Nathan Ferrandin-Schoffel
email: nathan.ferrandin-schoffel@mnhn.fr

Mohamed Haouas

Charlotte Martineau-Corcos

Odile Fichet

Anne- Laurence Dupont
email: anne-laurence.dupont@mnhn.fr

Modeling the reactivity of aged paper with aminoalkylalkoxysilanes as strengthening and deacidification agents

Keywords:

Aminoalkylalkoxysilane (AAAS) flexible copolymer networks can be used to deacidify and strengthen paper in a one-step operation. The treatment is versatile and can be tailored to convey a balanced combination of tensile strength and pliability to degraded paper from library and archives collections.

The treatment efficiency on bleached pulp and rag or cotton paper has been shown to be always suitable.

 Si, and 1 H-X (X = 13 C, 15 N, or 29 Si) heteronuclear multiple bond correlation (HMBC). The results showed that carbonyl and carboxyl groups react with the amine and silicate moieties of AM. These results are paralleled with the physicochemical properties of three oxidized and brittle newsprint papers to explain the variable degree of strengthening observed upon AAAS treatment.

 C NMR spectra (AM, methyl-β-D-glucopyranoside (MGPO), and AM/MGPO) and 1 H NMR spectra (p-benzoquinone and AM/p-benzoquinone).

Introduction

Paper slowly decays with time. Cellulose, the main biopolymer in paper, undergoes acid-catalyzed hydrolysis that decreases the degree of polymerization (DP), which in turn progressively leads to loss in mechanical resistance of the paper. In addition, the natural slow formation of carbonyl compounds among other degradation by-products increases the acidity. Late 19 th -early 20 th century lignocellulosic paper collections, especially newsprint paper of mediocre quality, degrade faster than more durable paper, and can become extremely brittle. As a preventive measure, mass treatments have been developed for the deacidification of libraries and archives collections. [START_REF] Baty | Deacidification for the Conservation and Preservation of Paper-Based Works: A Review[END_REF][START_REF] Carter | The Chemistry of Paper Preservation: Part 1. The Aging of Paper and Conservation Techniques[END_REF][START_REF] Cheradame | Mass Deacidification of Paper and Books. I: Study of the Limitations of the Gas Phase Processes[END_REF][START_REF] Ahn | Sustainability of Mass Deacidification. Part I: Concept, Selection of Sample Books and PH-Determination[END_REF] Nevertheless, these processes do not provide a comprehensive solution for the most degraded (acidic and brittle) documents, which additionally, would need to be mechanically strengthened. In order to overcome this problem, one-pot treatments based on the use of aminoalkylalkoxysilanes (AAAS), imparting simultaneous deacidification and strengthening to cellulosic objects, have been proposed. [START_REF] Rousset | Mass Deacidification of Paper and Books, II: Deacidification in the Liquid Phase Using Aminosilanes[END_REF][START_REF] Ipert | Mass Deacidification of Papers and Books III: Study of a Paper Strengthening and Deacidification Process with Amino Alkyl Alkoxy Silanes[END_REF][START_REF] Ipert | Mass Deacidification of Papers and Books. IV-A Study of Papers Treated with Aminoalkylalkoxysilanes and Their Resistance to Ageing[END_REF][START_REF] Dupont | Mass Deacidification and Reinforcement of Papers and Books VI-Study of Aminopropylmethyldiethoxysilane Treated Papers[END_REF][START_REF] Souguir | Chemical and Physicochemical Investigation of an Aminoalkylalkoxysilane As Strengthening Agent for Cellulosic Materials[END_REF][START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF][START_REF] Piovesan | Paper Strengthening by Polyaminoalkylalkoxysilane Copolymer Networks Applied by Spray or Immersion: A Model Study[END_REF] Previously published results showed the full potential of AAAS treatments in organic solvent phase, including fungistatic properties of the treated documents. [START_REF] Rakotonirainy | Mass Deacidification of Papers and Books: V. Fungistatic Properties of Papers Treated with Aminoalkylalkoxysilanes[END_REF] At the laboratory or the conservation workshop scale, the treatment can be applied by spray or by immersion, undiluted or diluted in an organic solvent, usually hexamethyldisiloxane (HMDSO). At the ''mass'' industrial scale, the use of a nonprotic solvent such as HMDSO is recommended to avoid the bleeding of water-soluble inks, dyes and binders, fiber swelling, and also to shorten the drying time. [START_REF] Baty | Deacidification for the Conservation and Preservation of Paper-Based Works: A Review[END_REF][START_REF] Dupont | Mass Deacidification and Reinforcement of Papers and Books VI-Study of Aminopropylmethyldiethoxysilane Treated Papers[END_REF][START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] More recently, flexible copolymer networks (co-AAAS) have been studied, with the aim of conveying a balanced combination of tensile strength and pliability to very degraded paper. [START_REF] Piovesan | Paper Strengthening by Polyaminoalkylalkoxysilane Copolymer Networks Applied by Spray or Immersion: A Model Study[END_REF] The treatment is versatile as its formulation can easily be modified by changing the monomers, their proportions and their concentration. Efficient deacidification and increase in the tensile breaking length has been obtained, [START_REF] Piovesan | Paper Strengthening by Polyaminoalkylalkoxysilane Copolymer Networks Applied by Spray or Immersion: A Model Study[END_REF][START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF] but an increase in folding endurance increase was less easily achieved, [START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF][START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] and slight yellowing was sometimes observed. [START_REF] Souguir | Chemical and Physicochemical Investigation of an Aminoalkylalkoxysilane As Strengthening Agent for Cellulosic Materials[END_REF][START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF] The interactions taking place between the AAAS and the paper are still unclear. The aim of this research was thus to investigate these interactions. It has been previously suggested that the amine group, which drives the deacidification, also favors the adsorption of the AAAS in the cellulosic material and forms hydrogen bonds. [START_REF] Dupont | Mass Deacidification and Reinforcement of Papers and Books VI-Study of Aminopropylmethyldiethoxysilane Treated Papers[END_REF][START_REF] Souguir | Chemical and Physicochemical Investigation of an Aminoalkylalkoxysilane As Strengthening Agent for Cellulosic Materials[END_REF][START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF] To account for the observed slight yellowing of the treated papers, the formation of Schiff bases through covalent bonding between the amine groups and carbonyl groups in oxidized paper was hypothethised. [START_REF] Rousset | Mass Deacidification of Paper and Books, II: Deacidification in the Liquid Phase Using Aminosilanes[END_REF][START_REF] Ipert | Mass Deacidification of Papers and Books III: Study of a Paper Strengthening and Deacidification Process with Amino Alkyl Alkoxy Silanes[END_REF][START_REF] Souguir | Chemical and Physicochemical Investigation of an Aminoalkylalkoxysilane As Strengthening Agent for Cellulosic Materials[END_REF][START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF] On the other hand, strengthening, which is thought to be imparted by the in situ polymerization, thus depends on the DP of the AAAS, i.e. on their ability to hydrolyze with the water naturally absorbed in the fibers in the first step of the reaction (eq. 1), and to subsequently condensate at room temperature. The condensation step can be hydrolytically-driven (eq.

2) in which case water is regenerated [START_REF] Rankin | Hydrolysis Pseudoequilibrium: Challenges and Opportunities to Sol-Gel Silicate Kinetics[END_REF] or alcohol-driven (eq. 3).

Si OR + O H 2 Si OH + ROH Si OH + Si O H Si O Si + O H 2 Si OH + Si RO Si O Si + ROH
 Hydrolysis:

(eq. 1)

 Hydrolytic polycondensation:

(eq. 2)

 Alcoholic polycondensation:

(eq. 3)

In acidic conditions and at room temperature the hydrolysis of alkoxysilanes is catalyzed, whereas the condensation is slower and can take up to several months. [START_REF] De Monredon-Senani | Interaction Organosilanes/Silice de Précipitation Du Milieu Hydro-Alcoolique Au Milieu Aqueux[END_REF][START_REF] Brochier Salon | Kinetics of Hydrolysis and Self Condensation Reactions of Silanes by NMR Spectroscopy[END_REF] In contrast, it has been shown that both the hydrolysis and the polycondensation steps are catalyzed in alkaline conditions. [START_REF] De Monredon-Senani | Interaction Organosilanes/Silice de Précipitation Du Milieu Hydro-Alcoolique Au Milieu Aqueux[END_REF][START_REF] Brochier Salon | Kinetics of Hydrolysis and Self Condensation Reactions of Silanes by NMR Spectroscopy[END_REF][START_REF] Hüsing | Formation and Structure of Gel Networks from Si(OEt)4/(MeO)3Si(CH2)3NR'2 Mixtures (NR'2 = NH2 or NHCH2CH2NH2)[END_REF] Hence, these reactions can be catalyzed by the amine functions on the AAAS molecules. [START_REF] Brochier Salon | Kinetics of Hydrolysis and Self Condensation Reactions of Silanes by NMR Spectroscopy[END_REF][START_REF] Hüsing | Formation and Structure of Gel Networks from Si(OEt)4/(MeO)3Si(CH2)3NR'2 Mixtures (NR'2 = NH2 or NHCH2CH2NH2)[END_REF] Acids continuously produced during the natural aging of paper would partly neutralize these bases in situ, gradually leading to near-neutral conditions and subsequently slower polycondensation kinetics. Likewise, the proposed formation of covalent bonds between the amines of the AAAS and carbonyl groups from the degraded paper could in all likelihood limit the size of the polymer. As another induced drawback, the alkaline reserve brought by the amine group would be lower than expected.

Previously, it has been suggested that several constituents of the paper were likely to affect the treatment's efficiency. Among them, alum-rosin sizing [START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF][START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] and groundwood pulp, [START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF][START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] which are present in significant amount in early newsprint paper made from unpurified wood. Mechanical pulp from softwood typically contains between 25 and 31 wt% of lignin in the near-native form. [START_REF] Smook | Handbook for Pulp & Paper Technologists[END_REF] Lignin is an amorphous tridimensional biopolymer largely responsible for the paper acidity and yellowing upon aging. [START_REF] Carter | The Chemistry of Paper Preservation: Part 2. The Yellowing of Paper and Conservation Bleaching[END_REF] More generally, reactions that occur at room temperature in the paper between AAAS and the biomacromolecules, i.e. cellulose, hemicelluloses and lignin, are difficult to characterize and are still not fully understood. For example, most routine spectroscopic techniques do not have enough resolving power to characterize precisely carbonyl groups in-situ. [START_REF] Potthast | Studies on Oxidative Modifications of Cellulose in the Periodate System: Molecular Weight Distribution and Carbonyl Group Profiles[END_REF] In this work, we identified possible reactions taking place between the AAAS and the paper, which could explain the variable degree of strengthening obtained with different papers. To this purpose, a model study was undertaken. Several organic molecules were chosen as their structure encompassed the main functional groups of cellulose and lignin. They were mixed to 3-aminopropylmethyldiethoxysilane, henceforth called AM, a bifunctional AAAS that condensates into a linear polysiloxane. Fourier-transform infrared (FTIR) and liquid-state Nuclear Magnetic Resonance (NMR) spectroscopies were used to study the molecular interactions between AM and the model organic molecules. Previous studies showed the capacity of AM to penetrate in the fibers network and in situ polymerize, yielding an increase in the mechanical properties, both in undegraded and degraded paper. [START_REF] Dupont | Mass Deacidification and Reinforcement of Papers and Books VI-Study of Aminopropylmethyldiethoxysilane Treated Papers[END_REF][START_REF] Souguir | Chemical and Physicochemical Investigation of an Aminoalkylalkoxysilane As Strengthening Agent for Cellulosic Materials[END_REF][START_REF] Piovesan | Paper Strengthening by Polyaminoalkylalkoxysilane Copolymer Networks Applied by Spray or Immersion: A Model Study[END_REF][START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF][START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] The study with the model compounds was thus paralleled with the results of the mechanical strengthening of three newsprint papers in various states of degradation, made of lignin-rich or lignin-free pulp mixtures.

Experimental

Paper samples

Naturally aged lignocellulosic newsprint paper was used. Three copies (called J1, J2 and J3) of the "Journal des Fabricants de Sucre", a newspaper published through late 19 th -early 20 th century, were provided by the Bibliothèque Nationale de France. Their constituents and some of their physicochemical properties are summarized in Table 1.

The fibers and the pulp of the three newsprint papers were determined by microscopic examination and histological staining with Herzberg, Lofton-Merritt and Graff-C solutions. [START_REF] Isenberg | Pulp and Paper Microscopy[END_REF] Microchemical test for rosin [START_REF] Browning | Analysis of Paper[END_REF] was performed to check for alum-rosin sizing. Kaolin fillers in the papers were identified by ATR-FTIR (Nicolet 6700 spectrophotometer). This clay mineral contains large amounts of kaolinite, an aluminosilicate which displays two typical absorption bands around 3690 cm -1 and 3620 cm -1 (O-H stretch) [START_REF] Derrick | Infrared Spectroscopy in Conservation Science[END_REF] (Figure S1 in the Supporting Information file). The ash content enabled to quantify the amount of mineral fillers in the papers (TAPPI T211 om-02). Two measurements were done per paper.

Paper treatment

AP/DIA (5/95) (wt/wt) mixture was directly sprayed on the paper samples with an airbrush (Airbrush Hobby Kit, Silverline). This formulation was previously established as the most efficient for the treatment of a sample from the same newspaper collection. [START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF] The uptake was calculated as follows:

UP(wt%) = (w f -w i ) w i × 100 (eq. 4)
where w i and w f are the paper weight before and after treatment, respectively. Each sample was dried for at least 3 h at room temperature, conditioned at 23 °C and 50% RH for a minimum of 24 h (TAPPI T402 sp-03) and weighted.

For relevant comparisons, the paper samples were treated so as to obtain similar uptakes (9 ± 0.5%).

Physico-chemical characterizations

The cold extract pH was measured with a pH-meter (Mettler Toledo MA235) according to TAPPI T509 om-02 standard adapted to a sample mass of 100 mg. Three repeat measurements were carried out per paper.

Prior to the mechanical testing measurements (tensile strength and folding endurance) the samples were conditioned according to TAPPI T402 sp-03. Tensile Breaking Length (BL) and Elongation at Break (EB) were measured with an Adamel Lhomargy instrument (DY-20N) according to TAPPI T494 om-01 standard. BL (km) is the limiting length of a strip of paper of uniform width, beyond which, if such a strip were suspended by one end, it would break of its own weight. It is related to the tensile strength T (kN m -1 ) according to:

BL = 102 × T R (eq. 5)
where R is the grammage of the paper (g m -2 ).

Folding endurance (FE) was determined with a Tinius Olsen instrument (applied force: 500 g) according to TAPPI T511 om-02 standard. Ten measurements were done per sample in the machine direction of the sheets.

The 1 H, [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C, [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si, and 1 H-X (X = 13 C, [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N, or [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si) heteronuclear multiple bond correlation (HMBC)

spectra were collected at room temperature for about 0.6 mL of the prepared solutions in deuterated dimethyl sulfoxide (DMSO-d 6 ) by using a H/X broadband probe with Z gradient installed in an Avance 400 NMR spectrometer. The 1 H spectra were recorded by using Larmor frequency at 400.13 MHz. A standard one-pulse sequence employing a 30-degree proton pulse was used. The spectral width and acquisition time were 5 kHz and 3 s, respectively. The relaxation delay was 1 s, and eight transients were signal-averaged. The 13 C NMR spectra were recorded at 100.62 MHz by using a power gated decoupling sequence with a 90-degree carbon pulse. The spectral width, acquisition time and recycle delay were 25 kHz, 1 s and 5 s, respectively. A total of 4096 transients were signal-averaged for a total measurement time of 7 h. The 29 Si NMR spectra were acquired at 79.49 MHz. The spectral width, acquisition time, and recycle delay were 6 kHz, 1.3 s, and 12 s, respectively. The 90-degree pulse length was 12.7 s and 4096 transients were signal-averaged. In the 1 H- [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C HMBC experiments, the proton dimension was acquired by using 4096 data points, 5 kHz spectral width, 32 scans, and 1 s recycle delay.

The [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C dimension was recorded by using 256 t1 increments, and 25 kHz spectral width. To collect 1 H- [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N HMBC spectra of the solutions, the spectrometer was operated at 40.55 MHz for [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N. The spectral width for the nitrogen dimension was 20 kHz, and 44 t1 increments were used. For the proton dimension, the acquisition time and the spectral width were 0.4 s and 5 kHz, respectively. The spectrum was acquired with 512 scans and 2 s recycle delay. For the 1 H- [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si HMBC experiments, the proton dimension was acquired by using 8192 data points, 5 kHz spectral width, 64 scans, and 1 s recycle delay. The [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si dimension was recorded by using 100 t1 increments, and 2.5 kHz spectral width. The 1 H, [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C, and [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si spectra were referenced to tetramethylsilane (TMS) at 0 ppm, whereas [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N spectra were referenced to nitromethane at 0 ppm.

A kinetic study of the model mixtures was performed by 1 H NMR at t 0 (i.e. time when the NMR solvent was added) and 7 days later with a Bruker Avance DPX-250 (250 MHz).

FTIR spectra of the newsprint papers and the model compounds in solution were obtained with a Nicolet 6700 spectrophotometer, equipped with a diamond Smart Endurance ATR macro-system scanning from 500 to 4000 cm -1 . For each spectrum, 64 scans were recorded, with a resolution of 4 cm - 1 . Each model compound was dissolved in dimethyl sulfoxide (DMSO). For each of the model mixtures, a spectrum was obtained at t 0 and 7 days later.

Results and discussion

First, a study of the chemical and mechanical properties of the three newsprint papers treated with AAAS copolymers was undertaken.

Physico-chemical measurements

As expected, each of the initially acidic newsprint papers was deacidified by the treatment (pH above 7.5) (Table 2). All papers also showed an increase in tensile strength, indicating improved inter and intra-fiber bonding via the breaking length (BL) and deformability through the elongation at break (EB).

It was noted that a larger increase of BL (J2: +40%, J1: +25%, J3: +26%) went along with a slightly smaller increase in EB (J2: +13%, J1: +20%, J3: +25%). Figure 1 shows the FE of the papers. J1 and J2 are very brittle, with an initial FE below 10 double folds (N < 10) whereas J3 is relatively less brittle (N ≈ 22). The papers reacted differently to the treatment, exhibiting a variable degree of strengthening. It was previously suggested that the initial FE impacted the reinforcement efficiency, with a critical value (N ≈ 10) below which the sample is hardly reinforced. [START_REF] Souguir | Strengthening of Degraded Cellulosic Material Using a Diamine Alkylalkoxysilane[END_REF][START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF] This limitation was indeed observed for J2, but in contrast, J1 was strengthened after treatment (N ≈ 27). This result is consistent with the higher elongation at break imparted by the treatment to J1 than to J2. The tests showed that, despite a different pulp composition, all three papers contain alum-rosin sizing and kaolin fillers and have similar grammage and thickness (Table 1). Lignin was evidenced in J2 and J3 using FTIR, by the absorption bands around 1590 cm -1 (aryl ring symmetric stretching) and 1510 cm - 1 (aryl ring asymmetric stretching). [START_REF] Calvini | FTIR -Deconvolution Spectra of Paper Documents[END_REF][START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies: Tables and Charts[END_REF] J1 does not contain lignin, as shown by the absence of these absorption bands on the FTIR spectrum (Figure S1 in the Supporting Information file). Another difference between the three papers appeared to be the proportion of kaolin, which was much lower for J2 (7.6% w ) than for J1 (17.8% w ) and J3 (16.3% w ) (Table 1). However, previous research showed that the quantity of fillers had no appreciable impact on the treatment efficiency. [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] Subsequently, it seems that the presence of lignin (from the groundwood pulp) could be a factor that most likely hampers the folding endurance improvement of the most brittle papers. This hypothesis was supported in previous research by the fact that an aged cotton paper, made of cellulose 100%, with a folding endurance of 20 double folds, could be strengthened. [START_REF] Piovesan | Strengthening Naturally and Artificially Aged Paper Using Polyaminoalkylalkoxysilane Copolymer Networks[END_REF] Thus, this raises the question of the chemical reactions possibly occurring between the AAAS, on the one hand, and cellulose and lignin, on the other hand that could account for the observed results. This interrogation prompted the need of modeling the system using simple molecules in the present research.

Model carbohydrate compounds

Cellulose is a polymer made of β-D-glucose bonded to each other with acetal functional groups linkages. A glycosidic bond links the equatorial -OH of the carbon C 4 from a unit to the carbon C 1 of the neighboring unit (Figure S2a in the Supporting Information file). In nature, the degree of polymerization (DP) of cellulose ranges from the hundreds to the thousands. [START_REF] Zhou | Mechanistic Understanding of Thermochemical Conversion of Polymers and Lignocellulosic Biomass[END_REF] D-glucose is a trivial molecular model of the cellulose monomer. D-glucose used in this research contains both α and β anomers, since pure β-D-glucose is not commercially available.

The other most abundant carbohydrate biopolymers in pulp are hemicelluloses. Unbleached and bleached woodpulp papers contain significant amounts of hemicelluloses. [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] They are heteropolysaccharides that act as a cohesive matrix between the cellulose microfibrils. [START_REF] Hon | Wood and Cellulosic Chemistry, Second Edition, Revised, and Expanded[END_REF] They are made of linear and branched chains of different sugar monomers. [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] The type of hemicelluloses and their proportion in the pulp depend on the origin of the fibers and on the pulping process. Their average DPs are low (about 200 for the softwood fibers [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF][START_REF] Jones | Chemical Compositions of Natural Fibres[END_REF] ) compared to cellulose. Although glucose is a recurring repeat unit in most hemicelluloses, other sugars are also present as for instance, galactose and mannose in galactoglucomannan. The latter are abundant in softwood fibers. [START_REF] Zhou | Mechanistic Understanding of Thermochemical Conversion of Polymers and Lignocellulosic Biomass[END_REF] D-glucose is the only saccharide monomer studied here. Until further investigation, a similar reactivity of the other sugar units will be assumed.

In order to study the interaction between the AAAS and cellulose, an AM/D-glucose equimolar mixture was dissolved in DMSO-d 6 , and the 2D 1 H- [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N HMBC NMR spectrum was recorded (Figure 2a). It showed a signal at about -354 ppm, corresponding to the nitrogen of unreacted AM, in addition to new signals at about -337 ppm and -326 ppm. These two new peaks were related to the nitrogen atoms N(α) of an α-glycosylamine and N(β) of a β-glycosylamine, respectively. The hydrogens H(α-2') (3.26 ppm) and H(β-2') (2.86 ppm) both correlate through 3 J couplings with the nitrogen atom of AM, which evidences the formation of a covalent bond between the amine group and D-glucose. In the experimental conditions, [START_REF] Cheradame | Mass Deacidification of Paper and Books. I: Study of the Limitations of the Gas Phase Processes[END_REF] J couplings showed stronger intensities than 2 J couplings, which may explain why a 2 J coupling was not detected between N(α) and H(α-1') (4.28 ppm), and was only moderately visible between N(β) and H(β-1') (3.62 ppm). The evidence of covalent bond formation is also provided by the 2D 1 H- [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C HMBC NMR spectrum with the 3 J couplings between C(α-a) (50.2 ppm) and H(α-1'), and between C(β-a) (49.8 ppm) and H(β-1') (Figure 2b).

The covalent bond between the amine of AM and the anomeric carbon of D-glucose was attributed to a Maillard reaction (Figure 3). This reaction involves an intermediate, which is the open ring structure of D-glucose, with an aldehyde group on C 1 . The first step of the reaction mechanism is the formation of a Schiff base, through a condensation reaction which releases water. Thereafter, glycosylamine is produced by Amadori rearrangement. Such addition of primary amines on glucose units has been reported in the literature. [START_REF] Roberts | Basic Principles of Organic Chemistry[END_REF][START_REF] Shendurse | Glucose: Properties and Analysis[END_REF] Figure 2. 2D NMR HMBC (a) 1 H- [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N and (b) 1 H- [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C spectra of AM/D-glucose (1/1) mixture. The hydrogens and carbons numbering is the same as in Figure 3.

It has to be noted that there is no such thing as a perfect molecular model of cellulose, and D-glucose is no exception. Indeed, the reaction occurs with the hemiacetal functional group of D-glucose, which is present only in terminal C 1 position (end-chain), i.e. under 1‰ considering an average DP of cellulose above 1000. Therefore, another model compound, MGPO (structure given in Figure S2b in the Supporting Information file) was used, which has a methoxy group in place of the -OH on C 1 and thus better represents a mid-chain glucose. The formation of a terminal aldehyde group is thus not possible.

The [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C NMR spectrum of AM/MGPO 1/1 mixture shows that no reaction was observed in solution after several days, as expected (Figure S3 in the Supporting Information file). This result confirms the proposed reaction mechanism between AM and D-glucose, and points out that only the reducing-end anhydroglucose C 1 of cellulose would react with AM in such way. As pointed out above, the average DP of cellulose being high, this indicates a very weak possible reactivity of the biopolymer. However, with a lower DP and branched chains, and hence a higher number of reactive sites, hemicelluloses are conceivably more reactive with AM. The same is true for degraded cellulose by acid catalyzed hydrolysis. [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Figure 3. Maillard reaction between AM and D-glucose. R-NH 2 refers to AM.

Model lignin derivatives

Lignin is another abundant biopolymer in mechanical pulp and unbleached pulp papers (i.e. J2 and J3). Lignin is a large polyphenol with a complex tridimensional amorphous structure, which builds a strong natural composite material with cellulose. [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Its structural units derive from three phenyl propane molecules, namely coniferyl, sinapyl and p-coumaryl alcohols, bound together by ether and carboncarbon bonds. [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF][START_REF] Jones | Chemical Compositions of Natural Fibres[END_REF][START_REF] Adler | Lignin Chemistry? Past, Present and Future[END_REF][35][36][START_REF] Sakakibara | Chemistry of Lignin[END_REF] In native softwood lignin, the structural elements are more than 95% coniferyl alcohol, [START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] which has only one methoxy group on the aromatic ring. The main monomer of grass lignin is coumaryl alcohol (no methoxy group), whereas hardwood lignin derives from sinapyl (two methoxy groups) and coniferyl alcohols. [START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] Lignins also contain other functional groups. In softwood lignin, the total number of hydroxyl groups lies between 1.3 and 1.5 per phenyl propane unit. [START_REF] Sakakibara | Chemistry of Lignin[END_REF][START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] It is estimated that there is only between 0.2 and 0.3 phenol group per phenyl propane unit, [START_REF] Adler | Lignin Chemistry? Past, Present and Future[END_REF][START_REF] Sakakibara | Chemistry of Lignin[END_REF][START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] thus aliphatic alcohols are predominant in the native biopolymer. There is about 0.2 carbonyl group per structural element of softwood lignin, [START_REF] Adler | Lignin Chemistry? Past, Present and Future[END_REF][START_REF] Sakakibara | Chemistry of Lignin[END_REF][START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] whereas only a few percent carboxyl groups are present. [START_REF] Brunow | Functional Groups and Bonding Patterns in Lignin (Including the Lignin-Carbohydrate Complexes)[END_REF] Vanilloid molecules can be used as relevant structural models of lignin. Vanillin, vanillic acid and vanillyl alcohol were chosen in this work. The first two molecules are cited as common degradation products of lignin and are also found in paper. 36,[START_REF] Tarabanko | Catalytic Oxidation of Lignins into the Aromatic Aldehydes: General Process Trends and Development Prospects[END_REF][START_REF] Nevell | Cellulose Chemistry and Its Applications[END_REF][START_REF] Strlič | The Effect of Volatile Organic Compounds and Hypoxia on Paper Degradation[END_REF][START_REF] Dupont | Comprehensive Characterisation of Celluloseand Lignocellulose-Degradation Products in Aged Papers: Capillary Zone Electrophoresis of Low-Molar Mass Organic Acids, Carbohydrates, and Aromatic Lignin Derivatives[END_REF][START_REF] Dupont | Capillary Electrophoresis with Electrospray Ionisation-Mass Spectrometry for the Characterisation of Degradation Products in Aged Papers[END_REF][START_REF] Shahani | Spontaneous Formation of Acids in the Natural Aging of Paper[END_REF] Natural aging of lignocellulosic paper also leads by oxidation to the formation of chromophore quinoid groups on the macromolecule. [START_REF] Carter | The Chemistry of Paper Preservation: Part 2. The Yellowing of Paper and Conservation Bleaching[END_REF]36 This was established as the main cause of yellowing of documents containing significant amounts of groundwood pulp. [START_REF] Carter | The Chemistry of Paper Preservation: Part 2. The Yellowing of Paper and Conservation Bleaching[END_REF] The reactivity of quinone structures was investigated here using p-benzoquinone.

3.3.1.

Vanillin. The nitrogen signal of AM at -354 ppm totally disappeared on the 2D 1 H- [START_REF] Piovesan | The Impact of Paper Constituents on the Efficiency of Mechanical Strengthening by Polyaminoalkylalkoxysilanes[END_REF] N HMBC spectrum of the equimolar mixture AM/vanillin (Figure 4a), whereas a new and unique signal appears at -61 ppm. This was attributed to the formation of an imine. A strong 2 J coupling was observed between the nitrogen N and the hydrogen H( 5) involved in the imine functional group (8.05-8.07 ppm). The 2D [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C-1 H spectrum (Figure 4b) confirmed this assignment, as proved by the coupling between the same H( 5) and the carbons C(2) (123.1 ppm), C(3) (127.3 ppm) and C(4) (109.9 ppm) of vanillin on the one hand, and C(6) (63.3 ppm) of AM on the other hand.

The FTIR spectrum of the equimolar mixture at t 0 (Figure S4 in the Supporting Information file) corroborated the rapid formation of an imine. Whereas the intense peak at 1675 cm -1 , characteristic of the conjugated C=O group (stretching) disappeared, a new band appeared at 1642 cm -1 , which indicates the production of a conjugated C=N bond (stretching). [START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies: Tables and Charts[END_REF] This demonstrates that the amine of AM reacted with vanillin, yielding a Schiff base (Figure 4c). The instant yellowing of the vanillin white powder upon addition of AM is consistent with this reaction, as Schiff bases are notoriously yellow. In addition, the production of ethanol (signals at 1.06 ppm and 3.44 ppm, Figure 4a) indicated a total hydrolysis of the ethoxy groups of AM (missing signals at 1.12 ppm and 3.68 ppm). This is consistent with the formation of water occurring with the condensation reaction.

The formation of the Schiff base was immediate and complete. 3.

Vanillic acid.

The pK A of amines are reported to be between 7 and 11, whereas the pK A of vanillic acid is about 4.5 at 25 °C. 47 An acid-base reaction between the two shall then proceed totally to form a salt. This was confirmed by the [START_REF] Hubbe | Deacidification of Acidic Books and Paper by Means of Non-Aqueous Dispersions of Alkaline Particles: A Review Focusing on Completeness of the Reaction[END_REF] C spectrum of the AM/vanillic acid (1/1) mixture (Figure 6b).

For AM, the nearest carbons of the ammonium group formed, C(10) (28. 6) showed frequency shifts. This is subsequent to the formation of a carboxylate involved in the molecule's conjugated system.

Table 3. Assignments of the [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si NMR signals for AM, with and without a vanilloid (see Figure 5). [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si NMR signal δ (ppm) Structure Formula A -4.9

D (OEt)(OEt) 0 (Me)(NH 2 C 3 H 6 )Si(OEt) 2 B -13.4 D (OEt) 1 D (OEt) 1 (OEt)(Me)(NH 2 C 3 H 6 )SiOSi(NH 2 C 3 H 6 )(Me)(OEt) C [-13; -14] D (OH) 1 D 2 (OH)(Me)(NH 2 C 3 H 6 )Si-O-Si D -19.3 (D 2 ) 4 [(Me)(NH 2 C 3 H 6 )SiO] 4 E [-21; -23] D 2 Si-O-(Me)(NH 2 C 3 H 6 )Si-O-Si F -5.8 D (OEt)(OEt) 0 (Me)(NH 3 + C 3 H 6 )Si(OEt) 2 G -6.1 D (OEt)(OH) 0 (Me)(NH 3 + C 3 H 6 )Si(OEt)(OH) H -14.1 D (OEt) 1 D (OEt) 1 (OEt)(Me)(NH 3 + C 3 H 6 )SiOSi(NH 3 + C 3 H 6 )(Me)(OEt) I -3.4 D (AV)(AV) 0 (Me)(NH 2 C 3 H 6 )Si(VA) 2 J -4.2 D (AV)(OEt) 0 (Me)(NH 2 C 3 H 6 )Si(VA)(OEt) K -12.6 D (AV) 1 D (OEt) 1 (VA)(Me)(NH 2 C 3 H 6 )SiOSi(NH 2 C 3 H 6 )(Me)(OEt) L -13.2 D (AV) 1 D (OEt) 1 (VA)(Me)(NH 2 C 3 H 6 )SiOSi(NH 2 C 3 H 6 )(Me)(OEt)
VA: substituted vanillyl alcohol. Solvent: DMSO-d 6 .

In the [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si NMR spectrum of the AM/vanillic acid mixture (Figure 5c), the D 0 signal F at -5.8 ppm corresponds to the protonated AM monomer, and can be compared to the signal A of the reference AM (Figure 5a). The D 1 signal B at -13.4 ppm was associated to the AM dimer naturally present in solution, which was identified from the absence of a D 2 peak relating to longer chains. The signal H at -14.1 ppm thus corresponded to the protonated dimer. The small D 0 signal G at -6.1 ppm is noteworthy, and attributed to a partially hydrolyzed AM monomer (see Table 3 for more details), which indicated the presence of residual water. This highlights the formation of an ammonium salt.

On the FTIR spectrum of the mixture at t 0 (Figure S5 in the Supporting Information file), the peak at 1695 cm -1 assigned to the conjugated C=O bond of the carboxylic acid (stretching) disappeared, whereas new bands appeared at 1544 and 1628 cm -1 , which correspond to C=C and C=O in an aryl carboxylate, respectively. [START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies: Tables and Charts[END_REF] This confirmed the fast acid-base reaction between vanillic acid and AM.

To conclude, an immediate and total acid-base reaction occurred between AM and vanillic acid, yielding an ammonium salt (Figure 6d). These results were consistent with the intended use of AAAS to deacidify paper (Table 2). They also hinted at interactions that possibly occur between lignin and AM in the paper through ionic bonding. Amine functions, if they immediately interact with carboxylated degradation products (of lignin and possibly of cellulose too), would then not be available to form hydrogen bonding with cellulose fibers. The consequence would be less strengthening and a lower reserve of the treated paper. L (-13.2 ppm) on the [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si spectrum (Figure 5d) indicated condensation reactions taking place with the ethyl silicates. The hypothesis of hydrolysis of these functional groups was excluded since the DMSOd 6 used was fresh and totally anhydrous, and also because if so, the signals would have been detected at lower frequencies than signals A and B of AM. [START_REF] Sugahara | 29Si-NMR Study of Hydrolysis and Initial Polycondensation Processes of Organoalkoxysilanes. I. Dimethyldiethoxysilane[END_REF] The presence of peaks A and B on the spectrum indicated that a portion of AM did not react. With the noticeable exception of hydrogens H(*), corresponding to H(1) of the model molecule which partly reacted with AM, no modification of the other signals' chemical shifts was noted on the 1 H spectrum of the mixture (Figure 7). The 2D HMBC 1 H- [START_REF] Area | Paper Aging and Degradation: Recent Findings and Research Methods[END_REF] Si spectrum of the mixture (Figure 7) clearly indicated the 3 J coupling between the silicon Si of AM and the hydrogen H(*) of VA, which confirmed the condensation mechanism proposed. The Si(I)/H(*) coupling corresponds to the AM monomer which has undergone two condensation reactions with vanillyl alcohol, whereas the Si(J)/H(*) coupling is associated to AM that has reacted only once.

This assignment was confirmed by the presence of a 3 J coupling between H(10) of the ethoxy group and Si(J), whereas it was absent for Si(I). The 3 J coupling between Si(K) and H(*) was attributed to the portion of AM dimers that reacted with vanillyl alcohol, since dimers were present in the commercial AM solution (signal B in Figure 5a). This also demonstrated that besides hydrolytic condensation, AM also underwent alcohol-driven condensation. Hence both water and alcohol are products of the AAAS polycondensation.

The study of vanillyl alcohol proved therefore that AM was able to react with other molecules, not only by the amine moiety, but also through the silicate groups. This work showed that AM can undergo successive esterification reactions with the primary aliphatic alcohol of the model molecule (Figure 7).

Such transesterification reaction is seldom reported and requires specific reaction conditions. [START_REF] Dakhili | Recombinant Silicateins as Model Biocatalysts in Organosiloxane Chemistry[END_REF] These reactions are limited despite the fact that condensation between alkoxysilanes and primary alcohols are catalyzed in alkaline conditions. 45 The silicon atoms numbering is the same as in Figure 5 and Table 3.

3.3.4.

Cyclohexanol. AM/cyclohexanol (1/1) was studied by 1 H NMR in order to investigate the impact of steric hindrance on the kinetics of condensation reactions, since cyclohexanol is a secondary alcohol. No reaction was detected in solution after seven days. It was thus assumed that the condensation reactions detailed above can take place mostly with the available primary aliphatic alcohols within lignin. Although they seem to be limited, and even more so in neutral conditions, it is essential to take these condensation reactions between AM and the lignin in paper into consideration, since they can ultimately result in limiting the polymerization of the AAAS thus leading to a low DP n ̅̅̅̅̅ . In situ, these reactions may compete with reactions between amines and carbonyls which, as seen previously, would have the opposite effect, namely increase the AM polycondensation.

p-benzoquinone.

The 1 H NMR spectrum (Figure S6 in the Supporting Information file) showed a doublet at 6.529 ppm and 6.525 ppm and the absence of the peak associated with p-benzoquinone (6.86 ppm). These two peaks were attributed to two different products. AM reacted with p-benzoquinone according to the reaction in Figure 8. [START_REF] Morrison | The Reaction of Benzoquinone with Amines and Proteins[END_REF][START_REF] Kutyrev | Nucleophilic Reactions of Quinones[END_REF] Evidencing this reaction demonstrated once more the ability of the amine groups of AM monomers to very quickly react and form covalent bonds with lignin and its degradation products. The amount of quinones arising from oxidized lignin in degraded papers can be substantial. However, this reaction mechanism can only occur with unsubstituted CH=CH bonds, which drastically reduces the number of available reactive sites. Aromatic groups of softwood lignin are rich in methoxy groups, the main precursor being coniferyl alcohol, as previously described.

3.3.6. Other models. Three other models of lignin, namely mequinol, phenol and anisole, and their reactivity with AM were studied because of the analogy to two other important functional groups in lignin: ether (methoxy group directly bonded to an aryl group) and phenol groups. No reaction was observed after 7 days in DMSO-d 6 . These two functional groups are most probably unreactive with AM.

Extrapolation to the mechanical properties of paper

As demonstrated, based on the modelling study, lignin and its degradation products can in all likelihood react with AM. Assuming that other AAAS, such as AP and DIA, have a similar reactivity than AM, the side reactions identified could occur when treating lignocellulosic papers with these compounds, and thus affect the strengthening efficiency. It is noteworthy that newsprint J2 is more brittle than J3 (Figure 1), and has a lower pH (Table 2). It is thus proposed that the numerous acidic and oxidized functions and degradation products of lignin in J2 reacted rapidly with the AAAS. This in turn could impact the mechanical reinforcement of J2 either by hindering hydrogen bonds formation or by limiting the polycondensation reaction. Both are assumed to be key parameters to the strengthening.

This could explain why the treatment was more efficient on the less degraded (and most likely less oxidized) lignocellulosic paper J3. Lastly, as J1 is as brittle as J2, but does not contain lignin, the absence of lignin could explain why its strengthening was more efficient.

Conclusion

The investigation using NMR enabled a better understanding of the reaction mechanisms that can occur between AM and key biopolymers of paper, namely cellulose (and indirectly hemicelluloses) and lignin. Cellulose being barely reactive towards AM, it seems that in order to effect strengthening, the interaction between AAAS and cellulose is mostly done through hydrogen bonds. In contrast, several characteristic and abundant functional groups of lignin were found very reactive, in particular the oxidized ones, such as aldehyde, ketone and quinone moieties, but also the numerous accessible aliphatic alcohols in the polyphenol. The reactions involving the amine moiety of AAAS modify the extent of the alkaline reserve, in addition to slowing the polycondensation. In the future, in situ analysis of treated papers will be carried out with solid-state NMR spectroscopy. This hopefully will allow corroborating the present work conclusions. The disparities in the treatment efficiency depending on the papers constituents will be more easily understood, with the aim to optimizing the treatment formulation.

Figure 1 .

 1 Figure 1. FE of J1, J2 and J3, before (red filled bars) and after (green hatched bars) treatment with AP/DIA (5/95).

Figure 4 . 2 𝐷 1 ) + 2 46 Figure 5 .

 4212465 Figure 4. 2D NMR HMBC (a) 1 H-15 N, (b) 1 H-13 C spectra of the AM/vanillin (1/1) mixture and (c) reaction between vanillin and AM: formation of a Schiff base. R refers to an ethyl group, a hydrogen or a polysiloxane chain.

  3 ppm) and C(9) (46.2 ppm), moved upfield leading to signals at 22.8 ppm and 42.9 ppm, respectively. The signals related to the farthest carbons C(11) (11.9 ppm), C(12) (-3.6 ppm), C(13) (58.6 ppm) and C(14) (19.5 ppm) remained at the same frequencies. Whereas C(8) of the -OCH 3 group of vanillic acid presented the same chemical shift with and without AM (56.5 ppm), each of the signals corresponding to the carbon atoms of the conjugated system (C(2) to C(7) in the [110; 180] ppm range in Figure

Figure 6 .

 6 Figure 6. 13 C NMR spectra of (a) AM, (b) AM/vanillic acid (1/1) mixture, (c) vanillic acid, and (d) reaction between vanillic acid and AM: formation of an ammonium salt.

Figure 7 .

 7 Figure 7. 2D NMR HMBC 1 H-29 Si of the AM/vanillyl alcohol (1/1) mixture, and reaction between vanillyl alcohol and AM. (1) First esterification. (2) Second esterification. * corresponds to C1 of vanillyl alcohol condensed with AM. The silicon atoms numbering is the same as in Figure 5 and Table3.

  First, AM attached to a C=C bond of the quinone, before the formation of a substituted hydroquinone by isomerization [step(1)]. The new product reacted immediately with the remaining p-benzoquinone through a redox reaction, yielding a free hydroquinone [step(2)]. AM reacted a second time with the other C=C bond, leading to the twice-substituted final product upon new isomerization [step(3)]. This reaction was instant, total, exothermic, and also occurred in anhydrous conditions. The reaction mechanism was confirmed by FTIR. The spectrum at t 0 of the equimolar mixture AM/ p-benzoquinone where the nearly total absence of characteristic peaks in the spectral range 1600-1700 cm -1 indicated the formation of the phenol groups (FigureS7in the Supporting Information file).

Figure 8 .

 8 Figure 8. Reaction between AM and p-benzoquinone. (1) First electrophile addition of AM and isomerization. (2) Redox reaction with the remaining p-benzoquinone. (3) Second electrophile addition of AM and isomerization.

Table 1 .

 1 Main constituents and characteristics of the newsprint papers.

		J1	J2		J3
	Date of publication	1911	1920		1923
	Pulp	BSP, softwood	GP/USP/BSP, softwood,	GP/USP/BSP, softwood,
			trace of straw	trace of straw
	Ash content (% w )	17.8	7.6		16.3
	Grammage (g m -2 )	57	49		52
	Thickness (µm)	59	64		63
	BSP: bleached sulfite pulp. GP: groundwood pulp. USP: unbleached sulfite pulp.
	2.2. Chemicals
	2.2.1. Aminoalkylalkoxysilanes
	AM (97%), 3-aminopropyltriethoxysilane	(AP) (>98%)	and N-(2-aminoethyl)-3-
	aminopropylmethyldimethoxysilane (DIA) (97%) were purchased from ABCR (Gelest, France) and
	used as received.

Table 2 .

 2 pH, BL and EB of the untreated and treated (AP/DIA 5/95) newsprint papers.

			J1	J2	J3
	pH	Untreated Treated	4.97 ± 0.03 7.76 ± 0.01	4.53 ± 0.06 7.58 ± 0.01	4.84 ± 0.03 7.57 ± 0.10
	BL (km)	Untreated Treated	2.09 ± 0.27 2.61 ± 0.30	2.76 ± 0.26 3.87 ± 0.46	2.70 ± 0.45 3.40 ± 0.41
	EB (%)	Untreated Treated	0.94 ± 0.15 1.13 ± 0.15	1.02 ± 0.15 1.15 ± 0.14	1.06 ± 0.17 1.33 ± 0.11

Acknowledgments

This work was supported by the Paris Seine Graduate School Humanities, Creation, Heritage, Investissements d'Avenir ANR-17-EURE-0021 -Foundation for Cultural Heritage Science, and by ANR-11-LABX-0039-grant (Labex CHARMMMAT). The participation of the Bibliothèque Nationale de France is acknowledged. Camille Piovesan, Alice Gimat, Oulfa Belhadj and Sabrina Paris-Lacombe (CRC), Isabelle Fabre-Francke (LPPI), Gérard Mortha (Grenoble INP -Pagora), Hervé Cheradame (Polytheragene) and Nathalie Steunou (Institut Lavoisier de Versailles) are warmly thanked for technical contribution and fruitful discussions.


Zaccharie Ramzi

Philippe Ciuciu

Jean-Luc Starck

Benchmarking MRI reconstruction neural networks on the fastMRI dataset

Deep learning is starting to offer promising results for reconstruction in MRI. A lot of networks are being developed, but the comparisons remain hard because the frameworks used are not the same among studies, the networks are not properly re-trained and the datasets used are not the same among comparisons. The recent release of a public dataset, fastMRI, consisting of raw k-space data, encouraged us to write a consistent benchmark of several deep neural networks for MR image reconstruction. This paper shows the results obtained for this benchmark allowing to compare the networks and links the open source implementation of all these networks in Keras. The main finding of this benchmark is that it is beneficial to perform more iterations between the image and the measurement spaces compared to having a deeper per-space network.

Introduction

Magnetic Resonance Imaging (MRI) is an imaging modality used to probe the soft tissues of the human body. As it is non-invasive and non-ionizing (contrary to X-Rays for example), its popularity has grown over the years, for example tripling between 1997 and 2006 according to [START_REF] Smith-Bindman | Rising use of diagnostic medical imaging in a large integrated health system[END_REF]. This is attributed in part to the technical improvements of this technique. We can for example mention higher field magnets (3 Teslas instead of 1.5), parallel imaging [START_REF] Roemer | The NMR phased array[END_REF] or compressed sensing MRI [START_REF] Lustig | Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging[END_REF] (CS-MRI). These improvements allow for better image quality and lower acquisition duration.

There is however, still room for improvement. Indeed, an MRI scan may last up to 90 minutes according to the NHS website [START_REF] Nhs | NHS website[END_REF], making it unpractical for • They are usually iterative involving the computation of transforms on large data, they therefore take a lot of time (2 min for a 512 × 512 -500 μm in plane resolution -slice [START_REF] Ramzi | Benchmarking proximal methods acceleration enhancements for CS-acquired MR image analysis reconstruction[END_REF], on a machine with 8 cores).

• The regularisation is usually not perfectly suited to MRI data (it is indeed very difficult to come up with a prior that perfectly reflects MR images).

This is where learning comes in to play, and in particular deep learning. The promise is that it will solve both the aforementioned problems.

• Because they are implemented efficiently on GPU, and do not use an iterative algorithm, the deep learning algorithms run extremely fast.

• If they have enough capacity, they can learn a better prior of the MR images from the training set.

One of the first neural networks to gain attention for its use in MRI reconstruction was AUTOMAP [START_REF] Zhu | Image reconstruction by domain-transform manifold learning[END_REF]. This network did not exploit a problem specific property except the fact that the outcome was supposed to be an image. Some more recent works [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF][START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF][START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] have tried to inspire themselves from existing classical methods in order to leverage problem specific properties, but also expertise gained in the field. However, they have not been compared against each other on a large dataset containing complex-valued raw data.

A recently published dataset, fastMRI [START_REF] Zbontar | fastMRI: An open dataset and benchmarks for accelerated MRI[END_REF], allows this comparison, although it is still to be done and requires an implementation of the different networks in the same framework to allow for a fairer comparison in terms for example of runtime.

Our contribution is exactly this, that is:

• Benchmark different neural networks for MRI reconstruction on 2 datasets: the fastMRI dataset, containing raw complex-valued knee data, and the OASIS dataset [START_REF] Lamontagne | OASIS-3: Longitudinal neuroimaging, clinical, and cognitive dataset for normal aging and Alzheimer's disease[END_REF] containing DICOM real-valued brain data.

• Provide reproducible code and the networks' weights1 , using Keras [13] with a TensorFlow backend [START_REF] Abadi | TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems[END_REF].

While our work focuses on classical MRI modalities reconstruction, it is to note that other works have applied deep learning to other modalities like MR fingerprinting [START_REF] Virtue | Better than real: Complex-valued neural nets for MRI fingerprinting[END_REF] or diffusion MRI [START_REF] Aggarwal | Modl-mussels: Model-based deep learning for multishot sensitivity-encoded diffusion mri[END_REF]. The networks studied here could be applied, but would not benefit from some invariants of the problem, especially in the fourth (contrast-related) dimension introduced.

Related works

In this section we briefly discuss other works presenting benchmarks on many different reconstruction neural networks.

In [START_REF] Minh Quan | Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss[END_REF], they benchmark their (adversarial training based) algorithms against classical methods and against Cascade-net (which they call Deep Cascade) [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] and ADMM-net (which they call DeepADMM) [START_REF] Yang | Deep ADMM-net for compressive sensing MRI[END_REF]. They train and evaluate quantitatively the networks on 2 datasets, selecting each time 100 images for train, 100 images for test:

• the IXI database2 (brains),

• the Data Science Bowl challenge3 (chests).

While both these datasets provide a sufficient number of samples to have a trustworthy estimate of the performance of the networks, they are not composed of raw complex-valued data, but of DICOM real-valued data. Still in [START_REF] Minh Quan | Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss[END_REF], they do evaluate their algorithms on a raw complex-valued dataset 4 , but it only features 20 acquisitions, and therefore the comparison is only done qualitatively.

In [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF], they benchmark their algorithm against classical methods. They train and evaluate their network on 3 different datasets:

• the brain real-valued data set provided by the Alzheimer's Disease Neuroimaging Initiative (ADNI) [START_REF] Petersen | Alzheimer's disease neuroimaging initiative (ADNI): clinical characterization[END_REF],

• 2 proprietary datasets with raw complex-valued data of brain data.

Again, the only public dataset they use features real-valued data. It is also to be noted that their code cannot be found online.

Models

In this section, we will first introduce what we call the classical models to do reconstruction in CS-MRI. The models we chose to discuss are in no way an exhaustive list of all the models that can be used without learning for reconstruction in MRI (think of LORAKS [START_REF] Haldar | Low-Rank Modeling of Local k-Space Neighborhoods (LO-RAKS) for Constrained MRI[END_REF] for example just to name this one), but they allow us to justify how the subsequent neural networks are built. These models are introduced in a second time.

Idealized inverse problem

In anatomical MRI, the image is encoded as its Fourier transform and the data acquisition is segmented in time in multiple shots or trajectory. This does not take possible gradient errors or B0-field inhomogeneities into account. Because each Fourier coefficient trajectory takes time to acquire, the time separating two consecutive shots, namely the TR or time of repetition, being potentially pretty long, the idea of CS-MRI is to acquire less of them. We therefore have the following idealized inverse problem in the case of single-coil CS-MRI:

y = F Ω x (1)
where y is the acquired Fourier coefficients, also called the k-space data, Ω is the sub-sampling pattern or mask, F Ω is the non-uniform Fourier transform (or masked Fourier transform in the case of Cartesian under-sampling), and x is the real anatomical image. Here we will only deal with Cartesian under-sampling, and there we have F Ω = M Ω F , where M Ω is a mask, and F is the classical Fourier transform. This model is also valid for 3D (volumewise) imaging, but in the following we only consider 2D (slicewise) imaging.

Classic models

The first (although unsatisfactory model) that can be used to perform the reconstruction of an MR image with an under-sampled k-space, is to simply use the inverse Fourier transform with the unknown Fourier coefficients replaced by zeros (zero-filled inverse Fourier transform). This method is called zero-filled reconstruction and we have:

xzf = F -1 y (2)
The second model we want to introduce makes use of the fact that MR images can be represented in a wavelet basis with only a few non-zero coefficients [START_REF] Lustig | Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging[END_REF] according to the sparsity principle. The reconstruction is therefore done by solving the following optimization problem:

xwav = arg min x ∈C n×n 1 2 y -F Ω x 2 2 + λ Ψx 1 (3)
where the notations are the same as in (1), and λ is a hyper-parameter to be tuned, and Ψ is a chosen wavelet transform. This problem can be solved iteratively using a primal-dual optimisation algorithm like Condat-Vù [START_REF] Condat | A Primal-Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms[END_REF] or Primal Dual Hybrid Gradient (PDHG) [START_REF] Chambolle | A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging[END_REF] (also known as Chambolle-Pock algorithm) or, if the wavelet transform is invertible (i.e. non-redundant or decimated), using a proximal algorithm like Fast Iterative Shrinkage Algorithm (FISTA) [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] or Proximal Optimal Gradient Method (POGM) [START_REF] Kim | Adaptive restart of the optimized gradient method for convex optimization[END_REF]. Since the problem is convex, all these algorithms converge to the same solution, only at different speeds.

The last model we choose to introduce is the dictionary learning model [START_REF] Ravishankar | Magnetic Resonance Image Reconstruction from Highly Undersampled K-Space Data Using Dictionary Learning[END_REF][START_REF] Caballero | Dictionary learning and time sparsity for dynamic MR data reconstruction[END_REF]. Its assumption is that an MR image is only composed of a few patches, and can therefore be expressed sparsely in a corresponding dictionary. This dictionary can be learned per-image, leading to the following optimisation problem:

xdl = arg min x,D,{αij } (i,j)∈I 1 2 y -F Ω x 2 2 + λ (i,j)∈I R ij x -Dα ij 2 2 subject to ∀(i, j) ∈ I, α ij 0 ≤ T 0 (4)
where the notations are the same as in (3) and I is the fixed set of patches locations, D is the dictionary, λ and T 0 are hyper-parameters to be set, and R ij is the linear operator extracting the patch at location (i, j). This problem is solved in 2 steps:

1. The dictionary learning step, where both the dictionary D and the sparse codes α ij are updated alternatively.

2. The reconstruction step, where x is updated. Since this subproblem is quadratic, it admits an analytical solution, which amounts to averaging patches and then perform a data consistency in which the sampled frequencies are replaced in the patch-average result.

Neural networks

The neural networks introduced here are all derived in a certain way from the classical models introduced before.

Single-domain networks

What we term single-domain networks, are network which only act either in the k-space or in the image (direct) space. They make use of the fact that we have a pseudo-inverse like in [START_REF] Smith-Bindman | Rising use of diagnostic medical imaging in a large integrated health system[END_REF]. They usually use a U-net-like [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] architecture. This network was originally built for image segmentation, but has since been used for a wide-variety of image-to-image tasks, mainly as a strong baseline. In [START_REF] Han | k-space deep learning for accelerated MRI[END_REF], they used a U-net to apply on the under-sampled k-space measurements before performing the inverse Fourier transform. In [START_REF] Hyun | Deep learning for undersampled MRI reconstruction[END_REF], they used a U-net to apply on the zero-filled reconstruction and correct the output of the U-net with a data consistency step (where they replace sampled values in the k-space). The network we implemented was however vanilla, without this extra data-consistency Figure 1: Illustration of the U-net from [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF]. In our case the output is not a segmentation map but a reconstructed image of the same size (we perform zero-padding to prevent decreasing sizes in convolutions).

step. Our implementation however only features the following cascade of number of filters: 16, 32, 64, 128. The original U-net is illustrated in Figure 1 where the number of filters used in each layer is 4 times what we used.

Cross-domains networks

The second class of networks we introduce, we term cross-domain networks. The key intuitive idea is that they correct the data in both the k-space and the image space alternatively, using the Fourier transform to go from one space to the other. They are derived from the optimisation algorithms used to solve the optimisation problems introduced before, using the idea of "unrolling" introduced in [START_REF] Gregor | Learning Fast Approximations of Sparse Coding[END_REF]. An illustration of this class of networks is presented in Figure 2.

Because these networks work directly on the input data (and not on a primarily reconstructed version of it), they need to handle complex-valued data. In particular, the classical deep learning frameworks (TensorFlow and Pytorch) do not feature the ability to perform complex convolutions off-the-shelf. The way convolution is performed in the original papers is therefore to concatenate the real and imaginary of the image (respectively the k-space), making it a 2-channel image, perform the series of convolutions, and have the output be a 2-channel image then transformed back in a complex image (respectively k-space).

The Cascade-net [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] is based on the dictionary learning optimisation prob-Figure 2: The common backbone between the Cascade net, the KIKI-net and the PD-net. US mask stands for under-sampling mask. DC stands for data consistency. (I)FFT stands for (Inverse) Fast Fourier Transform. N k,d is the number of convolution layers applied in the k-space. N i,d is the number of convolution layers applied in the image space. N C is the total number of alternations between the k-space and the image-space. It is to be noted that in the case of PD-net, the data consistency step is not performed, the Fourier operators are performed with the original under-sampling mask, and a buffer is concatenated along with the current iteration to allow for some memory between iterations and learn the acceleration (in the k-space net -dual net-it is also concatenated with original k-space input). In the case of the Cascade net, N k,d = 0, only the data consistency is performed in the k-space. In the case of the KIKI-net, there is no residual connection in the k-space. However, the k-space nets and image space nets could potentially be any kind of image-to-image neural network.

Figure 3: Illustration of the Cascade-net from [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF]. Here each C i is a convolutional block of 64 filters (48 in our implementation) followed by a ReLU non-linearity, n d is the number of such convolutional blocks forming a convolutional subnetwork between each data consistency layer DC, and n c is the number of convolutional subnetworks.

Figure 4: Illustration of the KIKI-net from [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF]. The KCNN and ICNN are convolutional neural networks composed of a number of convolutional blocks varying between 5 and 25 (we implemented 25 blocks for both KCNN and ICNN), each followed by a ReLU non-linearity and featuring between 8 and 64 filters (we implemented 32 filters). For both the varying numbers, the supplementary material of [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF] shows that the higher the better. The ICNN also features a residual connection.

lem [START_REF] Lustig | Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging[END_REF]. The idea is to replace the dictionary learning step by convolutional neural networks and still keep the data consistency step in the k-space. The optimisation algorithm is then unrolled to allow to perform back-propagation. [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] shows that we can perform back-propagation through the data consistency step (which is linear), and derive the corresponding Jacobian. The parameters used here for the implementation are the same as those in the original paper, except the number of filters which was decreased from 64 to 48 to fit on a single GPU. This network is illustrated in Figure 3.

The KIKI-net [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF] is an extension of the Cascade-net where they additionally perform convolutions after the data consistency step in the k-space. The parameters used here for the implementation are the same as those in the original paper. This network is illustrated in Figure 4.

The Primal-Dual-net (PD-net) was introduced by [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF] and applied to Figure 5: Illustration of the PD-net from [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF]. Here T denotes the measurement operator, which in our case is the under-sampled Fourier transform, T * its adjoint, g is the measurements, which in our case are the undersampled kspace measurements, and f 0 and h 0 are the initial guesses for the direct and measurement spaces (the image and k-space in our case). The initial guesses are zero tensors. Because we transform complex-valued data into 2-channel real-valued data, the number of channels at the input and the output of the convolutional subnetworks are multiplied by 2 in our implementation.

MRI by [START_REF] Cheng | Model learning: Primal dual networks for fast MR imaging[END_REF], is based on the wavelet-based denoising (3), and in particular the resolution of the corresponding optimisation problem with the PDHG [START_REF] Chambolle | A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging[END_REF] algorithm. Here the algorithm is unrolled and the proximity operators (present in the general case of PDHG) are replaced by convolutional neural networks. For our implementation, for a fairer comparison with Cascade-net and the Unet, we used a ReLU non-linearity instead of a PReLU [START_REF] He | Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification[END_REF]. This network is illustrated in Figure 5.

Training

The training was done with the same parameters for all the networks. The optimiser used was Adam [START_REF] Kingma | A Method for Stochastic Optimization[END_REF], with a learning rate of 10 -3 and default parameters of Keras (β 1 = 0.9, β 2 = 0.999, the exponential decay rates for the moment estimates). The gradient norm was clipped to 1 to avoid the exploding gradient problems [START_REF] Pascanu | On the difficulty of training Recurrent Neural Networks[END_REF]. The batch size was 1 (i.e. one slice) for every network except the U-Net where the whole volume was used for each step. For all networks, to maximise the efficiency of the training, the slices were selected in the 8 innermost slices of the volumes, because the outer slices do not have much signal. No early stopping or learning rate schedule was used (except for KIKI-net to allow for a stable training where we used the learning rate schedule proposed by the authors in the supporting information of [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF]). The number of epochs used was 300 for all networks trained end-to-end. For the iterative training of the KIKI-net, the total number of epochs was 200 (50 per sub-training). Batch normalization was not used, however, in order to have the network learn more efficiently, a scaling of the input data was done. Both the k-space and the image were multiplied by 10 6 for fastMRI and by 10 2 for OASIS, because the k-space measurements had values of mean 10 -7 (looking separately at the real and imaginary parts) for fastMRI and of mean 10 -3 for OASIS. Without this scaling operation, the training proved to be impossible with bias in the convolutions and very inefficient without bias in the convolutions.

Data

Under-sampling

The under-sampling was done retrospectively using a Cartesian mask described in the data set paper [START_REF] Zbontar | fastMRI: An open dataset and benchmarks for accelerated MRI[END_REF], and an acceleration factor of 4 (i.e. only 25% of the k-space was kept). It contains a fully-sampled region in the lower frequencies, and randomly selects phase encoding lines in the higher frequencies. It is to be noted that different under-sampling strategies exist in CS-MRI. Some of them are listed in [START_REF] Chauffert | Variable density sampling with continuous trajectories[END_REF], like for example spiral or radial. These strategies allow for a higher image quality while having the same acceleration factor or the same image quality with a higher acceleration factor. Typically, the spiral under-sampling scheme was designed to allow fast coronary imaging [START_REF] Irarrazabal | Fast Three Dimensional Magnetic Resonance Imaging[END_REF][START_REF] Meyer | Fast Spiral Coronary Artery Imaging[END_REF]. These under-sampling strategies must take into account kinematic constraints (both physically and safety based), but also should also be with variable density [START_REF] Chauffert | Variable density sampling with continuous trajectories[END_REF]. Recent works even try to optimize the under-sampling strategy under these kinematic constraints [START_REF] Lazarus | SPARKLING : variable-density k-space filling curves for accelerated T 2 * -weighted MRI[END_REF]. Others have tried to learn the under-sampling strategy in a supervised way. In [START_REF] Sanchez | Scalable Learning-Based Sampling Optimization for Compressive Dynamic MRI[END_REF], the under-sampling strategy is learned with a greedy optimisation. In [START_REF] Sherry | Learning the sampling pattern for MRI[END_REF], a gradient descent optimisation is used. Some approaches ( [START_REF] Aggarwal | Modl-mussels: Model-based deep learning for multishot sensitivity-encoded diffusion mri[END_REF][START_REF] Wu | Deep Compressed Sensing[END_REF][START_REF] Weiss | PILOT : Physics-Informed Learned Optimized Trajectories for Accelerated MRI[END_REF] even try to learn jointly the optimal under-sampling strategy along with the reconstruction.

fastMRI

The data used for this benchmark is the emulated single-coil k-space data of the fastMRI dataset [START_REF] Zbontar | fastMRI: An open dataset and benchmarks for accelerated MRI[END_REF], along with the corresponding ground truth images. The acquisition was done with a 15-channel phased array coil, in Cartesian 2D Turbin Spin Echo (TSE). The pulse sequences were proton-density weighting, half with fat suppression, half without, some at 3.0 Teslas (T) others at 1.5 T. The sequence parameters were as follows: Echo train length 4, matrix size 320× 320, in-plane resolution 0.5mm×0.5mm, slice thickness 3mm, no gap between slices. In total, there are 973 volumes (34, 742 slices) for the training subset and 199 volumes (7135 slices) for the validation subset.

Since the k-spaces are of different sizes, therefore resulting in images of different sizes, the outputs of the cross-domain networks were cropped to a central 320 × 320 region. For the U-net, the input of the network was cropped.

OASIS

The Open Access Series of Imaging Studies (OASIS) brain database [START_REF] Lamontagne | OASIS-3: Longitudinal neuroimaging, clinical, and cognitive dataset for normal aging and Alzheimer's disease[END_REF] is a database including MRI scans of 1068 participants, yielding 2168 MR sessions. Of these 2168, we select only 2164 sessions which feature T1-weighted sequences. 1878 of these were acquired on a 3.0 T, 236 at 1.5 T and the remaining are undisclosed (50). The slice size is majorly 256 × 256, and sometimes 240 × 256 (rarely it can be some other sizes).The number of slices per scan is majorly 176, and sometimes 160 (rarely it can be smaller).

The data was then separated in a training and a validation set. The split was participant-based, that is a participant cannot have a scan in both sets. The split was of 90% for the training set and 10% for the validation set. We further reduced the training data to make it comparable to fastMRI, to 1000 scans randomly selected for the training subset and 200 scans randomly selected for the validation subset.

Contrarily to fastMRI, the OASIS data is available only in magnitude and therefore is only real-valued.The k-space is computed as the inverse Fourier transform of the magnitude image.

Results

Metrics

The metrics we used to benchmark the different networks are the following:

• The Peak Signal-to-Noise Ratio (PSNR)

• The Structural SIMilarity index (SSIM) [START_REF] Wang | Image Quality Assessment: From Error Visibility to Structural Similarity[END_REF] • The number of trainable parameters in the network • The runtime in seconds of the neural network on a single volume The PSNR is computed as follows, on whole magnitude volumes:

P SN R(x, x) = 10 log 10 max(x) 2 1 n i,j,k (x i,j,k -xi,j,k ) 2 (5)
where x is the ground truth volume, x is the predicted volume (magnitude image), and n is the total number of points in the ground truth volume (same as the predicted volume). Since this metric compares very local differences it does not necessarily reflect the global visual comparison of the images. The SSIM was introduced in [START_REF] Wang | Image Quality Assessment: From Error Visibility to Structural Similarity[END_REF] exactly to take more structural differences or similarities between images into account. It is computed as in the original paper, per slice then averaged over the volume (the range however is computed volume-wise):

SSIM (x, x) = (2µ x µ x + c 1 )(2σ x σ x + c 2 )(cov xx + c 3 ) (µ 2 x + µ 2 x + c 1 )(σ 2 x + σ 2 x + c 2 )(σ x σ x + c 3 ) (6)
where x is the ground truth slice, x is the predicted slice, µ i is the mean of i, σ 2 i is the variance of i, cov ij is the covariance between i and j, c

1 = (k 1 L) 2 , c 2 = (k 2 L) 2 , c 3 = c2
2 , L is the range of the values of the data (given because computed over the whole volume), and k 1 = 0.01 and k 2 = 0.03.

While the 2 aforementioned metrics control the reconstruction quality, it is important to note that this is not the only factor to take into account when designing reconstruction techniques. Because the reconstruction has to happen fast enough for the MR physician to decide whether to re-conduct the exam or not, it is important for the proposed technique to have a reasonable reconstruction speed. For real-time MRI applications or dynamic MRI (e.g. cardiac imaging), it is even more important (for example in the context of monitoring surgical operations [START_REF] Horvath | Real-time Magnetic Resonance Imaging Guidance for Cardiovascular Procedures[END_REF]). The runtimes were measured on a computer equipped with a single GPU Quadro P5000 with 16GB of RAM.

Concurrently, the number of parameters has to stay relatively low to allow the implementation on the different machines with potentially limited memory, which will probably need to have multiple models (for different contrasts, different organs or different undersampling schemes including different acceleration factors). The quantitative results in Tables 1234show that the PD-net [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF] outperforms its competitors in terms of image quality metrics but also has the least amount of trainable parameters. It it slightly slower than the Cascade-net [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] though which can be explained by its higher number of iterations, involving therefore more costly Fourier transform (inverse or direct) operations. These results hold true on the 2 data sets, fastMRI [START_REF] Zbontar | fastMRI: An open dataset and benchmarks for accelerated MRI[END_REF] and OASIS [START_REF] Lamontagne | OASIS-3: Longitudinal neuroimaging, clinical, and cognitive dataset for normal aging and Alzheimer's disease[END_REF]. The only exception is that KIKI-net [START_REF] Eo | KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images[END_REF] is slightly better than the U-net [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] on the OASIS data set, but still far from the best performers. We can also note that the standard deviation of the image quality metrics is way higher in the fastMRI data set than in the OASIS data set. This higher standard deviation is explained by the fact that the 2 contrasts present in the fastMRI dataset, Proton Density with and without Fat Suprression (PD/PDFS), have widely different image metrics values. The standard deviations when we compute the metrics for each contrast separately are more in-line with the OASIS ones. The range of the image quality metrics is also much higher in the OASIS results.

Quantitative results

Qualitative results

The qualitative results shown in Figures 67confirm the quantitative ones on the image quality aspect. The PD-net [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF] is much better at conserving the high-frequency parts of the original image, as can be seen when looking at the reconstruction error, which is quite flat over the whole image.

Discussion

In this work we only considered one scheme of under-sampling. However, it should be interesting to see if the performance obtained on one type of undersampling generalizes to other types of under-sampling, especially if we do a

Reference Zero-filled KIKI-net U-net Cascade-net PD-net Reconstruction results for a specific slice (15th slice of sub-OAS30367 ses-d3396 T1w.nii.gz, part of the validation set). The top row represents the reconstruction using the different methods, while the bottom row represents the absolute error when compared to the reference. re-gridding step for non-Cartesian under-sampling schemes. On that specific point, the extension of the networks towards non-Cartesian sampling schemes is not easy because the data consistency cannot be performed in the same way, and the measurement space is no longer similar to an image (except if we re-grid). In a recent work [START_REF] Schlemper | Nonuniform Variational Network: Deep Learning for Accelerated Nonuniform MR Image Reconstruction[END_REF], some of the authors of the Cascade-net [START_REF] Schlemper | A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction[END_REF] propose a way to extend their approach to the non-Cartesian case, using a re-gridding step. The PD-net [START_REF] Adler | Learned Primal-Dual Reconstruction[END_REF] also has a straightforward implementation for the non-Cartesian case even without re-gridding, in what is called the learned Primal. In this case, the network in the k-space is just computing the difference (residual) between the current k-space measurements and the initial k-space measurements. Therefore there are no parameters to learn which alleviates the problem of how to learn them.

We also only considered a single-coil acquisition setting. As parallel imaging is primarily used in CS-MRI to allow higher image quality [START_REF] Roemer | The NMR phased array[END_REF], it is important to see how these networks will behave in the multi-coil setting. The difficult part in the extension of these works to the multi-coil setting will be to understand how to best involve the sensitivity maps (or even not involve them [START_REF] Gueddari | Calibrationless oscar-based image reconstruction in compressed sensing parallel MRI[END_REF]).

Regarding the networks themselves, the results seem to suggest that for cross-domain networks, the trade-off between a high number of iterations and a richer correction in a certain domain (by having deeper networks) is in favor of having more iterations (i.e. alternating more between domains). It is however unclear how to best tackle the reconstruction in the k-space, since the convolutional networks make an shift invariance hypothesis which is not true in the Fourier space where the coefficients corresponding to the high frequencies should probably not be treated in the same way as with the low frequencies. This leaves room for improvement in the near future.

Finally, this work has not dealt with recent approaches involving adversarial training for MRI reconstruction networks [START_REF] Minh Quan | Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss[END_REF][START_REF] Dragotti | DAGAN: Deep De-Aliasing Generative Adversarial Networks for Fast Compressed Sensing MRI Reconstruction[END_REF]. It would be very interesting to see how the adversarial training could improve each of the proposed networks.

Figure 6 :

 6 Figure 6: Reconstruction results for a specific slice (16th slice of file1000196, part of the validation set). The first row represents the reconstruction using the different methods, while the second represents the absolute error when compared to the reference.

Figure 7 :

 7 Figure 7:Reconstruction results for a specific slice (15th slice of sub-OAS30367 ses-d3396 T1w.nii.gz, part of the validation set). The top row represents the reconstruction using the different methods, while the bottom row represents the absolute error when compared to the reference.

Table 1 :

 1 Quantitative results for the fastMRI dataset. PSNR and SSIM mean and standard deviations are computed over the 200 validation volumes. Runtimes are given for the reconstruction of a volume with 35 slices.

	Network	PSNR-mean (std) (dB) SSIM-mean (std) #params Runtime (s)
	Zero-filled	29.61 ( 5.28)	0.657 ( 0.23)	0	0.68
	KIKI-net	31.38 (3.02)	0.712 (0.13)	1.25M	8.22
	U-net	31.78 ( 6.53)	0.720 ( 0.25)	482k	0.61
	Cascade net	31.97 ( 6.95)	0.719 ( 0.27)	425k	3.58
	PD-net	32.15 ( 6.90)	0.729 ( 0.26)	318k	5.55

Table 2 :

 2 Quantitative results for the fastMRI dataset with the Proton density fat suppression (PDFS) contrast. PSNR and SSIM mean and standard deviations are computed over the 99 validation volumes. Runtimes are given for the reconstruction of a volume with 35 slices.

	Network	PSNR-mean (std) (dB) SSIM-mean (std) # params Runtime (s)
	Zero-filled	28.44 (2.62)	0.578 (0.095)	0	0.41
	KIKI-net	29.57 (2.64)	0.6271 (0.10)	1.25M	8.88
	Cascade-net	29.88 (2.82)	0.6251 (0.11)	425K	3.57
	U-net	29.89 (2.74)	0.6334 (0.10)	482K	1.34
	PD-net	30.06 (2.82)	0.6394 (0.10)	318K	5.38

Table 3 :

 3 Quantitative results for the fastMRI dataset with the Proton density (PD) contrast. PSNR and SSIM mean and standard deviations are computed over the 100 validation volumes. Runtimes are given for the reconstruction of a volume with 40 slices.

	Network	PSNR-mean (std) (dB) SSIM-mean (std) # params Runtime (s)
	Zero-filled	30.63 (2.1)	0.727 (0.087)	0	0.52
	KIKI-net	32.86 (2.4)	0.797 (0.082)	1.25M	11.83
	U-net	33.64 (2.6)	0.807 (0.084)	482K	1.07
	Cascade-net	33.98 (2.7)	0.811 (0.086)	425K	4.22
	PD-net	34.2 (2.7)	0.818 (0.084)	318280

Table 4 :

 4 Quantitative results for the OASIS dataset. PSNR and SSIM mean and standard deviations are computed over the 200 validation volumes. Runtimes are given for the reconstruction of a volume with 32 slices.

	Network	PSNR-mean (std) (dB) SSIM-mean (std) # params Runtime (s)
	Zero-filled	26.11 (1.45)	0.672 (0.0307)	0	0.165
	U-net	29.8 (1.39)	0.847 (0.0398)	482k	1.202
	KIKI-net	30.08 (1.43)	0.853 (0.0336)	1.25M	3.567
	Cascade-net	32.0 (1.731)	0.887 (0.0327)	425k	2.234
	PD-net	33.22 (1.912)	0.910 (0.0358)	318k	2.758

https://github.com/zaccharieramzi/fastmri-reproducible-benchmark

http://brain-development.org/ixi-dataset/

https://www.kaggle.com/c/second-annual-data-science-bowl/data

http://mridata.org/list?project=Stanford\%20Fullysampled\%203D\%20FSE\ %20Knees