Delphine Talbot
email: delphine.talbot@sorbonne-universite.fr

Sebastien Abramson
email: sebastien.abramson@sorbonne-universite.fr

Nébéwia Griffete

Sébastien Abramson

Nebewia Griffete
email: nebewia.griffete@sorbonne-universite.fr

Agnès Bée
email: agnes.bee@sorbonne-universite.fr

pH-sensitive magnetic alginate/γ-Fe 2 O 3 nanoparticles for adsorption/desorption of a cationic dye from water

Keywords: Ferrofluid, alginate, magsorbent, magnetic nanoparticles, adsorption, water remediation, dye

pH-sensitive magnetic alginate/γ-Fe2O3 nanoparticles for adsorption/desorption of a cationic dye from water

Introduction

Among the different technologies used for the removal of pollutants from industrial effluents, adsorption is considered to be an efficient and economic process easy to implement. However, the preparation of adsorbents that can efficiently remove pollutants, while exhibiting green properties and facilities for industrial development still remains challenging [START_REF] Ali | Low cost adsorbents for the removal of organic pollutants from wastewater[END_REF]. Moreover, separating adsorbents from water after pollutants extraction remains a laborious and time-consuming step. Using magnetic adsorbents makes it possible to overcome these difficulties. Indeed, these materials can be easily extracted from the treated water by applying a simple magnetic process reducing the risk of secondary pollution of the environment by waste accumulation [START_REF] Mehta | Magnetic adsorbents for the treatment of water/wastewater-A review[END_REF].

Magnetic nanoparticles and more particularly iron oxide nanoparticles have been widely studied and used since a long time in many fields such as hyperthermia, magnetic drug delivery, magnetic resonance imaging, data storage, environmental remediation and so on [START_REF] Bayramoğlu | Kinetics of mercury ions removal from synthetic aqueous solutions using by novel magnetic p(GMA-MMA-EGDMA) beads[END_REF][START_REF] Chen | Comparison of methylparaben, ethylparaben and propylparaben adsorption onto magnetic nanoparticles with phenyl group[END_REF][START_REF] Su | Environmental implications and applications of engineered nanoscale magnetite and its hybrid nanocomposites: A review of recent literature[END_REF][START_REF] Caetano | Magnetic hyperthermia-induced drug release from ureasil-PEO-[gamma]-Fe2O3 nanocomposites[END_REF][START_REF] Neveu | Size-Selective Chemical Synthesis of Tartrate Stabilized Cobalt Ferrite Ionic Magnetic Fluid[END_REF][START_REF] Ngomsik | Magnetic solid-liquid extraction of Eu(III), La(III), Ni(II) and Co(II) with maghemite nanoparticles[END_REF][START_REF] Demarchi | Adsorption of the dye Remazol Red 198 (RR198) by Ocarboxymethylchitosan-N-lauryl/γ-Fe2O3 magnetic nanoparticles[END_REF][START_REF] Gupta | Application of response surface methodology to optimize the adsorption performance of a magnetic graphene oxide nanocomposite adsorbent for removal of methadone from the environment[END_REF]. These applications often need a surface engineering of the magnetic nanoparticles and the research of appropriate compounds to introduce additional functionalities to the nanoparticles attracts more and more attention. Nevertheless, it is still a challenge to synthesize functionalized magnetic nanoparticles in a controlled manner, especially when green and large-scale conditions are required [START_REF] Wu | Designed synthesis and surface engineering strategies of magnetic iron oxide nanoparticles for biomedical applications[END_REF].

Composites based on natural polymers are in focus as green adsorbents and among them, alginate seems to be a good candidate due to its strong adsorption properties. Alginate, mainly extracted from brown algae, is a water-soluble linear polysaccharide, which combines features of abundant resources, low-cost, biocompatibility and biodegradability. This natural polyelectrolyte is composed of β-D-mannuronate (M) and α-L-guluronate (G) units arranged in blocks rich in G units (G-block) or M units (M-block) separated by blocks of alternating G and M units (MG-block) [START_REF] Draget | Chemical, physical and biological properties of alginates and their biomedical implications[END_REF]. Alginates are extensively used in biotechnologies, medicine or food industry [START_REF] Remminghorst | Bacterial alginates: from biosynthesis to applications[END_REF][START_REF] Rehm | Bacterial alginates: biosynthesis and applications[END_REF]. It is mainly their gelling properties that are at the origin of their industrial applications. Due to this ability to form stable hydrogels, alginates can encapsulate different materials [START_REF] Zare | Development of zirconia nanoparticles-decorated calcium alginate hydrogel fibers for extraction of organophosphorous pesticides from water and juice samples: Facile synthesis and application with elimination of matrix effects[END_REF], which can strengthen their adsorption properties or confer them other properties, such as, for example, magnetic properties when magnetic nanoparticles are trapped in the polymer matrix [START_REF] Bunkoed | Polypyrrole-coated alginate/magnetite nanoparticles composite sorbent for the extraction of endocrine-disrupting compounds[END_REF][START_REF] Li | Synthesis of Magnetic Microspheres with Sodium Alginate and Activated Carbon for Removal of Methylene Blue[END_REF][START_REF] Pittermannova | Microfluidic fabrication of composite hydrogel microparticles in the size range of blood cells[END_REF][START_REF] Rocher | Removal of organic dyes by magnetic alginate beads[END_REF]. Therefore, alginate is currently a hot topic for environmental research and many polymer nanocomposites are developed for applications in this field [START_REF] Hu | Jinshui Yao Facile Preparation of Tannic Acid-Poly(vinyl alcohol)/Sodium Alginate Hydrogel Beads for Methylene Blue Removal from Simulated Solution[END_REF][START_REF] He | A novel 3D yttrium based-graphene oxidesodium alginate hydrogel for remarkable adsorption of fluoride from water[END_REF][START_REF] Wang | Yang-guang Wang and Xiao-kun Ouyang Fabrication of Composite Beads Based on Calcium Alginate and Tetraethylenepentamine-Functionalized MIL-101 for Adsorption of Pb(II) from Aqueous Solutions Polymers[END_REF][START_REF] Yi | The elucidation of surrounding alginate gels on the pollutants degradation by entrapped nanoscale zero-valent iron[END_REF][START_REF] Gjipalaj | Easy recovery, mechanical stability, enhanced adsorption capacity and recyclability of alginate-based TiO2 macrobead photocatalysts for water treatment[END_REF].

Recently, different methods have been reported for the synthesis of magnetic alginatenanoparticles [START_REF] Liao | Functionalized magnetic iron oxide/alginate core-shell nanoparticles for targeting hyperthermia[END_REF][START_REF] Ma | Preparation and characterization of superparamagnetic iron oxide nanoparticles stabilized by alginate[END_REF][START_REF] Mohammadi | Fast removal of malachite green dye using novel superparamagnetic sodium alginate-coated Fe3O4 nanoparticles[END_REF][START_REF] Peng | High drug loading and pHresponsive targeted nanocarriers from alginate-modified SPIONs for anti-tumor chemotherapy[END_REF][START_REF] Castelló | Chitosan (or alginate)-coated iron oxide nanoparticles: A comparative study[END_REF]. Two types of strategies are usually adopted: the magnetic nanoparticles are either synthesized in-situ in the alginate matrix, or synthesized separately (coprecipitation, hydrothermal treatment, etc.) and subsequently added to the alginate solution (so-called two-step method). However, in most of these materials, magnetic nanoparticles consist in magnetite (Fe 3 O 4 ), which is readily oxidizable and especially unstable in acidic environment [START_REF] Sidhu | Dissolution of iron oxides and oxyhydroxides in hydrochloric and perchloric acids[END_REF], which complicates the regeneration of this kind of materials. In addition, they are mainly used for biomedical applications [START_REF] Peng | High drug loading and pHresponsive targeted nanocarriers from alginate-modified SPIONs for anti-tumor chemotherapy[END_REF].

This study reports the synthesis of alginate/maghemite nanoparticles (AlgMNP) for the adsorption of water pollutants. The use of maghemite (γ-Fe 2 O 3 ) instead of magnetite as magnetic core is of particular interest because of the much larger stability of this phase. In addition to the green character and the adsorption sites of alginate, magnetic properties of the nanoparticles allow their fast extraction from complex environmental samples with a simple magnetic process. The two-step method described here consists first in the preparation of sizecontrolled maghemite nanoparticles (MNP) dispersed in diluted nitric acid to form a stable ferrofluid. Alginate solution is then added to the ferrofluid to obtain the magnetic adsorbent.

Alginate is bound to the surface of the MNP through its carboxylate functions; the remaining carboxylate groups can then be used to adsorb pollutants in water. This method, easy to implement, is an attractive approach for effective clean-up procedure because it does not use toxic products and can be developed at an industrial scale. Moreover, there is no need of centrifugation or filtration steps due to the magnetic properties of the materials used. The potential use of AlgMNP for water treatment was investigated through adsorption experiments, methylene blue being chosen as a model of pollutant. Dyes are compounds widely used in the industry and their presence in effluents is of serious concern because of their toxic effects for both environment and human health [START_REF] Kumar Gautam | Advanced Nanomaterials for Wastewater Remediation[END_REF]. The influence of pH, contact time, dye concentration on adsorption efficiency was studied. The reusability of the magnetic composite was demonstrated through adsorption/desorption cycles. We made a special effort to understand the interactions between the dye and the adsorbent. A good knowledge of our system and the control of the synthesis parameters enabled us to predict the number of active sites involved in the adsorption of the pollutant, which was then compared with the experimental results.

Experimental 2.1 Chemicals

Sodium alginate was purchased from Sigma Aldrich. The weight average molar weight (M w ) and the number average molar weight (M n ), obtained by gel permeation chromatography, are respectively 2.74x10 5 and 1.49x10 5 g/mol, leading to a polydispersity index (I P ) equal to 1.8.

The amount of carboxylate functions of alginate (pKa = 3.4-4.2 [START_REF] Lagoa | Evaluation of Dry Protonated Calcium Alginate Beads for Biosorption Applications and Studies of Lead Uptake[END_REF][START_REF] Lamelas | Quantifying Pb and Cd Complexation by Alginates and the Role of Metal Binding on Macromolecular Aggregation[END_REF]) was obtained from the dosage of their sodium counterions ([Na] alg ) by atomic absorption spectrometry with a Perkin-Elmer Analyst 100 apparatus. It is equal to 4.2 ± 0.1 mmol/g alg .

Methylene blue (MB) was purchased from Sigma-Aldrich (M=373.9 g/mol). Stock solutions of MB were obtained by dissolving the powder in distilled water. Their concentrations were measured by spectrophotometry at λ=664nm using a UV-Visible UVIKON XL Secoman apparatus; the extinction coefficient ε 664 is 83243 molL -1 cm -1 .

Preparation of magnetic nanoparticles

A two-step procedure was used to obtain AlgMNP. In the first step, maghemite (-Fe 2 O 3 ) nanoparticles (MNP) were synthesized according to the Massart's method [START_REF] Massart | Preparation of aqueous magnetic liquids in alkaline and acidic media[END_REF][START_REF] Bee | Synthesis of very fine maghemite particles[END_REF] by alkaline co-precipitation performed by rapid addition of concentrated ammonium hydroxide (20%, 1L) in an aqueous stoichiometric mixture of FeCl 2 and FeCl 3 obtained by mixing 180 g of Fe(II) salt (0.9 mol) and 715 mL of Fe(III) solution (1.5 mol) in acidic medium (HCl). The resulting dark black magnetite (Fe 3 O 4 ) was then stirred in nitric acid (360 mL, 2 moL/L), oxidized into maghemite by a boiling solution of Fe(NO 3 ) 3 (323g, 1.3 mol) and, after washing, dispersed into water leading to a stable magnetic fluid called ferrofluid. The pH value of the resulting suspension is about 2. At this time, nanoparticles are positively charged with nitrate counterions. In the second step, alginate solution (18 g/L) was added to the diluted ferrofluid to obtain the alginate-bound maghemite nanoparticles. Preliminary studies were carried out to optimize the ratio (m Alg being the introduced weight of alginate and m MNP , the weight of maghemite nanoparticles calculated from the iron concentration of the ferrofluid). Then, the reaction mixture was stirred for 30 min with an orbital shaker at 50°C. The alginate-bound maghemite nanoparticles were recovered from the reaction mixture by centrifugation (20min, 7500 rpm) or by using a permanent magnet with a surface magnetization of 800 G (typical value of the magnetic field at 1.5cm from a cylindrical Nd/Fe/B magnet with a diameter of 2.5cm and a height of 1cm). Then, nanoparticles were washed three times using distillated water and dried 48h in an oven at 70°C.

Characterization

The size and morphology of the magnetic nanoparticles were observed by transmission electron microscopy using a Jeol model JEM 100 CX at 100 kV. The magnetic size distribution of the nanoparticles, which can be approximated by a lognormal law, was obtained by fitting magnetization curves parameters and led to the mean diameter d Aim (ln d Aim = <lnd>) and distribution width  Aim of the nanoparticles [START_REF] Cabuil | Particle size determination in magnetic fluids[END_REF]. The experimental device for the magnetic measurements was based on the one described by S. Foner [START_REF] Foner | Very Low Frequency Integrating Vibrating Sample Magnetometer (VLFVSM) with High Differential Sensitivity in High dc Fields[END_REF]. X-ray diffraction (XRD) on dried samples using a Rigaku Ultima IV apparatus with a copper anode (Cu K α radiation,  = 1.54178Å) was performed to identify the crystallographic structure and size of the nanoparticles. The equivalent iron concentration of the ferrofluid ([Fe]) was determined by atomic absorption spectrometry using an air-acetylene flame (Perkin Elmer AAnalyst 100 apparatus). Before measurements, magnetic nanoparticles were heated to ebullition in concentrated hydrochloric acid (≈12 mol L -1 ) until total dissolution of the nanoparticles into iron (III) ions was achieved. The amount of alginate bound on nanoparticles was estimated from the thermogravimetric analysis (TGA) performed on dried nanoparticles in air atmosphere with a heating rate of 10°C/min (TA Instruments SDT Q600).

FT-IR spectra of pure alginate, MNP, and AlgMNP were recorded in KBr discs in the range of 4000-400 cm -1 on a Bruker Tensor 27 to check the binding of alginate to maghemite nanoparticles.

Adsorption/desorption experiments

Batch adsorption experiments were carried out at room temperature (≈20°C). 100 mg (m) of dried AlgMNP (R=0.14) were added to 10 mL (V) of MB at initial concentration C 0 , pH of the samples being adjusted using either HNO 3 or NaOH solutions. After stirring using an orbital shaker (IKA KS 260) (2h to reach equilibrium), nanoparticles were removed magnetically from MB solution. The concentration of MB remaining in solution at equilibrium (C eq ) or at a given time t (C t ) was measured by UV-Visible spectrophotometry at 664 nm. We checked for each measure that there was no change in the BM spectrum. In particular, no trace of alginate that could complex the BM appeared in the solution to be analyzed, which means that the AlgMNP retain their integrity during the various tests. The amount of adsorbed dye at equilibrium (Q eq ) or at a given time t (Q t ) was deduced using the mass balance equation .

For the desorption study, 100mg of AlgMNP were first added to 10mL of MB solution (C 0 = 200 mg/L) at pH 7. After two hours of stirring, the nanoparticles were magnetically recovered, gently washed with distilled water and then introduced into 10 mL of nitric acid 0.01 mol/L for 30 minutes. The particles were again separated from the solution. The quantity of desorbed MB was then determined by UV-Visible spectrophotometry. Before the next adsorption, the particles were washed to raise the pH to about 7. Ten adsorption/desorption cycles were carried out.

Results and discussion 3.1 Determination of the optimal concentration of polymer

At first, the optimal ratio needed to obtain an adsorbent with a high dye adsorption capacity was determined. For this, increasing amounts of alginate solution were added to a ferrofluid at an initial iron concentration [Fe] i =0.4 mol/L (m MNP =640mg), the total volume of the solution being 20 mL. Without alginate (R=0), the ferrofluid is a stable colloidal dispersion. Upon addition of alginate, a progressive flocculation of the ferrofluid occurs (Figure 1), which was followed by the determination of the iron concentration in the supernatant ([Fe] s ) (Figure 2). R=0 R=0.05 R=0.14 R=0.20 For R>0.14, flocculation is complete, there is no more nanoparticles in the supernatant, which appears colorless. This macroscopic phase separation was also followed by measuring the height of the brown precipitate (h). These measurements were carried out at equilibrium after leaving the samples on a magnet. As can be seen in Figure 2, h remains roughly constant for R > 0.14. The stability of the ionic ferrofluid used is due to electrostatic repulsions between the charged nanoparticles. Without alginate, the charges of the nanoparticles come from the ionization of hydroxyl groups at their surface. The surface charge is positive for a pH value below the point of zero charge (PZC) of the nanoparticles and negative for pH above the PZC. The ferrofluid flocculates for a pH value close to the PZC; for maghemite, the PZC is located around pH 7 [START_REF] Bée | Magnetic chitosan/clay beads: A magsorbent for the removal of cationic dye from water[END_REF]. The pH of the ferrofluid used in the samples being close to 2, nanoparticles are positively charged with nitrate counterions. By adding alginate, a self-assembly between negatively charged alginate and positively charged nanoparticles occurs involving the carboxylate functions of alginate; the surface charge of the nanoparticles decreases and the ferrofluid flocculates. In conclusion, we retained R=0.14 for the preparation of magnetic alginate nanoparticles.

Characterization of the magnetic nanoparticles

Magnetic nanoparticles were carefully characterized before investigating their adsorption properties. The iron concentration of the ferrofluid determined by atomic absorption spectrometry is 1.7 mol L -1 . A typical transmission electronic microscopy (TEM) picture of this diluted ferrofluid before alginate addition shows that the shape of the magnetic nanoparticles is roughly spherical. The analysis of the TEM pictures with the free software ImageJ leads to a mean diameter d TEM equal to 7.6 nm with a polydispersity of 0.5.

Figure 3 : A stable ferrofluid (left) and MNP picture obtained by transmission electron microscopy (right)

The mean diameter (d Aim ) and the polydispersity index ( Aim ) of the naked nanoparticles were also obtained from a two-parameter fit of the magnetization curve of diluted ferrofluid (39); they are 7.4 nm and 0.4 respectively. Evidence of the binding of alginate to the nanoparticles was provided by FT-IR analysis. The spectra of pure alginate, MNP and AlgMNP are shown in Figure 5. The corresponding characteristic bands and their attribution are listed in Table 1. The broad bands in the range 2600-3600 cm -1 , present in the three spectra, can be attributed to stretching vibrations of -OH bonds of alginate and sorbed water on nanoparticles. The weak peak at 2920 cm -1 on the alginate spectrum is assigned to -CH 2 groups. Alginate also displays characteristic bands at 1618 and 1420 cm -1 corresponding to the asymmetric and symmetric stretching vibrations of the carboxylate, respectively. The bands at 1094 and 1026 cm -1 are attributed to C-O bond in -C-OH groups. The bands at 581 and 631 cm -1 observed on the MNP spectrum are the signature of the Fe-O bonds. Moreover the intense narrow band at 1385 cm -1 is related to the nitrate counterions of the MNP. The small band at 1621 cm -1 is probably due to the deformation vibration of adsorbed water. The AlgMNP spectrum exhibits the characteristic bands from both alginate and MNP. The alginate carboxyl absorption bands are slightly shifted to 1609 and 1408 cm -1 probably due to the modification of the carboxyl environment, In order to calculate the amount of alginate bound to the MNP, we used thermogravimetric analysis (TGA). Figure 6 displays the curves from 20°C to 500°C for alginate, MNP and AlgMNP. TGA curve of MNP exhibits weight losses of 6% (up to 150°C) and 4% (between 150°C and 360°C) due to physically and chemically adsorbed water respectively. The alginate curve shows a first weight loss of 18%, between 50°C and 200°C, which corresponds to the water desorption and to the destruction of glycosidic bonds. Beyond 250°C, the glycosidic bonds were further destroyed [START_REF] Zhang | Pyrolysis products and thermal degradation mechanism of intrinsically flame-retardant calcium alginate fibre[END_REF]. The total weight loss for alginate is 60%. In the case of AlgMNP, a weight loss of about 7% was observed from 20°C to 180°C due to the loss of water. In the second stage (180-360°C), the weight loss is about 9%, which was assigned to the thermal degradation of alginate adsorbed onto the surface of the maghemite nanoparticles.

X-ray diffraction patterns (XRD) of MNP and AlgMNP are displayed in

These results show that AlgMNP are stable below 180°C and that the samples contain 84±1

wt.% of γ-Fe 2 O 3 and 9±1 wt.% of alginate. Thus the weight ratio of bound alginate to maghemite nanoparticles is 0.11. This value is slightly higher than reported values obtained for the coating of magnetite nanoparticles by alginate [START_REF] Castelló | Chitosan (or alginate)-coated iron oxide nanoparticles: A comparative study[END_REF][START_REF] Xu | The colloidal stability and core-shell structure of magnetite nanoparticles coated with alginate[END_REF]. From the density of the maghemite (ρ=5.074 g/cm 3 ) and the average molecular weight of alginate (M w =200,000 g/mol), it was estimated that 0.4 alginate chains on average were bound to each nanoparticle or about 2.6 nanoparticles were bound to each alginate chain. A similar result was obtained with PAA-magnetite nanoparticles (M w,PAA =180,000 g/mol); the authors have shown that about 2 PAA were bound to one particle [START_REF] Mak | Fast adsorption of methylene blue on polyacrylic acid-bound iron oxide magnetic nanoparticles[END_REF]. In our case, it is probably more accurate to consider alginate chains decorated with nanoparticles rather than an alginate coating around the nanoparticles. AFM images obtained by Ma et al. showing magnetite nanoparticles bound to the strands of alginate macromolecules corroborate this assumption [START_REF] Ma | Preparation and characterization of superparamagnetic iron oxide nanoparticles stabilized by alginate[END_REF].

Adsorption studies a. Effect of pH on MB adsorption

The charge and adsorption properties of the adsorbent are strongly affected by the pH of the solution. Hence, batch equilibrium studies were performed over a pH range from 2 to 11.2 to investigate the effect of pH on the adsorption of MB (C 0 =200mg/L) by AlgMNP (Figure 7). The amount of adsorbed MB increases from 0 to 98% adsorption with increasing pH from 2 to 5, and this can be attributed to the ionization of carboxylate groups of alginate, which interact with the positively charged dye via strong electrostatic interactions. In addition, electrostatic repulsions between carboxylate groups expand the alginate chains making the adsorption sites more accessible to the dye [START_REF] Mohammadi | Fast removal of malachite green dye using novel superparamagnetic sodium alginate-coated Fe3O4 nanoparticles[END_REF]. For pH<4.5, the carboxylate protonation becomes significant and the adsorption falls, no adsorption occurs at pH≤2 while 100% adsorption is obtained in a large pH range [4.5-11.2]. The shape of the curve is characteristic of the adsorption of positively charged species by adsorbents carrying carboxylate groups [START_REF] Mak | Fast adsorption of methylene blue on polyacrylic acid-bound iron oxide magnetic nanoparticles[END_REF][START_REF] Bée | Magnetic alginate beads for Pb(II) ions removal from wastewater[END_REF]. Due to the strong adsorption, the following studies were conducted at a fixed pH greater than 5.

b. Adsorption isotherm

The adsorption capacity of AlgMNP was determined by equilibrium adsorption experiments conducted at two pH values (7.1 and 10.1) (Figure 8). In both cases, the amount of adsorbed MB increases with increasing concentration of dye until reaching a plateau, which corresponds to the maximum adsorption capacity of AlgMNP.

A Langmuir equation was used to fit the experimental data. Its linearized form is , where Q eq , is the amount of adsorbed dye at equilibrium, Q max , the maximum adsorption capacity, C eq , the concentration of dye in the solution at equilibrium and, K L the Langmuir constant. Figure 8 and Table 2 The adsorption sites of AlgMNP originate from both the carboxylate functions of the alginate bound to the particles and the nanoparticles themselves, which bear positive charges in acid medium and negative charges in basic medium. The amount of carboxylate functions of alginate was estimated to be 0.57 mmol/g from the mass ratio of bound alginate to maghemite taking into account that each monomer of the alginate carries one carboxylate function. A part of these carboxylate functions is used to form bonds between the alginate chains and the nanoparticles while the remaining part is available to adsorb the dye. On the other hand, the amount of adsorption sites of alginate-free MNP for a given pH value had been estimated

previously by potentiometric measurements [START_REF] Ponton | Regeneration of thixotropic magnetic gels studied by mechanical spectroscopy: the effect of the pH[END_REF]. At pH 7.1, close to their point of zero charge, the naked nanoparticles are not charged. Therefore, at this pH value the adsorption sites of AlgMNP originate only from the carboxylate functions of the alginate bound to the nanoparticles (0.57÷0.05mmol/g). However, the maximum adsorption capacity (0.43 mmol/g) is lower than the amount of carboxylate functions because a part of them is used to form bonds with the nanoparticles. At pH 10.1, the magnetic nanoparticles, negatively charged, may also adsorb BM. This is why the maximum adsorption capacity is greater at pH 10 than at pH 7. At pH 10, the amount of negative adsorption sites of the MNP is 0.27 mmol/g [START_REF] Bée | Magnetic chitosan/clay beads: A magsorbent for the removal of cationic dye from water[END_REF].

As for alginate, some of these sites are involved in binding with alginate, the remaining part being available for adsorption of BM. At this pH, the estimated maximum number of adsorption sites of AlgMNP is therefore 0.84±0.05 mmol/g, which is close to the maximum adsorption capacity of AlgMNP deduced from the isotherm (0.73 mmol/g). The difference could be explained by the mobilization of a part of the sites for the adsorption of alginate on the nanoparticles. It could be noted that AlgMNP sites are about three times greater than for the magnetic nanoparticles without alginate.

The presence of available sites on the MNP agrees with our previous assumption of a polymer decorated with magnetic nanoparticles rather than a coating of alginate around the nanoparticles, which would mask their adsorption sites. Compared to other results from the literature, the maximum adsorption capacity of AlgMNP obtained in this work is either of the same order of magnitude or higher [START_REF] Li | Synthesis of Magnetic Microspheres with Sodium Alginate and Activated Carbon for Removal of Methylene Blue[END_REF][START_REF] Mohammadi | Fast removal of malachite green dye using novel superparamagnetic sodium alginate-coated Fe3O4 nanoparticles[END_REF][START_REF] Kannan | Sundaram Kinetics and mechanism of removal of methylene blue by adsorption on various carbons-a comparative study[END_REF][START_REF] Auta | Chitosan-clay composite as highly effective and low-cost adsorbent for batch and fixed-bed adsorption of methylene blue[END_REF].

c. Effect of contact time

The kinetics of MB adsorption by AlgMNP was investigated.

Reusability of AlgMNP

Reusability of AlgMNP is one of the key factors in evaluating the performance of an adsorbent. In order to investigate the reusability of AlgMNP, an adsorption/desorption cycle was repeated 10 times using the same nanoparticles. Carboxylate being protonated in acid medium, no more interactions occur between alginate and the dye and adsorption falls. Therefore, nitric acid at pH 2 was used as desorption agent. As it can be seen on the Figure 10, the amount of desorbed MB is greater than 99% and the adsorption capacity remains close to 100% even after 10 cycles. This result shows that the magnetic alginate nanoparticles could be used in a water treatment process without significant loss of their adsorption efficiency.

Conclusion

In this work, a magnetic adsorbent (magsorbent) based on an alginate nanocomposite with controlled particle size was prepared using a two-step method, which may be easily developed on an industrial scale. The combination of maghemite nanoparticles and a natural polymer such as alginate is one of the most promising methods to obtain a green adsorbent for the removal of pollutants from wastewater. This approach presents several advantages: maghemite has excellent stability with respect to oxidation; in addition, its magnetic properties makes it easy to recover after wastewater treatment by using a simple magnetic process; on the other hand, alginate is not only an abundant natural resource but its many carboxylate functions ensure both a efficient binding to the nanoparticles and a strong adsorption capacity for cationic pollutants. Synthesized nanocomposites were carefully characterized before being used for the removal of a cationic dye (methylene blue) from water. The adsorption efficiency was increased at higher pH values because of the strong electrostatic interactions between carboxylate functions of alginate and cationic dye. The highest adsorption is at a neutral or basic pH. The maximum adsorption capacity of MB by AlgMNP is 273mg/g. The contribution of both MNP and alginate to the adsorption of the dye has therefore been envisaged and an organization with magnetic nanoparticles bound to the alginate chains through their carboxylate functions rather than a coating of MNP by alginate was proposed. The reuse performance of AlgMNP was higher than 98% even after 10 adsorption/desorption cycles. We can conclude that this study could be helpful in the development of magnetic adsorbents. AlgMNP present a high application potential for the recovery of cationic pollutants from water due to its magnetic properties, its adsorption efficiency and its reusability.

Figure 1 :

 1 Figure 1: Effect on the amount of alginate on the stability of the ferrofluid.

Figure 2 :

 2 Figure 2 : Evolution of iron remaining in the supernatant ([Fe] s /[Fe] i ) and of the height of the precipitate (h) with R (inner diameter of the bottle = 2.7 cm).

Figure 4 .

 4 The characteristic peaks related to the crystal planes (220), (311), (400), (422), (511) and (440) are consistent with the data in JCPDS 39-1346 file corresponding to the maghemite structure. It is worth noting that the AlgMNP exhibit the same characteristic peaks as the naked MNP, indicating that alginate did not alter the crystalline phase of iron oxide. The average diameter of the nanoparticles, d RX , calculated through full width at half maximum of the strongest reflexion of the diffractograms, using the Scherrer equation, is 7.7 nm for both MNP and AlgMNP.

Figure 4 :

 4 Figure 4 : XRD of AlgMNP (a) and MNP (b)

  sodium ions being replaced by surface iron from MNP. However, no significant shifts are observed for Fe-O bands. The most notable change is the strong decrease of the nitrate band due to the replacement of nitrate ions by carboxylate from alginate. These results indicate that alginate was successfully bound to the MNP.

Figure 5 :

 5 Figure 5 : FT-IR spectra of alginate (a), MNP (b) and AlgMNP (c).

Figure 6 :

 6 Figure 6 : TGA curves of MNP (a), AlgMNP R=0.14 (b) and pure alginate (c).

Figure 7 :

 7 Figure 7: pH effect on MB adsorption (C 0 =200mg/L)

Figure 8 :

 8 Figure 8 : Adsorption isotherms of MB onto AlgMNP, solid line represents the Langmuir isotherm; pH= 10.1± 0.5 (), pH= 7.1± 0.3 ()

Figure 9

 9 illustrates the effect of contact time on the adsorption of MB with two initial concentrations C 0 equal to 2008 and 1479 mg/L at an initial pH close to 7. Adsorption is very rapid since 80% adsorption occurs in the first 15 minutes. Thereafter, adsorption becomes slower to reach an equilibrium time equal to 2h. The adsorbed amounts at equilibrium are in agreement with the adsorption isotherm at pH 7. The rapid adsorption is due to the availability of the active sites on the adsorbent at the initial stages. Moreover, as already mentioned above, at pH 7 the deprotonated carboxylate functions repel each other and allow easier access for the dye molecules. The gradual occupancy of the sites decreases the efficient interactions between MB and carboxylate and the adsorption becomes slower.

Figure 9 :

 9 Figure 9 : Influence of contact time on MB adsorption: pH=6.7±0.3, C 0 =2008mg/L (); pH=7.2±0.4, C 0 = 1479 mg/L ().

Figure 10 :

 10 Figure 10: successive sorption (black) /desorption (grey) cycles

Table 1 : FT-IR analysis of alginate, MNP and AlgMNP (25,26,29,40-42).

 1

		a
		b
		c
	Wavenumber cm -1	Assignment
		Alginate
	3431	O-H stretching vibration
	2920	Aliphatic C-H stretching vibration
	1618 and 1420	Asymmetric and symmetric stretching vibrations
		of carboxylate groups
	1094 and 1026	Stretching vibration mode of C-O bond in C-OH
		groups
		MNP
	3385	Stretching vibration of hydroxyl groups of
		sorbed water
	1621	Deformation vibrations of adsorbed water

  prove that adsorption of MB on AlgMNP is well described by the Langmuir model. The maximum adsorption capacities obtained from

	Langmuir equation, 0.73 mmol/g at pH 10.1 and 0.43 mmol/g at pH 7.1 are very close to the
	experimental values.
	pH	R 2	Q max		K L
		mmol/g	mg/g	L/mmol
	10.1± 0.5	0.999 0.73		273	11.6
	7.1± 0.3	0.989 0.43		161	2.3

Table 2 : Langmuir parameters

 2

Acknowledgements:

The authors are very grateful to Mohamed Hanafi for GPC determination of molecular weights of alginate.


N Guigui

C Philippe

A Gloaguen

S Karkar

V Guillemot

T Löfstedt

V Frouin

NETWORK REGULARIZATION IN IMAGING GENETICS IMPROVES PREDICTION PERFORMANCES AND MODEL INTERPRETABILITY ON ALZHEIMER'S DISEASE

Keywords: Imaging-Genetics, Networks, Structured constraints, Generalized Canonical Correlation Analysis

Imaging-genetics is a growing popular research avenue which aims to find genetic variants associated with quantitative phenotypes that characterize a disease. In this work, we combine structural MRI with genetic data structured by prior knowledge of interactions in a Canonical Correlation Analysis (CCA) model with graph regularization. This results in improved prediction performance and yields a more interpretable model.

INTRODUCTION

The joint study of brain images and genetic data is of growing interest to find associations between genetic variants and disease-related features that can be measured on brain MRI. The rationale behind this research avenue is that imaging endophenotypes stand as proxies to disease status and evolution, that can be measured with non-invasive methods to facilitate early diagnosis and follow-up. However, genetic information is expected to provide insights into the underlying disease mechanisms and valuable tracks to finding therapies.

Existing studies seek pairwise associations between a set of genetic variants and brain regions or at the whole genome and whole brain scale [START_REF] Shen | Genetic analysis of quantitative phenotypes in AD and MCI: imaging, cognition and biomarkers[END_REF]. In this work, we demonstrate how combining multiple sources and layers of information, i.e. imaging and genetic experimental data and prior knowledge, with data integration methods brings supplemental information, both in terms of performance and interpretability compared to univariate approaches. The aim of data integration methods is to account for known interactions between variables and data from each modality.

Nonetheless, they raise many methodological challenges as both data types are very high-dimensional (10 4 variables) [START_REF] Nathoo | A Review of Statistical Methods in Imaging Genetics[END_REF]. Canonical Correlation Analysis (CCA), and its regularized generalization to multiple blocks, RGCCA [START_REF] Tenenhaus | Regularized Generalized Canonical Correlation Analysis[END_REF], is a framework that allows to model and quantify interactions within and between blocks of variables of heterogeneous types, and with respect to the disease status. Variants of RGCCA and CCA allow to select the variables that are the most relevant in the model [START_REF] Tenenhaus | Variable selection for generalized canonical correlation analysis[END_REF][START_REF] Witten | A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis[END_REF][START_REF] Du | Structured sparse canonical correlation analysis for brain imaging genetics: an improved Graph-Net method[END_REF].

However, biomarker identification from genotyping data usually fails to provide biologically relevant associations. This is because, different alterations in the same biological pathway often lead to the same pathological phenotype, but they will all fail to pass significance thresholds. Moreover, empirical evidences show that many segregating variants affect multiple traits and a precise estimation of the proportion of such variants remains elusive [START_REF] Visscher | 10 Years of GWAS Discovery: Biology, Function, and Translation[END_REF]. Therefore including genetic interaction networks as prior knowledge to wisely concentrate the information from genome-wide disseminated single-nucleotide polymorhpisms (SNPs) is a promising approach.These can be used in a data pre-processing step or to compute a regularization constraint [START_REF] Azencott | Network-Guided Biomarker Discovery[END_REF], or both as in [START_REF] Hofree | Network-based stratification of tumor mutations[END_REF]. Azencott [START_REF] Azencott | Network-Guided Biomarker Discovery[END_REF] gives a review of statistical methods to regularize variable selection with graphs, however the focus is mainly on univariate regression.

Following the framework formulated in [START_REF] Löfstedt | Structured Variable Selection for Regularized Generalized Canonical Correlation Analysis[END_REF], we present here the use of the GraphNet penalty on a multiblock CCA model at a genome-wide scale. We use the graph Pathway Commons (PC) (http://www.pathwaycommons.org) as prior knowledge to predict the disease status and illustrate our method on the Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort (http://adni.loni.usc.edu/). In Section 2, we present the method and evaluation scheme, then in section 3 we present the data used to assess the method. In section 4 we present the results and discuss them in section 5.

METHOD

Let X ∈ R n×p and Z ∈ R n×q represent two blocks of variables, for n individuals with respectively p and q features (e.g. imaging and genetics data). We denote by y ∈ {0, 1} n the disease status or target. X and Z are scaled respectively by √ p and √ q to balance for the different number of features [START_REF] Tenenhaus | Regularized Generalized Canonical Correlation Analysis: A Framework for Sequential Multiblock Component Methods[END_REF] and we assume that X, Z, y have been centred and scaled to unit variance.

Structured Sparse CCA The SGCCA [START_REF] Tenenhaus | Variable selection for generalized canonical correlation analysis[END_REF] optimization problem in the context of two blocks and a target is the following:

max x∈R p z∈R q C(x, z) = x T X T y + z T Z T y + x T X T Zz (1) s.t. x 2 2 = z 2 2 = 1 and x 1 ≤ s 1 , z 1 ≤ s 2 (2)
The first two terms aim at maximizing the covariance between the two latent variables representing each block and the target, while the third term is the interaction between the two blocks.

The problem is constrained with an 1 -norm with parameters s 1 and s 2 , in order to obtain sparse loadings x and z, to select the key variables in the model. We propose to add a structured penalty to the objective function of SGCCA in order to select variables that are known to interact. Let G = (V, E) be a graph with q vertices (e.g. the genes), representing the known interactions between the q variables. Let A ∈ R q×q be its adjacency matrix, and L ∈ R q×q be its Laplacian matrix. The penalty is composed of two parts, the first enforces the weights of variables that are connected in G to be close to one another. The second part is an 2 -norm penalty for isolated nodes, as they would be more likely to be selected otherwise. Let 1 ≤ d < q be the number of isolated variables in G, and suppose variables are ordered so that these nodes are the last d columns of Z. Let J d be the q × q diagonal matrix with the last d coefficients equal to 1, all others to 0:

P λ,γ L (z) = λ 1 2 q i=1 q j=1 A ij (z i -z j ) 2 + γ q j=q-d z 2 j (3) = z T (λL + γJ d )z (4)
When used in combination with the 1 norm, this penalty is known as GraphNet [START_REF] Grosenick | Whole-brain Sparse Penalized Discriminant Analysis for Predicting Choice[END_REF]. The final optimization problem is thus:

max x∈R p z∈R q f (x, z) = C(x, z) -P λ,γ L (z) (5)
s.t. x 2 2 ≤ 1, z 2 2 ≤ 1 and x 1 ≤ s 1 , z 1 ≤ s 2 (6)
where λ, γ, s 1 , s 2 are regularization hyper-parameters. We henceforth call this model GraphNet-GCCA (GN-GCCA). Notice that the 2 constraints (2) are changed to inequality constraints in [START_REF] Du | Structured sparse canonical correlation analysis for brain imaging genetics: an improved Graph-Net method[END_REF] in order to make the feasible set convex.

For certain choices of parameters s 1 and s 2 , the 2 constraints will be active at the optimum solution as discussed in [START_REF] Witten | A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis[END_REF].

Model fitting

The problem does not have an analytical solution and is non-convex, but it is multi-convex in the sense that the loss function -f is convex when considered as a function of each individual block loading, while the others are kept fixed. Minimizing such functions under convex constraints can be achieved with a so called block relaxation algorithm. It consists in alternating between the blocks to minimize the partial loss w.r.t each vector, as in the procedure Algorithm 1 Block relaxation algorithm for GN-GCCA, with desired precision

Initialize: x (0) , z (0) 1: repeat 2:

for k = 1..max iter do 3:

grad x ← X T (y + Zz (t) )

4:

x (t+1) ← proj P (x (t) + α x • grad x )

5:

end for 6:

for k = 1..max iter do 7:

grad z ← Z T (y + Xx (t+1) ) -Lz (t)
8:

z (t+1) ← proj Q (z (t) + α z • grad z ) 9:
end for 10: until f (x (t) , z (t) ) -f (x (t+1) , z (t+1) ) ≤ 11: return x (t+1) , z (t+1) described in [10, alg. 2]. Let P ⊂ R p and Q ⊂ R q be the convex sets where the constraints (6) are satisfied, and proj P and proj Q the orthogonal projections on those sets. Details to compute these projections are given in [START_REF] Löfstedt | Structured Variable Selection for Regularized Generalized Canonical Correlation Analysis[END_REF]. Each partial constrained minimization problem is solved with a few FISTA iterations [START_REF] Beck | A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems[END_REF] with step sizes α x and α z . Algorithm 1 describes the full procedure. This algorithm was implemented using the pylearn-parsimony package (https://github. com/neurospin/pylearn-parsimony).

Model selection and evaluation

The previous model has four regularization hyperparameters: s 1 and s 2 control the sparsity, while λ controls the roughness of z over G, and γ controls the shrinkage for isolated features because of its interaction with λ, this parameter was set manually . To compare our method with SGCCA, we set aside a test set and a training set. The prediction performance is evaluated by fitting a Linear Discriminant Analysis (LDA) model on the projections t 1 = Xx and t 2 = Zz. This allows to seek the best hyperparameters, assessing them in a 5-fold cross-validation (CV) scheme, on the training set. We then choose the combination of parameters that achieved the highest area under the ROC curve (AUC) on the classification task with target y, and train the model on the whole training set with this choice of parameters. Finally the AUC is computed for this model on the test set.

DATA AND EXPERIMENTS

We used brain structural MRI data and genotype data from the ADNI database. The individuals are classified in four categories: Healthy Control (HC), Mild Cognitive Impairment (MCI), MCI who converted to AD in the 24 months following screening (MCIC), and Alzheimer's Disease (AD).

The Imaging data We used the computed mean cortical thickness for 75 ROI for each hemisphere and subcortical volumes for 63 brain regions with FreeSurfer on the Aparc2009 atlas, resulting in p = 213 features. Images for 406 individuals were available. We corrected the features for age and sex with a linear regression (the model is fitted separately on each fold of the CV).

The genetic data SNPs data were pruned for missing genotypes, Hardy-Weinberg test and minor allele frequency, and discarded when on chromosome Y or mitochondrial chromosomes, resulting in 491616 loci. As in PC, vertices are genes and not SNPs, we computed a score for each gene to reflect the burden of SNPs contained in this gene and its flanking regions (40kb upstream and downstream), thus accounting for cys-regulating SNPs. For an individual i, the state of the SNP s k is encoded as α ik ∈ {0, 1, 2} for the number of alternative alleles, w.r.t the reference genome. A multiple-SNP risk score for each gene is then computed as a weighted sum of the allelic counts α ik of each SNP:

X ig = pg k=1 β k α ik
for individual i and gene g, where β is the log odds ratio estimated in the IGAP meta-analysis [START_REF] Escott-Price | Common polygenic variation enhances risk prediction for Alzheimers disease[END_REF]. p g is the number of SNPs mapped to gene g. Genetics data where available for 757 patients and SNPs were mapped to q = 19217 genes.

The Network In this work, we used a network of physical interactions from Pathways Commons (PC), keeping only undirected, highly confident edges i.e. interactions supported by PubMed references with at least 2 different sources. It was extracted with XGR (eXploring Genomic Relations) [START_REF] Fang | XGR software for enhanced interpretation of genomic summary data, illustrated by application to immunological traits[END_REF].

RESULTS

Performance To probe the model's performance in learning early signs of the disease, we used the individuals with status HC and AD to train the models (n = 214 individuals), and individuals with status MCI or MCIC as test set (n = 157). The predictive power of the genetics data alone, and of the imaging data alone were assessed with an 1 , 2 regularized logistic regression (we refer to these models as LR Gen and LR Im). We also assessed the impact of the GraphNet penalty with a logistic regression and 1 regularization (LR GN Gen). The mean performance in the CV of the selected model and the performance on the test set of the best model trained on the whole training data are reported for the five models in Table 4. We also reported the number of selected features (signature). s 2 clearly stood out as a key parameter for performance and we observed better CV-mean AUC for smaller values. λ was searched in a range of small values (10 -3 -10 2 ) to avoid z = 0 from being an optimal but uninteresting solution.

Selected brain regions The brain ROI selected by GN-GCCA on the structural MRI data are represented in Figure 1. The regions selected consist in areas where atrophy is classically observed in the early stage of AD: hippocampus and temporal lobes, but not the exhaustive set of involved regions [START_REF] Lorenzi | Susceptibility of brain atrophy toTRIB3in Alzheimer's disease, evidence from functional prioritization in imaging genetics[END_REF]. These might be the regions where AD related atrophy is linked to a specific molecular mechanism.

Model

DISCUSSION AND CONCLUSION

We demonstrated a significant gain in prediction performance when integrating imaging and genetic data to build predictive models of AD. The gain is even more important when adding structured prior knowledge to the genetic data. Du et al. in [START_REF] Du | Structured sparse canonical correlation analysis for brain imaging genetics: an improved Graph-Net method[END_REF] and references therein investigated graph regularized CCA on ADNI on up to a few hundred candidate SNPs, and a few hundred imaging ROIs while our method is tested on the whole genome. Moreover, the graphs they used were weighted by sample correlations and did not leverage prior Their algorithm also considered the 1 norm to be smooth by adding a small constant to zero weights. Moreover their approach is not scalable to the whole-genome as it requires inverting a p×p matrix at each iteration. In comparison, our method uses an 1 projection that naturally results in sparse solutions. Lorenzi et al. [START_REF] Lorenzi | Susceptibility of brain atrophy toTRIB3in Alzheimer's disease, evidence from functional prioritization in imaging genetics[END_REF] instead applied a hard threshold on many bootstrapped samples to select SNP-sets that are then mapped back to genes, and did not use graph constraints. None of these methods jointly use the target and both data types to compute the canonical loadings. Finally we identified several pathways that seem to play a role in the disease at an early stage. Note that a similar regularization could be applied to the imaging data where the graph would encode spatial proximity or connections between regions.

Fig. 1 .

 1 Fig. 1. Visualization of the brain ROI selected by GraphNet-GCCA

Fig. 2 .

 2 Fig. 2. Visualization of the subgraph formed by the genes selected by GN-GCCA. Colors show different connected components and genes that belong to HSA 977225 are underlined

Table 1 .

 1 Results of the experiments for models with genetics data only (LR Gen, LR GN Gen), imaging only (LR Im) and both (SGCCA, GN-GCCA).

		Signature	CV Score (Acc/AUC)	Test Score (Acc/AUC)
	LR Gen	19200 genes 0.53/0.56	0.52/0.55
	LR GN Gen 35 genes	0.61/NC	0.57/0.62
	LR Im	42 ROI	0.88/0.94	0.68/0.72
	SGCCA	2 genes 23 ROI	0.90/0.95	0.68/0.74
	GN-GCCA	196 genes 18 ROI	0.86/0.94	0.70/0.75


Slim Karkar

Yann Le Guen

Cathy Philippe

Claire Dandine-Roulland

Morgane Pierre-Jean

Jean-Franc ¸ois Mangin

Edith Le Floch

Vincent Frouin

A study of feasibility for genome-wide haplotype association of complex traits in imaging genetics

Imaging-genetics cohorts allow to find associations between genotyped variants and brain imaging features. They are invaluable tools to evaluate the part of genetics and environment in the brain characteristic variance observed in normal and pathological populations. The present analyses were conducted using the 2018 UK Biobank (UKB) data release, and included 15,040 subjects for which sulcal opening, a measure of sulcus width, are extracted for 126 sulci using BrainVisa/Morphologist pipeline. Based on genetic maps, continuous blocks of high-confidence phase are extracted using the haplotype dataset of UK Biobank. The feasibility study analyzes 13,942 haplotypes of 1,756 blocks on chromosome 21, obtained with a stringent block definition, and shows that block-based test seems underpowered compared to haplotype-based association test.

INTRODUCTION TO IMAGING-GENETICS STUDIES

Imaging-genetics cohorts allow to find associations between genotyped variants and brain imaging features. They are invaluable tools to evaluate the part of genetics and environment in the brain characteristic variance observed in normal and pathological populations. Over the past years, studies using thousands of subjects like the one available from the Human Connectome Project (HCP) showed a strong genetic control over numerous cortical structures. Recently, measures of sulcal root position, sulci depth and grey matter thickness were shown by our group to be highly heritable [START_REF] Le Guen | Genetic influence on the sulcal pits: On the origin of the first cortical folds[END_REF], [START_REF] Le Guen | eqtl of kcnk2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF]. Traditional GWAS have shown a few new markers associated with these phenotypes [START_REF] Guen | The chaotic morphology of the left superior temporal sulcus is genetically constrained[END_REF].

Genetic basis of complex traits in neurosciences

Heritability studies have shown that the genetic basis for complex traits spans across the whole genome and we expect that causal variants will show only small size effects in the association [START_REF] Plomin | Common dna markers can account for more than half of the genetic influence on cognitive abilities[END_REF]. Hence, research consortia in imaging-genetics have built cohorts of expanding size, from HCP (≈ 1, 200 subjects) and IMAGEN (≈ 2, 000 subjects) to UKB (≈ 20, 000 subjects to this day) and even more sensitive association methods have been sought actively.

Haplotype association studies in neurosciences

The search for early biomarkers for pathologies like neurodegenerative diseases requires to better know the non-pathological variability and associated causal variants in control subjects to rapidly identify valuable biomarker candidates of a disease. In the case of psychiatric syndromes, the quantitative, complex traits derived from imaging data show a continuum with no clear borders between normal and pathological values. These examples are incentives to develop even more sensitive methods to detect associations in large cohorts of control subjects.

In this work, we aim to investigate the properties of association methods between complex phenotypes extracted from imaging data and genome-wide haplotypes, as defined by a block of contiguous markers of high-confidence phase.

I. MATERIAL AND METHOD

The present analyses were conducted under UK Biobank data application number 25251. The UK Biobank is a health research resource of about 500,000 people aged between 45 and 73 years old, that were recruited in the general population across Great Britain. In this work, we used the last release of imaging data of UKB, consisting of 20,060 subjects with a T1-weighted MRI and genotyping data [START_REF] Bycroft | The UK Biobank resource with deep phenotyping and genomic data[END_REF] A. Imaging data Using BrainVisa/Morphologist pipeline [START_REF] Riviere | Brainvisa: an extensible software environment for sharing multimodal neuroimaging data and processing tools[END_REF], [START_REF] Mangin | A framework to study the cortical folding patterns[END_REF], 126 sulci were automatically extracted from T1-weighted images. For each sulcus, several features such as sulcal width, cortical thickness, and pits (sulcal root) depth [START_REF] Le Guen | Genetic influence on the sulcal pits: On the origin of the first cortical folds[END_REF].

Sulci opening: We consider in this study the opening of a sulcus -a measure of its width -computed as the ratio of the volume of the cerebrospinal fluid it contains and the surface of the sulcus. In brain imaging, sulci widening is a phenotype used to study aging and can serve for example as a proxy for premature aging. Sulcal opening values observed on the UKB sample we used exhibit a lognormal distribution across all ages with few extreme values. Hence, opening values were log-transformed, and outliers were removed using interquartile range methods.

B. Genetic data

We used UKB genotype data of January 2018 release, that contains 658,720 SNPs spanning across the 22 autosomal chromosomes. We excluded from our analysis subjects with low QC index, first degree related individuals or sex mismatches. In total and out of the 20,060 subjects with neuroimaging, we retained 15,251 subjects with approximately 48% of males and 52% of females (see [START_REF] Le Guen | eqtl of kcnk2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF]). From this release, we used the phase information as computed by SHAPEit, made available for all genotyped SNPs in March 2018 release.

We define blocks of high confidence phase over the 22 autosomal chromosomes using the GRCH37 genetic recombination map in centimorgan (cM), based on the 1000 Genomes project. In order to keep block with high confidence phases, we retained in each block the SNPs being in linkage disequilibrium (LD) exclusively. In the approach proposed in [START_REF] Howard | Genome-wide haplotype-based association analysis of major depressive disorder in Generation Scotland and UK Biobank[END_REF], the genome is divided in overlapping blocks of fixed span of 0.25 to 1 cM on the genetic map. Here, we propose to define non-overlapping blocks. The end-SNP and the start-SNP of two consecutive blocks are separated by at least ∆ cM. This procedure guarantees that two consecutive SNPs in the same block are separated by less than ∆ cM. We investigate the effect of ∆ cM by using six various values ranging from 0.001 to 0.025 cM in Figure 2.

We proposed three different ad-hoc procedures for genome-wide haplotype association implemented in R.

Block-test using likelihood ratio test: In the first procedure, referred to as "block-test" hereafter, we use a linear regression of the phenotypes Y over the covariates X and all the haplotypes hi observed in a given block collated in a design matrix H. Each column hi contains codes from 0, 1, 2 in order to study the additive effects of the haplotype. The significance of the blocktest association is estimated using a Likelihood Ratio Test for nested Linear Models for Y : (i) the linear model that contains only the covariates Y ∼ βX (X), and (ii) the block model Y ∼ βX,H (X, H) that contains covariates and the design matrix .

Complete-test using a χ 2 test of the coefficients: A second procedure, referred to as "complete-test", uses the same linear model regression Y ∼ βX,H (X, H) as in the block-test. For this procedure, we compute a standard two-sided p-value of the t-statistic of the regression coefficient associated with each column hi ∈ H in the block.

Independent haplotype-test considering haplotypes as independent variables: Finally, we propose a third procedure, referred to as "Independent haplotype-test", where each haplotype is considered as an independent variable and is tested successively. In a similar way as the previous procedure, we use a linear modeling Y ∼ β X,h i (X, h i ) where hi is the additive effect of haplotype i. We compute a standard two-sided p-value of the t-statistic of the regression coefficient associated with the additive effect of hi. This procedure is similar to the quantitative linear association test implemented in PLINK for the SNPs.

The study of the properties of the proposed procedures was assessed using haplotype data from chromosome 21 and the opening of the Central Sulcus.

II. RESULTS

A. Genome-wide distribution of haplotype block

We explored six values of ∆ from 0.001 to 0.025 cM which defined six families of haplotype blocks. We studied the overall distribution of the genomic length L[h] of the blocks in the six configurations. These distributions present a log-normal shape as shown in Figure 2. The distribution median value is sensitive to ∆, and presents a heavy tail for lower values of ∆. For the most stringent criterion ∆ = 0.001, the distribution of L[h] seems homogeneous across all chromosomes. We chose this most conservative value because it ensures SNPs in LD in the block and this value was used for the following feasibility study of haplotype association.

B. Feasibility of genome-wide association studies for haplotypes

Across chromosome 21, we obtained 1,756 blocks of high confidence phase using ∆ = 0.001. These blocks led to 13,942 haplotypes. Selected blocks had a span on the genetic map of length ranging from 9.4e-07 to 1.2e-02 cM, with a median value of 6.7e-04 cM, gathering from 2 to 63 consecutive SNPs, with a maximum length on the chromosome of 126 kbp. We ran the three procedures to study haplotype association with opening values of the Calacrine Fissure. Previous univariate GWAS using genotyped data did not report any hit between this phenotype and chromosome 21. In the absence of association, we expect the p-values of the tests to be uniformly distributed between 0 and 1 and the empirical Cumulative Distribution Function to follow a linear trend with a slope of 1. We also expect the lowest p-value to be close to 1/NT where NT is the number of test. We compared the observed p-values distribution obtained with the three procedures with a null distribution as "expected" values using Quantile-Quantile plots (QQ-plot) reported in left panels of Figures 3,[START_REF] Plomin | Common dna markers can account for more than half of the genetic influence on cognitive abilities[END_REF] 1) "block-test" procedure: The block-test procedure produced 1,756 p-values for each block of high confidence phase along chromosome 21. The QQ-plot indicates a poor fit of the observed vs expected p-value distribution, with most significant p-values less significant than their expected values (see left panel of Figure 3). Defining 4 classes of blocks according to the length they span across the genetic map, we could not distinguish a particular trend for pvalues.

2) "complete-test" procedure: The complete-test produced 13,942 p-values from 1,756 linear model regressions. In the QQplot on Figure 4, p-values seem to fit the null distribution and seem also independent of the minor allele frequency (MAF) of the haplotypes as suggested by the MAF information on the QQ-plot. However, the p-value distribution seems influenced by the length in 3) "independent haplotype-test" procedure: The independent haplotype-test produced 13,942 p-values for each haplotype on chromosome 21, with one linear model fitted independently for each haplotype. In Figure 5, p-values seem to fit the null distribution on the Q-Q plot, also seem independent of the MAF, and the p-value distributions seem to be homogeneous across classes of increasing genetic length. .

III. DISCUSSION

In this study we used the most conservative value for ∆ in order to enforce as much as possible the high confidence of the phase of the markers along the block. Despite the unbalanced distribution of genetic length L[h] for the chosen ∆, we detected no effect on our analysis however. If this length would come to be an explanatory variable, one probably wants a more balanced distribution. Higher values of ∆ seems to produce distribution closer to the log-Normal form.

Regarding the procedures for testing association, the independent haplotype test seems to produce the most uniform distribution of p-values across all 4 classes of haplotype length on the genetic map, while the block-test seems more conservative, which might be explained by the high number of degrees of freedom of this test. As for the complete test procedure, it might not be the most suitable since it does nor produce uniform distribution of p-values for haplotypes of 2 to 4 or 5 SNPs in strong LD. For this study, we did not propose any correction for the p-values since we were not looking for significant associations, that is, we focused on finding the right null distribution.

CONCLUSION AND FURTHER WORK

In the context of imaging genetics, we proposed a study to test for association between haplotypes and quantitative traits relative to brain structures. A limitation of this work is the effect of haplotypes with low frequencies that could appears on larger scale. It is still unclear how haplotypes frequencies influence the results. Further work should include a procedure to filter out low-frequency haplotypes or alternatively, to filter out low-frequency SNPs.

Further work will define a procedure to correct p-values for multiple testing. The Bonferroni correction for the number of tests seems to be a too naive correction for p-values. However, [START_REF] Howard | Genome-wide haplotype-based association analysis of major depressive disorder in Generation Scotland and UK Biobank[END_REF] propose to correct by the estimation of the number of independent haplotypes. For the complete-test, that number should range between the number of blocks and the number of haplotypes. Finally, we could plan an extension of the complete-test using a mixed linear model.

Fig. 1 .

 1 Fig. 1. (left) Main brain structures : Sulci are the main furrows. For example, Central Sulcus is separating Frontal and Parietal lobes. (right) Sulcus opening, a measure of the sulcus width, is computed as the ratio of the volume of cerebrospinal fluid and the surface of the sulcus.

Fig. 2 .

 2 Fig. 2. (left) Distribution of block span on the genetic map, L[h], as defined by the genetic distance between bordering markers, in cM, over 22 autosomal chromosomes. ∆ is the parameter that accounts for the maximal length in cM between two consecutive SNPs in the same haplotype. (right) Distribution of haplotypes' span, L[h], across each autosomal chromosome. Using the most conservative value ∆ = 0.001, Distribution of blocks span appears homogeneous across the 22 autosomal chromosomes.

  Fig. 3. (left) QQ-plot for p-values obtained for 1,756 blocks of chromosome 21 with an ANOVA test with one p-value per block. (right) p-value distributions for 4 classes of L[h], the length of the span of the blocks on the genetic map, in cM.

Fig. 4 .

 4 Fig. 4. (left) QQ-plot for p-values obtained for 13,942 haplotype association tests using 1,756 linear model regressions; Color of points represent frequency f f of haplotypes. (right) p-value distribution for 4 classes L[h], the length of the span of the blocks on the genetic map, in cM.

Fig. 5 .

 5 Fig. 5. (left) QQ-plot for p-values of association test for 19,930 haplotypes, considered as independent variables, using linear regression (similar to PLINK quantitive association test). Color of points corresponds to the frequency of the corresponding haplotype. (right) p-value distribution for 4 classes of L[h], the length of the span of the blocks on the genetic map, in cM.

Neurospin, Institut Joliot, CEA, Universit Paris-Saclay, Gif-sur-Yvette, France

CNRGH, Institut Jacob, CEA, Universit Paris-Saclay, Evry, France

ACKNOWLEDGMENT

This work was funded by Fondation Pour la Recherche Medicale, FRM grant number DIC20161236445 awarded to J-F Mangin "Conception dun dictionnaire des motifs du plissement cortical pour la détection danomalies développementales"