To illustrate this, we show the number of analyzed brain scans for AD, MCI, and NC subjects in the first three columns of this table. The authors of [START_REF] Payan | Predicting Alzheimer's disease: a neuroimaging study with 3D convolutional neural networks[END_REF][START_REF] Cheng | Classification of MR brain images by combination of multi-CNNs for AD diagnosis[END_REF][START_REF] Khvostikov | 3D Inception-based CNN with sMRI and MD-DTI data fusion for Alzheimer's Disease diagnostics[END_REF] [31, [START_REF] Gupta | Natural image bases to represent neuroimaging data[END_REF][START_REF] Billones | DemNet: A Convolutional Neural Network for the detection of Alzheimer's Disease and Mild Cognitive Impairment[END_REF][START_REF] Valliani | Deep Residual Nets for Improved Alzheimer's Diagnosis[END_REF] use the whole brain scans. Our method remains "light" in the sense that we focus only on one ROI, which is the biomarker of AD, the Hippocampal ROI. Afterward, we do not use the 3D volume entirely, but only a light version of it, such as three slices. Even with this lightweight method, we get quite decent results, namely in the separation of AD/NC. However, for the most challenging classifications MCI/NC and AD/MCI, even though we earn some accuracy points, the results remain slightly weaker compared to AD/NC. This leads to the investigation of other brain structures and regions or even implies the entire brain for this study. Indeed, as our models focus on the hippocampus atrophy for the discrimination task, working on the hippocampus's limbus can involve more advantages to improve performances, where it encompasses the outermost surface of the hippocampus, which seems to be the most affected by the passage of the MCI stage. Going back to our approach "2-D+ε", we take only three slices where only a fraction of that surface, i.e., "the limbus" intersects, which could explain why MCI/AD and NC/MCI discrimination scores relatively low compared to a method that would be full 3D. Therefore, at the resolution at which we operate, the disease's characterization could be better determined at the hippocampus limbus level than with its whole internal structure [START_REF] Morra | Automated 3d mapping of hippocampal atrophy and its clinical correlates in 400 subjects with alzheimer's disease, mild cognitive impairment, and elderly controls[END_REF].

Despite our classification results still need to be improved by introducing other regions of interest or additional information, as mentioned above, the proposed cross-modal transfer learning definitely yields increased performances. It can thus be re-used in combination with other methods of classification, as those using whole brain or full 3D information. Specifically, in such a field as medical image analysis and classification where large corpora of annotated data are not available, proposed transfer learning will help in circumventing the lack of training data. The cross-domain transfer learning method presents good results in applications on natural images. On the contrary, cross-domain transfer from natural images to medical image domain remains limited as our results show. This is due to the large difference in terms on content between natural and medical images. In this work, we have shown the efficiency of implementing a cross-modal transfer in medical diagnostic applications. We hope that this finding will be successfully used by the research community fro medical image classification tasks.

Conclusion

In this paper, we have shown that intelligently initializing the network parameters, through transfer learning, allows to obtain better classification of AD stages by more that 5 points in some classification tasks (MCI/AD and NC/MCI). We compared various transfer learning schemes: cross-modal transfer learning using sMRI and DTI-MD brain images, cross-domain transfer learning from non-medical data to medical brain scans and a combination of both using a shallow LeNet network. Our approach remains light-weight in the sense that we used the "2-D+ε" scheme we previously developed on the hippocampal region, avoiding both 3D convolutions and full-brain usage. As an interpretation of our results, filters trained on a modality have similar geometrical characteristics that need a small adaptation when transferred to another modality. We think that this is due to the fact that underlying structures of the hippocampus present similar geometrical patterns, visual markers, on both modalities that characterize the progression of the disease. We think that proposed multi-modal transfer learning approach can be useful in other classification tasks on medical images.

Figure 1 :

 1 Figure 1: Schematic diagram of dataset preprocessing: 1) registration of all MRI scans on MNI space, followed with intensity normalization. 2) ROI selection process using the Atlas AAL for both hippocampal regions. 3) 2D-slice extraction from selected 3D-volume. 4) feeding the CNN networks.

Figure 2 :

 2 Figure 2: Illustration of the co-registration process includes spatial normalization and skull stripping.

Figure 3 :

 3 Figure 3: Illustration of the 2-D+ε Approach from each projection.

Figure 4 :

 4 Figure 4: Example of the hippocampal region with different projections for two Subjects: (A) -MD and (B) -sMRI.

Figure 5 :

 5 Figure 5: The scheme of Transfer Learning for parameters optimization from sMRI to MD-DTI modality. In the figure, an example of the proposed architecture for 2-way classification.

1 : 3 : 6 : 7 :

 1367 procedure PROCESS() //* Function to generate samples. *// 2: N = (max {card(#AD), card(#MCI), card(#NC)}) × F while N = 0 do 4: i, j, k, x ← random generate parameters() //* (i, j, k) ∈ -S, S and x ∈ [0, σ ] *// 5: Compute: element ← augmentaion function(i, j, k, x) //* for a given scan. *// Compute: roi ← Hippocampus cube(element) //* return the mean of the left and right of the ROI. *// Compute: patches extraction() //* Extraction of 2D patches. *// 8:

  with the late fusion by majority vote. Compared with the results from table 3, we have clearly augmentation of all metrics up to 5% for the most challenging classes AD/MCI and MCI/NC.

Figure 6 :

 6 Figure 6: Example of Transfer learning for single network -comparison of AD/NC: a) Transfer from sMRI to MD-DTI, b) Training from scratch on MD-DTI Dataset.

Figure 7 :

 7 Figure 7: Example of Transfer learning -comparison of AD/MCI: a) Transfer from sMRI to MD-DTI, b) Training from scratch on MD-DTI Dataset.

Figure 8 :

 8 Figure 8: Temporal loss curves comparison for AD/NC classification: a) From sMRI to MD-DTI transfer learning with reduced over-fitting -b) Training from scratch with little over-fitting.

Table 1 :

 1 Demographic description of the ADNI dataset group. Values are reported as mean and ± standard deviation (* Subjects with both modalities).

		Classes # Subjects	Age [range] / µ(θ )	Gender (#F/ #M) MMSE [range] / µ(θ )
	ADNI-1	AD	188	[55.18, 90.99] / 75.37 ± 7.52	99/89	23.3 ± 2.03
		MCI	399	[54.63, 89.38] / 74.89 ± 7.30	256/143	27.0 ± 1.78
		NC	228	[60.02, 89.74] / 75.98 ± 5.02	118/110	29.1 ± 1.00
	ADNI-2/Go AD	*48	[55.73, 90.87] / 75.60 ± 8.63	28/20	23.0 ± 2.42
		MCI	*108	[55.33, 93.62] / 74.40 ± 7.47	66/42	27.4 ± 1.99
		NC	*58	[59.91, 93.25] / 74.91 ± 5.90	28/30	28.9 ± 1.18
	ADNI-3	AD	*16	[55.26, 86.10] / 74.63 ± 9.92	4/12	-
		MCI	*165	[55.88, 95.93] / 75.01 ± 7.91	71/94	-
		NC	*341	[55.79, 95.39] / 73.52 ± 7.82	209/132	-

  (step 1) [16]. The alignment with MNI template is realized with a 3D affine transformation [53].

Table 2

 2 describes the split of samples before and after the augmentation process. Data are divided into Training, Validation, and Test subsets.

			Before Augmentation		After Augmentation
		AD	MCI	NC AD a	MCI a	NC a
	Dataset 1	Train 146 Valid 42 Test 64	482 126 64	446 48200 117 12600 64 640	48200 12600 640	48200 12600 640
		252	672	627 61440	61440	61440
	Dataset 2 *	Train 31 Valid 13 Test 20	198 55 20	299 29900 80 8000 20 200	29900 8000 200	29900 8000 200
		64	273	399 38100	38100	38100

Table 2 :

 2 Number of subjects for each class, with its corresponding augmentation, (

* Both modalities).

Table 3 :

 3 Classification results for each single projection and fusion by majority vote on sMRI dataset.

  Table 4 presents the final results of cross-modal transfer learning for each projection and

Table 4 :

 4 Binary classification results with Transfer Learning from sMRI to MD-DTI data and fusion (* both modalities).

	Tasks	Modalities Projection Acc (%) Sen (%) Spe (%) BAcc (%)
	AD vs. NC	SMRI	Sagittal	80.02%	81.95%	79.26%	80.60%
			Coronal	79.94%	80.59%	78.18%	79.38%
			Axial	79.11%	81.05%	79.42%	80.23%
		MD	Sagittal	81.85%	83.24%	79.49%	81.36%
			Coronal	79.22%	83.01%	78.56%	80.78%
			Axial	78.69%	82.44%	79.71%	81.07%
		Fusion (*)		86.83%	90.94%	87.14%	89.04%
	AD vs. MCI SMRI	Sagittal	65.32%	66.81%	64.52%	65.66%
			Coronal	64.57%	65.63%	63.74%	64.68%
			Axial	61.74%	63.05%	59.88%	61.46%
		MD	Sagittal	64.95%	70.19%	66.45%	68.32%
			Coronal	68.60%	72.51%	67.96%	70.23%
			Axial	62.36%	68.24%	62.47%	65.35%
		Fusion (*)		71.45%	78.66%	73.16%	75.91%
	MCI vs. NC SMRI	Sagittal	64.75%	62.35%	66.72%	64.53%
			Coronal	60.49%	58.62%	63.40%	61.01%
			Axial	60.15%	59.14%	62.63%	60.88%
		MD	Sagittal	63.59%	63.18%	66.93%	65.05%
			Coronal	67.14%	64.24%	69.86%	67.05%
			Axial	64.98%	63.91%	68.55%	66.23%
		Fusion (*)		69.85%	70.46%	75.73%	73.10%

Table 5 :

 5 Classification results with One-level scheme Transfer Learning: From MNIST to SMRI & From MNIST to DTI-MD data.

  use 3D convolutions. The authors of

	Tasks	Modalities Projection Acc (%) Sen (%) Spe (%) BAcc (%)
	AD vs. NC	MD	Sagittal	85.14%	87.95%	84.14%	86.04%
			Coronal	82.57%	84.55%	80.84%	82.69%
			Axial	81.21%	84.26%	81.10%	82.68%
		Fusion (*)		92.30%	93.95%	90.65%	92.30%
	AD vs. MCI MD	Sagittal	70.84%	77.25%	73.51%	75.38%
			Coronal	76.53%	78.39%	76.64%	77.51%
			Axial	69.21%	73.08%	68.52%	70.8%
		Fusion (*)		79.16%	82.72%	78.36%	80.54%
	MCI vs. NC MD	Sagittal	71.09%	70.15%	74.95%	72.55%
			Coronal	75.34%	72.41%	76.39%	74.40%
			Axial	70.21%	69.10%	73.64%	71.37%
		Fusion (*)		78.48%	77.72%	81.44%	79.58%

Table 6 :

 6 classification results with Two-level scheme Transfer Learning: From MNIST to DTI-MD crossed sMRI data.

http://www.oasis-brains.org/

https://wiki.cancerimagingarchive.net/display/ Public/LIDC-IDRI/

http://yann.lecun.com/exdb/mnist/

http://www.fil.ion.ucl.ac.uk/

Acknowledgments

This work was funded by The Franco-Moroccan TOUBKAL Research Grant AlClass, Project Code N • TBK/16/23, and the LaBRI, University of Bordeaux / CNRS / Bordeaux-INP. We thank Dr. Pierrick Coupé from LABRI, University of Bordeaux / CNRS / Bordeaux-INP for fruitful discussions.

Research :::: and ::::::::: Education, :::: and ::: the ::::: study :: is :::::::::: coordinated :: by ::: the ::::::::::: Alzheimer's :::::::::: Therapeutic ::::::::: Research ::::::: Institute :: at :: the ::::::::: University ::: of :::::::: Southern ::::::::: California. :::::: ADNI :::: data ::: are :::::::::: disseminated ::: by ::: the :::::::::: Laboratory ::: for :::::: Neuro ::::::: Imaging :: at :: the ::::::::: University :: of :::::::: Southern ::::::::: California. : .


Marie-Julie Demedardi

Claire Brechet

Edouard Gentaz

Catherine Monnier
email: catherine.monnier@univ-montp3.fr

EMOTIONAL UNDERSTANDING AND PROSOCIAL LYING 1 Prosocial Lying in Children between 4 and 11 years of age: The Role of Emotional Understanding and Empathy

Keywords: children, lie-telling, prosocial lies, emotional understanding, empathy

This study examined whether emotional understanding and empathy were predictors of prosocial lying in children aged 4 to 11 years. A total of 144 children participated in the study. To assess children's prosocial lying, we used the Helping Scenario, a classical experimental paradigm that allows children to lie to help others at their own expense.

Children's emotional understanding was assessed using the Test of Emotion Comprehension, a test that measures the nine components of emotional understanding. Children's empathy was assessed using the Griffith Empathy Measure, a questionnaire completed by parents, measuring affective and cognitive empathy. Results indicated that emotional understanding plays a crucial role in the development of children's ability to produce prosocial lies from 4 to 11. However, we found no significant relationship between empathy and prosocial lying.

Finally, our results confirm that as children grew up, they lied more and more effectively.

Overall, the results allow us to better understand the factors that contribute to the emergence and development of children's ability to produce prosocial lies.

Children tell lies (i.e., false statements made in order to deceive others). These lies are classified into two broad categories: antisocial (i.e., selfish lies) and prosocial (i.e., white lies).

Antisocial lies, told only for personal purposes, can be stated to avoid punishment or to obtain an undeserved reward [START_REF] Talwar | Lying in the elementary school years: Verbal deception and its relation to second-order belief understanding[END_REF][START_REF] Talwar | A punitive environment fosters children's dishonesty: A natural experiment[END_REF]. Prosocial lies, produced primarily for the benefice of others, are usually uttered to be polite or to avoid hurting the interlocutor (Talwar & Crossman, 2011). They may also be stated to protect or help others at the child's own expense [START_REF] Bottoms | Understanding children's use of secrecy in the context of eyewitness reports[END_REF][START_REF] Talwar | Children's lie-telling to conceal a parent's transgression: Legal implications[END_REF]2017).

Contrary to antisocial lies, prosocial lies are socially encouraged because they allow for the establishment and/or maintenance of harmonious social relationships (e.g., [START_REF] Depaulo | Lying in everyday life[END_REF]. Yet, in the judicial context, for example, lying to protect or help others escape a possible conviction can have serious repercussions. Despite this, little is known about the factors that contribute to the emergence and development of children's ability to produce prosocial lies.

For most children, prosocial lies appear from the age of 4 (see [START_REF] Demedardi | Le mensonge prosocial chez l'enfant : « Toute la vérité, rien que la vérité ! » [The prosocial lie in children: "The whole truth, nothing but the truth![END_REF], for a review of the literature on prosocial lies). Indeed, at this age, most children are able to lie about their appeal to an undesirable gift [START_REF] Lavoie | Polite, instrumental, and dual liars: Relation to children's developing social skills and cognitive ability[END_REF][START_REF] Talwar | White lie-telling in children in politeness purposes[END_REF], their opinion about a bad artwork [START_REF] Fu | Social grooming in the kindergarten: The emergence of flattery behavior[END_REF] or the physical appearance of others [START_REF] Talwar | Emergence of white lie-telling in children between 3 and 7 years of age[END_REF]. To a lesser extent, children aged 4 are also able to lie to let others win in their place [START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Talwar | Carving Pinocchio: Longitudinal examination of children's lying for different goals[END_REF] or to hide others' transgressions at the risk of being accused [START_REF] Talwar | Children's lie-telling to conceal a parent's transgression: Legal implications[END_REF]. Therefore, when there is a cost to themselves, children are less likely to lie for others [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Talwar | Carving Pinocchio: Longitudinal examination of children's lying for different goals[END_REF].

Although prosocial lies appear very early, they become more frequent with age -with or without cost- [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF]. Indeed, the majority of research has shown that school-aged children were more likely to produce prosocial lies than preschoolers [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | White lie-telling in children in politeness purposes[END_REF][START_REF] Warneken | Children tell white lies to make others feel better[END_REF]; but see [START_REF] Talwar | Children's lie-telling to conceal a parent's transgression: Legal implications[END_REF]. Moreover, with age, prosocial lies become more sophisticated and credible enough to deceive adults [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | White lie-telling in children in politeness purposes[END_REF][START_REF] Williams | The role of executive functions and theory of mind in children's prosocial lie-telling[END_REF]but see [START_REF] Lavoie | Polite, instrumental, and dual liars: Relation to children's developing social skills and cognitive ability[END_REF]. Researchers have shown that executive functions (i.e., working memory, inhibition and cognitive flexibility; [START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Williams | The role of executive functions and theory of mind in children's prosocial lie-telling[END_REF] and theory of mind [START_REF] Lavoie | Polite, instrumental, and dual liars: Relation to children's developing social skills and cognitive ability[END_REF][START_REF] Williams | The role of executive functions and theory of mind in children's prosocial lie-telling[END_REF] were involved in the development of prosocial lies. Indeed, lying requires the child to intentionally instill false beliefs in others, inhibit information about the truth and temporarily keep a lot of information in mind (Talwar & Crossman, 2011).

Very recently, studies have shown that empathy is also involved in prosocial lying in children [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF] and in adults [START_REF] Xu | Sadness empathy facilitates prosocial lying[END_REF]. Empathy is defined as an affective response that stems from the apprehension or comprehension of another's emotional state or condition [START_REF] Eisenberg | Emotion, regulation, and moral development[END_REF]; see Shamay-Tsoory, 2011, for a review). It is often considered essential to respond appropriately to the emotional state of others [START_REF] Eisenberg | The role of sympathy and altruistic personality traits in helping: A reexamination[END_REF]. Empathy is composed of an affective dimension (i.e., feeling and sharing an emotion similar to that felt by others, unconsciously and automatically) and a cognitive dimension (i.e., understanding what others feel by putting oneself in their place without losing one's identity). [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF] have shown that only the cognitive dimension of empathy was involved in the production of prosocial lies in children aged 7-11 years. According to [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF], prosocial lying is a cognitively complex behavior that requires more complex cognitive processes than simple automatic emotion sharing. Thus, children who are able to understand what a person is feeling would be more likely to engage in prosocial lying in order to help that person. Although it has been established that empathy plays an important role in the production of prosocial lies from the age of 7 [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF][START_REF] Xu | Sadness empathy facilitates prosocial lying[END_REF], little is known about its role in the production of prosocial lies during the preschool period.

Accordingly, this study examines the relationship between empathy and prosocial lying from the age of 4, the age at which lying emerges and begins to develop [START_REF] Lavoie | Polite, instrumental, and dual liars: Relation to children's developing social skills and cognitive ability[END_REF][START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Talwar | Carving Pinocchio: Longitudinal examination of children's lying for different goals[END_REF].

Studies exploring the factors involved in prosocial behavior have also reported the important role of emotional understanding [START_REF] Conte | Social cognition, language, and prosocial behaviors: a multitrait mixed-methods study in early childhood[END_REF][START_REF] Eggum | Emotion understanding, theory of mind, and prosocial orientation: Relations over time in early childhood[END_REF][START_REF] Ensor | You feel sad?' emotion understanding mediates effects of verbal ability and mother-child mutuality on prosocial behaviors: Findings from 2 years to 4 years[END_REF], which covers a wide range of sociocognitive skills, some of which are related to empathy [START_REF] Denham | Emotional development in young children[END_REF]. Emotional understanding can be defined as a set of skills that enable the child to understand the nature, causes, and consequences of emotions, whether these emotions are specific to the child or relative to another person [START_REF] Theurel | The integration of visual context information in facial emotion recognition from 5-to 15-year-olds[END_REF].

From a developmental perspective, emotional understanding evolves in several stages of increasing complexity between early childhood and pre-adolescence. [START_REF] Pons | Emotion comprehension between 3 and 11 years: Developmental periods and hierarchical organization[END_REF] developed an empirically-derived model, in which they identified nine components of emotional understanding hierarchically organized into three developmental stages (each comprising three components). In the external stage (around the age of 4), children acquire an understanding of the external aspects of emotions and the impact of external events on emotions. In the mental stage (around the age of 7), children are able to understand the impact of mental phenomena on emotions: they understand that beliefs, desires and intentions can produce emotional reactions, and they distinguish between felt and expressed emotions.

Finally, in the reflexive stage (around the age of 10), children are able to understand how an individual can reflect upon a given situation from various perspectives: they understand that emotions can be morally ambivalent, that emotions can be regulated in different ways and that two different emotions can be felt at the same time. Studies have shown that emotional understanding allows children to adapt their behavior in response to the emotions experienced by others during social interactions [START_REF] Denham | Emotional development in young children[END_REF][START_REF] Denham | Key considerations in assessing young children's emotional competence[END_REF] and that emotional understanding is a predictor of prosocial behaviors (i.e., behaviors that promote the wellbeing of others; [START_REF] Conte | Social cognition, language, and prosocial behaviors: a multitrait mixed-methods study in early childhood[END_REF][START_REF] Eggum | Emotion understanding, theory of mind, and prosocial orientation: Relations over time in early childhood[END_REF][START_REF] Ensor | You feel sad?' emotion understanding mediates effects of verbal ability and mother-child mutuality on prosocial behaviors: Findings from 2 years to 4 years[END_REF]. Because prosocial lies aim to promote the well-being of others, a relationship between prosocial lying and emotional understanding can also be hypothesized. Indeed, telling a prosocial lie requires the ability to recognize a person's emotions, to understand the cause of that emotion, and to understand that a lie can change that emotion.

The idea that emotional understanding might play a role in prosocial lying is consistent with the findings of [START_REF] Warneken | Children tell white lies to make others feel better[END_REF]. [START_REF] Warneken | Children tell white lies to make others feel better[END_REF] encouraged children aged 5 to 11 to lie about their opinion about the quality of an artist's artwork. They compared a "sad" condition in which the artist expressed sadness for the quality of his artwork to a "neutral" condition in which the artist was indifferent to the quality of his artwork. The results showed that 7-8 and 10-11 years-old children were more likely to lie than to tell the truth in the sad condition over the neutral one. According to the authors, these results suggest that children from the age of 7 understand the effect of prosocial lying on the emotions of others and use this knowledge appropriately. In the same vein, research in which children were asked to explain the reason for their lying showed that children progressively report lying to avoid hurting the feelings of others [START_REF] Heyman | Children's reasoning about lie-telling and truth-telling in politeness contexts[END_REF][START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Xu | Lying and truth-telling in children: From concept to action[END_REF]. Thus, children understand as they grow up that other people's emotions can be influenced by their lying behavior. However, these studies have not directly examined the relationship between emotional understanding and prosocial lying in children. The present study is intended to fill this gap.

The purpose of this study was to examine the relationship between emotional understanding, empathy and prosocial lying in children aged 4 to 11. We used the Helping Scenario experimental paradigm [START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Talwar | Carving Pinocchio: Longitudinal examination of children's lying for different goals[END_REF], in which the experimenter creates a real situation where the child can lie at his/her own expense to help an adult confederate win a gift, by reporting wrongly having lost the last round of a game. The Helping Scenario was chosen because it elicits lies intended solely to serve the interests of others without expecting rewards but at the expense of oneself [START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF][START_REF] Talwar | Carving Pinocchio: Longitudinal examination of children's lying for different goals[END_REF].

We also assessed children's empathy (affective and cognitive) through the Griffith Empathy Measure, a questionnaire completed by the parents (GEM; [START_REF] Dadds | A measure of cognitive and affective empathy in children using parent ratings[END_REF] and their level of emotional understanding using the Test of Emotion Comprehension (TEC; [START_REF] Pons | Test of emotion comprehension: TEC[END_REF]. Based on previous research, we expected that children with a higher emotional understanding would be more likely to lie for the benefit of others (H1). Furthermore, we expected that prosocial lying would appear more often in children with high cognitive empathy scores (H2) [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF][START_REF] Xu | Sadness empathy facilitates prosocial lying[END_REF]. Finally, we expected that as children grow up, they would lie more frequently (H3a) and more effectively (H3b) for the benefit of others [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | White lie-telling in children in politeness purposes[END_REF][START_REF] Warneken | Children tell white lies to make others feel better[END_REF]. This study was approved by the French INSERM Ethics Committee (IRB00003888, IORG0003254, FWA00005831). Criteria for inclusion were informed consent from parents and no developmental delay.

Method

Materials and procedure

Participants were observed individually in a quiet room in their school. Experience lasted on average 25 minutes for each child.

Lie-telling to help another

The Helping Scenario, based on [START_REF] Talwar | The role of executive functioning and theory of mind in children's lies for another and for themselves[END_REF], was used to create a real play situation where the child could choose to lie, at his/her own expense, to help an adult confederate win a gift. This game consisted of a die with different images on each side (rabbit, four-leaf clover, moon, heart, fish and bird), and a game board with several copies of the same images (i.e., 36 images, 6 copies of each image of the die). For attractiveness and credibility reasons, the game board was made more complex for older children. Thus, for children aged 7 to 9, the game board contained 64 images (i.e., 8 copies of each image of the die and 16 distractors). For children aged 10 to 11, the game board was identical to that of children aged 7 to 9, but the images were replaced by words. The goal of the game was to throw the die in turn and to get rid of all the tokens as quickly as possible by covering the images corresponding to the one on the die. For example, if the die landed on the heart's image, each player had to place one token on each image of the heart on the game board as quickly as possible. Each player had 10 tokens (except for players aged 7 to 11, who had 15 tokens). The first player to place all his tokens won the game and therefore a gift.

The experimenter (E) invited the child and the confederate (C) to play this game, which is played in four rounds. At the end of each round, the winner received a gift. Previously, C had been asked to pretend to lose each round and to look sad after each defeat.

In the fourth and final round, E excused himself and asked C and the child to continue playing. Although the child won the fourth round, C asked the child to tell that C won so that C could win the last gift. After the child has accepted or not, E came back into the room and C found an excuse to leave the room (i.e., four standardized excuses were created and used randomly). Then, E asked the child: "Who won the last round?". This question was used to determine if the child would lie to help the adult. Based on their response to this question, children were categorized as "truth-tellers" (coded 0) or "helping liars" (coded 1).

Immediately after, E asked: "How many tokens did you have left?" and "How many tokens did he have left?". These questions were used to assess the liar's ability to maintain his/her lie.

The liars' responses were classified into two categories: "does not maintain the lie" (e.g., "I had no tokens left" or "C had 2 tokens"; coded 0) or "maintains the lie" (e.g., "I had 4 tokens left and C had no tokens left"; coded 1). At the end of the experiment, children all received four gifts and a debriefing of the study was done.

Emotional understanding

The Test of Emotion Comprehension (TEC; Pons & Harris, 2000, French version) was used to measure the nine components of emotional understanding: (1) recognition, (2) external cause, (3) reminder, (4) desire, (5) belief, (6) hiding, (7) regulation, (8) mixed, and (9) morality. The experimenter presented several short stories to the child, accompanied by illustrations. For each story, the main character was represented with four possible emotions ("happy", "sad", "angry", "scared", and "just alright"). At the end of each story, the child was asked to point to the image corresponding to the emotion felt by the main character. For each successful component, the child scored 1 point (ranging from 0 to 9, α = .60).

Empathy

Parents completed the Griffith Empathy Measure (GEM [23 items]; Dadds et al., 2008, French version) to measure the affective and cognitive components of empathy. For each item (i.e., statement describing ways to act), parents rated their agreement on a 9-point Likert scale (-4 = strongly disagree to +4 = strongly agree). A high score indicated a high level of empathy (GEMaffective ranging from -36 to +36, α = .65; GEMcognitive ranging from -24 to +24, α = .47).

Results

Preliminary analyses revealed that gender did not significantly affect the results. Thus, gender was not included as a factor in the reported analyses. In addition, preliminary analyses revealed that there was no significant correlation between emotional understanding and affective (r = -.121) or cognitive (r = .154) empathy, even when controlling for age (r = -.039 and r = .141 respectively). However, although very low, there was a significant negative correlation between affective, and cognitive empathy (r = -.186, p <.05). Then, hierarchical logistic regressions were conducted to examine whether (1) children's age, empathy and emotional understanding were predictors of prosocial lies, and (2) children's age was a predictor of prosocial lie maintenance.

Among the 144 children who participated, 50 (35%) lied to help the confederate win the last gift. A hierarchical logistic regression was conducted to determine which factors might predict children's prosocial lie-telling. For this hierarchical logistic regression analysis, children's age (continuous predictor variable) was entered on the first step, empathy scores (i.e., GEMaffective and GEMcognitive) on the second step and emotional understanding scores (i.e., TEC) on the third step. The results are presented in Table 1. According to H3a, with age, children were more likely to lie to help others at their own expense (p = .001). In addition, in line with H1, emotional understanding significantly predicted prosocial lie-telling (p = .003), with children with higher scores of emotional understanding lying more than those with lower scores (see Table 2). However, contrary to H2, cognitive empathy did not predict children's prosocial lying (p = .283). Affective empathy was not a significant predictor either (p = .106).

Among the 50 children who lied to help the confederate, 21 (42%) maintained their lie in response to the follow-up questions. A logistic regression was conducted to determine whether children's age predicted the maintenance of their lie. Thus, for this logistic regression analysis, children's age (continuous predictor variable) was entered as the only predictor. The model was significant, 2 (1, 50) = 8.865, p = .003, Nagelkerke R 2 = .219. According to H3b, with age, children were more likely to maintain their lies, B = .522, SE = .195, Wald = 7.157, p = .007, odds ratio = 1.686; 95%CI = [1.150, 2.471].

Discussion

This study examined the role of emotional understanding and empathy in prosocial lying in children aged 4 to 11. We used the Helping Scenario to assess children's willingness to lie for others. Children also completed an emotional understanding measure (i.e., TEC) and the parents completed a questionnaire (i.e., GEM) measuring the affective and cognitive empathy. Three main results were observed.

Consistent with our hypothesis, the first main result showed for the first time that improved ability in emotional understanding strongly predicted prosocial lies. Specifically, children who decided to tell a lie scored higher on the TEC than non-liars. Thus, our results suggest that in order to decide to tell a prosocial lie to help a person in distress, the child must be able to recognize the emotion of the person to whom the lie benefits and understand the cause of this emotion. In addition, the child must be able to understand that by deciding to lie, he/she can change that person's emotional state. In this study, for example, children had to recognize that the confederate was sad, to understand that this emotional state was due to not winning any gifts, and to understand that lying to the experimenter (by falsely claiming to have lost the last round) could change the confederate's emotional state. As a result, children with a high emotional understanding could have been more inclined to lie to improve the emotional state of the confederate at their own expense. Our findings are consistent with previous research highlighting the role of emotional understanding in children's prosocial behaviors [START_REF] Conte | Social cognition, language, and prosocial behaviors: a multitrait mixed-methods study in early childhood[END_REF][START_REF] Eggum | Emotion understanding, theory of mind, and prosocial orientation: Relations over time in early childhood[END_REF][START_REF] Ensor | You feel sad?' emotion understanding mediates effects of verbal ability and mother-child mutuality on prosocial behaviors: Findings from 2 years to 4 years[END_REF], suggesting that emotional understanding may play an important role in children's overall prosocial orientation.

The second main result was that children's level of cognitive empathy did not significantly predict their prosocial lying, which did not confirm our hypothesis as well as the results of studies that have shown a significant relation between these two variables [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF][START_REF] Xu | Sadness empathy facilitates prosocial lying[END_REF]. Several reasons may explain this result. First, the contradictory results could be due to differences in the age range studied (i.e., 4-11 years in the present study, compared to 7-11 years for [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF]and 18-36 years for Xu et al., 2019). Indeed, compared to [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF], our age range also included preschool children and it might be that the relationship between cognitive empathy and prosocial lying becomes more robust from the school period onwards. Second, it is possible that the tool we used to assess children's empathy (i.e., GEM) have limited the relationship between this ability and prosocial lying. Indeed, in our sample the internal consistency of the items measuring cognitive empathy was very low (α = .47) compared to that of [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF]α = .62), which may limit its explanatory power. Third, the paradigm used in the study by [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF] was somewhat different from that used in the current study. The main difference was that in [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF], the child and the confederate had to get to know each other better before playing. During the conversation, the confederate informed the child that he/she enjoyed collecting stickers as a hobby (a sticker book was one of the prizes to be won). Thus, having this information could have made it easier for the child to imagine how the confederate felt about the loss of the sticker book and to act accordingly, thereby increasing the relationship between empathy and prosocial lying.

Although our results are different from those of [START_REF] Nagar | The role of empathy in children's costly prosocial lie-telling behaviour[END_REF] and [START_REF] Xu | Sadness empathy facilitates prosocial lying[END_REF], they are consistent with other studies that have shown that cognitive empathy did not always predict altruistic behavior (e.g., [START_REF] Edele | Explaining altruistic sharing in the dictator game: The role of affective empathy, cognitive empathy, and justice sensitivity[END_REF]. These mixed findings as well as the limited internal consistency of the tool used to measure cognitive empathy highlight the need for further research to improve our knowledge of the exact relationship between empathy and prosocial lying, especially over a large age range. Future studies may need to use more direct and reliable measures to assess children's cognitive empathy. Finally, consistent with previous studies [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | White lie-telling in children in politeness purposes[END_REF][START_REF] Warneken | Children tell white lies to make others feel better[END_REF], our results confirmed that the frequency and effectiveness of prosocial lying were significantly related to children's age. Thus, as children grow older, they become more engaged in effective lying behavior aimed at helping a person in distress at their own expense.

In everyday life, prosocial lies are necessary to establish and/or to maintain harmonious social relationships. Thus, from an early age, children are encouraged to produce prosocial lies. However, the production of prosocial lies in the judicial context, to protect or help others to escape conviction, can have serious repercussions. It is therefore important to better understand the factors that contribute to the emergence and development of children's ability to produce prosocial lies. Our results contribute to the literature on children's prosocial lying, by showing the crucial role of emotional understanding in the development of prosocial lying and by confirming that as children grow older, they become more engaged in effective prosocial lying behavior. Nevertheless, this study is not without limitations. First, although we have shown that improved emotional understanding promotes the production of prosocial lying in children, it is not clear whether children lied with the intention of alleviating the confederate's emotional distress or out of fear of being reprimanded if they did not. For this, children should have been asked to explain the reason for their lies [START_REF] Popliger | Predictors of children's prosocial lietelling: Motivation, socialization variables, and moral understanding[END_REF][START_REF] Talwar | Emergence of white lie-telling in children between 3 and 7 years of age[END_REF][START_REF] Xu | Lying and truth-telling in children: From concept to action[END_REF] with the caveat that it is difficult for young children to make their motive explicit [START_REF] Talwar | Emergence of white lie-telling in children between 3 and 7 years of age[END_REF]. In addition, their compassion for the confederate could have been examined. Compassion is a related construct in the empathy domain that refers to being emotionally motivated to relieve a person's suffering or emotional distress and involves an action tendency to help that person [START_REF] Lupoli | Lying because we care: Compassion increases prosocial lying[END_REF]. [START_REF] Lupoli | Lying because we care: Compassion increases prosocial lying[END_REF] showed that compassion increased prosocial lying behaviors in adults. Thus, in future research, it would be interesting to examine the role of emotional understanding, empathy, and compassion in the production of children's prosocial lying while asking them to justify their lying in order to increase our knowledge about the sociocognitive mechanisms underlying the development of prosocial lying in children.

  ParticipantsA total of 144 children aged 4 to 11 (66 boys, M = 7.7 years, SD = 1.98) participated in the study. Children were recruited from four elementary schools in the South of France.

Table 1

 1 Hierarchical logistic regression model for children's prosocial lie-tellingStep 1: Nagelkerke R 2 = .169; Step 2: Nagelkerke R 2 = .197; Step 3: Nagelkerke R 2 = .267

	Predictors	B	SE	Wald	OR [95% CI]	2	p
	Step 1					18.856	.001
	Age	.410	.101	16.539	1.507 [1.237-1.837]		.001
	Step 2					3.351	.187
	Cognitive empathy	.024	.022	1.155	1.024 [.981-1.069]		.283
	Affective empathy	.029	.018	2.616	1.029 [.994-1.066]		.106
	Step 3					8.753	.003
	Emotional understanding	.430	.156	7.586	1.537 [1.132-2.087]		.006
	Note.

Table 2

 2 Means (SDs) scores on emotional understanding and empathy as a function of children's

	lying behavior
			Children's lying behavior
	Predictors
		Truth-teller n = 94	Lie-teller n = 50	Total n = 144
	understanding	4.97 (1.97)	6.52 (1.18)	5.51 (1.88)
	Cognitive empathy	3.68 (7.87)	5.38 (10.21)	4.27 (8.75)
	Affective empathy	5.48 (11.29)	6.98 (11.29)	6 (11.28)

Formatting of funding sources

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.