Xiaoxi Pan

Mouloud Adel
email: mouloud.adel@univ-amu.fr

Caroline Fossati

Thierry Gaidon

Julien Wojak

Eric Guedj

Multiscale spatial gradient features for 18F-FDG PET image-guided diagnosis of Alzheimer's disease

Keywords: Multiscale Spatial Gradients, Ensemble Classification, 18 F-FDG PET, Alzheimer's Disease

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

Introduction

There is a new case of dementia in the world every 3 seconds [START_REF] Patterson | World Alzheimer Report 2018: the state of the art of dementia research: new frontiers[END_REF], and the most common form of dementia, which accounts for 60%-70%, is Alzheimer's Disease (AD) [START_REF] Burns | Alzheimer's disease[END_REF]. It affects one in nine over 65s [START_REF]Alzheimer's Association, 2015 Alzheimer's disease facts and figures[END_REF] and one in three over 85s [START_REF] Hebert | Alzheimer disease in the United States (2010-2050) estimated using the 2010 census[END_REF]. It is estimated that 131 million people will be living with the disease in 2050 [START_REF] Prince | World Alzheimer report 2016: improving healthcare for people living with dementia: coverage, quality and costs now and in the future[END_REF]. At present, AD is incurable and irreversible, even if promising developments for treatments are expected to be achieved in the near future. Diagnosis at early stage of Mild Cognitive Impairment (MCI) is of importance in the therapeutic perspective to stop or slow down the neurodegenerative process. 18 F-FluoroDeoxyGlucose Positron Emission Tomography ( 18 F-FDG PET, referred to as FDG-PET hereafter) has proved to be one of the effective modalities for identifying AD or MCI since it can detect the subtle changes in the cortex [START_REF] Davison | A comparison of FDG-PET and blood flow SPECT in the diagnosis of neurodegenerative dementias: a systematic review[END_REF][START_REF] Mosconi | Pre-clinical detection of Alzheimer's disease using FDG-PET, with or without amyloid imaging[END_REF][START_REF] Rice | The diagnostic value of FDG and amyloid PET in Alzheimer's disease-A systematic review[END_REF]. Therefore, a variety of studies have been published in either predicting MCI conversion or differentiating AD from Normal Control (NC) using FDG-PET modality. Among them, considerable attention has been dedicated to developing Computer-Aided Diagnosis (CAD) approaches for AD. Their basic idea is to investigate characteristic patterns in FDG-PET images through combing medical image processing and machine learning techniques, thereby achieving diagnosis goals.

A fundamental step in CAD methods is feature extraction from images. There have been many studies on feature extraction for Alzheimer Disease and its prodromal stage [START_REF] Rathore | A review on neuroimaging-based classification studies and associated feature extraction methods for Alzheimer's disease and its prodromal stages[END_REF]. Feature extraction approaches from FDG-PET images could be roughly grouped into two main categories: voxel-wise and region-wise. Voxel-based methods could include the whole voxels of each brain image, as has been done by Hinrichs et al. [START_REF] Hinrichs | Spatially augmented LPboosting for AD classification with evaluations on the ADNI dataset[END_REF] who utilized all the voxels to predict AD under a boosting framework. In this case, even though using the whole voxels preserves all the information, it can lead to feature redundancy, which is a burden for training a classifier. Voxel-based methods could also use a set of discriminative voxels in order to avoid feature redundancy. Cabral et al. [START_REF] Cabral | Predicting conversion from MCI to AD with FDG-PET brain images at different prodromal stages[END_REF] selected typical voxels through using mutual information, which obtained promising performance for predicting MCI conversion. Padilla et al. [START_REF] Padilla | NMF-SVM based CAD tool applied to functional brain images for the diagnosis of Alzheimer's disease[END_REF] applied non-negative matrix factorization to select and extract the most relevant voxels for the purpose of diagnosing AD. Region-based methods generally use a pre-defined anatomical atlas to segment the FDG-PET image into different Regions of Interest (ROIs) in order to compute statistical or other parameters that can represent glucose rates within ROIs or connections between them. Pagani et al. [START_REF] Pagani | Volume of interest-based [ 18 F] fluorodeoxyglucose PET discriminates MCI converting to Alzheimer's disease from healthy controls. A European Alzheimer's Disease Consortium (EADC) study[END_REF], Li et al. [START_REF] Li | Gaussian mixture models and model selection for [18F] fluorodeoxyglucose positron emission tomography classification in Alzheimer's disease[END_REF], Cheng et al. [START_REF] Cheng | Domain transfer learning for MCI conversion prediction[END_REF] and Gray et al. [START_REF] Gray | Multiregion analysis of longitudinal FDG-PET for the classification of Alzheimer's disease[END_REF] used the mean intensity value of each ROI as the feature, whereas Garali et al. [START_REF] Garali | Histogram-Based features selection and volume of interest ranking for brain PET image classification[END_REF] used first order statistics (mean, variance, skewness and kurtosis) and entropy extracted from the grey-level histogram of each ROI as features to identify AD. Pan et al. [START_REF] Pan | Multi-level Feature Representation of FDG-PET Brain Images for Diagnosing Alzheimer's Disease[END_REF] computed connectivities among ROIs and exploited these connections as features to tackle the problem of AD diagnosis.

Another important part of CAD is machine learning techniques. Traditional machine learning methods, such as Support Vector Machine (SVM) [START_REF] Cortes | Support-vector networks[END_REF], random forests [START_REF] Breiman | Random forests[END_REF], are usually applied to do the classification task. Cabral et al. [START_REF] Cabral | Predicting conversion from MCI to AD with FDG-PET brain images at different prodromal stages[END_REF] and Padilla et al. [START_REF] Padilla | NMF-SVM based CAD tool applied to functional brain images for the diagnosis of Alzheimer's disease[END_REF] used SVM and FDG-PET images to predict MCI conversion and identify AD, respectively. Gray et al. [START_REF] Gray | Random forestbased similarity measures for multi-modal classification of Alzheimer's disease[END_REF] exploited random forests to derive the pairwise similarity measures from features and then made a classification by using random forests as well for multi-modality, in which FDG-PET was included. Recently, deep learning techniques have shown impressive performance in recognition and classification tasks [START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF]. Therefore, a great amount of research have turned to the use of neural networks to address the problem of AD diagnosis and obtained satisfactory results [START_REF] Lu | Multiscale deep neural network based analysis of FDG-PET images for the early diagnosis of Alzheimer's disease[END_REF][START_REF] Liu | Classification of Alzheimer's Disease by combination of convolutional and recurrent neural networks using FDG-PET images[END_REF][START_REF] Ding | A deep learning model to predict a diagnosis of Alzheimer disease by using 18F-FDG PET of the brain[END_REF]. Even though deep learning models have made significant progress, their interpretability still needs to be investigated.

In this paper, the feature is represented from another point of view, the spatial gradient of FDG rates in PET brain images, instead of voxel-wise and ROI-wise features as many studies have done previously. This work is motivated by the observed differences of FDG rates between NC and AD subjects. The spatial gradient is quantified by a 2D histogram of orientation, which is similar to Histogram of Oriented Gradient (HOG) [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] that has been successfully applied for object detection in 2D images. Our contributions can be summarized into three aspects: 1) 1D HOG descriptor, used in natural scene images originally, is improved to 2D HOG to quantify spatial gradients, thereby characterizing 3D FDG-PET brain images. Moreover, 2D HOG is expressed as Small Scale HOG (SSH) and Large Scale HOG (LSH), which proves to be more effective than the commonly used features, voxelwise and ROI-wise; 2) a region ranking method is proposed to select distinctive ROIs by using multiple SSH features; 3) an ensemble classification framework is designed through considering both small and large scales 2D HOG for individual and concatenated regions, which enhances the diagnosis accuracy for both binary and multiple classifications.

The rest of the paper is organized as follows. In Section 2, multiscale spatial gradient features for diagnosing AD is described. Section 3 presents and analyzes experimental results, and in Section 4, a discussion is given. Finally, a conclusion of this work is given in Section 5.

Methods

In this paper, FDG-PET images are characterized by spatial gradients for AD diagnosis.

First, the spatial gradient of FDG-PET image is computed and then 90 ROIs are extracted from the gradient image through Automated Anatomical Labeling (AAL) [START_REF] Tzourio-Mazoyer | Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain[END_REF] atlas, in which the cerebellum is not considered. Next, some distinctive ROIs are selected through a proposed ROI ranking method which considers multiple SSH descriptors of each region.

Finally, an ensemble classifier is trained under the selected ROIs by using SSH and LSH features. Figure 1 illustrates the flowchart of the proposed diagnosis method.

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 MRI, PET, other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of MCI and early AD.

In ADNI dataset, there are four types of FDG-PET data 1 , which are 1) Co-registered Dynamic; 2) Co-registered, Averaged; 3) Co-reg, Avg, Standardized Image and Voxel Size; 4) Co-reg, Avg, Std Img and Vox Siz, Uniform Resolution. The proposed method is evaluated on FDG-PET images from type 3). After downloading the baseline data, images are spatially normalized to MNI template using SPM12 [START_REF] Penny | Statistical parametric mapping: the analysis of functional brain images[END_REF] with 2 × 2 × 2 mm 3 voxel size and 91 × 109 × 91 tensor dimension. This normalization is based on two steps: a global affine transformation followed by a non-rigid spatial transformation. For each step, a minimization of the mean squared difference between the image to be normalized and a linear combination of one or more template images. The general affine transformation assumes a 12-parameter model whereas the non-rigid spatial transformation is based on the combination of the lowest frequency components of the three dimensional cosine transform. Next, the intensity normalization is performed through dividing each voxel intensity by the average value of global gray matter which is extracted under the aid of AAL template, and the data is with 32-bit depth. Thereafter, images are further smoothed by a Gaussian kernel with a full width at half maximum of 8 mm following many related studies [START_REF] Hinrichs | Spatially augmented LPboosting for AD classification with evaluations on the ADNI dataset[END_REF][START_REF] Li | Gaussian mixture models and model selection for [18F] fluorodeoxyglucose positron emission tomography classification in Alzheimer's disease[END_REF][START_REF] Garali | Histogram-Based features selection and volume of interest ranking for brain PET image classification[END_REF][START_REF] Liu | Classification of Alzheimer's Disease by combination of convolutional and recurrent neural networks using FDG-PET images[END_REF][START_REF] Zhu | Subspace regularized sparse multitask learning for multiclass neurodegenerative disease identification[END_REF]. Through the pre-processing pipeline, experimental data are converted to a standard space, among which 741 images are selected according to the diagnostic information2 (LONI → Download → Study Data → Assessments → Diagnosis → Diagnostic Summary). Specifically, there are 247 AD subjects who stay in AD stage without conversion, 246 NC subjects who stay in NC stage without progression, 123 progressive MCI (pMCI) subjects who have progressed from MCI stage to AD in the available scan time and 125 stable MCI (sMCI) subjects who have not converted to AD in the follow-up period which lasts at least 24 months. A full list of subject IDs used in this paper can be found in the Supplementary Material. The demographic and clinical information of subjects is provided in Table 1, in which MMSE stands for the Mini-Mental State Examination.

Histogram of Oriented Gradient for 2D images

Histogram of Oriented Gradient (HOG) is a descriptor which was proposed in for human detection [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. The fundamental idea is that the local object appearance and shape within an image can be represented by the distribution of intensity gradients or edge directions.

Generally, the image is divided into small connected regions, and for the pixels within each region, a histogram of gradient directions is computed. The descriptor is the concatenation of these histograms. HOG feature is an effective hand-crafted descriptor for object detection since it can capture the edge or gradient structure which is discriminative for local shape [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF].

2D Histogram of Oriented Gradient for FDG-PET images

Figure 2 shows the difference between an NC (top row) and an AD (bottom) subjects. is for an NC subject and the bottom row is for a subject with AD.

Spatial Gradient Computation

For a 2D image, the gradient is computed from the horizontal and vertical directions, and the corresponding orientation is determined by one angle. Similarly, the spatial gradient of a 3D image is calculated in the x, y and z directions and the orientation is decided by two angles.

For a voxel with an intensity f (x, y, z) at the position (x, y, z), its numerical gradient can be computed as:

g x = 0.5 × (f (x + 1, y, z) -f (x -1, y, z)) g y = 0.5 × (f (x, y + 1, z) -f (x, y -1, z)) g z = 0.5 × (f (x, y, z + 1) -f (x, y, z -1)) (1)
where g x , g y and g z are gradients in the x, y and z directions, respectively. The magnitude g is obtained through:

g = g x 2 + g y 2 + g z 2 (2)
and the orientation is represented by the polar angle, φ, and the azimuth angle, θ, as shown in Figure 3 φ = arctan

g y g x θ = arccos g z g ( 3
)
where φ is in the range [-180

Spatial Gradient Quantification

In this step, a 2D histogram is constructed based on gradient orientations (φ and θ),

and the magnitude is used to count the occurrence of a certain orientation. Specifically, φ and θ are viewed as two properties of the 2D histogram, and then are evenly divided into several intervals or bins, respectively. Last, if the gradient orientation is within a certain interval, the value for that interval is accumulated. According to [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], the value is computed via a function of the gradient magnitude. Considering the magnitude of each voxel is small, the exponential function of magnitude, g e , is applied as the counting value in each interval,

g e = exp( g ) (4)
Consequently, the FDG-PET image can be represented by a 2D histogram, and meanwhile, the representation will vary with the number of bins in the histogram. Figure 4 shows histograms with different numbers of bins for the same FDG-PET image, where the top row is for an NC subject and the bottom row is a subject with AD. It can be seen that for a subject, either NC or AD, a change in the number of bins can result in different representations (each row). Moreover, the difference between NC and AD is obvious under the same number of bins (each column), which also implies that it is reasonable to use 2D HOG as the feature to diagnose AD. Compared with Global HOG which is computed on the whole subject without segmen-

h = h h (5)
where h is 2D HOG vector for each scale of any ROI in a subject, h stands for 2D HOG descriptor after normalization.

ROI Ranking

The probability of AD occurring in each ROI is not consistent, which means that some ROIs are more likely to be affected by AD, while others are not. In practice, doctors pay more attention to key areas as well. Therefore, it is necessary to select typical ROIs which are more susceptible to AD. A region ranking method is developed to achieve the goal of region selection. Specifically, 2D HOG features are applied to characterize each ROI and then fed into a linear SVM to compare each ROI's classification accuracy, thereby ranking ROIs. In order to obtain a robust and reliable result, multiple SSH features are considered, and the average accuracy is used for ranking ROIs. The reason LSH is not taken into account is that the feature dimension of LSH is relatively low, for example, the dimension of 1 × 1 LSH is only 1. Figure 5 shows the framework of the ranking method.

A region with a higher classification accuracy implies its stronger ability to recognize AD.

At last, some ROIs can be selected through the ranking order. The ranking results are analyzed in Section 3.3.

Ensemble Classification

After ranking ROIs, the top N ROIs with higher performance are selected as candidate regions for the classification. SVM is a popular and effective classifier in AD diagnosis [START_REF] Padilla | NMF-SVM based CAD tool applied to functional brain images for the diagnosis of Alzheimer's disease[END_REF][START_REF] Gray | Multiregion analysis of longitudinal FDG-PET for the classification of Alzheimer's disease[END_REF][START_REF] Liu | Relationship induced multi-template learning for diagnosis of Alzheimer's disease and mild cognitive impairment[END_REF]. In this study, SVM is applied in an ensemble classification framework which considers both SSH and LSH features of selected ROIs. The motivation of designing an ensemble framework is inspired by the idea that weak classifiers can be combined into a strong classifier. Although SVM is usually treated as a strong classifier, we make it become a relatively weak model through controlling input features (ROI by ROI, scale by scale) and setting the parameter C which is introduced in Section 3.1. In addition, concatenating all the scales of 2D HOG and all the regions is not an efficient way to train a model due to too many inputs. Therefore, the selected regions guided by the ROI ranking method are utilized to train a set of weak classifiers, and then multiple classifiers are integrated to make a prediction. The framework of the ensemble classifier is briefly presented in Figure 6. Specifically, five types of SSH features (18 × 18 (φ × θ), 12 × 18, 18 × 9, 12 × 9 and 12 × 6 bins) are extracted from each selected ROI and then are used to train five classifiers, respectively. The average score of five classifiers is considered as the corresponding ROI's output, S, which is expressed as:

S = 1 T T t=1 s t (6)
where s t is the output score of SVM with t-th scale of SSH descriptor and T is the number of scales of SSH, here T = 5. For LSH, all the features of candidate ROIs are concatenated to feed into a classifier because of the low feature dimension. The final decision, Y, is made through an addition strategy of two parts' outputs, small and large scales,

Y = sgn   N i=1 S i + M j=1 L j   (7)
where sgn(•) is a sign function, N is the number of candidate ROIs, M is the number of scales of LSH, M = 2, and L is the score of LSH-based classifier. Therefore, the proposed ensemble classifier not only considers the performance of each individual ROI, but also considers the performance of cascade regions.

Results

Setup

In order to test the effectiveness of the proposed method, the experiments are mainly conducted on two classification tasks, AD vs. NC and pMCI vs. sMCI. We apply four metrics, namely classification accuracy (ACC), sensitivity (SEN), specificity (SPE), and area under curve (AUC), to evaluate the corresponding performance. For all the metrics, a higher value indicates better performance. Specifically, ACC is the proportion of samples that are correctly predicted. SEN implies the proportion of correctly classified AD or pMCI samples. SPE represents the proportion of NC or sMCI samples that are correctly classified. of all the SVMs used in the ensemble classifier is set to 0.5 in order to construct relatively weak classifiers. In this paper, the SVM algorithm is implemented with the LIBSVM toolbox [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF].

Evaluation on Spatial Gradient Feature for FDG-PET

The spatial gradient feature is compared with commonly used voxel-wise and ROIwise features. The comparison results of different representations are presented in Table 2 and Table 3 for AD vs. NC and pMCI vs. sMCI, respectively. The terms 'Voxel' and 'ROI' stand for classification results which obtained by using voxel intensity and region's mean intensity, respectively. 'Global HOG' means 2D HOG descriptor is computed on the subject without parcellation. The results of Global HOG and SSH shown in the following two tables are achieved based on a histogram with 18 × 9 bins, while for LSH, its results are computed on a histogram with 5 × 5 bins. As for classifying pMCI from sMCI, which is reported in Table 3, voxel-wise feature is inferior to ROI-wise feature, which is different to their performance in AD classification.

The reason could be that the dimension of voxel-wise feature is far greater than that of ROIwise feature, which can cause the problem of feature redundancy. Such a problem would harm the classifier training, especially for the case that the classifier could not work well.

As can be seen from Table 3, all the features (voxel-wise, ROI-wise, Global HOG, LSH and SSH) cannot perform as well as in AD diagnosis, thus prediction of pMCI is a challenging task and it is easily influenced by redundant features. Nevertheless, SSH descriptor still has dominant performance with an accuracy of 66.53%, which is 2.07% higher than the ROIwise feature, and LSH has comparative performance with the ROI-wise features. The other three metrics also indicate SSH is superior to the other features. Therefore, characterizing FDG-PET images by 2D HOG locally is effective and feasible. The reason could be that the spatial gradient is calculated at voxel level and the 2D histogram is computed at ROI level, which makes the 2D HOG descriptor become a bridge to link voxel-wise feature and ROI-wise feature.

Evaluation on ROI Ranking Method

Since different regions have different abilities to diagnose AD, a simple ROI ranking method using multiple SSH features and SVM is proposed. Specifically, for each region, five scales of SSH features are extracted and then fed into five SVMs, respectively. The average accuracy of classifiers is considered as the ranking metric, and the higher the accuracy is, the stronger the ability of the region to distinguish AD from NC. Multiple SSH features are applied to ensure the reliability and robustness of the results. Figure 7 shows the maximum difference of accuracy in five scales of SSH, ∆acc, for each ROI. As can be seen, the difference is obvious, especially for identifying pMCI from sMCI, and the highest ∆acc is 17.84%, which proves that it is rational to use multiple SSH features to rank ROIs.

Moreover, the change of maximum difference in pMCI prediction is larger than that of AD classification, which is because that the later task is easier than the former one. For AD classification, different scales of effective features usually achieve high-level accuracies but with a little differences. While for pMCI prediction, it is challenging, so different scales of features may not be that effective and may achieve unstable performance, which could cause a larger difference within different scales of features. Table 4 presents the top 20 regions ranked by the proposed method for two tasks, AD vs. NC and pMCI vs. sMCI, which can be regarded as potential FDG-PET indicators for the subsequent classification tasks.

Evaluation on Ensemble Classification

An ensemble classification framework is designed by considering SSH and LSH descriptors together since these multiple scales of 2D HOG contain both specific (SSH) and general information (LSH). Figure 8 shows the accuracies under different number of regions for AD vs. NC and pMCI vs. sMCI. It should be noted that the x axis denotes ROIs which have the SSH-based classifier, which proves that the integration strategy is effective. Moreover, the accuracy trend is a result of a trade-off between informative and redundant features.

The highest accuracy is attributed to the optimal balance between effective features and a proper dimension, while for the decline of the accuracy trend, it is caused by the feature redundancy overwhelming its effectiveness.

Comparison with the State-of-the-art Methods

The proposed method is compared with other methods which also address the problem and pMCI vs. sMCI. It can be seen that the proposed method outperforms others in terms of ACC and SPE for AD diagnosis, but it is inferior to Lu's method in respect of SEN and Li's method in AUC, with a slight difference of 0.32% and 0.07%, respectively. For the case of distinguishing between pMCI and sMCI, our method is not as effective as Lu's method, which utilized multiscale deep neural networks in terms of ACC and SPE, but it should be noted that the data used in Lu's method for the task of pMCI vs. sMCI is unbalanced (pMCI: 112, sMCI: 409), which can enhance the accuracy to a certain degree.

Moreover, we also report the results of multiple classifications, AD vs. MCI (pMCI + respectively in order to achieve balance data for the four-class classification task. Clearly, the proposed method still outperforms voxel-based and ROI-based approaches, especially for the task of four-class classification, where the improvement is significant. Besides, our method is also superior to the comparison methods, and the differences with the best will motivate us to explore the effects of other descriptors used in natural scene images for FDG-PET images, such as LBP (Local Binary Pattern) [START_REF] Ojala | A comparative study of texture measures with classification based on featured distributions[END_REF], SIFT (Scale Invariant Feature Transform) [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Furthermore, 2D HOG can characterize different regions with the same feature dimension, even if each ROI contains a different number of voxels, which may be significant for some research and applications. As for the dataset, a key issue needs to be taken seriously. In fact, most state-of-the-art methods are not comparable because the experimental dataset is different. Even though most methods use the ADNI dataset, the image processing procedures and subject IDs are different, which greatly weakens the significance of comparison. Considering not all methods are open-source, evaluating methods over the same dataset is more practical.

Conclusion

We introduce 2D HOG to quantify spatial gradients thereby characterizing FDG-PET images instead of the common features, such as voxel-wise and ROI-wise features. Experimental results suggest that 2D HOG is more efficient and effective than voxel-wise and ROI-wise features, respectively. Besides, a ROI ranking method is proposed by applying multiple SSH descriptors and according to which, a set of candidate ROIs are selected to assist the diagnosis. Furthermore, an ensemble classification framework is designed over the selected regions through using SSH and LSH features. The ensemble classifier is effective and outperforms other methods according to the evaluation on the ADNI dataset.

Meanwhile, we test the proposed method on multiple classification tasks as well, and improvements are also significant.

Acknowledgements

Data collection and sharing for this project was funded by the Alzheimer's Disease

Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904)

and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI

Figure 1 :

 1 Figure 1: The flowchart of the proposed method.

Figure 2 (

 2 Figure 2(a) displays one of the slices and the circled area belongs to region Parietal Inf R in AAL template. The enlarged areas are shown in Figure 2(b) (different colors indicate different intensities). It can be clearly seen that the intensities are different between NC and AD PET scans. In addition, the gradients are also different, as shown in Figure 2(c) (different colors indicate different gradient magnitudes), and the gradient change of the AD scan (bottom) is more obvious than that of NC (top). This observation drove us to investigate the effectiveness of spatial gradients in diagnosing AD. In this paper, we

Figure 2 :

 2 Figure 2: An instance to differentiate between NC and AD via intensity and gradient, where the top row

Figure 3 :

 3 Figure 3: An instance of the polar angle φ and azimuth angle θ.

(a) 18 ×

 18 18 (φ × θ) bins (b) 18 × 9 bins (c) 12 × 6 bins

Figure 4 :

 4 Figure 4: Improved 2D Histogram of Oriented Gradient (HOG) for an NC subject (top row) and an AD subject (bottom row) without segmentation.

Figure 5 :

 5 Figure 5: The framework of region ranking method.

Figure 6 :

 6 Figure 6: The framework of the ensemble classification, circles with different colors indicate different regions.

Figure 7 :

 7 Figure 7: Maximum difference of accuracy in five scales of SSH for each ROI

Figure 8 :

 8 Figure 8: Performance under different numbers of ROIs.

Table 1 :

 1 Demographic and clinical information of subjects. SD * ) 75.09 ± 8.01 74.41 ± 6.09 74.01 ± 6.87 74.45 ± 7.86 MMSE(Mean ± SD) 23.17 ± 2.13 29.04 ± 1.18 26.81 ± 1.64 27.66 ± 1.69

	Characteristic	AD	NC	pMCI	sMCI
	Number of subjects	247	246	123	125
	Female/male	103/144	119/127	51/72	44/81
	Age(Mean ±

* SD-Standard Deviation

  10 tation, calculating locally is also worth considering. To achieve this goal, each subject is divided into 90 regions by using AAL atlas, and a histogram can be constructed for each region. It should be noted that the 2D histogram is computed in irregular regions which contain different numbers of voxels. Since the number of bins in a histogram is adjustable, multiple scales of 2D HOG features are extracted from each ROI to make the features informative, including 18 × 18 (φ × θ), 12 × 18, 18 × 9, 12 × 9 and 12 × 6 bins, which are denoted Small Scale HOG (SSH) because of the small interval used in gridding angles for constructing a histogram. Correspondingly, some Large Scale HOG (LSH) descriptors, in which a large interval is used to grid angles, are also extracted from each ROI, including 5 × 5 (φ × θ), 1 × 1 bins. The features from different ROIs exhibit different ranges, so normalization is essential in order to achieve good performance. In this paper, L 2 norm is used to do the normalization.

Table 2 :

 2 Performance of different type of feature for AD vs. NC(%)

	Feature	Dimension ACC	SEN	SPE	AUC
	Voxel	160990	91.25 90.44 92.05 96.43
	ROI	90	86.47 85.23 87.71 93.40
	Global HOG	162	63.75 62.37 64.95 68.89
	LSH	2250	92.64 91.45 93.95 97.31
	SSH	14580	93.37 91.77 95.10 97.49

Table 3 :

 3 Performance of different type of feature for pMCI vs. sMCI(%)It can be seen from Table2that voxel intensity is a kind of effective feature in classifying AD from NC, which achieves an accuracy of 91.25%. But the feature dimension is too large, which is a drawback for training a model. Even though the dimension of ROI-wise feature is small, its performance is not satisfactory and it is not as effective as voxel-wise feature, with an accuracy rate of 86.47%. Global HOG, with 63.75% accuracy, cannot achieve a noteworthy result. It can be explained that: 1) Global HOG is computed on the whole subject without considering local details; 2) the dimension, 162 (18 × 9), seems to be a satisfactory one, but the effective information is much less than that because of a lot of zero values, like Figure4(b), which is less informative. LSH and SSH descriptors can guarantee the performance and meanwhile tackle the problem of large dimension. Even though 14580 (18 × 9 × 90) is not an absolutely desirable dimension, it is still acceptable compared to the dimension of the voxel-wise feature. In addition, owing to the adjustable number of bins in a histogram, a smaller dimension can be obtained if setting an appropriate number of bins.

	Feature	Dimension ACC	SEN	SPE	AUC
	Voxel	160990	62.79 61.80 64.76 68.94
	ROI	90	64.46 63.29 65.84 69.75
	Global HOG	162	50.57 52.63 50.39 51.90
	LSH	2250	64.74 64.55 65.74 69.83
	SSH	14580	66.53 65.24 68.18 72.80

Table 4 :

 4 Top 20 ROIs for AD vs. NC and pMCI vs. sMCIIn the case of pMCI vs. sMCI, as illustrated in Figure8(b), the performance of SSH is still superior to LSH and both of them reach the best accuracy with the top 10 regions, which are 73.92% and 68.81%. Furthermore, the ensemble classifier raises the accuracy to 75.38% with 10 ROIs, and can significantly improve performance under different numbers of ROIs. In summary, for the two tasks (AD vs. NC and pMCI vs. sMCI), LSH descriptor does not perform as well as SSH feature. It is because LSH contains more general features, which makes it less informative than SSH. Besides, the ensemble classifier is better than

	Rank	AD vs. NC	pMCI vs. sMCI
	1	Cingulum Post R	Cuneus L
	2	Cingulum Post L	Cingulum Post L
	3	Hippocampus L	Temporal Inf L
	4	Hippocampus R	Precuneus L
	5	Precuneus R	Temporal Mid R
	6	Cuneus R	Rectus L
	7	Fusiform R	Parietal Inf R
	8	Temporal Inf L	Lingual R
	9	Cuneus L	SupraMarginal L
	10	Parietal Inf R	Rectus R
	11	Precuneus L	Temporal Inf R
	12	Paracentral Lobule R	Cuneus R
	13	Paracentral Lobule L	Hippocampus R
	14	Temporal Mid L	SupraMarginal R
	15	Fusiform L	Parietal Inf L
	16	Parietal Sup R	Frontal Sup L
	17	Occipital Mid R	Supp Motor Area L
	18	Temporal Sup L	Parietal Sup R
	19	Occipital Sup R	Occipital Mid L
	20	Angular L	Angular R

  of AD/pMCI diagnosis under the modality of FDG-PET, including methods of Hinrichs et al.[START_REF] Hinrichs | Spatially augmented LPboosting for AD classification with evaluations on the ADNI dataset[END_REF], Padilla et al.[START_REF] Padilla | NMF-SVM based CAD tool applied to functional brain images for the diagnosis of Alzheimer's disease[END_REF], Li et al.[START_REF] Li | Gaussian mixture models and model selection for [18F] fluorodeoxyglucose positron emission tomography classification in Alzheimer's disease[END_REF], Gray et al.[START_REF] Gray | Multiregion analysis of longitudinal FDG-PET for the classification of Alzheimer's disease[END_REF], Pan et al.[START_REF] Pan | Multi-level Feature Representation of FDG-PET Brain Images for Diagnosing Alzheimer's Disease[END_REF], Zhu et al.[START_REF] Zhu | Subspace regularized sparse multitask learning for multiclass neurodegenerative disease identification[END_REF], Lu et al.[START_REF] Lu | Multiscale deep neural network based analysis of FDG-PET images for the early diagnosis of Alzheimer's disease[END_REF] and Liu et al.[START_REF] Liu | Classification of Alzheimer's Disease by combination of convolutional and recurrent neural networks using FDG-PET images[END_REF]. Lu's and Liu's methods applied the neural network technique, and other approaches were developed under the traditional classification framework. Noting that Pan's method is evaluated on the same dataset with the proposed method for the sake of a fairer comparison.

Table 5 and

 5 Table 6 present the comparison results of two binary tasks, AD vs. NC

Table 5 :

 5 Performance comparison for AD vs. NC(%)

	Method	Subjects	ACC	SEN	SPE AUC
	Hinrichs et al. [10]	89AD + 94NC	84	84	82	87.16
	Padilla et al. [12]	53AD + 52NC	86.59 87.50 85.36	--
	Li et al. [14]	25AD + 30NC	89.1	92	86	97
	Gray et al. [16]	50AD + 54NC	88.4	83.2	93.6	--
	Pan et al. [18]	247AD + 246NC 90.55 90.46 90.92 95.22
	Zhu et al. [30]	51AD + 52NC	93.3	--	--	--
	Lu et al. [24]	226AD + 304NC 93.58 91.54 95.06	--
	Liu et al. [25]	93AD + 100NC	91.2	91.4	91.0	95.3
	Our method	247AD + 246NC 93.65 91.22 96.25 96.93

Table 6 :

 6 Performance comparison for pMCI vs. sMCI(%) .38 74.84 77.11 80.70 sMCI) vs. NC and AD vs. pMCI vs. sMCI vs. NC, and compare them with the results of voxel-based, ROI-based and state-of-the-art methods, which are shown in Table7 and

	Method	Subjects	ACC	SEN	SPE	AUC
	Gray et al. [16]	53pMCI + 64sMCI	63.1	52.2	73.2	--
	Pan et al. [18] 123pMCI + 125sMCI 71.10 68.04 74.38 74.78
	Zhu et al. [30]	43pMCI + 56sMCI	69.9	--	--	--
	Lu et al. [24]	112pMCI + 409sMCI 81.55 73.33 83.83	--
	Our method	123pMCI + 125sMCI 75

Table 8 .

 8 It is worth noting that the number of subjects of AD and NC is reduced to 123

http://adni.loni.usc.edu/methods/pet-analysis-method/pet-analysis/

https://ida.loni.usc.edu/

result are 9.2% and 13.75% regarding three-class and four-class classifications, respectively.

The results of multiple classifications are lower than those of binary classifications, which indicates the multiple classification is a very challenging problem and more investigations need to be carried out in the future.


Ph Mokhtar

Turcry

K Namoulniara

P-Y Mahieux

J Lux

A Aït-Mokhtar

Ph Turcry

Efficiency of water repellent surface treatment: Experiments on low

Introduction

Corrosion due to aggressive agents such as chloride is the most frequent pathology of reinforced concrete structures [Costa, 2002], [Poupard et al., 2003]. Corroded structures need expensive repairs. Ollivier et al. pointed out that the United States of America should dedicate a budget of more than $ 100 billion during ten years for repairs [START_REF] Ollivier | La durabilité des bétons, 2 e éd[END_REF]. The use of mortar patches is a common method to repair of corroded structures [Canisius, 2004], [START_REF] Soufi | Influence of polymer proportion on transfer properties of repair mortars having equivalent water porosity[END_REF]. Alternative solution to avoid spending time and money in maintenance is to protect newly cast concrete structures with surface treatments, which limit the diffusion of aggressive agents through the concrete cover. The European standard EN 1504 defines three types of surface treatments [START_REF] Nf En | Products and systems for the protection and repair of concrete structures[END_REF][START_REF] Nf En | Products and systems for the protection and repair of concrete structures[END_REF]: coatings, impregnations and hydrophobic impregnations. Coatings produce a hydrophobic layer on the exposed concrete surface. Impregnations reduce the cover porosity by filling partially or completely capillary pores. Hydrophobic impregnations, also called water repellent agents, are surface treatments designed to protect concrete against water penetration. Their chemical components reduce surface tension and turn concrete into a hydrophobic material. Unlike coatings, no film forms on the cover surface and the concrete appearance remains unchanged: only the surfaces of capillary pores are coated with the hydrophobic substances.

Since the 1970's, many researches were conducted to evaluate the performance of repair products. Basheer et al. give an overview of the scientific studies conducted on the methods of assessment and performance of surface treatments [START_REF] Basheer | Surface treatments for concrete: Assessment methods and reported performance[END_REF]. More recently, other works complete this literature review [Schueremans et al., 2007], [Medeiros, 2008 and2009], [Pigino et al., 2012], [Pan et al., 2017].

Surface treatments tend to improve the performance of concrete against the penetration of aggressive agents and service-life of concrete structures. Johansson shows that the penetration of chloride ions is reduced from 70% to 80% [Johansson, 2006]. Medeiros shows that the diffusion coefficient of chloride is reduced by 80%, while the capillary absorption coefficient is reduced from 73% to 98% [Medeiros, 2008]. Zhao et al. confirm that the dissolved ions cannot penetrate the porous network because the water penetrates only in vapor form [START_REF] Zhao | Water repellent surface treatment in order to establish an effective chloride barrier, Hydrophobe IV[END_REF]. Basheer et al. state that it is important to study the behavior of concrete treated in their environment and also for a long term period because it seems that the water repellents are sensitive to climate conditions (UV attacks, freeze-thaw cycles, etc.) and their ability to protect the concrete can be reduced [START_REF] Basheer | Surface treatments for concrete: Assessment methods and reported performance[END_REF].

However, Schueremans et al. [START_REF] Schueremans | Chloride penetration in RC-structures in marine environment -long term assessment of a preventive hydrophobic treatment[END_REF] show, through a study focused on the quality and quantity of water repellent agents, that it is possible to obtain good results in the long term with respect to the penetration of aggressive agents and thus in protection of reinforced concrete structures.

Putting aside the eventual degradation of water repellents, many studies agree that these protections are effective for reducing the transfer process and particularly the water absorption. One can, however, wonder how long the protective effects last, since only a small depth of the cover is usually impregnated by the hydrophobic treatment. Thus, one can wonder if protection remains efficient when aggressive agents have diffused beyond the treated depth.

In order to answer this question, the long term efficiency of two standard water repellent agents were investigated in the present study. An experimental campaign was carried out on a concrete with poor mechanical performances. This kind of concrete was chosen in order to reduce the time of the mass transfer phenomenon investigated in our laboratory work, namely water capillary absorption. Microstructure of treated and non-treated concrete specimens was also investigated. In parallel, numerical simulations based on a pore network model were also carried out in order to complete our investigation on the effects of the porous network connectivity reduction and the penetration depth on the capillary absorption.

Water repellents

Two water repellents agents were studied: the first agent (denoted H1) is composed of siloxanes molecules, while the second one (denoted H2) is composed of silanes molecules.

Silanes are a family of monomers molecules derived from different alkyl groups linked to each other by a silicone molecule (Figure 1). The waterproofing properties of silanes come from a double chemical reaction (hydrolysis and condensation) between water repellent and concrete. Silanes molecules react with the interstitial water to form silanols groups (reaction in an alkaline solution at high pH value) and ethanol. Silanols groups then react with siliceous phases (aggregates and cementitious matrix) to form a hydrophobic layer on the surface of concrete pores (alkyl group). Generally, larger the molecule of the alkyl group is, better the repellency of the silane is. Silanes are volatile and some active components can evaporate during the application, especially if the operation does not take place under ideal conditions. Siloxanes are oligomer molecules derived from partially hydrolyzed molecules of silanes. Siloxanes react with concrete silica to form a hydrophobic layer. In concrete pores, the reactions are identical to those described for the silane molecules, with a residual hydrolysis and polycondensation. [Johansson, 2010] Water repellents were applied on the surface of the tested specimens (described in the following section) by paintbrush in two coats for an overall quantity equal to 300 g/m 2 (i.e. at 150 g/m²/coat). To ensure a complete polymerization, the treated specimens were conserved 14 days in room conditions before furthers investigations. [Tahlaïti, 2010], [START_REF] Soufi | Influence of polymer proportion on transfer properties of repair mortars having equivalent water porosity[END_REF] Among all the specimens, some of them (denoted C15) were non-treated and used as reference, others were treated with the water repellent H1 and H2 (denoted C15H1 and C15H2, respectively). Note that the water repellents were applied only on one sawn surface of the specimens (i.e. one plane circular surface). The experimental program is summarized in Figure 2.

Experimental procedures

To measure the penetration depths of the water repellents, the 30 mm thick slices were split into two parts then split areas were sprayed with water, according to the European standard [START_REF] Nf En | Products and systems for the protection and repair of concrete structures[END_REF][START_REF] Nf En | Products and systems for the protection and repair of concrete structures[END_REF]. The penetration depths were measured with a ruler and correspond to the thickness remained dry due to the hydrophobic effect of water repellent.

The water porosities were assessed according to the French standard NF P 18-459 [NF P 18-459, 2010]. Specimens were water-saturated during 24 hours under vacuum (around 2.5 kPa) and then oven-dried at 105°C. Volumes were determined by hydrostatic weighing. The average of water porosities values were calculated form three results for each study case.

Mercury intrusion porosimetry (MIP) was carried out with the Micrometrics Autopore III 9420 porosimeter, according to a procedure recommended by the AFPC-AFREM [START_REF] Arliguie | GranDuBé, grandeurs associées à la durabilité des bétons, 1 ère éd[END_REF]. The test was performed on 15 mm side cubes sawn from the 30 mm thick slices. In the case of treated slices, these cubic samples contained the treated zone (determined previously with the penetration depth). Before mercury intrusion, the cubic samples were dried during 7 days at 45°C in a ventilated oven.

Capillary water absorption tests were performed following the procedure proposed by the AFPC-AFREM [START_REF] Arliguie | GranDuBé, grandeurs associées à la durabilité des bétons, 1 ère éd[END_REF]. The lateral faces of the 50 mm thick specimens were sealed with a resin coating to insure unidirectional transport from bottom to the top during absorption. Before testing, specimens were dried at 80°C until weight stabilization. They were cooled and then put in contact with water (Figure 2). Specimens were regularly weighted during the test.

A FEI/Philips Quanta 200 SEM was used to make images on non-polished layer samples in environmental conditions, in order to locate the water repellents and observe their microstructures.

3 Experimental results

Penetration depths

As shown in Figure 3, the average penetration depth of C15H1 is lower than the penetration depth of C15H2, with 2.5 and 9.5 mm average depths, respectively.

When applied on a concrete surface, the water repellent goes into the cement matrix by capillary suction. This absorption follows the classical Washburn law [START_REF] Aït-Mokhtar | On the applicability of Washburn law: Study of mercury and water flow properties in cement-based materials[END_REF] during 6 to 8 hours according to Gerdes et al. [START_REF] Gerdes | Decisive Factors for the Penetration of Silicon-Organic Compounds into surface near Zones of Concrete[END_REF]. After that, the absorption slows down due to chemical reactions taking place in the porous network (hydrolysis and polycondensation, see section 2.1). Over time, these chemical reactions and the consecutive condensation prevent any additional penetration of water repellent.

The absorption of a water repellent depends, on the one hand, on the chemical nature (type of cement, nature of aggregates, temperature, pH), the physical properties (i.e. porosity) and the water saturation degree of the concrete substrate, and, on the other hand, on the application, the quantity and the quality of the water repellent agent. In this study, the main reason that can explain the difference between the penetration depths is the molecular structure of the studied products and the chemical reactions that occur during the absorption phase, since all other parameters remain constant.

The water repellent H1 is a siloxane solution in solvant phase. The siloxane molecule is itself a silane molecule partially hydrolyzed. When siloxane contacts the interstitial solution, a residual hydrolysis takes place very quickly causing the formation of polysiloxanes. These polymers create a hydrophobic barrier that reduces and even cancels the additional absorption of the product, which explains the lower penetration depth of H1. This assumption seems to be confirmed by SEM analyses. A thin film can be seen on the concrete surface (image 3, Figure 3) revealing the presence of H1 repellent on the surface, while no film can be detected on the surface of the sample treated by H2 (image 4, Figure 3).

The water repellent H2 is a pure silane solution (99 % of actives substances). The time required for complete hydrolysis and poly-condensation is longer than that of H1. Furthermore, the smaller size of the H2 molecules allows the water repellent to penetrate the pores network inaccessible to the H1 siloxane molecules. The combination of these two parameters allows the H2 water repellent having a better penetration depth in comparison with H1.

In addition, images 5 and 6 in Figure 3 show that microstructures of water repellents after condensation are identical in both cases with agglomerated forms, partially distributed on the pores surface.

Water porosities

Table 2 gives penetration depths and water porosities obtained on the reference concrete C15, C15H1 and C15H2. Note that C15H1 and C15H2 samples are composed both of treated concrete and non-treated concrete layers, since the penetration depths are less than the smallest dimension of samples (30 mm). The reference C15 concrete has the highest water porosity with 14,9%. As planned, with this high value, this concrete can be classified under the low durability category structure materials [START_REF] Afgc | Conception des bétons pour une durée de vie donnée des ouvrages -Indicateurs de durabilité, document scientifiques et techniques[END_REF]. The lowest porosities were determined on C15H1 and C15H2 concretes with 13,2% and 8,1%, respectively. Note that porosities of the treated specimens are inversely proportional to the penetration depth of water repellents. In view of these results, we can question the effectiveness of this method.

Indeed, if we use the measured water porosities in the two treated samples as a basis for calculating the porosity in the treated layer, we found negative values, even when taking into account the measurement uncertainties. This can be explained by two things: first, it is possible that zones that are only partially filled by water repellent could not be detected visually. The boundary between the treated and untreated zone is probably more gradual than what is observed on the photographs. Secondly, it is likely that water will not be able to enter the porous network partially or totally treated with a water repellent, despite the vacuum imposed in the desiccator. The saturation time imposed during 24 hours was not sufficient to obtain a complete saturation of the treated layer.

Although the treated layers have a hydrophobic behavior, the problem encountered in the determination of the water porosity can also be related to the reduction of the pores connectivity. Water repellent cover partially or totally the pore surfaces. This behavior occurs when the size of polymerized water repellent molecule is lower than the pores diameter. The siloxane molecular structure is close to polymerized molecule. We can assume that the pores diameter ranging from 3 to 30 nm (the size of siloxane molecules) in the case of C15H1 and between 0.4 and 30 nm in the case of C15H2 are no longer accessible to water. This hypothesis is confirmed by MIP curves shown in Figure 4. For the pore size distribution range from 5 to 19 nm and 3

to 19 nm respectively for the C15H1 and C15H2, the volume of mercury that has penetrated the sample is really smaller than in the case of the reference concrete. Finally, it seems that the treated layers can be only very partially water-saturated.

However, as shown in the following section, the water-absorption is sufficient to ensure a percolation through the treated layer.

Capillary water-absorption

Figure 5 gives the time-evolution of the specimen mass during the absorption test.

The non-treated specimen has a classic kinetic of capillary absorption [START_REF] Ollivier | La durabilité des bétons, 2 e éd[END_REF], with a linear evolution of the absorbed water mass per unit volume in function of the square root of time until saturation. This law does not apply to the water repellent concretes, particularly during the first four hours of capillary absorption, where the amount of absorbed water is lower than the one observed for the reference concrete. Capillary absorption curves of the treated concretes can be divided into three parts, as follows:

Phase I: when the treated concrete sample is placed on capillary absorption test, water does not immediately penetrate the porous network because the external surface is hydrophobic, as shown by the capillary contact angle test (Figure 6). In both cases, the initial contact angle is clearly greater than 90°, revealing a low wetting surface, with 92° and 122° respectively for the C15H1 and C15H2. However, on the surface of treated samples, the hydrophobic effect is temporary, since the contact angle is not constant with time: it decreases quickly. In the best case, it takes up to 45 minutes for a drop of water to completely penetrate the material. In fact, the water repellent does not prevent the water vapor diffusion [START_REF] Johansson | Protection of concrete with Water Repellent Agents -What Is Required to Achieve a Sufficent Penetretion Depth? Concrete Repair[END_REF][START_REF] Nf En | Products and systems for the protection and repair of concrete structures[END_REF], 2005]. Under a moisture gradient, the water molecules will be adsorbed on the internal surface of the concrete pores and on the water repellent alkyl groups.

Over time, it creates a continuous water film, which gradually reduces the effect of the water repellent and consequently increases capillary absorption.

This phenomenon is observed in both cases and corresponds to the significant change in the slope of the absorption curves. This change indicates the time required for the continuous film to form and it is sufficient to create a percolation through the untreated material. For the C15H1 and C15H2, it occurs after 45 minutes and 4 hours, respectively. This second water repellent H2 protects much longer than the H1, although these two products only protects very temporarily. This latest observation is very remarkable and highlights the limit of these two water repellents for the protection of concrete structures. In addition, the required amount of water to ensure a percolation is similar in both cases, i.e. about 10 kg/m 3 . This means that the water fills about 1% of the overall porosity. It is unlikely that this amount of water will fill the entire the porous network of the treated layer.

Phase II: The capillary absorption curves follow a conventional kinetics (similar to the reference concrete), indicating that water is in the untreated layer. However, the absorption kinetics are still lower. The repellent layer provides greater hydraulic resistivity thereby reducing the overall conductivity of the treated samples. This effect is greater when the depth of penetration increases (i.e. for C15H2).

Phase III: This phase corresponds to the moment when the water reaches the upper face of the sample. Unlike the C15, C15H1 and C15H2 need about respectively 24 hours and 4 days to reach their maximum absorption. In the case of C15H1, the amount of absorbed water is the same as the C15. The thin penetration depth of this water repellent is insufficient to observe a significant decrease of the total amount of water absorbed. However, we notice a significant decrease of water absorbed in the C15H2 sample.

Therefore, the water repellent increases the time required to reach the maximum absorption but not enough to protect over time the concrete structures against the aggressive agents. These results show that the effectiveness of water repellents depends mainly on the depth of penetration. These results do not challenge the quality of the water repellents used, but rather their effectiveness to protect over time the concrete structures against the capillary absorption. In the literature, researchers focused their studies on concretes specially formulated for aggressive environments and have not highlighted this phenomenon.

All are unanimous to say that these surface treatments have a real effect on the absorption and hence are effective to limit the transport of aggressive agents but none of them quantify the temporary effect of this maintenance system. For a predictive approach to sustainability maintenance by surface treatments, our results highlight the necessity to extend existing studies since it seems that this phenomenon observed on a low-quality concrete is also valid on all types of concrete.

Modeling of capillary absorption

In order to complete the experimental study, numerical simulations of capillary absorption using a pore network approach were carried out. This numerical approach aims at simulating the effects of the porous network connectivity reduction and the penetration depth on the capillary absorption.

Modeling principle

Pore network models were used successfully to study a wide range of multiphase flows and to predict effective properties like permeability [START_REF] Amiri | Tri-dimensional modelling of cementitious materials permeability from polymodal pore size distribution obtained by mercury intrusion porosimetry tests[END_REF]. In this work, we use a dynamic pore network model derived from generic "ball-and-stick" models [START_REF] Blunt | Simulation and theory of two-phase flow in porous media[END_REF], [START_REF] Leventis | Capillary imbibition and pore characterisation in cement paste[END_REF], [Valavanides, 2001], [Gielen, 2007] within which pore bodies and throats are represented by spheres and straight cylinders with circular cross-sections, respectively. These models only allow "piston type" motion, in which the invading wetting fluid advances in a connected front that occupies the bulk of the pore space. A detailed description of the model is available in [Lux & Anguy, 2012].

We consider 3D networks made of pores arranged on a regular cubic lattice and connected by throats. In this kind of networks, the coordination number (i.e. the number of throats connected to each pore) is 6.

Construction of pore networks

The pore size distribution for the reference network, which represents the untreated sample, is partially based on the MIP results. The objective here is not to create a realistic pore network, which is a quite difficult task (see for instance [START_REF] Tsakiroglou | [END_REF]), but to create a network with a correct range of pore sizes. We use two Gaussian distributions noted G1 and G2 in order to generate two distinct populations of pores. Table 3 summarizes the properties of the chosen distributions and their respective probability, as well as the prescribed minimum and maximum radii. The radius Rij of a throat connecting two pores i and j of radii Ri and Rj respectively, is uniformly distributed between min(Ri,Rj) and min(Ri,Rj)/2. Figure 7a and 7b show the pore and throat radius distributions measured on a realization of a pore network made of 10x25x10 pores along x, y and z-axes, respectively. In this work, we consider that water repellent enters preferentially into the smallest pores and fills them entirely. We use a simple invasion percolation algorithm is used in order to simulate the penetration of the water repellent into the network and the subsequent reduction of the porosity [START_REF] Ferer | Pore-level modeling of immiscible drainage: validation in the invasion percolation and DLA limits[END_REF]. This algorithm works as follows: the smallest pore located at the bottom surface is first marked as filled. Then, at each iteration, the algorithm looks at all neighbors of the currently filled pores and immediately fills the smallest of all the neighbors. As filled pores are non accessible to water, they are therefore deleted from the network. The algorithm stops when the porosity of the layer is lower or equal to the porosity objective, which is arbitrarily set to 1/3 of the initial network porosity. Note that connectivity checks are performed at each iteration to ensure that the deletion (i.e. filling) of an element does not lead to the creation of two disconnected networks. Thus, all the empty pores remain connected to the whole network and to the inlet (i.e. there is always at least one path that connects two pores in the network). The porosity objective in the layer being very small (around 5%), this constraint is necessary to avoid the disconnection of all the remaining pores from the inlet.

Figure 8 shows a realization of a network made of a 10 x 25 x 10 pores and the results of our algorithm for a layer height of 30% of the total network height. Note that the connectivity in the layer partially filled by the water repellent is drastically reduced.

Simulations of water capillary absorption

Simulations of free imbibition are carried out on several realizations of network of size 8.9 x 22 x 8.9 µm³ made of 10 x 25 x 10 pores. In order to study the influence of the layer thickness on the imbibition kinetics, the base network is modified using the aforementioned procedure to generate layers of thickness equal to 5%, 10%, 20% and 30% of the total height of the base network. Porosity is fixed to 15% in the base network and is reduced to approximately 5% in the layer where the water repellent penetrates.

Water invades the network from the bottom (y=0), and air can escape from the opposed face (top). The other sides are impermeable to water and air. Initially, all pores are filled with air, except for the pores in contact with the water reservoir, which are half filled. The water pressure at the inlet is constant and fixed to a value close to the atmospheric pressure (100kPa) like in previous experiments, and the same applies for the air pressure at the outlet. Simulation ends when all air is trapped.

Figure 9 shows different stages of the imbibition in a realization of a pore network with a treated layer thickness equal to 20% of the sample height. Figure 10 shows the evolution of the water saturation for each network versus the square root of time.

Note that the water saturation stabilizes only after a very short time. This is due to the small dimensions of the studied network. What is important here is the relative difference between all numerical experiments.

Although air trapping is more important in the numerical simulations than in the experimental results, the general behavior is very similar. In the modified part of the studied networks (treated layer), the imbibition kinetic is indeed far slower than for the reference network until the height of the water front exceeds the layer thickness.

In a second part, the slope of the curves greatly increases and remains constant until the water front reaches the top of the network, which indicates that the imbibition follows a classical Washburn law. Note that the slopes of the first and second part are both smaller when the thickness of the treated layer increases.

The slow kinetic observed in the first part is the consequence of:

• The reduction of the macroscopic hydraulic conductivity in the layer, due to the lower porosity but also to the lower connectivity of the pore network.

• The reduction of the macroscopic capillary pressure, due to the fact that the layer contains far fewer small pores than in the reference network.

When the fluid enters the unmodified part of the network, i.e. the part where the water repellent has not penetrated, the capillary flow is only affected by the reduction of the hydraulic resistance in the layer, which explains that the imbibition kinetic becomes slower than in the reference network when the thickness of the hydrophobic layer increases. Note however that the flow pattern can be altered by the low connectivity in the layer. For example, Figure 8b shows a typical capillary fingering pattern, which partly explain the important trapping observed in the simulations.

Pore network simulations allow reproducing qualitatively the observed experimental behavior. It is a good indication that the observed kinetics may be partly explained by the filling of the smaller pores by the water repellent. However, one can notice that the experimental kinetics in the layer is slower than in simulations. This could be explained by other effects, which are not taken into account in the numerical simulations. In particular the contact angle is assumed to be zero and constant in simulations. Even if Figure 5 shows that the contact angle at the surface decreases quite quickly, we have no indication about the evolution of the wettability inside the concrete sample. It is likely that the average contact angle is greater inside the treated layer than in the rest of the network, which decrease the capillary pressure inside the layer. In this paper, we have studied the effects of two water repellent agents on the water imbibition kinetics and on the microstructure of a relatively low performance concrete.

Numerical simulations based on a pore network approach were also carried out to support our experimental findings. Based on our results, the following conclusions can be drawn:

• The water porosity of concrete samples is reduced in the treated layer by water repellent penetration. The internal face of concrete pores is coated with the hydrophobic water repellent, which fills primarily the smallest pores and reduces consequently the connectivity of porous network. This explains why it is difficult to access the global porosity of specimens treated with water repellent.

• The efficiency of the water repellents to reduce capillary absorption kinetics depends of their penetration depth.

• The repellent effect is quite limited in time. When water exceeds the limit of the treated layer, the absorption kinetics become quite similar to those of the non-treated concrete. The capillary absorption coefficient is however lower due to a lower hydraulic conductivity in the treated layer.

• Pore network simulations allow reproducing qualitatively the experimental behavior. This supports the fact that the observed kinetics may be partly explained by the filling of the smallest pores by the water repellent.

These results confirm the importance of an exhaustive preliminary study in view to protect a reinforced concrete structure in its environment. Particular attention must be paid to the perfectibility of water repellent against the water absorption kinetics in view to improve the efficiency and durability of these systems.

Figure 1 :

 1 Figure 1: Silane and siloxane molecule[Johansson, 2010]

Figure 2 :

 2 Figure 2: Summary diagram of the experimental procedure

Figure 3 :

 3 Figure 3: Multi-scan approach of water repellents penetration (image 1 and 2: transversal sections, images 3 and 4: SEM scans of concretes surface, images 5 and 6: SEM scans of the pores surface

Figure 4 :

 4 Figure 4: Pores size distribution of C15, C15H1 and C15H2 concretes

Figure 5 :

 5 Figure 5: Capillary absorption of C15, C15H1 and C15H2 concretes

Figure 7 :

 7 Figure 7: Pore radius distribution (a) and throat radius distribution (b)

Figure 8 :

 8 Figure 8: A realization of a 10 x 25 x 10 pore network before (a) and after (b) removing small pores in a layer of thickness equal to 30% of the total height of the network.

Figure 9 :

 9 Figure 9: Visualization of the pore network at different stages of the imbibition: (a) water begins to fill the unmodified part of the network, (b) final state in the modified layer, (c) percolation and (d) final state. Empty pores are colored in white while filled pores are either colored in blue or in red. Red pores are located in the layer where the water repellent penetrates, while blue pores are located in the unmodified part of the network.

Figure 10 :

 10 Figure 10: Imbibition kinetics for different layer thicknesses expressed in percentage of the total height.

Table 2 :

 2 Measured penetration depths of water-repellent and water porosities of

	C15, C15H1 and C15H2
		Penetration	Water
		depth (mm)	porosity (%)
	C15	0	14.9 ± 0.5
	C15H1	2.5 ± 0.5	13.2 ± 0.1
	C15H2	9.5 ± 2.5	8.1 ± 0.5

Table 3 :

 3 Pore size distributions for the reference network

		Average	Standard	Probability	Minimum	Maximum
		radius (nm)	deviation (nm)		radius	radius
	G1	400	100	40%	1	1000
	G2	10	5	60%	1	100

6 Acknowledgements E. Conforto, research engineer (LaSIE UMR 7356 CNRS), is acknowledged for her support in the scanning microscopy investigations.