Enikő Molnár

Balázs Váradi

Zoltán Garda

Richárd Botár

Ferenc K Kálmán

Éva Tóth

Imre Toth

Ernő Brücher

Gyula Tircsó
email: gyula.tircso@science.unideb.hu

Imre Tóth

Remarkable differences and similarities between the isomeric Mn(II

Keywords: Magnetic resonance Imaging (MRI), Contrast Agents (CAs), Manganese(II) complexes, Stability, Inertness, Solvent exchange, Relaxivity

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

Introduction

The linear and macrocyclic aminopolycarboxylate (APC) complexes are widely used in medical diagnosis and therapy for the complexation of metal ions applied in the field of nuclear medicine, optical and magnetic resonance imaging (MRI) [1][2][3][4][5][6][START_REF]The Chemistry of Contrast Agents in Medical Magnetic Resonance Imaging[END_REF]. These different fields have some special claims regarding the properties of complexes, but the general requirements, what should be accomplished for all the complexes, are the high stability and kinetical inertness. In other words, it means that the complexes must not dissociate in the body of patients during and after the examinations are performed. Considering the very complex nature of biological fluids which contain a great number of metal ions and complex forming ligands, this is a very strict requirement. Preventing any interaction between the APC complexes in use and the endogenous metals or ligands, the structure of APC ligands play very important role. So the ligands have to be specially designed for the robust complexation of metal ions. To design a ligand, the chemical properties of metal ions, its ionic size, electronic structure and coordination number are considered first to find out how many and which kind of donor atoms are optimal for the complexation. Some special structural elements may also be beneficial e.g. the rigidity of the ligand and/or pre-organize it for the metal ion by the use of suitable functional group(s) and/or by introducing some cyclic or macrocyclic building blocks.

In MRI several Gd(III) -APC complexes are clinically used as contrast agents (CAs) to increase the contrast between the healthy and diseased tissues or organs by differently increasing the relaxation rates of protons (mainly water protons) in the body. The complex forming ligands used in the CAs are the linear dtpa, the macrocyclic dota and their derivates [6,[START_REF]The Chemistry of Contrast Agents in Medical Magnetic Resonance Imaging[END_REF]. Recently there are 6 linear and 3 macrocyclic CAs in clinical use but in the last decade some concerns arose with the use of the linear agents. It was recognized that a new disease, called nephrogenic systemic fibrosis (NSF), was associated with the use of the Gd 3+ -based CAs. NSF was developed in patients with severe renal impairment, when the elimination of the CAs was slow and during the longer residence time of the Gd(III) complex in the body of patients a small amount of Gd(III) remained deposited, that presumably triggered the development of NSF [START_REF] Grobner | [END_REF][9][10]. More recently the presence of Gd has been detected in different organs, also in the brain of patients with normal renal function after a few repeated MR scans using CAs [11][12][13][14]. The presence of traces of non-chelated Gd(III) in the body of patients caused anxiety because the strong toxicity of Gd(III) ion. These problems with the use of the linear Gd(III)-based CAs turned the attention of researchers to the complexes of Mn(II) as potential contrast agents [15][16]. The relaxation effects of Mn(II) ion are known to be similar to that of Gd(III), but on the other hand, Mn(II) is an essential metal ion and some amount of the intravenously administered Mn(II) is eliminated relatively rapidly through the hepatobiliary system. Since the coordination number of Gd(III) is 8-9, while that of Mn(II) is 6-7, the ligands used in the Gd(III)-based CAs can not directly be used for the complexation of Mn(II). Some examples are known, when the Mn(II) complexes with ligands performing well with Gd(III) are stable and inert, but the absence of inner sphere water molecule(s) results in negligible enhance in the contrast) [17]. Another critical point is the lability/inertness of the complexes (the terms lability/inertness in this manuscript describe dissociation kinetics parameters of the complexes rather than the solvent exchange kinetics).

The majority of the open-chain ligands form very labile complexes with the Mn(II). The most socking example is that of [Mn(dtpa)] 3-which was found to dissociate within the dead-time of the stopped-flow instrument (ca. 8 msec) in the presence Cu(II) ion as a ligand scavenger [18]. In order to find suitable ligands for Mn(II), a number of new linear and macrocyclic ligands were synthesized during the last years and the properties of their complexes were investigated extensively [15][16][17][18][19][20][21][22][23][24]. Amongst the hexadentate linear ligands edta forms relatively stable [Mn(edta)] 2-complex in which a water molecule occupies the seventh coordination site, but the kinetic inertness of the complex is also low, because of the flexible structure of the ligand [18]. The edta analogue trans-1,2-cdta (abbreviated as t-cdta through the text, Figure 1) ligand is more rigid and it forms more stable and more inert complex, [Mn(tcdta)] 2-, than [Mn(edta)] 2-, therefore it seems to be suitable as a potential CA and recently was evaluated even in preclinical studies [25][START_REF] Ussov | Tomsk/RU In ECR[END_REF]. The behaviour of the cdta derivative ligands are likely similar to the parent ligand, so their Mn(II) complexes are promising for the preparation of Mn(II)-based MRI CAs [18,22].

While the complexation properties of the t-cdta ligand have been studied extensively, there is very few knowledge about the complexes of the other stereoisomer, the cis-1,2-cdta ligand (abbreviated as c-cdta in the text). The donor atoms of the t-cdta form a semi-rigid preorganized coordination cage, where metal ions of different size may enter and generally form complexes of high stability [START_REF]Critical Stability Constants[END_REF]. The maximum distance between the two N-donor atoms in t-cdta is 4 Å, while that in the c-cdta is 3.1 Å only [START_REF] Charlier | [END_REF], so the coordination cage formed by the donor atoms of cis-isomer is relatively small, particularly for larger metal ions, that predicts low stability complexes with c-cdta. Since the size of the Mn(II) ion is not too large and there are only very few data in the literature regarding the complexation of c-cdta, we decided to study its complexes with Mn(II) and with some other endogenous metal ions.

N N N N C C O O 2 2 H H C C O O 2 2 H H H H O O 2 2 C C H H O O 2 2 C C N N N N C C O O 2 2 H H C C O O 2 2 H H H H O O 2 2 C C H H O O 2 2 C C e e d d t t a a c c - -c c d d t t a a N N N N C C O O 2 2 H H C C O O 2 2 H H H H O O 2 2 C C H H O O 2 2 C C t t - -c c d d t t a a N N N N O O N N N N N N H H O O C C O O 2 2 H H C C O O 2 2 H H C C O O 2 2 H H C C O O 2 2 H H 4 4 - -h h e e t t - -t t - -c c d d t t a a
Figure 1. Formulae of the edta, c-cdta, t-cdta, 4-het-t-cdta ligands.

Results and Discussion

The synthesis of c-cdta ligand (Scheme 1) was accomplished in a two-step reaction sequence starting from cis-cyclohexane-

Equilibrium studies

In order to characterize the metal complexes as potential MRI contrast enhancement agents, we have to know their equilibrium and kinetic behaviour in biofluids, the proton relaxation effects, together with another parameters which determine the proton relaxation rates in the presence of these complexes. For the assessment of the equilibrium behaviour we have determined the stability constant of the Mn(II) complex and those of the Mg(II), Ca(II), Zn(II) and Cu(II) complexes that may compete with the Mn(II) ion for the c-cdta ligand in biofluids. To obtain information about the kinetic inertness of [Mn(c-cdta)] 2-complex, the rates of the exchange reactions occurring between the complex and Cu(II) ion have been investigated, vide infra. These properties are already known for the [Mn(t-cdta)] 2-complex, so the comparison of the behaviour of the two complexes is possible.

The protonation constant of the c-cdta ligand have been determined by pHpotentiometric titration. The protonation constants (Ki H ) defined by Equation ( 1), are presented in Table 1.

] H ][ L H [ ] L H [ 1 i i H i + - = K
] L ][ M [ ] ML [ ML = K (eqn. 2)
In Equation (2) [M], [ML] and [L] are the equilibrium concentrations of the M(II) metal ion, ML complex and L 4-ligand, respectively (the charges of ions will be used only when it is necessary).

The metal ions and protons compete for the ligand, so at lower pH values protonated complex can be formed. The protonation constant of complexes is defined as follows:

] H ][ L MH [ ] L MH [ 1 i i L MH i + - = K (eqn. 3)
where i = 1 or 2, [M(HiL)] and [M(Hi-1L)] are the concentration of the protonated complexes.

Table 1. Protonation constants of the c-cdta, t-cdta and and edta ligands and stability constants of their Mg(II), Ca(II), Zn(II), Cu(II) and Mn(II) complexes (log K values, 25°C, 0.15 NaCl).

[a] this work; [b] Ref. [START_REF]Critical Stability Constants[END_REF]; [c] 0.15 M NaCl, 25°C from the diploma work of Veronika Józsa, 2015, University of Debrecen, Debrecen, Hungary; [d] Ref. [18]; [e] calculated by using cMn=cLig=0.01 mM at pH=7.4 according Tóth et al. [29]; [d] determined by simultaneous fitting of the UV-vis and pH-potentiometic data.

The protonation constants of c-cdta determined in this work at 25°C in 0.15 M NaCl solution (Table 1) agree relatively well with those reported in the literature [START_REF]Critical Stability Constants[END_REF] (20°C and 0.1 M KCl) indicating that the interaction of c-cdta with the Na(I) and K(I) ions is negligible. The t-cdta ligand behaves differently, since its logK1 H value in 0.15 M KNO3 is much higher (11.70) than in 0.15 M NaCl (9.54), because the stability constant of the [Na(t-cdta)] 2- complex is relatively high (log KNaL = 4.4) [START_REF]Critical Stability Constants[END_REF], while the interaction of the ligand with K(I)

ion is very week. The different behaviour of the two ligands indicates that the coordination c-cdta [a] c-cdta [b] t-cdta [c] t-cdta [b] edta [c] H 2.8 2.95 [d] log KMnH2L --1.89 [d] -- [d] pMn [e] cage formed by the two iminodiacetate (ida) groups of the c-cdta is too small to accommodate the larger M(I) metal ions, while in the case of t-cdta the coordination cage is larger, where even a Na(I) ion can enter.

The stability constant of the Mg(II)-and [Ca(c-cdta)] 2-complexes obtained in this work are comparable with those, published in the literature (Table 1). For the c-cdta complexes of Mn(II), Zn(II) and Cu(II) there are not any stability constant data in literature. The stability constants of complexes formed with the c-cdta, t-cdta and edta ligands in 0.15 M NaCl (Table 1) do not differ considerable, because the formation of [Na(t-cdta)] 2-complex. Considering the formation of Na(I) complexes, it is more correct to compare the log KML values of the ccdta with those log KML values which were obtained for the t-cdta complexes at 0.1 M KCl ionic strength. This comparison shows that the stability constants of the [M(c-cdta)] 2- complexes are lower by 2-3 log units than those of the [M(t-cdta)] 2-complexes (Table 1).

These differences in the stability constants are related to the different size of the coordination cages formed by the donor atoms of the two ligands. 1 H-NMR studies indicated that the two ida groups in t-cdta occupy equatorial position [30] and this is probable for the case of the ccdta, too. The distance between the two N atoms is significantly larger and so the coordination cavity is also larger for the t-cdta, in which the accommodation of metal ions of different size is more probable. The narrower size of the coordination cage in the c-cdta results in the stretching of the ligand structure by the entrance of a metal ion that leads to the lower stability constant.

The paramagnetic metal ions used as a component of the MRI CAs result in high proton relaxation rates. Since the free metal ions are toxic, their complexes are used as CAs. The longitudinal relaxation rate of water protons (1/T1) in the presence of Mn 2+ ions is significantly higher than in the presence of [Mn(c-cdta)] 2-or [Mn(t-cdta)] 2-. (T1 is the longitudinal relaxation time.) The relaxation effect of paramagnetic ions and complexes are expressed by their relaxivity values. The relaxivity (r1, mM -1 s -1 ) is the increase in the water proton relaxation rate resulted in by 1.0 mM increase in the concentration of the paramagnetic species. The relaxivity of the Mn(II).aq ion at 25°C and 20 MHz is 7.94 mM -1 s -1 , while that of the [Mn(c-cdta)] 2-complex is 3.85 mM -1 s -1 . This latter r1 value is similar to that of the [Mn(edta)] 2-(3.23 mM -1 s -1 ) and [Mn(t-cdta)] 2-(3.62 mM -1 s -1 ) complexes, in which one water molecule is coordinated in the inner sphere of the Mn(II).

The formation of [Mn(c-cdta)] 2-complexes could be followed by measuring the water proton relaxation rates in equilibrium solutions of 1.0 mM Mn(II) and 1.0 mM c-cdta as a function of pH. At pH<2 only Mn(II).aq is present, when the 1/T1 values are high (Fig. 2).

With the increase of pH, MnHL and MnL complexes are formed, when the water molecules are displaced from the inner sphere of Mn 2+ ion and the proton relaxation rates decrease. After the formation of [Mn(L)] complex is complete (at about pH>5) the 1/T1 values are constant (this constant 1/T1 value is the relaxivity of the MnL complex). In Figure 2 As it was underlined several times in the literature reliable stability data for Cu(II) -APC complexes can be obtained either by competition method with the use of auxiliary ligand (the equilibrium of the Cu(II) -auxiliary ligand system must be well known), or it can be accessed by using multiple methods simultaneously (e.g. UV-vis spectrophotometry applicable also in very acidic conditions and pH-potentiometry) [31]. The absorption spectra of the [Cu(c-cdta)] 2-complex as a function of H + ion concentration is shown on Figure 3. The spectrophotometric data were fitted simultaneously with the pH-potentimetric titration data and the stability constants determined are shown in Table 1. In line with the equilibrium data obtained for other metal ions studied, the stability constant of the [Cu(c-cdta)] 2-complex is smaller than the value of the corresponding [Cu(t-cdta)] 2-complex determined under identical conditions (the difference is ca. 0.7 log units).

Dissociation kinetics of [Mn(c-cdta)] 2-complex

The fate of the MRI CAs in biological fluids is strongly influenced by their kinetic inertness, that is an important property in determining the extent of dissociation of the complex while in the body. For the Gd(III)-based CAs it was generally assumed that the deposition of Gd in the body was the result of the dissociation of Gd(III) complexes, that occurred through transmetallation reactions with the endogenous metal ions like Zn(II), Cu(II)

and Ca(II) [33,34]. To obtain information about the kinetic properties of CAs, the rates of the metal-exchange reactions are generally studied between the Gd(III) or Mn(II) complex and an exchanging metal ion, that is very often Cu(II), because in this case the reaction can easily be followed by spectrophotometry. The kinetic behaviour of [Mn(c-cdta)] 2-was studied similarly by following the exchange reactions depicted in Equation ( 4):

[Mn(c-cdta)] 2-+ Cu(II) [Cu(c-cdta)] 2-+ Mn(II) (eqn. 4)
Since the stability constant of [Cu(c-cdta)] 2-is much higher than that of [Mn(c-cdta)] 2- (Table 1), reaction (4) takes place to completeness in the presence of Cu(II) excess (10-40 fold of Cu(II) excess was used). Under such conditions the reactions can be treated as pseudofirst-order ones kinetically. The rate of the exchange reaction ( 4) is directly proportional to the concentration of the complex and can be expressed as follows:

t obs t )] L ( Mn [ dt )] L ( Mn [ d k = - (eqn. 5)
In Equation ( 5) kobs is a pseudo-first-order rate constant, [Mn(L)]t is the total concentration of complexes containing the [Mn(c-cdta)] 2-species. The rates of the exchange reactions ( 4 where presumably one acetate group of [Mn(c-cdta)] 2-is coordinated to the Cu(II) ion and so the probability of protonation decreases (in fact [Mn(c-cdta)Cu] is a "dead-end" complex.)

Since the reaction rate does not increase with the increase of the Cu(II) concentration, the exchange reactions do not take place with the direct attack of the Cu(II) ion. On the basis of the experimental data we came to the conclusion, that the exchange reactions between the [Mn(c-cdta)] 2-and Cu(II) ion occur through dissociation mechanism. After the rate determining dissociation of the complex, the released c-cdta ligand rapidly react with the Cu(II) ions by the formation of [Cu(c-cdta)] 2-. The dissociation of the [Mn(c-cdta)] 2-complex can take place spontaneously and by the assistance of H + ions. The spontaneous dissociation is presumably very slow, while the proton assisted dissociation occurs relatively rapidly at pH values around 4-5. The protonation of [Mn(c-cdta)] 2-presumable occurs at a carboxylate group (log KMHL = 2.85), but for the dissociation of the complex the proton must be transferred to a N atom, when the protonated iminodiacetate (ida) group is not coordinated anymore and this intermediate may dissociate (or the complex may be re-formed by the dissociation of the proton). Considering the possible reaction pathways, the rate of dissociation of the complex can be expressed by Equation ( 6)

[Mn(HL)] [Mn(L)] t d [Mn(L)] MnHL 0 t k k + + = - (eqn. 6)
where k0 and kMnHL are the rate constants, characterizing the dissociation of the MnL and MnHL complexes, respectively.

Comparing the total concentration of the complex [Mn(c-cdta)] 2-([Mn(L)]t = [Mn(L)] + [Mn(HL)] + [Mn(L)Cu]), Equation ( 5) and ( 6), for the kobs value Equation ( 7) is obtained:

] [Cu ] [H 1 ] [H 2 MnLCu MnHL MnHL MnHL 0 obs + + + + + ⋅ ⋅ + = K K K k k k (eqn. 7)
The term KMnHL [H + ] in the denominator of Equation ( 7) is very small in the pH range 4.03 -5.03, so it can be neglected. For the calculation of the unknown parameters Equation [START_REF]The Chemistry of Contrast Agents in Medical Magnetic Resonance Imaging[END_REF] has been used as follows:

] [Cu 1 ] [H 2 MnLCu 1 0 obs + + + ⋅ + = K k k k (eqn. 8)
where k1 = kMnHL KMnHL.

By fitting the kobs values presented in Figure 4 to Equation (8) the k0, k1 and KMnLCu values have been calculated and presented in Table 2. The k0 value is very low, so it can be neglected.

Table 2. Rate and equilibrium constants characterizing the dissociation rates of the [Mn(ccdta)] 2-, [Mn(t-cdta)] 2-and [Mn(edta)] 2-complexes.

c-cdta

t-cdta [a,b] edta [a,b] k1 (M -1 s -1 ) (1.02±0.09)×10 5 4.0×10 2 5.2×10 4

k2 (M -2 s -1 ) - - 2.3×10 8 k3 (M -1 s -1 ) - - 45±8 log KMnLCu 79±23 79 - log KH 2.85 [c] 2.90 [c] 2.65 [c]
t1/2 [d] (h) 0.47 12 0.076

[a] Ref [18]; [b] for [Mn(t-cdta)] 2-k1 = 3.2×10 2 M -1 s -1 and t1/2 = 15 h while for [Mn(edta)] 2-k3 = 3.0×10 -1 M -1 s -1 , k4 = ~4.8×10 1 M -2 s -1 and log KMnHL =3.10 were found in Ref. [36]; In Table 2, the k1 rate constants and KMnLCu stability constants obtained for the [Mn(ccdta)] 2-complex are compared with those of the [Mn(t-cdta)] 2-. The presented data indicate that the k1 dissociation rate constant of [Mn(c-cdta)] 2-is approximately 250 times higher than that of [Mn(t-cdta)] 2-. This difference is shown in the values of the half-life of dissociation, that is much longer for complex [Mn(t-cdta)] 2-. The results of these studies indicate that the kinetic inertness of the Mn(II) complex formed with the c-cdta isomer is significantly lower than that of the t-cdta isomer.

O and NMRD measurements

The nuclear magnetic relaxation dispersion (NMRD) profile of the [Mn(c-cdta)] 2-was measured at three different temperatures (298, 310 and 323 K) in the frequency range 0.01-80

MHz to gain information on the relaxation properties of the complex (Figure 5). The NMRD profiles show a single dispersion at 1-10 MHz which is typical of low molecular weight Mn 2+ complexes. Their temperature dependence indicates that the relaxivity of the complex is limited by fast rotation. Several physico-chemical parameters affect the relaxivity of a paramagnetic metal complex, such as the water exchange rate, the electron spin relaxation parameters and the rotational correlation time. The temperature dependence of the transverse and longitudinal 17 O relaxation rates can provide information on the water exchange rate (kex) and the rotational correlation time (τR), respectively, while the chemical shift is related to the number of the water molecules directly coordinated to the paramagnetic metal center (q).

Variable temperature transverse and longitudinal relaxation rates and chemical shifts were measured for an aqueous solution of [Mn(c-cdta)] 2-and for a diamagnetic reference at 9.4 T.

The T1 values showed negligible difference between the Mn(II) complex and the reference, thus were not included in the calculations. The reduced 17 O transverse relaxation rates, 1/T2r and chemical shifts, ∆ωr, have been fitted according to the Swift-Connick equations, by assuming a simple exponential behaviour for the electron spin relaxation. The water exchange rate, kex 298 , its activation enthalpy, ∆H ≠ , and 1/T1e 298 were calculated (the activation energy of electron spin relaxation was fixed to 1 kJ/mol). The hydration number was fixed to q=1 based on analogy to [Mn(t-cdta)] 2-and on the relaxivity value measured at 20 MHz (r1=3.85 mM -1 s - 1 , r1,[Mn(t-cdta)]2-=3.62 [18]). To characterize the rotational dynamics of the complex, the 1 H NMRD data were analyzed by the Solomon-Bloembergen-Morgan (SMB) [37] and Freed models [38] (inner-(IS) and outer-sphere (OS) relaxation mechanisms) and the Swift-Connick [39,40] equations related to the solvent exchange (see Supporting Information). The parameters for water exchange were fixed to those determined from the 17 O NMR study. The distances between the metal ion and the inner and outer sphere water protons were also fixed (rMnH = 2.83 Å and aMnH = 3.6 Å), as well as the diffusion coefficient and its activation energy (DMnH = 26 × 10 -10 m 2 s -1 and EMnH = 18 kJ/mol -1 ) and the activation energy of the modulation of the zero-field-splitting (Ev = 1.0 kJ/mol). The values of the electron relaxation parameters τv 298 and ∆ 2 were found to be 7.6±0.9 ps and (4.1±0.8)×10 20 s -2 , respectively. The best-fit parameters obtained are given in Table 3 and compared with those of [Mn(t-cdta)] 2-and [Mn(4-het-cdta)] 2-complexes [41,42]. The experimental data and the fitted curves are shown in Figures 5 and6. Table 3. Parameters obtained from the analysis of 17 O NMR and 1 H NMRD data for the [Mn(c-cdta)(H2O)] 2-complex.

Parameter [Mn(c-cdta)] 2- [Mn(t-cdta)] 2-[c] [Mn(4-het-cdta)] 2-[d]
kex 298 (x10 7 s -1 ) [a] 22.5 ± 0.5 14.0 17.6

∆H ‡ (kJ mol -1 ) [a] 42.0 ± 0.8 42.5 36.2

∆S ‡ (J K -1 mol -1 ) [a] +56 ± 4 --AO/ (10 6 rad s -1 ) [a] 42.7 ± 0.9 4.2 40

1/T1e 298 (x10 7 / s -1 ) [a] 27.0 ± 0.5 0.65 7

ErH (kJ mol -1 ) [b] 21.8 ± 2.0 -26.2

τrH 298 (ps) [b] 52 ± 5 -104.9

[a] from 17 O NMR; [b] from NMRD; [c] Ref. [41]; [d] Ref. [42].

As Figure 5. shows, the reduced transverse 17 O relaxation rates undergo a changeover at ~290 K between the fast and intermediate water exchange regime. Importantly, the water exchange rate has a significant contribution in the overall correlation time (1/τc = kex + 1/T1e; kex varying from 10 to 80% in the temperature range 273-338 K) which makes possible to accurately determine kex. The kex 298 value characterizing the water exchange of [Mn(c-cdta)] 2- is ca. 60% higher than that of [Mn(t-cdta)] 2-, which can be likely related to the structural differences between the two complexes. The smaller coordination cage of the cis derivative induces more steric crowding around the metal ion. The importance of steric crowding to accelerate water exchange has been previously demonstrated in the case of a dissociatively activated mechanism [43]. The high positive activation entropy points to a dissociative mechanism, thus the [Mn(c-cdta)] 2-complex can be expected to have a faster water exchange, as it is indeed observed.

On the other hand, kex 298 is the half of the value determined for [Mn(edta)] 2-(47.1×10 7 s -1 ) [44]. This is related to the more rigid structure of the cdta 4-ligands which detains the rearrangement of the coordination environment during the water exchange process. The water exchange is also several times slower on [Mn(c-cdta)] 2-than on the 1,4-do2a or do1a complexes, having a very different, macrocyclic structure [45]. The rotational correlation time (τrH 298 ) obtained from the analysis of the relaxivity data is in the usual order of magnitude of typical low molecular weight complexes.

Summary

Our efforts to find suitable ligand for Mn(II) complexation started from equilibrium, kinetic and relaxometric characterization of Mn(II) complexes formed with commercially available open-chain ligands a few years ago [18]. The studies performed revealed that the rigid t-cdta forms relatively inert complex with Mn(II) ion (the half-life of the dissociation is expected to be 12 hours near to physiological conditions). However, to best of our knowledge Mn(II) complexes of other rigid edta derivative ligands such as cis-1,2-cyclohexanediamine or o-phenylenediamine tetraacetates (e.g. c-cdta or phdta) were not studied in detail in this respect. The distance between the two nitrogen atoms of the ida groups in these ligands are expected to differ substantially form that in t-cdta (where it is accepted to be 4 Å) and so the coordination properties of the ligands are likely to be affected by this structural diversity.

The thermodynamic, kinetic (solvent and dissociation kinetic data) and relaxation parameters expressed in terms of water exchanges rates and relaxivity values (at 25 °C) are collected in the following table. The data collected for these hexadentate chelators indeed indicate that the Mn(II) complexes of these ligands differ notably. The most pronounced difference is seen for the dissociation kinetic data as the Mn(II) complex formed by the cisderivative ligand shows properties similar to those seen for other labile complexes of linear edta-type ligands [18] whereas the t-cdta forms relatively inert Mn(II) complex. Moreover, the stability (especially the conditional stability) behaves alike by making the trans-1,2cyclohexanediamine unit to be an excellent "building block" when tailoring inert Mn(II) complexes for safe MRI applications. r1p 298 [e] 3.85 3.62

[a] Ref. [18]; [b] pMn values were calculated at pH=7.4 by using 0.01 mM Mn(II) and ligand concentration as suggested by É. Tóth and co-workers [29]; [c] the half-lives (h) of dissociation were extrapolated to pH=7.4 by using 0.01 mM Cu(II) ion concentration; [d] form Ref. [41]; [e] at 25 o C and 20 MHz.

Experimental section

Synthesis

All commercially purchased reagents (Sigma-Aldrich) and solvents (Scharlab) were used as received and without further purification. NMR spectra were acquired on Bruker Avance 360 and 400 MHz spectrometers and processed/analyzed using MestreNova 8.1 The combined organic fraction was dried over anhydrous Na2SO4 and evaporated under reduced pressure. The purification of the title product by flash chromatograpy on silica gel by using petrolether and ethyl acetate (increasing the ethyl acetate content of the eluent gradually form 10:1 to 5:1 by volume) yielded the pure substance (0.82 g which corresponds to 20 % yield). Rf in 5:1 petrolether : ethyl acetate is 0.38.

Equilibrium studies

(Na + +H + )Cl -). The samples, whose acidity was varied in the concentration range of 12.1 -462 mM, were equilibrated for 1 day before recording the absorption spectra at 25 o C in Peltier thermostated semi-micro 1 cm Hellma® cells using a Varian CARY 1E, UV-Vis spectrophotometer. The molar absorptivity of the [Cu(c-cdta)] 2-, complex was determined at 21 wavelengths (600 -800 nm range) by recording the spectra of 1.72×10 -3 , 2.55×10 -3 and 3.48×10 -3 M complex solutions, while for the Cu(II) ion, previously published molar absorptivity values (determined under identical conditions) [48] were used for the data fitting.

The molar absorption coefficients of monoprotonated [CuH(c-cdta)] 2-were calculated during the simultaneous data refinement (UV-vis and pH-potentiometric titration curves). The protonation (ligand and complexes) and stability constants (complexes) were calculated from the titration data with the PSEQUAD program [START_REF] Zékány | Computational Methods for the Determination of Formation Constants[END_REF].

Kinetic studies

The dissociation rates of the [Mn(c-cdta)] where At, A0 and Ae are the absorbance at time t, at the start and at the equilibrium of the reaction, respectively. The calculations were performed with the computer program Micromath Scientist, version 2.0 (Salt Lake City, UT, USA) by using a standard least-squares procedure [50].

H-, and 17 O-NMR relaxometry

Longitudinal (1/T1) and transverse (1/T2) relaxation rates and chemical shifts of an aqueous solution of the Mn(II) complex (pH=7.4, 10.3 mM) and of a diamagnetic reference (HClO4 acidified water, pH = 3.3) were measured in the temperature range 273 -338 K using a Bruker Avance 400 (9.4 T, 54.2 MHz) spectrometer. The temperature was determined according to previous calibration by means of ethylene glycol and methanol as standards [START_REF] Raiford | [END_REF].

1/T1 and 1/T2 values were determined by the inversion -recovery and the Carr -Purcell -Meiboom -Gill spinecho technique, respectively [52]. The technique of the 17 O NMR measurements has been described previously [53]. To avoid susceptibility corrections of the chemical shifts, a glass sphere fitted into an 10 mm NMR tube was used to contain the samples. To improve sensitivity, 17 O enriched water (10% H2 17 The least-squares fit of the 17 O NMR and of the NMRD data was performed using Visualiseur/Optimiseur [54,55] running on a MATLAB 8.3.0 (R2014a) platform.

Relaxivity determination

The longitudinal water proton relaxation rate (r1 = 1/T1 -1/Tw) was measured at 20

MHz with a Bruker Minispec MQ-20 relaxometer (Bruker Biospin, Germany). Samples were thermostated by using a circulating water bath at 25.0 ± 0. chelator have been investigated by pH-potentiometry, UV-vis spectrophotometry, 1 H and 17 O NMR relaxometry. pMn as well as the inertness of the Mn(II) complex formed with c-cdta ligand were found to differ substantially form those found for the trans-derivative while the water exchange rate and the relaxivity do not differ remarkably. Our results confirm, that the t-cdta platform has better potential for further ligand development owing to better dissociation kinetic parameters of its Mn(II) complex.

  the species distribution curves (calculated from the equilibrium data) are also presented (together with the 1/T1 values), calculated for the 1.0 mM Mn(II) -1.0 mM c-cdta system. The data in Figure2indicate that the decrease of the proton relaxation rates with the increase of the pH occurs parallel with the formation of the MnHL and MnL complexes. The good agreement in the trend of the data obtained by pH-potentiometry and relaxometry, indicates the reliability of the equilibrium data.

Figure 2 .

 2 Figure 2. Species distribution curves and the relaxation rate (1/T1-1/T1w) profile (red points) of 1.021 mM [Mn(c-cdta)] 2-as a function of pH (T = 25 o C, I=0.15 M NaCl, 20 MHz).

Figure 3 .

 3 Figure 3. Absorption spectra of the [Cu(c-cdta)] 2-system as a function of [H + ]. Conditions: [c-cdta 4-] = 3.528 mM, [Cu(II)] = 3.410 mM, 25 o C, 1.0 M (Na + +H + )Cl -, cH+ in the samples are as follows: 12.1; 18.8; 33.9; 49.0; 75.9; 101.0; 156.8; 205.0; 253.3; 349.8 and 462.4 mM (from top to bottom).

Figure 4 .

 4 Figure 4. The rate constants (kobs) characterizing the exchange reactions of [Mn(c-cdta)] 2-(0.4 mM) with Cu(II) measured by stopped-flow method. The Cu(II) concentration are 4.0 mM (red), 8.0 mM (green), 12.0 mM (blue) and 16.0 mM (wine-red) (25°C, 0.15 M NaCl).

  [c] determined by pH-potentiometry and fixed in the data fitting; [d] calculated at pH = 7.4 in the presence of cCu2+ = 1×10 -5 M.

1 )Figure 5 .

 15 Figure 5. Variable temperature reduced transverse 17 O relaxation rates (top) and chemical shifts (bottom) recorded for [Mn(c-cdta)] 2-(B = 9.4 T). The solid lines correspond to the fit of the data.

Figure 6 . 1 H

 61 Figure 6. 1 H NMRD profiles of [Mn(c-cdta)] 2-at 25 °C (♦), 37 °C () and 50 °C ().

(

  Mestrelab Research) or Topspin (Bruker) softwares. The ESI QTOF (Electro Spray Ionization Quadrupole Time of Flight) spectrometry was performed at the Department of Applied Chemistry of the University of Debrecen.a). Synthesis of cis-1,2-CDTA-tetraethyl ester:Ethyl-bromoacetate (6.14 g, 36.8 mmol, 4.2 equiv.) was dissolved in dry acetonitrile and added dropwise to the mixture of 1.00 g (8.76 mmol) cis-1,2-diaminocyclohexane, 5.66 g N,N-diisopropylethylamine (43.8 mmol, 5.0 equiv.) and 1.31 g (8.76 mmol, 1.0 equiv.) of NaI prepared in dry acetonitrile (30 ml). The reaction mixture was then heated to reflux in N2 atmosphere and kept at this temperature for 7 hours. The precipitate formed in the course of reaction was filtered off from the hot solution and the filtrate was evaporated under reduced pressure. The resulting oily residue was dissolved in 40 mL of bi-distilled water and extracted with portions of chloroform (3×40 mL).

1 H

 1 -NMR: (360.13 MHz, CDCl3): δ = 1.23 (12H, t, J = 7.11 Hz, -O-CH2-CH3); 1,29 (2H, m, -CH2-); 1.43 (2H, m, -CH2-); 1.62 (2H, m, -CH2-); 1.79 (2H, m, -CH2-); 3.16 (2H, d, J = 6.88 Hz, -CH-); 3.70 and 3.79 (8H, AB doublet, 2 JAB = 17.90 Hz, -N-CH2-COO-); 4.11 (8H, q, J = 7.11 Hz, -O-CH2-CH3). 13 C-NMR: (90.55 MHz, CDCl3): δ = 14.34 (4C, s, -O-CH2-CH3); 23.44 (2C, s, -CH2-CH2-); 26.88 (2C, s, -CH2-); 53.08 (4C, s, -N-CH2-COO-); 60.27 (4C, s, -O-CH2-CH3); 60.55 (2C, s, -CH-); 172.58 (-N-CH2-COO-). MS (ESI-Q-TOF): m/z [M+] calculated for C22H38N2ONa: 481.252; found: 481.251 b). Synthesis of cis-1,2-CDTA: 0.40 g NaOH (10 mmol, 6 equiv.) was dissolved in 1,0 mL of bi-distilled water and was added portionwise to the solution of 0.75 g (1.64 mmol) cis-1,2-CDTA-tetraethyl ester dissolved in 15 mL of ethanol. The reaction mixture was heated to 78•C and refluxed at this temperature for 24 h. The white precipitate formed during the reaction (Na4(c-cdta)) in the reaction mixture was filtered off and washed twice with cold ethanol (2×10 mL), diethyl ether (2×10 mL) and dried to constant weight. The solid obtained was dissolved in minimal amount of water required to dissolve it (3.0 mL) and pH of the sample was set to pH=2.02 by addition of concentrated HCl. The white precipitate formed as a result of acid addition was filtered off and washed with small portions of acidified distilled water (pH=2.50), cold ethanol (2×5 mL), diethyl ether (2×5 mL) and dried until the weight of the product become constant. Yield: 0.22 g (39 %). 1 H-NMR: (360.13 MHz, D2O): δ = 1,51 (2H, m, -CH2-); 1.74 (2H, m, -CH2-); 1.83 (2H, m, -CH2-); 2.00 (2H, m, -CH2-); 3.51 (2H, m, -CH-); 3.83 and 4.01 (8H, AB doublet, 2 JAB = 17.39 Hz, -N-CH2-COO-). 13 C-NMR: (90.55 MHz, D2O): δ = 22.11 (2C, s, -CH2-CH2-); 23.19 (2C, s, -CH2-); 54.76 (4C, s, -N-CH2-COO-); 61.26 (2C, s, -CH-); 172.81 (-N-CH2-COO-). MS (ESI-Q-TOF): m/z [M+] calculated for C14H21N2O8NaNa: 391.109; found: 391.109

  2-chelate were studied at 25 o C (Peltier thermostated) and 0.15 M NaCl ionic strength by stopped-flow method monitoring the formation of the Cu(II) complex at 300 nm using an Applied Photophysics DX-17MV instrument. All dissociation reactions were performed under pseudo-first order conditions where the exchanging metal ion (Cu(II)) was in 10-40-fold excess relative to the complex (ccomplex = 2.0×10 -4 M, pH range 4.25-5.10). The kinetic studies were carried out in a noncoordinating buffer to maintain the pH in the samples constant (0.05 M N,-methylpiperazine (nmp) with log K2 H = 4.34 under the conditions applied). The pseudo-first-order rate constants (kobs) were calculated by fitting the absorbance-time data series to equation 9:

  O, CortecNet) was added to the solutions to reach around 1% enrichment. Proton NMRD profiles of the [Mn(c-cdta)] 2-complex (1.00 mM, pH = 7.4) were recorded in aqueous solution on a Stelar SMARTracer Fast Field Cycling relaxometer (0.01 -10 MHz) and a Bruker WP80 NMR electromagnet adapted to variable field measurements (20 -80 MHz) and controlled by a SMARTracer PC-NMR console. The temperature was monitored by a VTC91 temperature control unit and maintained by a gas flow. The temperature was determined by previous calibration with a Pt resistance temperature probe.

2 -

 2 2 o C. The longitudinal relaxation times (T1) were measured by using the inversion recovery method (180 o -τ -90 o ) by averaging 5 -6 data points for each concentration point obtained by using 14 different τ values (τ values ranging between 0 to at least 6 times the expected T1). cdta trans-1,2-Diaminocyclohexane-N,N,N′,N′-tetraacetic acid cis-1,2-cdta cis-1,2-Diaminocyclohexane-N,N,N′,N′-tetraacetic acid DIPEA N,N-diisopropylethylamine ida Iminodiacetate NMRD Nuclear magnetic relaxation dispersion 4-het-cdta 4-((1-(2-Hydroxyethyl)-1H[1,2,3]triazol-4-yl)methoxy)methyl-trans-1,2diaminocyclohexane-N,N,N',N') complexes formed with c-cdta (cis-1,2-diaminocyclohexane-N,N,N′,N′-tetraacetic acid).

Table 4 .

 4 Comparison of the most important physico-chemical data of the Mn(II) complexes formed with t-cdta and c-cdta ligands.

	Parameter	[Mn(c-cdta)] 2-	[Mn(t-cdta)] 2-[a]
	log K[Mn(L)]	14.19	14.32
	pMn [b]	7.82	8.68
	t1/2 (h) [c]	0.47	12
	kex 298 (x10 7 s -1 )	22.5	14.0 [d]

Acknowledgements

This research was funded by the Hungarian Scientific Research Fund (OTKA K-120224 project), Le Studium, Loire Valley Institute for Advanced Studies (Gy.T. and F.K.K.) and the János Bolyai Research Scholarship (Gy.T. and F.K.K.) of the Hungarian Academy of Sciences. E. T. acknowledges support of the Ligue contre le Cancer (France). The research was also supported by the EU and co-financed by the European Regional Development Fund under the project GINOP-2.3.2-15-2016-00008.

The chemicals (MCl2 salts) used in the studies were of the highest analytical grade.

The concentration of the stock solutions was determined by complexometric titration using a standardized Na2H2edta solution and appropriate indicators (Patton & Reeder (CaCl2), Eriochrome Black T (MgCl2 and MnCl2), xylenol orange (ZnCl2), murexid (CuCl2)) [START_REF] Schwarzenbach | Complexometric titrations[END_REF].

The pH-potentiometric titrations were carried out with a Methrohm 888 Titrando titration workstation, using a Metrohm 6.0233.100 combined electrode. The titrated solutions (6.00 mL) were thermostated at 25 °C. The samples were stirred and kept under inert gas The stability constant and solution speciation of the [Mn(c-cdta)] 2-complex was confirmed using 1 H relaxometry in order to support the stability constant value obtained by pH-potentiometry (for details about T1 measurements, see the "Relaxivity determination" section below). 3.00 mL of a 1.02 mM [Mn(c-cdta)] 2-sample were titrated with solid NaOH or gasous HCl (in order to avoid sample dilution) to adjust the pH in the range 1.97 -7.12, followed by recording and averaging of 5 -6 T1 values for each pH data point.

Owing to the high conditional stability of [Cu(c-cdta)] 2-, the formation of the complex was complete (nearly 100%) even at pH = 1.54 (starting point of the pH-potentiometric titration). For this reason, 12 out-of-cell (batch) samples containing a slight excess of ligand and the Cu(II) ion were prepared (cL = 3.528 mM, cCu2+ = 3.410 mM, 25 o C, 0.5 M

Abbreviations

MRI

Magnetic resonance imaging APC aminopolycarboxylate

CA Contrast Agent

Appendix A. Supplementary data

Supplementary data associated with this article ( 1 H-, 13 C-NMR and MS data of the pure compounds, and the equations related to the solvent exchange kinetics data refinement) can be found, in the online version, at http://dx.doi.org/10.1016/xxxxxxxxxxxxxxxxx .

Highlights

-Stability constants, water exchange rates and the relaxivities of the isomeric [Mn(cdta)] 2- complexes are similar -The conditional stability and the inertness of the [Mn(t-cdta)] 2-is higher than that of [Mn(ccdta)] 2- -The trans-1,2-cyclohexanediamine unit is an excellent "building block" when designing Mn(II) complexes as safe MRI agents


C Feutry

P Piantanida

P Duhamel

LEARNING SEMI-SUPERVISED ANONYMIZED REPRESENTATIONS BY MUTUAL INFORMATION

This paper addresses the problem of removing from a set of data (here images) a given private information, while still allowing other utilities on the processed data. This is obtained by training concurrently a GAN-like discriminator and an autoencoder. The optimization of the resulting structure involves a novel surrogate of the misclassification probability of the information to remove. Several examples are given, demonstrating that a good level of privacy can be obtained on images at the cost of the introduction of very small artifacts.

INTRODUCTION

The problem we address, i.e. removing private information from a set of images, while still allowing any other search in the set is a problem that is supervised with respect to the private label, but unsupervised with respect to the original image. Hence the name of semi-supervised anonymization.

Related work. Some previous works addressed similar problems, very often in a more constrain setup. Reference [START_REF] Chen | Vgan-based image representation learning for privacypreserving facial expression recognition[END_REF] proposes a tool that allows to perform some emotion preservation while transferring identities using a variational generative adversarial network. This work is fully supervised contrary to our work, that only use private labels during training. Several other works, such as [START_REF] Edwards | Censoring representations with an adversary[END_REF] or [START_REF] Raval | Protecting visual secrets using adversarial nets[END_REF] address the problem of hiding tags (text label, QR code,...) in images. This problem is somewhat different from ours, because their target is to hide some regions of the image while we focus on hiding inherent private data. Another very related approach is image to image translation, such as [START_REF] Zhu | Unpaired image-to-image translation using cycle-consistent adversarial networks[END_REF][START_REF] Liu | Unsupervised image-to-image translation networks[END_REF]. Instead of transferring the images to another domain, we provide a single "identity domain" representation where it is as hard as possible to track the initial identity (assuming that the private information is the identity).

De-identification is similar to anonymization but consists in the process of preventing someone personal ID from being revealed. The main difference between this concept and data anonymization is that some identifying information can be preserved in order to be relinked only by a trusted party or by the original data operator, whereas in the case of anonymization, no re-identification should be possible (e.g., [START_REF] Blaz Meden | Face deidentification with generative deep neural networks[END_REF][START_REF] Li | De-identification without losing faces[END_REF]).

Contributions. This paper builds on previous ideas from [START_REF] Feutry | Learning anonymized representations with adversarial neural networks[END_REF][START_REF] Louizos | The variational fair autoencoder[END_REF] to introduce a semi-supervised framework for remov-

x Autoencoder x x Discriminator Private task S y q Ŷ |V q Ẑ| X v
Fig. 1: Architecture (S is the selector block).

ing a private information from image datasets, while keeping the distortion on the resulting images as low as possible.

Our model was applied to two image datasets: handwritten digits and facial expression. It provided interesting results in the context of anonymization [START_REF] Hukkelås | Deepprivacy: A generative adversarial network for face anonymization[END_REF] without a target task. Two cases should be distinguished: the case where the targeted private features are strongly connected to the other constitutive features of the image (denoted as intertwined) or not. Both cases are addressed but the later is less difficult.

PRESENTATION OF THE PROBLEM

Since we wish to be able to perform the same classifications (except the private one) on the processed images and on the original images, the metric used to evaluate the output must be a mix between a fidelity metric (to ensure as much information as possible remains) and a private metric. However, the similarity metric cannot be a fixed metric (e.g., a traditional distortion). Indeed, a metric that performs well on a kind of data may not be as efficient on another data. Preliminary tests using fixed metrics yielded poor results. This directed us to use a GAN-like discriminator as the flexible metric that is trained concurrently with an autoencoder. This has the advantage of keeping a wide range of applications and yet to obtain accurate and neat results. This flexibility is also useful to train an anonymizing autoencoder where images will need to change significantly to remove private features. The resulting architecture is displayed on Fig. 1. Random variables are in capital letters; • means computed over empirical distribution or estimate when over respectively a function or a random variable. Let T n be a training set of size n, where each element (x i , z i ) is composed of x i ∈ X , a real vector and of z i ∈ Z, the private label of this sample.

Our goal is to learn the parameters of the three networks: (i) q X|X the deep autoencoder; (ii) q Ŷ |V the discriminator; and (iii) q Ẑ| X the private classifiers. The selector S is before the discriminator and it allows a selection between x and x. The input v of the discriminator is the output of S which is driven by a variable

Y . If y = 1 then v = x otherwise v = x.
Learning with anonymization constraints. Consider the following constrained classification problem: min (q X|X ,q Ŷ |V )∈F P e q X|X , q Ŷ |V : min q Ẑ| X : X →P(Z)

P e q X|X , q Ẑ| X ≥ 1 -ε , (1)
for a prescribed probability 1/|Z| ≤ ε < 1, where the minimization is over the set of restricted encoders and classifiers (q X|X , q Ŷ |V ) ∈ F according to a model class F.

The above expression requires representations with (1ε)-approximate guarantees (over all possible classifiers) w.r.t. the misclassification probability of the private labels. Obviously, ε can be replaced by a suitable positive multiplier λ ≡ λ(ε) yielding a relaxed version of the objective.

min P e q X|X , q Ŷ |V -λ • P e q X|X , q Ẑ| X , [START_REF] Edwards | Censoring representations with an adversary[END_REF] where q Ẑ| X is the minimizer of P e q X|X , q Ẑ| X . Expression (2) does not lead to a tractable objective for training (q X|X , q Ŷ |V ). Yet, it suggests a competitive game between two players: an adversary trying to infer the private labels Z from our representations x, by minimizing P e q X|X , q Ẑ| X over all possible q Ẑ| X , and a fidelity learner predicting the labels Y (i.e. the representations' fidelity), by optimizing a classifier q Ŷ |V over a prescribed model class F. We can trade-off these two quantities via the autoencoder model q X|X . This idea will be further developed in the next section.

Cross-entropy loss: Given two distributions q X|X : X → P(X ) and q Ŷ |V : V → P(Y), define the average (over representations) cross-entropy loss as:

q X|X (•|x), q Ŷ |V (y|•) := q X|X (•|x), -log q Ŷ |V (y|•)
(3) As usual, we shall measure the expected performance of (q X|X , q Ŷ |V ) via the risk:

L(q Ŷ |V , q X|X ) := E p XY q X|X (•|X), q Ŷ |V (Y |•) . (4)
It is not difficult to show from Fano's inequality that the lower bound on the misclassification task of the private labels is a monotonically decreasing function of the mutual information I(Z; X). This implies that any limitation of the mutual information between private labels Z and representations X will bound from below the probability of misclassification of private labels, whichever the chosen classifier q Ẑ| X . Besides, it is well-known that the cross-entropy provides a surrogate to optimize the misclassification probability of Ŷ , which motivates the cross-entropy loss. These information-theoretic bounds provide a mathematical objective in order to browse the trade-off (1) between all feasible misclassification probabilities P e q X|X , q Ŷ |V as a function of the prescribed (1-ε) probability. Therefore, the learner's goal is to select an autoencoder q X|X and a classifier q Ŷ |V by minimizing jointly the risk and the mutual information. Yet, since p XY Z is unknown one cannot directly measure neither the risk nor the mutual information. It is usual to measure the agreement of a pair of candidates using a training dataset whose empirical distribution pXY Z is known. This yields an informationtheoretic objective, being a surrogate of expression of eq. ( 2):

min L emp (q Ŷ |V , q X|X ) + λ • I(Z; X) , (5)
for a suitable multiplier λ ≥ 0, where L emp (q Ŷ |V , q X|X ) denotes the empirical risk of eq (3) (i.e. averaged w.r.t. pXY ). The mutual information must be empirically evaluated using qZ| X as being the posterior according to q X|X pXZ . Eq. ( 5) may be independently motivated by a different problem studying distortion-equivocation trade-offs [START_REF] Villard | Secure multiterminal source coding with side information at the eavesdropper[END_REF].

Reconstruction learning with anonymization. The initial experiments performed with a similar training objective as the one introduced by [START_REF] Ganin | Unsupervised domain adaptation by backpropagation[END_REF] led to an unstable training and a poor trade-off between the degree of anonymity and the quality of the representation. This guides us to use instead a new adversarial training objective given below.

An examination of expression [START_REF] Liu | Unsupervised image-to-image translation networks[END_REF] shows that it cannot be optimized since the posterior distribution qZ| X is not computable in high dimensions. We will looser this surrogate by upper bounding the empirical mutual information I(Z; X) = H(Z) -H(Z| X). The empirical entropy of Z can be upper bounded as follows:

H(Z) ≤ E pZ -log q Ẑ (Z) (6)
≤ E pZ E q X -log q Ẑ| X (Z| X)

≡ E pZ E pX q X|X (•|X), q Ẑ| X (Z|•) (7)
:= L obj emp (q Ẑ| X , q X|X ), (8)
where ( 6) follows since the relative entropy is non-negative; [START_REF] Li | De-identification without losing faces[END_REF] follows by the convexity of t → -log(t) and ( 8) follows from the definition of the cross-entropy loss. We will also resort to an approximation of the conditional entropy H(Z| X) by learning an adequate empirical cross-entropy risk:

H(Z| X) ≈ E pXZ q X|X (•|X), q Ẑ| X (Z|•) ≡ L emp (q Ẑ| X , q X|X ), (10)
which assumes a well-selected classifier q Ẑ| X , i.e., the resulting approximation error given by D qZ| X q Ẑ| X |q X w.r.t. the exact q Ẑ| X is small enough. By combining expressions ( 9) and [START_REF] Hukkelås | Deepprivacy: A generative adversarial network for face anonymization[END_REF], and taking the absolute value, we obtain :

I(Z; X) L obj emp (q Ẑ| X , q X|X ) -L emp (q Ẑ| X , q X|X ) , (11)
that leads to our tractable learning objective, which is an approximation of expression ( 5), being the surrogate of (2):

L λ (q Ŷ |V , q Ẑ| X , q X|X ) := L emp (q Ŷ |V , q X|X ) + λ • L obj emp (q Ẑ| X , q X|X ) -L emp (q Ẑ| X , q X|X ) ,

for a suitable classifier q Ẑ| X and multiplier λ ≥ 0 which tunes the trade-off between the discriminator and the private task. The data representations provided by the autoencoder q X|X must blur the private labels Z features from the raw data X while preserving original data's relevant features. Note that (9) corresponds to the loss of a 'random guessing' classifier in which the representations X are independent of private labels Z. Thus, training encoders q X|X to minimize (12) enforces the best classifier q Ẑ| X to get closer -in terms of loss -to the random guessing classifier.

Training objectives. The terms of loss function ( 12) are successively the binary cross-entropy of discriminator, the cross-entropy of the private branch and the objective crossentropy which expressions follow:

L emp (q Ŷ |V , q X|X ) = - 1 n n i=1 y i log q Ŷ |V (•|v i ) +(1 -y i ) log(1 -q Ŷ |V (•|v i )) , (13)
L emp (q Ẑ| X , q X|X ) = 1 n n i=1 e(z i ), -log q Ẑ| X (•|x i ) , (14)
L obj emp (q Ẑ| X , q X|X ) = 1 n n i=1 pZ , -log q Ẑ| X (•|x i ) . ( 15
)
e(z i ) are "one-hot" vectors (z i component is 1 and the others 0) of the true labels of sample i = [1 : n]. pZ is the empirical distribution of the private labels. Each branch, i.e., q Ŷ |V and q Ẑ| X , is trained to minimize the associated cross-entropy loss (respectively ( 13) and ( 14)), whereas the autoencoder q X|X is trained to minimize eq. ( 12) (minimizing the cross-entropy loss with respect to Ŷ predictor while maximizing the new adversarial loss defined with respect to the Z predictor).

Training procedure. First the autoencoder is pre-trained using a pixel to pixel metric loss for a few epochs. Then the discriminator is pre-trained by alternating between: (i) updates on the discriminator parameters while autoencoder's parameters are frozen, with inputs going through the selector being a combination of autoencoded and original samples; (ii) updates on the autoencoder while discriminator's parameters are frozen. This allow the discriminator to learn the difference between both distributions. Finally, updates between the autoencoder on one side and the discriminator and the private branch on the other side are done alternatively. It is during this step that the privacy/fidelity trade-off occurs in the autoencoder.

SIMULATION RESULTS

The datasets used are doubly labelled: one is the private task classification and the other is used for the regular task classification in the performances assessment.

The Pen-digits dataset, from [START_REF] Alimoglu | Methods of combining multiple classifiers based on different representations for pen-based handwriting recognition[END_REF] provides the coordinates of digitally acquired pen movements of 44 persons (30 are in the training set and 14 in the test-set) writing digits from 0 to 9. We only used the training set (only 30 identities) which was randomly split into training, validation and test sets (size 5494, 1000 and 1000, respectively), sharing images of the same 30 persons. After being drawn, each image was down-sampled into a 20x20 grey-scale image.

The FERG (Facial Expression Research Group) dataset [START_REF] Aneja | Modeling stylized character expressions via deep learning[END_REF] contains 55767 annotated face synthetic images of six stylized characters modeled with the MAYA software. These images depict the seven following facial expressions: 'neutral', 'anger', 'fear', 'surprise', 'sadness', 'joy' and 'disgust'. Original 256x256 colour images have been pre-processed into re-sized 8-bit grey-scale 50x50 images.

Pen-digits results The private task labels in this dataset correspond to the writer identity. Hence, we want to transform handwritten digit images into other handwritten digit images while removing identity features and keeping relevant features about the original image digit.

Once the model is trained, original images are autoencoded into transformed images. Assessing the performance in a fair and automated manner is hard for it implies comparing and judging pictures. We processed images of the whole dataset and associated them with their original labels. Two assessments networks are then trained on the processed images, trying to recover both labels (ID and digit). The evaluation, plotted on Fig. 2, gives the accuracy on both tasks.

This assessment method judges a semi-supervised trans- formation in a fully supervised manner, yet it is a good compromise to quantify the effect on the ID features and on some other features that we wanted to keep. This testing is realistic since we used all the available ID information to train a ID features extractor from the transformed images. It is the best case scenario for an attacker wanting to recover the ID from the processed dataset. Note that even if testing on only one non-private feature is not ideal, the fidelity assessment is performed on a feature not used during the training of the model. FERG dataset. The private features to remove from the representations are the users identity labels.

Intertwined task method. The standard procedure explained before failed on the FERG dataset. Despite producing images that fool human eyes, they fail to fool a neural network. This is caused by the intertwined nature of the data. Identity traits and emotions are quite mixed. Our previous method led to a network that can produce visually anonymized results, but not a computer validated anonymization. To improve the anonymization on this kind of datasets, we derived a second method which specificity is restricting to one chosen identity the discriminator's train samples. This method although providing a small gain in anonymization was not sufficient. To perform a good anonymization we provided a protocol of substitution. Once all images are transformed into the chosen identity, each is compared to the chosen identity's real samples. Using pixel to pixel euclidean distance images are substitute by the closest chosen identity's training set image. The substitution is made really easy because the encoder shifts the representation of the original images very close to the selected identity while maintaining the underlying structure of the emotion features.

FERG Results. As stated above, two methods are associated with this particular framework. The first method restricts the discriminator to only one chosen ID and produces qualitative results shown on the Figs. 3 and4. Whereas the substitution method produces measurable results shown in Table 1.

Remark. One could argue that even if the dataset is anonymized, the chosen identity is still disclose because every sample wears the face of one preexisting identity. This is not a problem at all considering one can add an artificially identity in the databse to be the template of the anonymized faces, while still using our framework. Indeed the fact that the intertwined method allows to choose the final identity

CONCLUSION

We succeed in providing an anonymization framework where only the private labels are used to train the model. This is possible via the introduction of a new loss (expression [START_REF] Ganin | Unsupervised domain adaptation by backpropagation[END_REF]), an adapted architecture and training protocol. To train this loss, we took advantage of and reused widely known and efficient methods such as GAN discriminator and autoencoder, and we managed to perform a smart combination of both to design a working architecture that was suited to our loss.

The results of this paper are made even more impressive because of the visual representation. The network we designed manages to affect the hidden representation of the data in the autoencoder to shift private bit of information toward a dataset universal template with minimum cost to the other parts of the information. The generalized information that we can deduce is that this method allows a structure adapted spatial shift in the learnt features space: shifting from identity to another without major loss of sub-cluster structure (here emotions or digits). The usage of the learnt discriminator allows a neat output image, pixel perfect in most cases.

Fig. 2 :

 2 Fig. 2: Accuracy as a function of λ ∈ [0, 4.0] on Pen-digits dataset. Horizontal black dashed line is the random guessing classifier over the user-ID (3.33%).

Fig. 3 :

 3 Fig. 3: Evolution of the reconstructed images for a training and a validation samples (respectively top and bottom line). The resulting identity is fixed by the operator. The first image is the end of the pre-training phase: the autoencoder is trained to reproduce its entry (due to the pre-training's performance, this image is identical to the input sample). Going right follows the increase of the training epochs one by one. The most impressive changes occur over these first epochs. The following epochs (not show here) only refine the details of the encoded image to make it look genuine, resulting in the far right images that are the end training results.

Fig. 4 :

 4 Fig. 4: 24 couples of images composed of the sample and corresponding model output. All are randomly taken from the test set. Some representations are showing artifacts (missing an eye), e.g., representation at positions (line,column): (3,4),(3,2). Other shift emotions (1,4). from the training set make it possible.

Table 1 :

 1 Accuracies for anonymization (supervised during training) with different targeted IDs and with the accuracies of emotions recognition (unsupervised during training) to measure conservation of unlabeled intertwined features. These are obtained by training a classifier on the processed dataset.

	targeted identity	0	2	3	4	5
	identity accuracy (%)	42.47 38.25 36.21 32.77 31.62
	emotions accuracy (%)	72.45 77.46 72.32 66.79 72.95