Arcadie Fuior

Akram Hijazi

Olga Garbuz

Valentina Bulimaga

Liliana Zosim

Diana Cebotari

Mohamed Haouas

Ion Toderaş

Aurelian Gulea
email: guleaaurelian@gmail.com

Sébastien Floquet
email: sebastien.floquet@uvsq.fr

Screening of biological properties of MoV2O2S2-and MoV2O4-based coordination complexes: Investigation of antibacterial, antifungal, antioxidative and antitumoral activities versus growing of Spirulina platensis biomass

Keywords: Molybdedum, cluster, complex, cytotoxicity, antibiotic activity, spirulina platensis

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

Introduction

Molybdenum is an essential microelement since it is found at the heart of molybdoenzymes, so much vital for the living organisms. [START_REF] Maia | Molybdenum Tungsten Enzym[END_REF][START_REF] Hille | [END_REF][3][4] Nitrogen assimilation by plants, [START_REF] Seefeldt | Molybdenum Tungsten Enzym[END_REF][START_REF] Kappler | [END_REF] vitamin A formation in mammals, [7] and many other crucial biological redox processes in nature are efficiently performed by these metalloproteins that incorporate a Mo(+VI) or a Mo(+V) cation usually surrounded by O, N and S atoms, which can finely tune its electronic properties and the redox processes, passing through Mo(+V) and Mo(+IV) atoms. [8] Since this metal plays an active and important role in the metabolism of the majority of Earth's life forms, this makes molybdenum very interesting for biochemical and biomedical research and many authors have developed numerous very simple coordination complexes with molybdenum cores in the seventies and eighties to mimic such enzymes. [9][10][11][12][13][14][15][16] Developing new Mo-based compounds is indeed a good strategy to achieve promising drugs or biomimetic molecules. For this purpose, coordination compounds of dimolybdic [Mo (V) 2O2E2] 2+ (E = O or S) cores with simple ligands containing carboxylate and amine functions like EDTA, H3NTA, H2IDA, L-cysteine, L-histidine and others have been widely investigated. [17][18][19] In particular, since Mo (+V) binuclear complexes were interesting as models for redox active centers of molybdoenzymes, detailed electrochemical studies were performed in aqueous or in organic solvents to show the reversible or quasi-reversible cyclic processes of step-by-step one-electron reduction of the two Mo (+V) atoms to Mo (+IV) in the [Mo2O2E2] 2+ core in non-aqueous media. [20][21][22][23] However, surprisingly, to the best of our knowledge, no direct biological properties of [Mo2O2S2] 2+ and [Mo2O4] 2+ -based complexes were reported until a recent paper of Gretarsdóttir et al. in which six simple complexes combining the [Mo2O2S2] 2+ core and ligands such as disulfides, tetrasulfides, cyclopentadiene Cp, DMF, Serine, Cysteine or threonine have been suggested to have biological potential. Their convenient low cytotoxicity on cells was considered useful for possible therapeutic applications like the development of catalytic drugs. [24] In the last two decades, we acquired a strong expertise in the chemistry of such dimolybdic cores, which are used in the domain of polyoxometalates as building block for the synthesis of supramolecular assemblies, [25,26] some of them being capable of exhibiting electrocatalytic properties for the reduction of protons. [27,28] Furthermore, more recently we have patented results showing that such complexes can stimulate activity of microalgae Spirulina platensis. [START_REF] Floquet | Hidrat de bis[(iminodiacetat)oxotiomolibdat(V)] de tetrafenilfosfoniu şi procedeu de cultivare a microalgei Spirulina platensis cu utilizarea acestuia[END_REF] The compound [Mo2O2S2(IDA)2] 2-was indeed proved to regulate the ratio of acid and sulfate polysaccharides (up to 46% and respectively 100% more compared to a MoO3 solution at the same mass concentration) in the total carbohydrate composition of the cultivated spirulina cyanobacteria. [START_REF] Floquet | Hidrat de bis[(iminodiacetat)oxotiomolibdat(V)] de tetrafenilfosfoniu şi procedeu de cultivare a microalgei Spirulina platensis cu utilizarea acestuia[END_REF] These promising results prompted us to investigate more widely the biological activities of two series of [Mo2O2E2] 2+ coordination complexes either with E = S or E = O. Herein we first present the synthesis and characterization of 13 complexes (see Figure 1) combining such dimolybdic cores with ligands like oxalates (Ox), L-cysteine (L-cys), L-histidine (L-his), Iminodiacetate (IDA), Nitrilotriacetate (NTA) or ethylenediamine tetraacetate (EDTA) with a new synthetic route followed for sulfurated compounds. The complexes were characterized in the solid state by FT-IR and elemental analysis and in solution by ESI-MS and NMR which bring interesting results about the dynamics of such compounds in solution. A second part is devoted to biological screening tests allowing to investigate their potentialities for applications.

Figure 1 Structures of [Mo2O2S2] 2+ and [Mo2O4] 2+ complexes with H2Ox, H2IDA, H3NTA, H4EDTA, H2L-cys and HL-his ligands.[30-38] Color code: Mo(blue), O(red), S(yellow), N(green), C(black), H(white)

Results and discussion

Syntheses

The synthesis of Mo V 2O4-based complexes has followed the classical strategy of sodium molybdate Na2MoO4 interacting with ligands H2Ox, H3NTA, H4EDTA, H2-L-cys, H-L-his ligands in the presence of a reducing agent (usually hydrazine or dithionite) in aqueous solution at room temperature or at moderate heating. [17] On the other hand, compounds containing the di-μ-sulfido bridged unit [Mo V 2O2S2] 2+ were synthesized according to a method very different from those usually described in the literature which are often complicated and require strict precautions (long hydrosulfuration, specific reflux conditions in organic solvents, slow crystallization under inert atmosphere). The one pot synthesis used here avoids these problems by employing the K1.5(NMe4)0.5[I2Mo10O10S10(OH)10(H2O)5].20H2O solid precursor which generates [Mo2O2S2] 2+ species in acidic aqueous by hydrolysis of the bridging hydroxo groups. [39] The resulting red-orange solution then reacts with the ligand, which is introduced in excess in order to avoid the formation of cyclic compounds, well-known with this cluster. [26,28,33,[39][40][41] Mo2O2S2-based complexes are obtained preferably as K + salts, while the oxo homologues are more easily prepared in the form of Na + salts. The corresponding synthetic procedures are detailed in the experimental section given in the Supporting Information (SI). The compounds are isolated in the solid state and characterized by FT-IR, EDX, Elemental analysis, TGA, ESI-MS and NMR. In the following of this paper, for clarity, they will be denoted by the general notation "cation-core-ligand". Some are obtained as single crystals and the new structures are discussed below.

Structure of TBA-Mo2O2S2-IDA

The [Mo2O2S2(IDA)2] 2-complex was obtained with difficulty in solid state, compared to the other Mo2O2S2-based complexes and also to its oxo analogue which couldn't be isolated. A powder has been isolated only as PPh4 + salt, while single crystals were collected from a highly concentrated solution close to dryness containing an excess of TBA + cations. In contrast with the other complexes, the molecular structure of the anionic complex [Mo2O2S2(IDA)2] 2-depicted in Figure 2 was not known. The single dinuclear unit {Mo2O2S2} contains two iminodiacetate ligands (IDA 2-) coordinated to the Mo(V) centers, but the coordination modes of both iminodiacetato ligands are not equivalent and the resulting complex is chiral. Both ligands coordinate to Mo V centers through two oxygen atoms of the two carboxylate functions of IDA 2-and by the central nitrogen atom. The difference comes from the coordination mode of the latter. In a first ligand, N atom occupies the apical position in trans configuration with the Mo=O bond. In the other ligand, this atom coordinates Mo in an equatorial position, which constitutes an unusual coordination mode in this family of complexes to our knowledge. As shown in Figure 2b, in the solid state, both enantiomers are found in the crystal cell to give a racemic crystal. According to literature data for [Mo2O2S2]-based architectures, [26]

Solution studies

The complexes were characterized in solution by 1 H-NMR spectroscopy and by 1 H DOSY-NMR technique, in deuterated water and DMSO (see Supporting Information) at a concentration of 5 mM. Note that, since the complex Mo2O2S2-L-his is neutral and that the oxalate containing complexes Cs-Mo2O2S2-ox and Na-Mo2O4-ox do not have protons, they are logically excluded from this 1 H NMR study.

The 1 H-NMR spectra of Mo2O4 and Mo2O2S2 based complexes in D2O are given in figures 3 and 4, while the NMR spectra recorded in DMSO are given in Figure S4 in SI. In aqueous solution, complexes K-Mo2O2S2-EDTA and K-Mo2O2S2-L-cys give spectra with fine peaks which can be assigned to the protons of coordinated ligands in frozen geometries. The signals of methylenic protons of L-cysteine ligands in K-Mo2O2S2-L-cys have shifted from around 3 ppm in the free ligand to 2.9 ppm and 3.5 ppm, and the proton on the tertiary carbon atom initially found at 3.9 ppm is now downfield by 0.6 ppm. In the case of K-Mo2O2S2-EDTA, chelation of EDTA ligand to the metal centers leads to a more intricate spectrum, from only 2 singlets in the free ligand to 3 pairs of doublets of the same intensity. The NMR spectrum evidences a symmetry lower in solution than the expected C2v of the frozen structure depicted in Figure 1. Anyway, both complexes show no sign of free ligand, excluding the possibility of decoordination.

In contrast, the identification of peaks seems impossible for K-Mo2O2S2-HNTA. For the latter, the peak at 3.8 ppm assigned to a small proportion of uncoordinated ligand and the large signals suggest partial decoordination of the complex and also dynamic properties of the HNTA ligands. This result is consistent with the NMR study published in 2010, where it was suggested that the methylene groups give broad and poorly resolved signals because of dynamic equilibria between the cis and trans isomers passing through partial decoordination of one ligand and thus the presence of free ligand and consequently of monocoordinated complex [Mo2O2S2(HNTA)(H2O)3] + . [33] The presence of additional isomers such as those obtained with TBA-Mo2O2S2-IDA cannot be ruled out either. When used as building blocks in supramolecular polyoxometalate assemblies, the [Mo2O2S2] 2+ core is usually considered much more labile than its oxo analogue [Mo2O4] 2+ . [25,40] The same phenomenon is expected in coordination complexes. However, the spectrum of Na-Mo2O4-EDTA shows something that looks like two very broad signals corresponding to the 12 protons of coordinated EDTA ligand, in contrast to its sulfido bridged analogue. The dynamics evidenced for the EDTA ligand in Na-Mo2O4-EDTA preclude any confident assignment of the peaks, even if we can suppose the presence of the peaks of the 6 pairs of non-equivalent protons strongly overlapped. In contrast also with sulfide analogues, the 1 H-NMR spectrum of K-Mo2O4-HNTA displays well resolved signals assigned to coordinated ligands but an important signal of free ligand at 3.8 ppm suggests, as for K-Mo2O2S2-HNTA, a decoordination process even at 5 mM concentration (Figure 4). Finally, the case of Na-Mo2O4-L-cys shows a mixture of free ligand and the target complex, in contrast with K-Mo2O2S2-L-cys, which displays only the signals of coordinated ligands. Changing the solvent can dramatically affect the dynamics of such complexes. By replacing water with DMSO, the dynamics in solution are indeed strongly modified as shown in Figure S4 in SI. In such a medium, the decoordination process no longer exists, which renders easier the characterization of these complexes where the dynamics is clearly attenuated, except for Na-Mo2O4-EDTA for which the 1 H NMR spectrum is particularly broad. Finally, 1 H DOSY NMR is a powerful tool to describe this type of compounds in solution in terms of size and molecular mass and for evidencing dynamic processes and supramolecular interaction in solution. [33,41,42] Besides, as part of a larger project on molybdenum complexes and their associations with cyclodextrins and/or proteins, these values can constitute good reference values. 1 H DOSY NMR spectra were recorded in D2O and in DMSO. For example, the spectrum obtained for K-Mo2O2S2-EDTA is depicted in Figure 5

Biological activities

The solution studies evidence different behaviors in solution in the two series of coordination complexes prepared for this study. It was thus interesting to compare biological activities on different targets and associated either to alkali or to organic cations. The compounds were subjected to multiple in vitro biological tests to evaluate their antioxidant potential as well as their action on various cells and microorganisms as described in experimental section.

Antitumoral properties

One of the leading causes of cancer-related mortality is pancreatic cancer, while among women specifically it is the cervical cancer. Rhabdomyosarcoma is the most common soft tissue sarcoma affecting children and adolescents. Generally, these types of tumors are not detected at an early stage and are difficult to treat. [43][44][START_REF] Ashford | Preventing cervical cancer worldwide[END_REF][START_REF] Hinson | [END_REF][47] In this study, the in vitro tests were performed three types of human cancer cell lines originating from these specific malignant organs, namely: HeLa (cervical cancer), BxPC-3 (human primary pancreatic adenocarcinoma), and RD (human rhabdomyosarcoma) in comparison with mammalian healthy cells (MDCK). The results, as IC50 values, are compiled in the Table 2.

Table 2 Cytotoxicity of compounds on healthy (MDCK) cells, pancreatic cancer (BxPc

Antioxidative properties

The [Mo (V) 2O2E2] 2+ compounds have served for many years as small molecular models for the active centers of many molybdoenzymes. [12,20,35,[49][50][51][52][53][54][55][56] Such molybdenum-sulfur systems are indeed involved in certain enzymatic electron transfer processes and therefore an antioxidant activity can be expected. Here we have tested the possibility of such biological application by measuring the radical scavenging capacity against two common oxidant radicals: ABTS •+ and DDPH • . In the first method, ABTS •+ radical cation is inhibited by reduction in a one electron transfer process, whereas the second method supposes neutralization of DDPH • by a hydrogendonating antioxidant. This will provide additional clues regarding the mechanism behind the in vitro antiradical property of our complexes. The IC50 values are given in Table 3. Out of the 13 measured compounds, 7 of them show a significant antioxidant activity. The table 3 indicates better results in the case of ABTS method, suggesting the appropriate electron transfer mechanism or the slow reaction kinetics with DDPH • (incubation time dependent activity). Clearly the compounds bearing cysteine ligands are the most powerful antioxidants of the series, being more active than reference antioxidant Trolox (water soluble analogue of Vitamin E) (K-Mo2O2S2-Cys almost 14 times and Na-Mo2O4-Cys almost 10 times respectively) and even the natural flavonoid Rutin (K-Mo2O2S2-Cys almost 11 times and Na-Mo2O4-Cys almost 8 times respectively). Somehow, cysteine ligands help in this process. Indeed, cysteine molecule is known to have antioxidant properties due to oxidizable thiol -SH group, which becomes more reactive in ionized thiolate form and has the ability to form disulfide bonds, like in the oxidative dimerization of cysteine into cystine. [57,58] Oxalate and Histidine are also known to be able to be oxidized and the activities of the corresponding complexes are also probably related to the activity of the ligands.

In contrast, IDA, NTA or EDTA ligands are not expected to be oxidized. The chelate-type [Mo2O2E2(EDTA)] 2- complexes have no activity at all as for the other coordination complexes formed around the Mo2O4 core, while the Mo2O2S2 complexes formed with IDA and HNTA display some activity. It suggests that his antioxidant ability depends on the redox properties of the Mo(V) centers which can be tuned with the nature of the coordinated ligand and by its oxo or sulfurated environment and more importantly from the ligand's redox properties itself. [20,21,53,59] The stability of the EDTA could also play a role by protecting the Mo(V) centers from oxidation, a process which leads to the breaking of the Mo-Mo bond and possibly to the decomposition of the compound, [60] while coordination of two separate ligands on each side of the Mo2O2E2 core leads to more labile structures and with higher ability to be oxidized. The combination of the aforementioned factors made K2[Mo2O2S2(L-cys)2].5H2O and Na2[Mo2O4(L-cys)2].5H2O to act as good small molecular-weight nonenzymatic antioxidants.

Antibacterial and antifungal activities

In the context of the worrying rise of antibiotic resistance levels worldwide, treating common diseases requires unceasing demand for new drugs combining efficiency and safety. Here, our molybdenum complexes were tested for their action deliberately on bacterial and fungal strains with medium to high antibiotic resistance. As clearly seen from the results in the tables 4 and 5 below, the majority of complexes have a weak antibacterial and antifungal activity against all tested G+ and G-bacteria and fungi, except for two of them. PPh4-Mo2O2S2-EDTA and PPh4-Mo2O4-EDTA are indeed significantly active against Gram positive bacteria with minimum inhibitory concentrations (MIC) of 0.002 mg/mL (S. aureus, B. cereus), 0.004 mg/mL (B. subtilis) and respectively 0.004 mg/mL (S. aureus, B. cereus), 0.008 (B. subtilis), being twice as active as the reference antibiotic Doxycycline or even pretty active when Doxycycline has no activity at all. Against fungi our compounds have barely any effect. Only those with tetraphenylphosphonium (PPh4-Mo2O2S2-EDTA, PPh4-Mo2O4-EDTA) have a better activity, but still ~15 times less active than Fluconazole fungi inhibitor drug.

Structure-activity correlation shows an activity dependency on the nature of the cation associated to the coordination compounds. Apparently PPh4 + cation is responsible for this inhibitory effect on bacteria and fungi, since other salts of the same complexes do not show any significant activities. This seems logical since tetraphenylphosphonium -a lipophilic membrane-permeable cation is considered biocide, acting upon several cellular targets inside microorganisms. [48] It is actually an excellent strategy to obtain multi-target compounds by combining bioactive organic cations to our bioactive coordination complexes. One inspiring example is the association of antibacterial tetraalkylammonium cations with [α-SiW11O39] 8-to create POM-based ionic liquids as coatings with anticorrosion and antimicrobial properties. [61]

Table 4 Activity of Mo2O2E2 complexes against G+ and G-bacteria

Influence on unicellular microorganisms

Toxicity bioassay using infusoria microorganisms is a simple and very effective method to investigate the effect of chemical compounds on living organisms. Paramecia are aquatic single-celled eukaryotes, called protozoans, having short life cycle and rapid population growth through asexual reproduction. Their behavior and development are strongly dependent on the quality parameters of the environment, therefore they serve as good bioindicators for toxicological studies in laboratory conditions. [START_REF] Chubik | METHOD OF DETERMINING TOXICITY OF CHEMICAL SUBSTANCES IN AQUEOUS MEDIA[END_REF][START_REF] Madoni | [END_REF][START_REF] Лотова | Студенческая Наука XXI Века[END_REF][START_REF] Roșcov | [END_REF][66] Here, we test the toxicity of our compounds by exposing Paramecia caudatum Ehrenberg to different concentrations of each abiotic factor (chemical stressor) and the median lethal concentration (LC50) is evaluated based on their viability results. Li-Mo2O4-EDTA ≥100 ≥100

PPh4-Mo2O4-EDTA ≥100 ≥100

The majority of compounds are non-toxic since they have LC50 ≥ 100 µM. Once again, the non-natural cations TBA + and PPh4 + salts show inhibition action against microorganisms, as exposed anteriorly on bacteria and fungi, especially when associated to the Mo2O2S2 2+ core. The K-Mo2O4-HNTA substance was identified as the most toxic, killing half of the paramecia at very low concentrations (3.3 µM after 24h or 1.7 µM after 48h). Thereby, most of the compounds of this class prove to be generally inoffensive to model unicellular systems and could eventually qualify for biological tests on superior organisms.

Growth-stimulating properties

Spirulina platensis is a multicellular and filamentous blue-green microalga that draw considerable attention as a source of bioactive substances like proteins, vitamins, essential amino acids, β-carotene, etc. and has a great potential for development, especially as a small-scale crop for nutritional enhancement in humans and animals and as drugs. [START_REF] Habib | Review on culture, production and use of Spirulina as food for humans and feeds for domestic animals and fish, Food and Agriculture Organization of the United Nations[END_REF][START_REF] Djur | [END_REF] Research in recent decades have shown that the use of inorganic salts as well as coordination compounds of Zn(II), Fe(III), Cr(III), Cu(II) and other transition metals as chemical regulators contributes to the increased synthesis of bioactive components in the cyanobacterium Spirulina platensis, as well as to the biotransformation of the elements by their incorporation into bioorganic components. [START_REF] Cepoi | Recent Adv. Trace Elem[END_REF] The use of chemical regulators in phycobiotechnology has resulted in the development of a series of procedures to increase the productivity of Spirulina and to trigger the biosynthesis of specific metabolites of the biomass, including bioaccumulation of elements. [START_REF] Rudic | Phycobiotechnology-fundamental research and practical achievements[END_REF] Investigation of Mo2O2E2 compounds as new chemical regulators containing a molybdenum source for cultivation of Spirulina platensis could present interest, given the important role of this oligoelement in the metabolism of prokaryotic and eukaryotic organisms. Therefore, their effect on this microalga's productivity and biochemical composition is examined hereafter. The results are given in table 6.

This study was focused on the three compounds K-Mo2O2S2-HNTA, TBA-Mo2O2S2-IDA and PPh4-Mo2O4-EDTA. As seen in table 6, the results do not manifest any toxic effect towards Spirulina in the investigated concentration ranges. Productivity tends to rise proportionally with concentration in the case of K-Mo2O2S2-HNTA, TBA-Mo2O2S2-IDA, reaching maximum values at 50 mg/L. For K-Mo2O2S2-HNTA we get the higher growth rate of +6.94%, which equals to 1.26 g/L in the 10 th day. Beyond 50 mg/L concentration, a minor decrease is observed, but the productivity is still comparable to the reference sample. The repartition of biochemical components of spirulina seems to be reshaped also. The protein quantity is diminished by 3.1% for the HNTAbearing compound while in the presence of TBA-Mo2O2S2-IDA it increased by 8.5%. The biomass' peptide content, separated from the total protein fraction increases within the following concentration limits: 40-70 mg/L (K-Mo2O2S2-HNTA) and 50-70 mg/L (TBA-Mo2O2S2-IDA), reaching values that exceed the reference by 11.04-29.22%, 34.09-77.92% and 16.23-87.01%, respectively. Concerning the two compounds containing the Mo2O2S2 core, the gain in carbohydrates and lipids goes up to 19.63% and 14.04% for the one with HNTA ligands, whereas 9.56% and 23.71% for the one with IDA ligands. At first look, PPh4-Mo2O4-EDTA seems unsuitable for spirulina growing as productivity tends to decrease by 17.7% in the presence of 10 mg/L of this compound. However, this compound shows excellent results in selectively boosting the peptide ratio to 222.7%. So, despite the 1/6 biomass loss, production of peptide-rich spirulina is an interesting feature.

Experimental section

Experimental details are given in Electronic Supporting Information.

Conclusion

In this paper, we have investigated two series of dinuclear complexes based on [Mo2O2S2] 2+ and [Mo2O4] 2+ cores. Almost all of them have been obtained in a one-step synthesis in water, consistent with green chemistry principles and a new synthetic procedure has been applied for sulfurated ones.

The solution studies revealed totally different behaviors for both series of complexes in terms of dynamics and lability depending on the nature of the central core and of the ligand. Although, the major part of these complexes was already known in the literature as models for enzymes, the biological activities of such series of complexes was in fact poorly explored. In the present study, strong evidences have been acquired for the biological efficiency of these coordination compounds. Although some compounds display cytotoxicity, we have evidenced that it is probably due to the organic counter cations PPh4 + , which are non-innocent. Therefore, we can say that these molybdenum complexes appear non-toxic.

Complexes [Mo2O2E2(EDTA)] 2-associating PPh4 + cations have been found with significant activity against all tested Gram-positive bacteria Staphylococcus aureus, Bacillus cereus and Bacillus subtilis with as low as 1.9 µg/mL of compound and also against all tested fungi Candida albicans, Candida krusei/parapsilosis and Cryptococcus neoformans with as low as 3.9 µg/mL of compound due to the antimicrobial effect of the cation. Such compounds can be considered as simple but relatively potent bacterial and fungal inhibitors. The majority of compounds are harmless in vitro and seem promising for application in biology as antioxidant agents, with minimal IC50 1.9 µM.

At the same time, the first three tested Mo complexes of our series are able to promote growth of Spirulina platensis culture, building up to 7% more biomass and they modulate its biochemical composition, diversifying even more its possible uses to a food supplement with rich peptide content (+123%) for instance. Concerning this class of complexes, novel biological tests are envisioned to eventually extend the arsenal of beneficial properties to other possible applications.

Figure 2 .

 2 Figure 2. (a) Molecular Structure of the complex [Mo2O2S2(IDA)2] 2-, (b) View of the two isomers of the complex [Mo2O2S2(IDA)2] 2-in the crystal cell

Figure 3 1

 3 Figure 3 1 H-NMR spectra of Mo2O2S2-based complexes in D2O. * indicates the presence of decoordinated ligand.

Figure 4 1

 4 Figure 4 1 H-NMR spectra of Mo2O4-based complexes in D2O. * indicates the presence of decoordinated ligand.In summary, these 1 H-NMR spectra in water evidence a dynamic behavior which is at the opposite of what is usually admitted for these clusters in polyoxometalate chemistry. More dynamics means higher lability, and thus more decoordination/coordination process are observed in [Mo2O4]-based coordination complexes. Besides, HNTA complexes exhibit a noticeable presence of free ligand at a concentration of 5 mM, which suggest a low chemical stability, as for Na-Mo2O4-L-Cys. Changing the solvent can dramatically affect the dynamics of such complexes. By replacing water with DMSO, the dynamics in solution are indeed strongly modified as shown in FigureS4in SI. In such a medium, the

  , while the values of self-diffusion coefficient are gathered in the table 1. The signals give more or less the same value with an average of 515 ± 6 μm 2 /s in D2O. It means that all the signals come from the same molecule, which exclude any decoordination process. Moreover, from Stokes-Einstein law this value allows calculation of a hydrodynamic radius around 0.4-0.5 nm, rather consistent with the expected molecular complexes. Similar values are obtained with the other complexes studied in D2O, while D logically decreases to an average of 191 µm².s -1 in more viscous solvent such as DMSO according to the Stokes-Einstein equation.

Figure 5 1

 5 Figure 5 1 H DOSY NMR spectrum of compound K-Mo2O2S2-EDTA in D2O, 5 mMIt is noteworthy for both K-Mo2O2S2-HNTA and K-Mo2O4-HNTA complexes that two sets of D values are identified in D2O, suggesting the existence of two species with essential difference in size and mass. The peak at 3.8 ppm for K-Mo2O2S2-HNTA diffuses at 603 μm 2 /s and the one at 3.75 ppm for K-Mo2O4-HNTA has a diffusion speed of 623 μm 2 /s, which is significantly high, in agreement with a free ligand in equilibrium with the complex, while the average D value for this family of compounds with no big differences in size is around 500 in D2O. The situation is totally changed in DMSO. As shown in figureS5(SI), only one set of D values is obtained in agreement with a decoordination/coordination process stopped.

- 3 )

 3 , cervical cancer (HeLa) and rhabdomyosarcoma cancer (RD) cells. (NM = Not measured) Mo2O4 complexes with different ligands and in form of different salts serves as an indication of their toxicity on cells, and allows identification of the effect of each parameter i.e. cation, molybdenum core and ligand. The data in the table 2 show no toxicity or very weak toxicity in comparison to the reference cancer drugs Cis-platin and Doxorubicin. Only two complexes of EDTA show non-negligible values: PPh4-Mo2O2S2-EDTA has IC50 = 59.81 ± 26.45 μM on BxPc-3 cancer cells and PPh4-Mo2O4-EDTA has IC50 = 25.09 ± 7.02 μM on BxPc-3 cancer cells and 60.72 ± 12.1 μM on normal MDCK cells, which means that all other complexes are non-toxic or almost non-toxic.The comparison of these two complexes in their alkali salts which are not at all toxic suggests that it is not the [Mo2O4(EDTA)] 2-and [Mo2O2S2(EDTA)] 2-complexes that induces toxicity for these two compounds but the tetraphenylphosphonium counter-cations.[48] Hence, Mo2O2S2 and Mo2O4 complexes with Histidine, Cysteine, IDA, HNTA and EDTA ligands are surely safe in vitro as long as the cation is alkaline. These findings reveal the importance of the counter cation on the overall toxicity and show that molybdenum(V) [Mo2O2S2] 2+ and [Mo2O4] 2+ moieties coordinated with biocompatible ligands are non-toxic. This is in good agreement with other cytotoxicity results related to Mo2O2S2-based complexes associated to NEt4 + or alkali cations reported in the literature.[24]

Table 1 .

 1 Self-diffusion coefficients D measured for complexes containing L-Cys, HNTA and EDTA ligands in D2O and in DMSO (5 mM concentration)

	Compound	D (μm 2 /s) in D2O	D (μm 2 /s) in DMSO
	K-Mo2O2S2-L-Cys	503 ± 14	193 ± 6
	K-Mo2O2S2-HNTA	499 ± 147	173 ± 6
	K-Mo2O2S2-EDTA	515 ± 6	216 ± 6
	Na-Mo2O4-L-Cys	502 ± 14	183 ± 7
	K-Mo2O4-HNTA	526 ± 43	Not soluble
	Na-Mo2O4-EDTA	533 ± 8	Not soluble

Table 3

 3 Antioxidant activities for Mo2O2S2 and Mo2O4 complexes against ABTS and DPPH

	Compound	IC50 (µM)	IC50 (µM)
		ABTS	DPPH
	Cs1.5Na0.5-Mo2O2S2-Ox	42.5±7.10	≥100
	K-Mo2O2S2-L-cys	1.9±0.12	36.4±6.10
	TBA-Mo2O2S2-IDA	27.2±3.69	-
	K-Mo2O2S2-HNTA	20.1±0.25	-
	K-Mo2O2S2-EDTA	-	-
	PPh4-Mo2O2S2-EDTA	-	-
	Mo2O2S2-L-his	7.0±0.52	-
	TBA0.5Na1.5-Mo2O4-Ox	74.9±3.78	≥100
	Na-Mo2O4-L-cys	2.73±0.65	-
	K-Mo2O4-HNTA	-	-
	PPh4-Mo2O4-EDTA	-	-
	Na-Mo2O4-EDTA	-	-
	Li-Mo2O4-EDTA	-	-
	TROLOX	26.3±0.70	48.9±0.80
	RUTIN	20.7±0.12	64.8±2.10
	NM = Not measured

Table 6

 6 Toxicity of Mo2O2E2 complexes on Paramecia

	Compound	24h	LC50 (µM)	48h
	K-Mo2O2S2-L-Cys	≥100		≥100
	Mo2O2S2-L-His	28.72 ± 5.5		≥100
	TBA-Mo2O2S2-IDA	56.98 ± 5.6		48.66 ± 2.2
	K-Mo2O2S2-HNTA	≥100		≥100
	K-Mo2O2S2-EDTA	≥100		≥100
	PPh4-Mo2O2S2-EDTA	40.25 ± 2.6		≥100
	Na-Mo2O4-L-Cys	≥100		≥100
	K-Mo2O4-HNTA	3.3 ± 0.2		1.7 ± 0.01
	Na-Mo2O4-EDTA	≥100		≥100

Table 6

 6 Activity of Mo2O2E2 complexes on Spirulina platensis growth

	Compound	Conc.,	Producti	Proteins*	Peptides*	Carbohydrates*	Lipids*
		mg/L	vity* in	% of	%	% of	%	% of	%	% of	%
			the 10 th	dry	compared	dry	compared	dry	compared	dry	compared
			day, g/L	weight	to	weight	to	weight	to	weight	to
					reference		reference		reference		reference
	Control	0	1,180	65,95	100,00	3,08	100,00	11,61	100,00	10,33	100,00

Acknowledgements

SF gratefully acknowledges Mrs. Aurelie Avenel for her contribution for the syntheses of complexes, Dr Nathalie Leclerc for ATG and EDX measurements and Dr Clément Falaise for his help for X-ray diffraction studies. This work has been supported by a public grant overseen by the French National Research Agency (ANR) under the contract POMEAH•ANR-08-JCJC-0097 and the LABEX CHARMMMAT ANR -11-LABX-0039 (Project COMPA). We also thank University of Versailles, the "Institut Universitaire de France, IUF" and the CNRS for financial support. AF and DC gratefully acknowledge Campus France for Eiffel Doctorate grant as well as IUF and State University of Moldova.

Supporting Information

The Supporting Information contains, experimental details followed for syntheses, characterizations (FT-IR and ESI-MS spectra) and protocols for biological tests. It contains also, additional NMR spectra


Human Neck Finite Element Model Development and Validation against Original Experimental Data

Keywords: Modal analysis, Finite element neck model, Validation

This study proposes a detailed FEM of a human volunteer's neck and proceeds to an original model validation against experimental data recorded with this human volunteer. In order to evaluate the new model against existing data a successful temporal validation of the model was obtained under frontal, lateral, oblique and rear impact. New validation parameters are based on an experimental test proceeded in the frequency domain in order to extract the volunteer's Head-Neck system's modal characteristics. In deep validation of the head neck FEM is then performed by superposing the numerical and experimental frequency response function. Model optimisation in the frequency domain permitted after significant properties modification to reproduced accurately both, the neck extension mode at 1.4 Hz and the head retraction mode at 8.8 Hz. Finally the "frequency domain optimised" FEM response was superimposed with the temporal corridors provided in the literature. It must be mentioned that the model's response in the temporal domain remains inside existing corridors after this model optimisation in the frequency domain illustrating that the temporal validation is not accurate enough. This study proposes a neck model with improved geometry description and biofidelity with special attention paid to the retraction mode, a phenomenon which is often masked in the temporal domain.

INTRODUCTION

In spite of a large number of research projects in the field of cervical spine biomechanics, current protection systems are still not capable of significantly reducing low-AIS lesions, lesions that generate considerable costs to society [START_REF] Deng | Kinematics of Human Cadaver Cervical Spine During Low Rear-End Impacts[END_REF]. The main reason for this failure is the inability of the medical profession to diagnose the origin of the pains that victims of these accidents complain of. This is why scientists, who have over many years undertaken experimental and theoretical studies to try to better understand the mechanical behaviour of the head-neck system during an impact, have encountered so many difficulties in their approach. First of all, series of tests designed to describe the response of the cervical column under impact have been undertaken. As far as these experiments characterising the behaviour of this complex segment under impact are concerned, we will not list here all the characterisations that have been carried out. On the other hand, a review of the data most commonly used to validate the Finite Element Models (FEM) of the cervical column will be done.

The tests carried out by the N.B.D.L (National Biodynamics Laboratory) reported by [START_REF] Ewing | Dynamic response of the head and neck of the living human to -Gx impact acceleration[END_REF][START_REF] Ewing | Dynamic response of the human head and neck to +Gy impact acceleration[END_REF] with frontal, lateral and oblique impacts are frequently used as a basis for the validation for finite element models of the neck : [START_REF] Dauvilliers | Modélisation Tridimensionnelle et Dynamique du Rachis Cervical[END_REF], [START_REF] Jager | A Global and a Detailed Mathematical Model for Head-Neck Dynamics[END_REF], [START_REF] Nitsche | Validation of a Finite-Element-Model of the Human Neck[END_REF], [START_REF] Happee | A Mathematical Human Body Model for Frontal and Rearward Seated Automotive Impact Loading[END_REF], [START_REF] Yang | Development of a Finite Element Model of the Human Neck[END_REF] and [START_REF] Astori | Multy-Body Numerical model of the Human Neck[END_REF]). The advantages of such a validation include two important aspects : the first lies in the fact that the tests were carried out on volunteers subjected to high intensity impacts, thus leading to the non-linearity of the neck. Secondarily, these tests were very well instrumented, thus allowing the recording of the accelerations at the first thoracic vertebra T1, which constitutes the initial conditions in the modelling. As for the recording of the six head accelerations components, three linear and three rotational accelerations were fixed at head level and this constitutes the basis for the validation of the FEMs. However, it is clear that in spite of the complete instrumentation of the test subject, the FEM reproduce correctly only the kinematics of the head.

Over the last few years we have seen the emergence of new experiments determining the movements relating to the cervical vertebrae during a rear impact (movements of the articular surfaces as well as the bodies of the vertebrae). In this line of research, two studies are essential, that of [START_REF] Ono | Cervical Injury Mechanism Based on the Analysis of Human Cervical Vertebral Motion and Head-Neck-Torso Kinematics During Low Speed Rear Impacts[END_REF] on 12 volunteers and that of [START_REF] Deng | Kinematics of Human Cadaver Cervical Spine During Low Rear-End Impacts[END_REF] on 6 cadavers. These studies are a significant step in the attempt to correctly reproduce the behaviour of the spine from a local point of view. These experiments allow us not to limit ourselves to only the movements of the anatomical centre of the head, unlike the classic tests. However, at the present time, none of these investigations have succeeded in determining firmly the reasons why patients complain of neck pain, nausea or muscular pain after a low intensity rear impact [START_REF] Ovadia | Whiplash injury-a retrospective study on patients seeking compensation[END_REF]. Alongside the experiments considering the whole of the head-neck system, local approaches have attempted to characterise behavioural laws concerning the intervertebral discs, ligaments and muscles.

Furthermore, in spite of all these highly instrumented experiments that give access to the kinematics of the head, or to the relative movements of the vertebrae, these validation data are always expressed in the form of an experimental corridor where the minimum and the maximum recorded values define the range of validity of an FEM. Most often these corridors are wide, and the response of the model may present inclines with very variable origins, oscillations or acceleration peaks whilst remaining within the area considered, but implying very varied characteristics of the model. These factors are in fact of considerable importance for the identification of the mechanical systems, but cannot be taken into account in this temporal approach, which is always difficult in impact situations.

The aim of this study is first to construct an FEM of the upper spine with a high degree of geometrical accuracy, to evaluate it in comparison to the tests generally accepted by the scientific community considering the kinematics of the head in front, lateral and oblique impact situations (N.B.D.L.) and a rear impact situation [START_REF] Prasad | Biofidelity of Anthropomorphic Test Devices for Rear Impact[END_REF].

Impulse experiments carried out in the sagittal plane will then allow the modal characteristics of the headneck system to be extracted. A numerical frequency analysis will be carried out on the neck FEM with a view to comparing the natural frequencies of the model and of the volunteer. The optimisation of the model will allow us to numerically reproduce the experimental modal behaviour. The temporal validation of the optimised model in the frequency domain will show that the adjustment of the mechanical properties on the temporal response alone is not enough to ensure biofaithful behaviour.

METHODS

Model geometry

It is possible to distinguish in the bibliography two families of FEM of the cervical column. The models that represent an average cervical spine [START_REF] Jager | A Global and a Detailed Mathematical Model for Head-Neck Dynamics[END_REF], [START_REF] Nitsche | Validation of a Finite-Element-Model of the Human Neck[END_REF], Dauvillier (1994), [START_REF] Astori | Multy-Body Numerical model of the Human Neck[END_REF]) and those obtained by generating a mesh from scanner or IRM sections of volunteers, a method used by [START_REF] Camacho | Experimental Flexibility Measurements for the Development of a computational Head-Neck Model Validated for Near-Vertex Head Impact[END_REF][START_REF] Yang | Development of a Finite Element Model of the Human Neck[END_REF] For our study and because we are focusing on a specific volunteer, it seemed to us essential to base our work on the geometry of a living human subject of average size and close to 50th percentile male : [Height: 1.72 m, weight: 72 Kg, age: 33 years]. This approach allows us not to make any approximation regarding the geometry of the cervical vertebrae. From a modelling point of view, we have taken millimetric scanner 1 sections of the subject. These scanner sections then underwent grey level processing in order to extract the bone part of the cervical column and of the skull. A rough tria mesh in STL format was then constructed as shown in figure 1.

This file was then imported under the Hypermesh V6.0 meshing software so that the cervical vertebrae could be completely meshed. Indeed, the explicit finite element calculation codes used in the crash field, such as Radioss, require regular meshing that checks certain geometrical criteria influencing the calculation time pitch. Table 1 recapitulates the 1 The files of the CT scan used were obtained from a ct scan performed on a medical member of our team. The scan was initially performed for diagnosis purpose. The results of the ct scan being normal, the idea appeared to use the files obtained. The files were then anonymized and transmitted to our institution. We insist on the fact that the ct scan was performed for medical purpose and not for our study. We simply jumped on this opportunity. Furthermore it was performed on a doctor who had a perfect knowledge of the consequences of this imaging technique and the possible benefit of the ct scan was greater than the risks of ignoring a severe injury of the cervical spine.

values of these criteria used throughout the meshing process. The cervical vertebrae were modelled using shell elements, the intervertebral discs with brick elements, the ligaments using spring elements and finally the muscles as well as the soft tissues with brick elements. This choice of modelling may be justified as follows : For the cervical vertebrae, shell elements offer the possibility of strictly respecting the anatomical surface. This choice involves declaring this part as a rigid body in order to respect the geometry of the articular surfaces and to correctly reproduce the inertias. This simplifying hypothesis is not restrictive, however, insofar as the model is not interested to reproduce bone fractures, but to simulate more moderate lesions.

The modelling of the ligaments using spring elements means that parametric studies can easily be made both on stiffnesses and damping.

As far as the lower cervical spine is concerned, we have distinguished five types of ligaments : The anterior longitudinal ligament (ALL), the posterior longitudinal ligament (PLL), the flavum ligament (FL), the interspinal ligament (ISL) and finally the capsular ligaments (CL). This ligamentary system is shown in figure 2a.

For the upper cervical spine, we have modelled the posterior common ligament (C2-C0; C2-C1; C1-C0), the atloidien-axoidien anterior ligament, the transverse ligament, the yellow ligament (C2-C1), the transverse axoid ligament, the anterior occipito-atloid ligament, the alar ligament, the posterior occipitoatloidien ligament, capsular ligament C2-C1, capsular ligament Head-C1, membrane tectaria, the median occipito-odontoid ligament as well as the lateral occipito-atloidien ligament as illustrated in figure 2b.

Moreover, concerning the ligaments, we have deliberately introduced different numbers of springs for a given ligament out of a concern for biofaithfulness. The number of springs per ligaments is given in Appendix A.

The choice of modelling the intervertebral discs using volume elements in three layers was found to be indispensable to reproducing their behaviour correctly, given the 3D nature of this structure.

Finally, the soft tissues and the muscles are modelled using brick elements. This type of element offers the advantage of taking account of muscular mass, stiffness and the damping characteristics of this continuous medium. Particular attention was paid to the continuity of the meshing between the head, the cervical vertebrae and the muscles. Our will was to proceed a continuous meshing between the surface meshing (head and cervical spine) and the volume meshing (muscles and soft tissues). Therefore the only parts that are not in the continuous meshing are the articular process and the spinous process. Here, we placed an interface.

The finite element model of the head-neck system thus defined consists of 73 185 elements divided into 13 458 shell elements, 504 spring elements and 59.223 volume elements. A representation of this model is given in figures 3 and 4.

Mechanical properties

The ligaments were modelled using non-linear spring elements with a damping coefficient of η=900 Nm/s (De [START_REF] Jager | A Global and a Detailed Mathematical Model for Head-Neck Dynamics[END_REF], η=300Nm/s, Dauvilliers 1994, η=2000Nm/s). To define the behaviour laws of each ligament in both the lower and upper cervical spines, we referred to two complementary studies by [START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF] and [START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF]. The Chazal et al. study (1985) highlights the non-linear viscoelastic behaviour of ligaments whereas [START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF] gives information on their failure properties. The overall behaviour of the ligaments can then be characterised by three pairs of coefficients α 1 , α 2, α 3 determining the zone of low rigidity or neutral zone, the linear part, and finally the plastic behaviour. The coefficients used for our model are described in Table 2 and a representation of the typical behaviour of the five ligaments of the lower cervical spine is illustrated in figure 5. [START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF] and [START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF].

In order to take into account the initial lengths of the ligaments in the model as well as those measured anatomically by Yoganandan et al. on the lower cervical spine (2001) we calculated the laws as follows :

3 * * 1, 2,3 * * m i i i m i spring L d L L i F L F N L α α    =       =     =       (1)
with d i is the spring elongation, F i the force, N spring number of springs, L the experimental ligament length and L m the mean length spring in the model.

For the upper ligaments the initial experimental lengths are not given by [START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF], so the ratio between the initial length of the model and experimental are equal to 1.

All the mechanical properties of the ligaments are detailed in the summary table in Appendix A.

Intervertebral disks have been poorly studied. [START_REF] Moroney | Load-displacement properties of lower cervical spine motion segments[END_REF] have measured a value of 500 MPa for compression solicitations. [START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF] have measured a Young modulus of elasticity for the anulus ground substance ranged from 3.4 to 4.7 MPa , and a linear Young modulus of 450 to 500 MPa for the anulus fibers.

In our model the hypothesis of a homogeneous linear elastic isotrope material was considered with a Young modulus of 100 Mpa and a Poisson's ratio of 0.3. These values are situated between the extreme values related in the literature which represents a global behaviour of this structure [START_REF] Kleinberger | Application of finite element techniques to the study of cervical spine mechanics[END_REF], [START_REF] Dauvilliers | Modélisation Tridimensionnelle et Dynamique du Rachis Cervical[END_REF]).

The cervical vertebrae were declared as rigid bodies. The mechanical characteristics in terms of masse and inertias are taken from the work of [START_REF] Deng | Response of a human head/neck/upper-torso replica to dynamic loading-II. Analytical/numerical model[END_REF] and detailed in Table 3.

A passive law (Eq.2) has been set to model the muscle behaviour. This law was taken from the work of [START_REF] Knudsen | Torsiona elasticity of muscles[END_REF].

( )

0 1 E e αε σ α = - (2)
The Young Modulus E 0 was calibrated with the Myers et al. experimental data (1995) and set to 0.15 Mpa and α set to 1.

For the anterior muscles an elastic law was applied. The Young modulus is E=0.05 Mpa and the Poisson's ratio is set to 0.4.

TABLE 2. Coefficients used to define the ligaments constitutive laws [START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF]. The rupture strengths are taken from [START_REF] Yoganadan | Biomechaics of the cervical spine part 2. cervical spine soft tissues responses and biomechanical modelling[END_REF]. For remaining data see Appendix A. (1997). This temporal analysis allows us to validate the model in accordance with the classic validation procedures systematically chosen in the literature [START_REF] Dauvilliers | Modélisation Tridimensionnelle et Dynamique du Rachis Cervical[END_REF], [START_REF] Jager | A Global and a Detailed Mathematical Model for Head-Neck Dynamics[END_REF], [START_REF] Astori | Multy-Body Numerical model of the Human Neck[END_REF]). The advantage of the N.B.D.L tests is that they are well instrumented tests carried out on volunteers and quite violent (15g for 100 ms). First of all, they constitute an interesting source of validation of the head kinematics. Of course, the same mechanical properties of the model were retained for the four impact simulations. The imposed speeds at T1 are given in figures 6 to 9. The evaluation of the model is then obtained by superposing the acceleration and the displacement of the anatomical centre of the head (AC). In figures 10 through 13 we report the results obtained with the four impact configurations (front, rear, oblique and lateral). In order to compare the response of the model with the experimental data, a quantitative analysis was carried out for the four impacts (frontal, rear, oblique and lateral). Each curves were sampled at 2000 Hz like the experimental data. This statistical method consists then to determine the number of points at the interior of the experimental corridor. The values obtained in term of percentage are given in appendix C. We can observe in figure 10b a good correlation in terms of y-angular acceleration (74.3%).

α 1 α 2 α 3 Ligament ε 1 /ε 3max F 1 /F 3max ε 2 /ε max F 2 /F 3max ε 3max C2-C5 F 3max C2-C5 ε 3max C5-T1 F 3max
55% of the response in terms of linear accelerations is include in the experimental corridor (figure 10a and10c). In addition, the model correctly reproduces displacement along X-axis to 67.9% (figure 10d).

One can nevertheless observe that displacement along Z-axis is underestimated (9%) as illustrated in figure 10f.

The details of the statistics results is presented in table C1 (Appendix C)

The view shown in figure 10g illustrates realistic kinematics with regard to the inclination of the column. Ewings et al. 1968Ewings et al. , 1977)).

Rear impact. This validation, less well instrumented than that carried out by the N.B.D.L, nevertheless allows us to estimate the behavior of the model during a rear impact. Figures 11a to 11e compare the model and the cadaver responses in terms of accelerations, displacement and rotation of the AC.

The FEM correctly reproduces rotation along the Yaxis (100%), displacement of the centre of gravity of the head following Z-axis (70.2%) as well as the angular acceleration along the axis Y (60.3%).

However, only 5.8% of the points calculated concerning displacement along axis X is inside the corridor (figure 10c). All results are given in table C4 (Appendix C). The configuration of the finite element model for the rear impact is shown in figure 11f. Oblique impact. Kinematics' response under an oblique impact is very close to that of the volunteers throughout the duration of the impact (66%, table C3 Appendix C). Only the z-axis displacement is slightly underestimated (27.3%) as illustrated in figure 12l. , Ewings et al. 1968, Ewings et al. , 1977)).

Lateral impact. In the same way as for the oblique impact configuration, there is globally a good correlation between the FEM and the volunteers throughout the duration of the impact (73.3%) as it can be observed in figure 13 and calculated in table C2 (Appendix C). , Ewings et al. 1968, Ewings et al. , 1977)).

In view of these results, it is commonly accepted that the model correctly reproduces the behaviour of the head-neck system in spite of the wide corridors which allow non-significant acceleration peaks or inclines. Figure 14 clearly illustrates that these "corridor validations" allow an FEM to have several fundamentally different responses that absolutely do not translate the behaviour. It is, however, impossible in the temporal domain to analyse what mechanical significance they represent.

Experimental modal analysis of the head neck system

In this section, modal characterization of the headneck system in vivo is described for five healthy volunteers (25-48 years) without cervical disease 2 . All subjects underwent a medical check prior to the experiments, and were asked if any symptoms of any kind have appeared after the experiments. None of them complained any discomfort.

The experimental device, represented in figure 15, consists on a pendulum (length of 0.5 m) articulated on a gantry which impacted frontally the volunteers' forehead with a basketball. The mass of the impactor is about 3.6 kg and the angular displacement of 20° can produce an impulsive force of 60 N to 150 N and head accelerations of 2g to 3g. The volunteer is seated on a rigid seat without headrest. His shoulders are strongly maintained against the backrest of the seat in order to avoid the contribution of the torso movement. In this experimental configuration, the volunteers have been asked to close eyes and to have the muscles of the neck relaxed during the entire test, in order to avoid the muscles contribution. It is hypothesized that the head motion remains in the sagittal plan with an amplitude sufficiently small (a few degrees) that the applied frontal force and the recorded linear acceleration can be assumed as unidirectional in the antero-posterior direction. The head acceleration is measured using nine accelerometers (Entran EGA ±10 g) arranged in the well-known 3-2-2-2 (figure 16) in order to calculate the linear component of the head acceleration at vertex and at atlanto-occipital joint. The impulsive force is recorded using a force sensor (PCB 208A02 11.432 mV/N).

2 The experimentation being non invasive and done within normal human physiological limits (the impacts to the head being largely under tolerance limits) our methodology was conducted in accordance with the practice of the responsible governing authority as described by the Ethics Committee of the French Centre National de Recherche Scientifique (CNRS). Figure 18 illustrates the temporal signals of force input and acceleration output at vertex level. After impact, the transfer function between force and point S acceleration is estimated in terms of apparent mass. Special attention has been paid to the management of noise, as well as the checking of linearity, ergodicity and the stationary nature of the signals.

In order to minimize noise linked to the digitizing effects, the Standard Normalized Error is calculated by repeating each experiment ten times for each volunteer. The estimated transfer function and the standard deviation can then be calculated with a 95% confidence. However, the first step is to check the linearity of the head-neck system by figuring the coherence function between the input force signal (x(t)) and the output acceleration signal (y(t)). The equation of the coherence function can be written as : Where G XX , G YY and G XY are the autospectrums and interspectrum of the signals obtained in the frequency domain [START_REF] Bendat | Random Data: Analysis and measurement procedure[END_REF].

The linearity of the system is generally accepted if the coherence function remains between 0.9 and 1 in the analysed frequency range. The response signal contains not only the response due to the measured excitation, but also the response due to the ambient random excitation. We can therefore characterize this typical measurement as having noise in the measured output signal. Using the principle of least squares, to minimize the effect of noise at the output, the best Frequency Response Function (FRF) estimator is :

1 ˆ( ) ( ) ˆ( ) ( ) . ( ) j w XY XX G w H w H w e G w -Φ = = (4)
Accordingly, the transfer function between force and accelerations have been calculated in terms of apparent mass by using equation (4) for a total of five volunteers. The five subjects are all males with different masses and sizes. Figure 20a represents the average of the five transfer functions between force and point S acceleration with their standard deviations. Figure 20b shows that the coherence functions are close to 1 in the analysed frequency range, which proves the linear behaviour of the headneck system under the conditions of the study.

All the subjects present a similar frequency response characterized by two natural frequencies : the first mode at 1.3 ± 0.1 Hz and the second mode at 8 ± 0.7 Hz. The transfer function between atlanto-occipital junction and force for all subjects was calculated but not reported. Therefore the simplest model which can simulate this transfer function is a two mass system connected with a set of springs and dashpots. This is provided by the classical two pivots neck model. This two pivots model is justified by the studies of Wen andLavaste (1993a and1993b) which explain the main mobilities of the head-neck system are located at the C7-T1 jonction and the atlanto-occipital joint.

A single punctual transfer function between acceleration at vertex and the input force at forehead cannot contain all information relating to a system with two degrees of freedom. A second transfer function is needed in order to extract the mode shape relating to each identified natural frequency. The horizontal linear acceleration at atlanto-occipital joint (point O H ) is selected for this purpose and the transfer function between this parameter and the input force is established in a way similar to the transfer function between point S acceleration and force. We have represented the transfer function in terms of apparent mass for each volunteer and tabulated the frequencies responses in figure 19. The mode shapes obtained with the imaginary part of the dynamic stiffness, are drawn in figure 21a and the extracted eigenvector are shown in figure 21b where we can observe clearly that the first mode at 1.3 ± 0.1 Hz is an extension mode and the second mode at 8 ± 0.7 Hz a retraction mode (figure 21c). It can be remarked that the standard deviation on natural frequencies obtained for the five volunteers is very low for the biomechanical domain as stipulated in Table 4 (15% for the first natural frequency and 17.5 % for the second natural frequency). In figure 20, the error on the modulus and phase is explained by the intersubject variability (head and neck masses, length, etc.). These results constitute new validation parameters for the following FE model validation in the frequency domain.

RESULTS

Numerical modal analysis of the head-neck system

The above descript experimental modal analysis was performed numerically by simulating the experience with the head-neck FE model followed by a results analysis in the frequency domain as illustrated in figure 22. The experimental results presented in this modal analysis correspond to the response of the subject modelled (RECVS01). We assumed that the first thoracic vertebra was fixed and reproduced the impact load in its amplitude and duration. To simulate the force of the impact between the ball and the head a Young's modulus of E=0.3 Mpa (ν=0.49) was implemented for the modelling of the ball and an initial horizontal speed of 0.6 m/s. The acceleration at the vertex and at the occipital joint was computed over 5 s.

The superposition of the experimental and digital curves in terms of force and acceleration is illustrated in figure 23b. The conformity of the simulation with the experiment is based on the ball-head interaction force 23a. After calculation of the acceleration at the vertex and at the atlanto-occipital joint the same signal processing is carried out as in the experimental tests between the force and the acceleration (Matlab software). In order to analyse the frequency response of the model, we therefore expressed the amplitude in terms of apparent mass amplitude, phase and imaginary part.

(a)

According to the amplitude of the apparent mass, reported in figure 24, we are able to assert that model does not faithfully reproduce the behaviour of the head-neck system. This frequency response establishes that the model reproduces more or less correctly the first mode, namely flexion-extension at a natural frequency of 2.8 Hz as against 1.4. Hz for the volunteer. On the other hand, as far as the second mode is concerned, namely the retraction of the neck or the translation of the head, the calculation shows

(b)
that the model has a natural frequency of 40 Hz as against 8.8 Hz in-vivo as illustrated with the imaginary part in figure 25. This poor result can be explained by the fact that the temporal validation, under front and rear impacts, focused on the flexionextension motion of the head and less on its translation. This latter motion is in fact much more difficult to analyse from the temporal data at the very beginning of the impact. It is therefore understandable that the modal behaviour of the neck FE model is realistic for extension motion of the head (100% error), but reproduces very badly the second mode relating to retraction (350% error). Moreover, in view of the apparent mass, we can see that the amplitudes are clearly too low, which expresses a too high a damping of the ligaments and in particular in the upper cervical spine. It should be noted that the ligament damping is open to discussion in the FEM of the neck for the simple reason that no experimental data are available. In the literature it is the overall trajectory of the model during an impact that serves to adjust the value of this damping.

In order to reproduce a realistic modal behaviour of the neck FEM, characterised by the two natural frequencies (1.4 Hz and 8.8 Hz) of the RECVS01 volunteer , a parametric study concerning the rigidity as well as the dumping has been performed. It appeared that the model whose frequency behaviour comes closest to that of the volunteer is obtained with a reduction in the rigidity of the upper ligaments of 20% with a damping of 9 Nm/s against 900 Nm/s for the initial model, and a damping of the lower ligaments remaining at 900 Nm/s. The apparent mass of the optimised model is superposed with that of the volunteer in figure 26.

In spite of a difference between the model and the volunteer, the frequency response of the model presents resonance frequencies very close to the volunteer. So, after optimisation the calculated values obtained from our FEM were f 1 =1.6Hz (against 1.4Hz) and f 2 =8Hz (against 8.8Hz) respectively corresponding to a 14% difference in the first mode and a 9% difference in the second mode.

This result demonstrates the importance of the mechanical properties of the ligaments and in particular their damping coefficients, which have a major influence on the resonance frequencies as well as on the volunteer's apparent mass amplitude. As the signals recorded during the temporal validation are too complex, it is quasi-impossible to observe the influence of these parameters on the retraction phenomenon. Only a modal analysis will finally have allowed us to adjust this parameter in a realistic way. However, we can observe in figure 27, in view of the imaginary part at the head-neck junction, that the amplitude is still too low, which expresses a slightly too low flexion motion possibly due to approximation in head mass and inertia estimation. The modal analysis being restricted to the linear domain, the optimised model was finally confronted with a classic validation in the temporal domain by the simulation of the four impacts simulated earlier with the initial mechanical properties. The goal of this study was not to fit the temporal response but to demonstrate the compatibility of a frequency approach and temporal approach. The results of this temporal domain check of the frequency optimised model are reported in figures 28 to 31. Statistical values have been calculated in Appendix C.

For the frontal impact, only the displacement in the z plane is slightly underestimated (26% against 9% for the initial model) as illustrated in figure 28f. Globally, 45.6% of the FEM response is in the experimental corridor Concerning the rear impact, responses of the FEM are lower than cadavers for xdisplacement (36.9%) and higher for y-rotation (50%). In addition for the oblique impact the globally response of the model is very similar to the volunteers as illustrated in figure 30 with 62.1%. Finally for the lateral impact there is a good correlation between the model and the volunteers with a statistical value about 73.3% (figure 31).

Temporal results of the neck FEM during the frontal impact after its frequency validation. Temporal results of the neck FEM during the rear impact after its frequency validation. [START_REF] Prasad | Biofidelity of Anthropomorphic Test Devices for Rear Impact[END_REF].

Temporal results of the neck FEM during the oblique impact after its frequency validation. ( (

(b) (c) (a) (d) (e) (f) (g) h) i) (j) (k) (l) FIGURE 30.
Temporal results of the neck FEM during the oblique impact after its frequency validation : X-axis (a), y-axis (b) and z-axis (c) linear accelerations of the AC respectively, x-axis (d), y-axis (e) and z-axis (f) angular accelerations of the AC respectively, x-axis (g), y-axis (h) and z-axis (i) rotation of the AC respectively and x-axis (j), y-axis (k) and z-axis (l) displacement of the AC respectively (N.B.D.L. Ewings et al. 1968Ewings et al. , 1977)).

Temporal results of the neck FEM during the lateral impact after its frequency validation.

(b) (c) (a) (d) (e) (f) (g) (h) (i) (j) (k) (l) FIGURE 31.
Temporal results of the neck FEM during the lateral impact after its frequency validation : X-axis (a), y-axis (b) and z-axis (c) linear accelerations of the AC respectively, X-axis (d), y-axis (e) and z-axis (f) angular accelerations of the AC respectively, X-axis (g), y-axis (h) and z-axis (i) rotation of the AC respectively and X-axis (j), y-axis (k) and z-axis (l) displacement of the AC respectively (N.B.D.L. Ewings et al. 1968Ewings et al. , 1977)).

DISCUSSION

The main discussion concerning the method is the hypothesis of linearity, doe to the assumption made at the transfer function definition level, but also in relation to the low impact energy involved in the experimental impact. It is therefore important to remember that the methodology is well designed to describe the neck behavior for low energy impact, or before non-linearity. Under these restrictions, how can modal analysis techniques, inform us about the complex properties of the human neck? Let us first recall that linear behavior has been checked in our experiments through the coherence function that remains between 0.9 and 1. Results illustrate the importance of mass distribution and damped elastic properties at the beginning of the motion. Resonance frequencies and mode shapes give the dynamic deformation initialization that may eventually continue until non-linearity appears in case of energetic impact.

The experimental vibration study under impulsive condition led to the identification of two natural frequencies at 1.3±0.1 Hz and 8±0.7 Hz associated with two separate deformation mode shapes. The new issue at this level is that the first mode at 1.3±0.1 Hz is associated with the neck extension and the second mode at 8±0.7 Hz a anterior-posterior retraction of the neck. It is important to mention here that five more human volunteer males have been submitted to the same test procedure and conduced to very similar results.

To the authors knowledge it is the first time that a head-neck FE model is validated against modal characteristics recorded in vivo. It was possible to fit both natural frequencies and the related mode shape by drastically modifying the ligament rigidity and dumping, two mechanical parameters reported in the literature with a very wide discrepancy.

With regard to the temporal validation of the FEM, we compared the response of our model with those of the literature. The figure B1 (Appendix B) superimposes the various responses of the models for the frontal impact. A statistical analysis was carried out for each authors (Appendix C, C1 Concerning the oblique impact, the response of the optimized model is better than the initial model in terms of displacement, however we can note a reduction in the statistics for the angular accelerations. Globally, the response of the model remains acceptable with 66% of fidelity for the initial model and 62.1% for the model optimized in the frequency domain.

Finally, the results obtained in rear impact with the initial model and the optimized model do not present a great difference (53.4% and 51.9% respectively). However, we can note that the optimized model answers better in term of x-displacement (36.9% against 5.8%).

Thus, it appears that this model improvement or model optimization in the frequency domain has no significant influence on the temporal validation but permitted the adjustment of the anterior-posterior retraction mode. This illustrating the difficulty to validate correctly models under impact condition only in the temporal domain.

CONCLUSION

This study proposes the construction of a very close geometrical model of the cervical column of a given volunteer subject. With the mechanical parameters available from the literature it was possible to validate this model in the temporal domain as regards to the kinematics of the head in rear, front, lateral and oblique impact situations. The lack of precision in the mechanical properties of the models and the "width" of the validation corridors led us to undertake a new characterisation of the cervical column in vivo based on modal analysis. This modal analysis based on the recording of the apparent mass in an impulse situation is restricted to the sagittal plane and reveals for the first time a flexion-extension mode at 1.3 Hz ± 0.1 and a retraction or translation mode at 8 Hz ± 0.7.

The option was then taken to carry out a numerical modal analysis of the model reproducing the experimental analysis. It appeared that for the initial model's mechanical parameters the flexion mode was appropriately reproduced, but the retraction motion was 16 times too rigid since its natural frequency was 4 times higher. An optimisation of the model in the frequency domain was obtained by reducing the damping and the rigidity of the upper ligaments. The two natural frequencies of the head-neck system were then faithfully reproduced by the model as well as the related mode shapes.

A classical temporal validation of the frequency domain optimised neck FE model was also performed and revealed no significant difference with the initial model. This result illustrates how difficult and how inefficient it is to validate complex systems against only temporal responses to impacts.

APPENDIX A TABLE A1. Material properties of the lower ligaments [START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF][START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF].

Model [START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF]

FIGURE 2 .

 2 FIGURE 2. Ligamentary system of the lower cervical spine (a) and of the upper cervical spine (Atlas-Axis) (b).

FIGURE 3 .

 3 FIGURE 3. Surface meshing of the cervical column (C1-T1) with its ligamentary system.

FIGURE 4 .

 4 FIGURE 4. Complete finite element model of the headneck system including modelling of the muscles using volume elements.

FIGURE 5 .

 5 FIGURE 5. Behaviour laws of the anterior longitudinal ligaments, ligament (ALL C2-C5), posterior longitudinal ligament (PLL C2-C5), flaval ligament (FL C2-C5), interspinous ligament (ISL C2-C5), capsular ligament (CL C2-C5).[START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF] and[START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF].

FIGURE 6 .FIGURE 8 .

 68 FIGURE 6. Imposed speed at T1 in the case of a front impact.

Frontal impact .

 impact Figures 10a to 10f show the response of the finite element model comparing to the volunteers in terms of accelerations, displacement and rotation.

  FIGURE 10. Results of the frontal impact : (a) X-axis linear acceleration of the anatomical centre (AC) of the head, (b) Y-axis angular acceleration of the AC, (c) Z-axis linear acceleration of the AC, (d) X-axis displacement of the AC, (e) Rotation of the AC, (f) Z-axis displacement of the AC and (g) Kinematic response of the finite element model to a frontal impact (N.B.D.L.,Ewings et al. 1968Ewings et al. , 1977)).

  FIGURE 11. Results of a rear impact : (a) Resultant acceleration of the AC, (b) angular acceleration of the AC., (c) X-axis displacement of the AC, (d) Rotation of the AC, (e) Z-axis displacement of the AC and (f) Kinematic response of the finite element model to a rear impact[START_REF] Prasad | Biofidelity of Anthropomorphic Test Devices for Rear Impact[END_REF]).

  FIGURE 12. Results of an oblique impact : X-axis (a), y-axis (b) and z-axis (c) angular accelerations of the AC respectively, Xaxis (d), y-axis (e) and z-axis (f) linear accelerations of the AC respectively, X-axis (g), y-axis (h) and z-axis (i) rotation of the AC respectively, X-axis (j), y-axis (k) and z-axis (l) displacement of the AC respectively and (m) Kinematic response of the finite element model to an oblique impact (N.B.D.L.,Ewings et al. 1968Ewings et al. , 1977)).

  FIGURE 13. Results of a lateral impact : X-axis (a), y-axis (b) and z-axis (c) angular accelerations of the AC respectively, X-axis (d), y-axis (e) and z-axis (f) linear accelerations of the AC respectively, X-axis (g), y-axis (h) and z-axis (i) rotation of the AC respectively, X-axis (j), y-axis (k) and z-axis (l) displacement of the AC respectively and (m) Kinematic response of the finite element model to a lateral impact (N.B.D.L.,Ewings et al. 1968Ewings et al. , 1977)).

FIGURE 14 .

 14 FIGURE 14. Example of three different dynamic responses, all of them acceptable within a corridor.

FIGURE 15 .

 15 FIGURE 15. Experimental test device for the modal analysis of the head neck system.

FIGURE 16 .

 16 FIGURE 16. Detail of the 3D accelerometer setup fitted to the volunteer's head.

Force

  and acceleration signals are digitised via a National Instrument (NI) data acquisition and signal conditioning is provided with a PXI-6070E 12 bit card. Signal acquisition is performed under LabView (NI) program and data processing is written using Matlab software. Antero-posterior linear accelerations are calculated at the vertex level (point S) as well as at the atlas to occipital jonction point O H , as illustrated in figure 17.

FIGURE 17 .

 17 FIGURE 17. Representation of the head-neck system in the sagittal plan.

FIGURE 19 .

 19 FIGURE 19. Representation of the five apparent mass at vertex and at atlanto-occipital joint.

FIGURE 20 .

 20 FIGURE 20. (a) Average of the five transfer functions between vertex acceleration and force, and (b) Coherence function.

FIGURE 22 .

 22 FIGURE 22. (a) Experimental configuration of the experimental modal analysis, and numerical replication with the finite element model of the neck (b).

  FIGURE 23. (a) Experimental and digital interaction force between the head and the ball and (b) acceleration curves for the head at the vertex.

FIGURE 24 .

 24 FIGURE 24. Apparent mass and of the neck FEM with the identical mechanical properties as during the multidirectional temporal validation superimposed to the volonteer one.

FIGURE 25 .

 25 FIGURE 25. Superimposition of the numerical and volunteer imaginary part of the transfer function at the vertex and atlanto-occipital junction. It appears clearly that the second natural frequency is over estimated by the model.

.

FIGURE 26 .

 26 FIGURE 26. Apparent mass and of the optimise neck FEM superimposed to the volonteer one.

FIGURE 28 .

 28 FIGURE 28. Temporal results of the neck FEM during the frontal impact after its frequency validation: (a) X-axis linear acceleration of the anatomical centre AC of the head, (b) Y-axis angular acceleration of the AC, (c) Z-axis linear acceleration of the AC, (d) X-axis displacement of the AC, (e) Rotation of the AC and (f) Z-axis displacement of the AC.

FIGURE 29 .

 29 FIGURE 29. Temporal results of the neck FEM during the rear impact after its frequency validation : (a) Resultant acceleration of the AC, (b) angular acceleration of the AC., (c) X-axis displacement of the AC, (d) Rotation of the AC and (e) Z-axis displacement of the AC[START_REF] Prasad | Biofidelity of Anthropomorphic Test Devices for Rear Impact[END_REF]).

TABLE 1 .

 1 Meshing criteria allowing the calculation time to be optimised.

	Criterion	Values
	Length min.	2.25 mm
	Length max.	3 mm
	aspect ratio	[1-2]
	warpage	[0-5]
	angle quad (°)	[70-110]

angle tria (°)

[50-80]

Jacobien [0.7-1]

% of trias 6 FIGURE 1. 3D reconstruction of the spine based on scanner sections in stereolithographic format.

C5-T1

	ALL	0.21	0.11	0.78	0.87	0.308	92.8	0.354	145.2
	PLL	0.25	0.12	0.77	0.89	0.182	71.1	0.341	188.2
	FL	0.28	0.21	0.76	0.88	0.77	121.5	0.884	129.1
	ISL	0.3	0.17	0.75	0.87	0.609	38.6	0.681	38.6
	CL	0.26	0.15	0.76	0.88	1.41	119.7	1.16	181.1
	Average for the upper	0.26	0.15	0.76	0.88	1	-	1	-
	cervical spine

TABLE 3 .

 3 Inertial properties of the cervical vertebrae and head applied to the centres of gravity.

Name Mass [Kg] Ixx [Kg.m²] Iyy [Kg.m²] Izz [Kg.m²]

	T1	-	-	-	-
	C7	0.22	2.2	2.2	4.3
	C6	0.24	2.4	2.4	4.7
	C5	0.23	2.3	2.3	4.5
	C4	0.23	2.3	2.3	4.4
	C3	0.24	2.4	2.4	4.6
	C2	0.25	2.5	2.5	4.8
	C1	0.22	2.2	2.2	4.2
	Head	4.69	181	236	173

Temporal validation of the finite element model

A temporal validation of the new model has been carried out in comparison to the N.B.D.L tests

[START_REF] Ewing | Dynamic response of the head and neck of the living human to -Gx impact acceleration[END_REF]

) with front, oblique and lateral impacts as well as a rear impact validation based on Prasad's tests on cadavers

Table 4 :

 4 Eigenfrequencies of the five volunteers and the average.

		Mode 1	Mode 2
	RECVS01	1.4 Hz	8.8 Hz
	RECVS02	1.7 Hz	8.3 Hz
	RECVS03	1 Hz	7.8 Hz
	RECVS04	1.2 Hz	8.4 Hz
	RECVS01	1.2 Hz	6.9 Hz
	Means	1.3±0.1 Hz	8±0.7 Hz
	Errors	15 %	17.5 %
	(a)	(b)

table )

 )

	displacement of the CG of the head are not available,
	but these kinematics' parameters are most difficult to
	validate.
	Moreover, during the experimental test the initial
	condition at the first thoracic vertebrae was not
	clearly estimated. Ewing et al. (1969) reported that a
	rotation appears during the test that can explain the
	underestimated result for the z-axis displacement (9%
	for the initial model and 26% for the optimized
	model).
	The figure B2 (Appendix B) shows the various
	curves of validations of several authors (Dauvilliers
	(1994), De Jager et al. (1996) and Astori et al.
	(1998)) for a lateral impact. No difference is
	observed between the initial model and the model
	optimized in frequency domain (73.3%). Moreover
	the response of our model is better than the authors'
	responses for this impact (Dauvilliers' model 39.3%,
	De Jager's model 58.6% and 62.8% for the model of
	Astori).
	. The
	comparison of the statistical data in term of
	accelerations shows a better response of our model
	(initial and optimized) with 61.7% for the initial
	model, 53.9% for the optimized model, 34.3% for the
	Dauvilliers' model, 48.9% for the De Jager's model,
	51% for the Astori's model and finally 36.85% for the
	Happee's model. Globally, the models of Astori and
	De Jager's model present better statistics, but it
	should be noted that for these two authors the

, Yoganandan et al. 2001) Ligament Abbreviation

	APPENDIX A Continued.
			Level	Yoganadan (Exp)	Initial length (Exp)	Average of length (model)	Number of springs		α 1		α 2		α 3
				Fc [N] Dc [m] Lo [mm] Lo [mm]	-	F [N]	D [mm]	F [N]	D [mm]	F [N]	D [mm]
			C2-C3				5.37	7	0.42	0.34	3.29	1.29	3.78	1.65
	Anterior		C3-C4	92.76	5.79	18.8	7.15	7	0.55	0.46	4.38	1.72	5.04	2.20
	ligament longitudinal	ALL	C4-C5 C5-C6				5.95 5.17	8 7	0.40 0.64	0.38 0.38	3.19 5.09	1.43 1.43	3.67 5.86	1.83 1.83
			C6-C7	145.2	6.47	18.3	5.69	10	0.49	0.42	3.92	1.57	4.15	2.01
			C7-T1				7.57	12	0.55	0.56	4.35	2.09	5.00	2.68
			C2-C3				4.91	5	0.44	0.22	3.26	0.69	3.67	0.89
			C3-C4	71,07	3,45	19	2.88	6	0.21	0.13	1.59	0.40	1.79	0.52
	Posterior ligament longitudinal	PLL	C4-C5 C5-C6				3.46 3.24	6 6	0.26 0.68	0.16 0.28	1.92 5.05	0.48 0.85	2.16 5.68	0.63 1.10
			C6-C7	188,16	6,10	17.9	3.11	7	0.56	0.26	4.16	0.82	4.67	1.06
			C7-T1				5.34	8	0.84	0.45	6.24	1.40	7.02	1.82
			C2-C3				5.18	12	1.29	1.12	5.42	3.03	6.17	3.98
			C3-C4	121.44	6.54	8.5	6.83	14	1.46	1.47	6.13	3.99	6.97	5.26
	Flavum ligament	FL	C4-C5 C5-C6 C6-C7	9,370 129,09	10.6	6.72 4.88 4.71	12 12 13	1.68 1.04 0.92	1.45 1.21 1.17	7.04 4.36 3.88	3.93 3.28 3.17	8.00 4.95 4.41	5.17 4.31 4.16
			C7-T1				7.38	10	1.89	1.83	7.90	4.96	8.98	6.52
			C2-C3				2.44	24	0.26	0.94	1.55	2.74	1.75	3.61
			C3-C4	10,24 119,63	6.92	3.39	28	0.31	1.30	1.84	3.81	2.09	5.01
	Capsular ligament	CL	C4-C5 C5-C6				3.38 3.72	30 28	0.29 0.53	1.3 1.12	1.71 3.15	3.80 3.28	1.95 3.58	5.00 4.31
			C6-C7	7,79	181,05	6.72	2.64	31	0.34	0.79	2.02	2.33	2.29	3.06
			C7-T1				3.84	34	0.45	1.16	2.68	3.38	3.04	4.45
			C2-C3				11.29	5	1.42	2.06	7.29	5.17	8.38	6.87
			C3-C4	6,33	38,61	10.4	9.94	5	1.25	1.81	6.42	4.54	7.38	6.05
	Interspinous ligament	ISL	C4-C5 C5-C6				7.91 14.04	5 4	0.99 2.32	1.44 2.86	5.11 11.9	3.61 7.17	5.87 13.68	4.81 9.56
			C6-C7	6,74	38,59	9.9	8.74	6	0.96	1.78	4.94	4.46	5.67	5.95
			C7-T1				14.70	6	1.62	3.00	8.30	7.50	9.55	10.01

TABLE A2 .

 A2 Material properties of the upper ligaments[START_REF] Chazal | Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction[END_REF][START_REF] Yoganandan | A. Single rear impact produces lower cervical spine soft tissue injuries[END_REF]).

	L igament	Level	Yoganadan (Exp)	Value implement	Average of length (model)	Number of springs	Model (Chazal et al. 1985, Yoganandan et al. 2001) α1 α 2 α3
			Fc [N]	Dc [mm]	Fc [N]	Lo [mm]		D [mm]	F [N]	D [mm]	F [N]	D [mm]	Fc [N]
	Capsular ligament	C2 -C1	314	9.3		3,83	28	2,34	0,86	6,84	5,03	9,00	5,71
	Ligament posterior	C2 -C0			83	40,38	3	5,20	4,15	15,20	24,35	20,00	27,67
	Atloidien -axoidien anterior	C2 -C1	263	11.8		11,93	6	3,07	6,58	8,97	38,57	11,80	43,83
	Transv ersal ligament	C2 -C1			200	5,40	6	1,30	5,00	3,80	29,33	5,00	33,33
	Ligament flavum	C2 -C1	111	9.6		13,62	14	2,50	1,19	7,30	6,98	9,60	7,93
	Ligament posterior	C2 -C1			71	22,62	3	1,30	3,55	3,80	20,83	5,00	23,67
	Transversal axoidien	C2 -C1			436	9,78	7	3,25	9,34	9,50	54,81	12,50	62,29
	Occipital transversal	C2 -C0			250	8,34	4	1,30	9,38	3,80	55,00	5,00	62,50
	Ligament posterior	C1 -C0	83	18.1		16,74	4	4,71	3,11	13,76	18,26	18,10	20,75
	Occipito -atloidien anterior	C2 -C0			232	44,14	6	4,91	5,80	14,36	34,03	18,90	38,67
	Ligament flavum	C0 -C1			83	15,23	5	4,71	2,49	13,76	14,61	18,10	16,60
	Alar ligament	C0 -C2	357	14.1		7,28	6	3,67	8,93	10,72	52,36	14,10	59,50
	Membrane tectoria	C0 -C2	76	11.9		27,48	3	3,09	3,80	9,04	22,29	11,90	25,33
	Capsular ligament	C1 -C0	320	9.9		5,26	26	2,42	0,92	7,07	5,42	9,30	6,15
	Apical ligament	C2 -C0	214	8		3,36	2	2,08	16,05	6,08	94,16	8,00	107,00
	Occipito -atloidien lateral	C1 -C0			100	8,66	6	5,20	2,50	15,20	14,67	20,00	16,67

AKNOWLEDGMENT

The authors wish to thank I.R.C.A.D. (Institut de Recherche contre les Cancers de l'Appareil Digestif) for their contribution and the French ministry of research for their financial support (PREDIT n°00A008601).

FIGURE B1

. Different models' responses during the frontal impact [START_REF] Dauvilliers | Modélisation Tridimensionnelle et Dynamique du Rachis Cervical[END_REF], [START_REF] Astori | Multy-Body Numerical model of the Human Neck[END_REF]), DeJager et al. (1996), [START_REF] Happee | A Mathematical Human Body Model for Frontal and Rearward Seated Automotive Impact Loading[END_REF], and the initial model).

APPENDIX B Continued.

FIGURE B2

. Different models' responses during the lateral impact [START_REF] Dauvilliers | Modélisation Tridimensionnelle et Dynamique du Rachis Cervical[END_REF], [START_REF] Astori | Multy-Body Numerical model of the Human Neck[END_REF]), DeJager et al. (1996), and the initial model).

APPENDIX C.