Raquel Sendon

Juana Bustos
email: jbustos@msps.es

José Juan Sánchez

Mª Perfecto Paseiro

Eugenia Cirugeda

Sánchez

Raquel Sendón

Jose Juan Sánchez

Mª Eugenia Cirugeda

VALIDATION OF A LIQUID CHROMATOGRAPHY-MASS SPECTROMETRY METHOD FOR THE DETERMINATION OF PRIMARY AROMATIC AMINES MIGRATING FROM COOKING UTENSILS, AND ITS APPLICATION TO REAL SAMPLES

Keywords: Food contact materials, Food simulants, Migration Food contact materials, Primary aromatic amines, Migration, LC-MS/MS, Nylon cooking utensils

whether they are published or not. The documents may come

MS/MS method for the determination of eight primary aromatic amines (m-PDA, 2,6-TDA, 2,4-TDA, 1,5-DAN, ANL,4,4, in the aqueous food simulant 3 % acetic acid (w/v). The detection limits calculated were adequate with respect to the present legislation. The method was validated at four concentration levels (2, 5 10 and 20 µg kg -1 ). The global internal reproducibility was in the range 5.6-21.4 % (RSD R ) depending on the compound and concentration. Mean recoveries for all levels varied between 89-100%, depending on the amine. A total of 39 samples of cooking utensils were analyzed using the described method and the results obtained after the 3 rd migration test were not compliance in approximately half of the samples.

Introduction

Food contact materials are all materials and articles intended to come into contact with foodstuffs, including packaging materials but also cutlery, dishes, processing machines, containers etc. In last years, the use of plastic utensils, such as turners, whisks and spoons for cooking and frying have increased because they are cheap, unbreakable (resist high temperatures), and do not scratch other surfaces.

Most of these utensils are made of nylon, which is the generic name for polyamides that have characteristic amide groups in the backbone chain. These amide groups are very polar and can interact with each other by hydrogen bond. Because of this, and because the polyamide backbone is regular and symmetrical, nylons are often crystalline, and make very good fibers.

There are two classes of synthetic nylons. One of these is formed from cyclic monomers, for example nylon 6, which has six carbon atoms per repeat unit and is normally manufactured from the monomer ε-caprolactam. Another example in this class is nylon 12, having 12 atoms in the repeat unit, and made from the lactam of 12-amino docecanoic acid. The single index used in describing these nylons indicates the number of carbon atoms in the repeat unit. The second class of synthetic nylon is formed from various diamines and diacids, and their nomenclature involves two indices indicating the number of carbon atoms in the diamine and diacid units respectively, e.g., nylon 6,6, made from hexamethylene diamine and adipic acid. [START_REF] Gupta | Handbook of Polymer Science and Technology[END_REF]). These materials could incorporate azodyes (synthetic organic colorants with characteristic chromophoric azo groups) where primary aromatic amines (PAAs) are the starting substances. Moreover, pigments and dyes may contain aromatic amines as technical impurities. polyurethanes. These polyurethanes are used as adhesives in laminated films but also in printing inks and lacquers. If the ratio between the monomers is well balanced and the reaction is complete all monomers disappear to form the polymeric network. But if free isocyanates are present and reach the food, this leads to the formation of PAAs.

In brief, PAAs can occur in food contact articles as residual monomers, as hydrolysis products of isocyanates or as contaminants of azodyes. They may migrate or be chemically formed in foodstuffs coming into contact with these articles. Furthermore, if these food contact materials are not well manufactured or made with inappropriate raw material, chemical substances can migrate from the articles to the food and thus contribute to food contamination. Consequently, PAAs have been implicated in several alerts related to the migration from kitchen utensils made of nylon imported mainly from China. (RASFF). Some PAAs are classified as "possibly carcinogenic to humans" by the International Agency for Research on Cancer, and thus their presence in foodstuffs should be avoided [START_REF] Brede | Determination of primary aromatic amines in water food simulant using solid-phase analytical derivatization followed by gas chromatography coupled with mass spectrometry[END_REF].

European Legislation stipulates that the quantity of PAAs in food or in food simulant shall not be detectable (i.e.: 0.01 mg kg -1 of food or food simulant, for m-PDA the specification is 0.02 mg kg -1 analytical tolerance included) (EC 2007). To reach this detection limit the European Committee for Standardization has proposed a photometric method (based in a diazotisation of free PAA) to be used only for screening purposes, and if PAAs levels are above 2 µg aniline equivalents kg -1 , a confirmation method is needed. (Moteresen 2005).

The published literature shows that a number of different analytical methods have been employed for PAAs, although not necessarily applied to food contact materials. Aniline has been analysed by Bornick et al (Bornick 2001) using solid phase extraction and liquid chromatography (LC) while their iodo-derivatives by liquid-phase microextraction and gas chromatography GC); LC/MS has been selected by [START_REF] Sakai | LC-MS determination of urinary toluenediamine in workers exposed to toluenediisocyanate[END_REF] to determine toluenediamine, and LC-MS/MS by Sutthivaiyakit et al. (Sutthivaiyakit 2005) to analyze 2,4-toluendiamine, 4,4'-diaminodiphenylether 4,4'-methylenedianiline and 3,3'-dimethylbenzidine. Regarding food contact materials, Brede et al [START_REF] Brede | Determination of primary aromatic amines in water food simulant using solid-phase analytical derivatization followed by gas chromatography coupled with mass spectrometry[END_REF] applied solid-phase analytical derivatization followed by GC/MS for the determination of PAAs in water food simulant. LC with ultraviolet detection has been used in the analysis of aniline in plastic materials subjected to repeated testing [START_REF] Brede | Determination of primary aromatic amines in water food simulant using solid-phase analytical derivatization followed by gas chromatography coupled with mass spectrometry[END_REF][START_REF] Brede | Migration of aniline from polyamide cooking utensils into food simulants[END_REF]) and for the determination of eight PAAs in laminates aqueous migration solution [START_REF] Ellendt | Analysis of laminates -Determination of Isocyanate Residues and Primary Aromatic Amine Migration[END_REF].

Previous work has demonstrated the high potential of electrospray ionization MS/MS for the analysis of methylene-dianiline (MDA) and 2,4-toluenediamine (TDA) by direct analysis of some food simulants [START_REF] Palibroda | Direct analysis of some diamines in food simulants used for migration tests by electrospray ionization tandem mass spectrometry[END_REF], and subsequent work has enhanced the capabilities of LC methods coupled to MS. Moteresen et al [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF]) developed an LC-MS/MS method to determine several PAAs in aqueous food simulants. Also in a recent Food Standards Agency project [START_REF] Burch | Development and validation of an LC-MS-MS method for the determination of primary aromatic amines, and the use of this for assessment of migration models for foods with high packaging:food mass ratios[END_REF] another LC-MS/MS and its applicability to the quantification of these PAAs in food simulants, foodstuffs and in the evaluation of migration models has been described. The aim of this work was to develop a simple LC-MS/MS method for the determination of eight primary aromatic amines and to validate the method in the aqueous food simulant 3 % acetic acid (w/v). This simulant is considered to be the most aggressive for nylon. The selected amines were: m-phenylenediamine (m-PDA), 2,6-and 2,4-toluenediamine (2,6-TDA, 2,4-TDA) 1,5diaminonaphthalene (1,5-DAN), aniline (ANL),4,4,4'methylenedianiline (4,. Method validation should demonstrate that it is fit for purpose, taking into consideration the restrictions or specific migration limit (SML) established in Directive 2002/72/EC and amendments [START_REF] Ec | [END_REF], which in this case, as mentioned before, for all amines selected was not detectable. Furthermore, and in view of the numerous alerts appeared in the last years (RASFF), the developed method was applied to several real samples to evaluate the potential migration of these PAAs from nylon cooking utensils.

Materials and methods

Chemicals and reagents

Methanol was LC-MS CHROMASOLV grade, supplied by Riedel-de Haën (Seelze, Germany) and ultrapure water was prepared using a Milli-Q filter system (Millipore, Bedford, MA, USA). 101-80-4, (99.9%)], were purchased from Riedel-de Haën (Seelze, Germany), and aniline hydrochloride [ANL, CAS Registry No. 142-04-1, (99.6%)] was from Sigma (Schnelldorf, Germany).

Glacial acetic acid was purchased to Probus (Badalona, Spain), and ammonium hydroxide (solution ≥ 25 % in H 2 O) was from Fluka Chemie AG (Buchts, Switzerland). PTFE filters (0.2 µm internal diameter) used were from Waters (Milford, MA, USA).

PAAs individual stock solutions containing 500 mg L -1 were prepared in methanol. These solutions were kept protected from light in the refrigerator up to six moths. Mixed intermediate standards solutions in methanol and 3 % acetic acid (w/v) were prepared by dissolving appropriate amounts of each individual solution to yield concentrations of 5 mg L -1 and 0.1 mg L -1 respectively. Mix solution of 5 mg L -1 was prepared monthly and the 0.1 mg L -1 solution weekly. Calibration solutions in 3 % acetic acid (w/v) were prepared daily in the range of 2-20 µg L -1 . All solutions were kept at 4ºC and protected from light.

Equipment and chromatographic conditions

The LC-MS/MS system comprised an Alliance 2795 liquid chromatograph coupled to a Micromass Quattro Premier mass detector controlled by Mass Lynx software. The chromatographic separation was performed with an XTerra RP18 column (150 × 4.6 mm, 5µm particle size) at 22ºC. All was from Waters, (Milford, MA, USA).

The mobile phase was ammonium acetate 10 mM/methanol 95/5 in an isocratic mode for 0.8 min, followed by a gradient to 25% methanol for two min and another gradient to 67% methanol for forty min and finally an isocratic elution during seven min. The flow rate was 0.6 ml min -1 and the injection volume was 50 µL. MS/MS detector settings: positive electrospray ionization (ESI) mode, probe temperature 400 ºC, ionization source temperature 120 ºC, electron multiplier voltage 660 V, desolvation gas nitrogen at 700 L h -1 , cone gas nitrogen at 50 L h -1 and collision gas argon at 0.6 mL min -1 . The acquisition was in multiple reaction monitoring (MRM) See Table 1 for cone voltages and collision energies.

Infrared spectrometry (Fourier Transformed IR)

The FTIR of all samples were performed on a Perkin Elmer (Boston MA) Spectrum One apparatus running under Spectrum ® (v. 6.1.0) software and equipped with an universal attenuated total reflectance (UATR). ATR spectra were measured with a resolution of 4 cm -1 in the range 4000-650 cm -1 . Each spectrum was the result of 16 co-added scans. For identification purposes, sample spectra were compared against IR libraries (Know It All Informatics System, IR edition v. 7.9 and HaveItAll ® IR databases, all from Bio-Rad Laboratories, Inc).

Samples and method procedure

Different cooking utensils were bought in several local supermarkets. A brief description is given in Table 2. All information displayed in this table was obtained from the label (if included) or from the utensil itself. The samples were bought in two different periods: The first batch was acquired at the end of 2005 (Samples 1 to 22) and the second one in the middle of 2007 (Samples 24 to 41). In all samples, migration experiments were carried out in the same way. PAA migration was determined by immersing the cooking utensils in 3% acetic acid (w/v). Each sample was placed in a beaker or flask and filled with a given volume of simulant, covered with aluminum foil and transferred to a preheated oven. The aqueous simulant was previously heated and then put into contact with the sample, ensuring that the sample surface (that is intended to use in food contact) is immersed in the simulant (including 2 cm of handle in some cases). The contact area was estimated by geometric calculation. Migration conditions were 2 hours at 100 ºC following what is established in Directive 97/48/EC (EC 1997) and assuming a > 5 < 30 min contact time in the worst foreseeable use. Taking into consideration that these articles are intended for repetitive uses, tree migration experiments were performed using new simulant each time, and at least 24 hours were allowed to go by between each exposure.

For the LC analysis, an aliquot of the migration solution was neutralized to pH 6-7 with ammonia, filtered through a 0.2 µm PTFE syringe filter and injected into the chromatograph. In any case no clean up was applied.

Validation experiments

Calibration was performed by external calibration using series of standards of known concentrations prepared in fresh simulant, neutralized in the same way as samples and injected in duplicate or triplicate. Each calibration curve was constructed at four calibration levels ranging from 2 to 20 µg L -1 and assessed by linear regression. The curves were prepared in three no consecutive days, spread over 20 days. Residual plots were examined to detect obvious patterns and the ANOVA lack of fit test was used to assess linearity. Other parameters estimated were the correlation coefficient and the linearity value calculated according to the following equation:

            - × = b Sb Cm 1 100 %
being Sb the standard deviation of the slope and b the slope of the calibration curve.

Detection limits (DLs) were determined in accordance with the American Chemical Society (MacDougall 1980), analyzing solutions of decreasing concentration until the minimum concentration required to generate a signal-to-noise ratio of 3 was achieved. Trueness and precision of the method were determined by running spiking experiments on migration solutions with each batch of samples. Data from 12 days, over a period of three and a half months, were collected for validation. The number of withinday replicates ranged from 2 up to 10. Four fortification levels (2, 5 10 and 20 µg kg -1 ) were assayed using several migration solutions from different samples.

Between 8 and 22 data, depending on the amine and concentration, were taken for validation.

Precision (internal reproducibility) and trueness are expressed in terms of relative standard deviation and recovery for each level and amine, respectively.

Selectivity was based on the comparison of the chromatograms obtained for PAAs standards solutions, PAAs-free migration solutions from nylon utensils and spiked migration solutions.

Finally, the stability of PAAs in the food simulant was determined by the analysis of solutions of 3% acetic acid (w/v) spiked with a combination of PAAs (n=5 replicates) followed by storage for 2 hours at 100 ºC and 2 hours at 70 ºC. The mean results in each case were compared with those obtained for the same spiked solutions not subjected to the storage conditions (control solutions).

Results and discussion

Mass Spectrometry

In order to obtain the more characteristic ions (parent and daughter ions, Table 1), individual standard solutions of each PAA (5 µg mL -1 ) were infused directly to the mass spectrometer using a built-in syringe pump and acquired in full scan mode (range from 60 to 300 m/z). Cone voltages were adjusted to the maximum sensitivity for each parent ion. The ESI source parameters were also optimized, i.e. probe temperature, ionization source temperature, electron multiplier voltage, desolvation and cone gas. Once the best conditions for the parent ions were selected, daughter ion scans were acquired. Several collision voltages were tested and those yielding the more intense and stable fragments were chosen. In this way, the cone and collision voltages were fixed to obtain the highest signal/noise ratio for each transition.

The multiple reaction monitoring transitions (MRM) selected for each PAA do not differ from those reported in the literature [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF][START_REF] Burch | Development and validation of an LC-MS-MS method for the determination of primary aromatic amines, and the use of this for assessment of migration models for foods with high packaging:food mass ratios[END_REF]. In all cases the more intensive parent ion corresponds to the protonated molecular ion [M+H] + , and no other ions of significant intensities were observed. For each PAA, two different daughter ions were selected; one for quantification purposes (Table 1) and the second only for confirmation reasons. For m-PDA, ANL and 3,3-DMB the quantifier daughter ion corresponds with the loss of the primary amine group; 2,6 and 2,4-TDA loose the methyl group [M+H-CH 3 ] + ; 4,4'-DPE and 4,4'-MDA yield the [M+H-C 6 H 7 N] + ion, i.e. lose one aniline group. The case of 1,5-DAN is different; its ionization yields the [M+H] + , but also an intense [M • ] + is observed when the daughter ion scan is acquired (Figure 1), and the same occurs when the molecule looses one of its primary amine group.

Moreover, the specificity of the transitions used for quantification purposes was examined analyzing individual standard solution of each PAA. Thus, no all transitions were used all the time because this greatly influence the sensitivity and therefore ions selected as qualifiers were only monitored in those samples that exhibited results over the restriction limit. For 2,6 and 2,4-TDA the

Chromatography

Initially, a previous validated method [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF] was checked for in-house validation in order to analyze the samples, but not satisfactory results were obtained for some amines. The separation of 2,6 and 2,4-TDA was not adequate when simulant 3% acid acetic (w/v) was used and also the sensitivity of 3,3'-DMB was compromised. After several trials with different combinations of mobile phase /column, ammonium acetate 10 mM/methanol using a C18 column were selected as they produce better peak resolutions for all PAAs. The gradient used is one based in the proposed method for the LC-UV determination of these PAAs, [START_REF] Brauer | Spezifische bestimmung primärer aromatischer amine in wässrigen migraten von bedarfsgegentäden aus kunststoff[END_REF] but some adjustments were made to make it suitable for the column used in the MS/MS method. Figure 2 shows the chromatograms obtained. In spite of the possibility of scheduled acquisition for each PAA individually, acquisition of all compounds in all run time was decide because preliminary experiments on some samples exhibited unknown peaks with the transitions selected but at different retention time of the respective PAA. This could be used to try to detect/identify other compounds that are migrating or that are formed into the simulant, but this was not within the scope of the present work. Further effort is needed in this field.

Method validation

Linear regression was applied to the calibration curves, obtained in three different days. The examination of the residual plots did not show any obvious pattern. Residuals were normally distributed when tested by Shapiro-Wilks normality test (p>0.05). ANOVA test showed a high Detection limits for the individual PAAs are given in Table 3. As can be seen in the table, DLs were calculated as amount of analyte present in the food simulant (µg kg -1 ) but also they have been referred to the cooking utensil surface. Due to the different shapes of cooking utensils, each sample has its own contact-surface/simulant-volume ratio. So, in two opposite and real situations these DLs were calculated; the lowest ratio represents an utensil with low contact surface (Slotted turner, 1.114 dm 2 ) but for which a high amount of simulant was needed (425 mL) to ensure that all the contact surface was tested, and the highest ratio represents an utensil with medium surface (Sauce spoon, 1.3781 dm 2 ) but for which less simulant was required (134 mL). In the above mentioned table DLs were also calculated using the standard assumption of 6 dm 2 in contact with 1 kg of food/food simulant (ratio 6 dm 2 L -1 ). In all cases the detection limits calculated were adequate regarding the present legislation. Previously published works also reached these levels, but only in one case no pre-concentration step was needed (Motersen 2005). The present work achieves lower DLs for all amines selected (except for ANL), because less compounds are monitored at the same time, thus increasing the sensitivity. Other works [START_REF] Brede | Determination of primary aromatic amines in water food simulant using solid-phase analytical derivatization followed by gas chromatography coupled with mass spectrometry[END_REF] The global internal reproducibility, including all concentration levels and all amines (Table 4) was in the range 5.6-21.4 % (RSD R ) depending on the compound and concentration. These values are appropriate since they are below the value estimated by the Horwitz function modified by Thompson (Thompson 2000) i.e. RSD 22% for concentrations below 1.2 10 -7 . Mean recoveries were good for all the amines and at all concentrations, ranging between 93 and 100% (including all levels), and are also according to CEN TS 15356 (CEN 2006). The lowest validation level guarantees the method utility for routine analysis regarding compliance or not with Directive 2002/72/EC [START_REF] Ec | [END_REF].mThe lowest validation level guarantees the method utility for routine analysis regarding compliance or not with Directive 2002/72/EC [START_REF] Ec | [END_REF].

No certified reference material is available for the determination of PAAs migrating from cooking utensils. To prove the validity and also to support the trueness of the method, the laboratory participated in a proficiency test organized by FAPAS ® . The testing sample was acetic acid 3% containing 2,4-TDA, ANL and 4,4´-MDA with assigned values of 1.02, 16.0 and 2.00 µg aniline eq. kg -1 simulant, respectively. Thus the trueness of the method was assessed regarding these compounds. The obtained z-score according to FAPAS protocol [START_REF] Fapas® | Food analysis performance assessment scheme, Protocol for the Organization and Analysis of Data[END_REF] method. These conditions were 2 hours at 100 ºC and 2 hours at 70 ºC in 3% acetic acid (w/v). The mean results (n=5 replicates) in each case were compared with the control solutions (not subjected to the storage conditions). For all amines tested the recoveries found after 2h, 100ºC were ≥ 83%, except for 1,5-DAN (6%). Statistical evaluation of the results (t-Test, 95% confidence level) showed no difference between means before and after 2 hours at 100ºC for ANL, 4,4´-DPE, 4,4´-MDA and 3,3´DMB. After 2h, 70 ºC the recoveries found were ≥ 92 %, except for 1,5-DAN (65 %). Statistical evaluation of the results (t-Test, 95% confidence level) showed no difference between means before and after 2h at 70ºC for 2,4-TDA¸ ANL, 4,4´-DPE, 4,4´-MDA and 3,3´-DMB. Previous work reported that most of the PAAs were stable to heating (121ºC 2h) in water, 3% acetic acid and 10% ethanol, but not in olive oil [START_REF] Burch | Development and validation of an LC-MS-MS method for the determination of primary aromatic amines, and the use of this for assessment of migration models for foods with high packaging:food mass ratios[END_REF] except 1,5-DAN.

After heating and storage (121ºC plus 10 days, 40ºC) only the early-eluting PAAs (m-PDA, 2,6 and 2,4 TDA) were poorly recovered, which agrees with the results here presented.

Sample analysis

According to our knowledge only a couple of works have investigated the occurrence of PAAs from cooking utensils [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF][START_REF] Perharic | Primary aromatic amines in kitchen utensils in Slovenia[END_REF]), while different national enforcement campaigns on PAAs detected several non-compliances with the EU limit. RASFF has reported several food alerts since four years ago (2005: 22 notifications; 2006: 30 notifications; 2007: 23 notifications; 2008: 28 notifications) (RASFF). Because of this, several samples were bought in different retail stores in Madrid (Spain). The first batch was obtained at the beginning of the method development, and the second one, at the end as a complement to the first one.

Initially, and before migration experiments were carried out, contact area of each utensil was calculated and also IR spectra were obtained for all samples. IR spectroscopy is the most direct way of identification of polymers; when using the UATR accessory no sample preparation is needed.

FTIR spectra were collected just by placing the cooking utensil over the diamond and lowering the showed that the majority of samples were made of Polyamide 6 (77 % of samples tested) or Polyamide 66 (23 % of samples). The strongest peaks in the IR spectra are those corresponding to the Amide I and Amide II bands found at 1630 (C=O stretch) and 1530 cm -1 (C-N stretch and CO-N-H bend), respectively. Also, the N-H stretch in 3300 cm -1 is very intense [START_REF] Enlow | Discrimination of Nylon polymers using attenuated total reflection mid-infrared spectra and multivariate statistical techniques[END_REF].

Next step was to carry out migration experiments. As it has been mentioned in materials and methods (section 2.5), samples were exposed during 2h to 100ºC to the aqueous food simulant 3% acetic acid. Mortensen et al [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF]) also used these conditions. Because of the hydrophilic nature of these compounds and the type of polymer, 3% acetic acid was selected for migration experiments as the worst scenario regarding PAA migration. Migration in olive oil was reported elsewhere (Brede 2004, Burch and[START_REF] Burch | Development and validation of an LC-MS-MS method for the determination of primary aromatic amines, and the use of this for assessment of migration models for foods with high packaging:food mass ratios[END_REF]) to be a difficult task. Lower migration values were obtained compared with those in aqueous simulants perhaps due to degradation of PAAs or reaction with oil components. These utensils are intended for cooking, thus to be used with hot food; furthermore some of them are labeled with the expression Heat resistance 210/220 ºC. EU legislation (EEC 1982) establishes for aqueous food simulants a contact temperature of 100 ºC when the contact temperature in real use is ≥ 100 ºC. The migration time selected was bases on the assumption that contact time in real use (worst scenario) would be 30 min, therefore and according to what is established, the migration time should be 4 times 30 min, that is 2 hours. Moreover, as nylon utensils are intended for repeated use, for compliance purposes only the result of the third migration was taking into account. Results are summarized in Table 5. In some samples very high amounts of PAAs were found and dilution of the migration solutions was necessary prior the analysis. A total of 39 samples were analyzed and 54% of them did not fulfill current legislation. Furthermore from 1 st to 3 rd migration, samples 11,18,19,36,37,38,40,41 gave to the simulant an evident brown color. These samples corresponded with those that exhibited higher migration values. The two main compounds identified were 4,4'-MDA and ANL, as previously found by other authors [START_REF] Mortensen | Specific determination on of 20 primary aromatic amines in aqueous food simulants by high performance-electrospray ionizationtandem mass spectrometry[END_REF][START_REF] Brede | Migration of aniline from polyamide cooking utensils into food simulants[END_REF][START_REF] Perharic | Primary aromatic amines in kitchen utensils in Slovenia[END_REF], and as appeared in several RASFF notifications (RASFF). In some samples 2,4-TDA and 3,3-DMB were also identified but by far in less amounts than 4,4'-MDA and ANL. Always the higher values were attributed to 4,4'-MDA. It has been suggested that this compound is used in the manufacture of some types of polyamide, apparently as a comonomer, to increase the stability of the plastic at high temperatures.

Other authors suggest that 4,4`-MDA is used to produce the azodyes that give the utensils this black color [START_REF] Petersen | Memorandum for the Danish Veterinary and Food Administration: An acute case of primary aromatic amines migrating form cooking utensils[END_REF]). Slight differences were found regarding the number of samples that exceeded the SML depending on the year that were bought. Samples acquired during 2005 were 57% no compliant while those acquired in 2007 produced a 50% of no compliant results. Of all the samples giving PAAs values above the restriction limit, only one had been identified by IR as PA66 (sample 5), wich represents 11 % of all samples identified as PA66, against the 67 % of samples identified as PA6. Samples labeled as "duplicated" in table 2 were those bought at the same time, in the same store, sometimes sold as a set and apparently identical, but no information was available e.g. a batch number confirming that the two samples came from the same lot.

Duplicate samples gave quite different results, e.g. samples 3 and 18. This suggests a lack of homogeneity in these types of materials.

Although only the result of the third exposure was taking into account to compare it with the legal limits, also the first and second migration solutions were analyzed. Figure 3 shows graphically results obtained. As can be seen, for sample 11, the result obtained in the first migration is not completely shown due to the high value (6416 µg dm -2 ). In most cases the migration levels decreased between the first and second migration, otherwise for samples 17 and 22 the levels found in the second migration were slightly higher. Also a decrease in PAA migration values was observed in most of the samples (73%) between second and third migration. However, this was not the case for samples 4, 11, 18, 19, 37, 39; in particular this increase was considerable in samples 18 and 37.

It is worth noting, that in 20 out of the 22 samples exceeding the restriction limit, the values found were significantly above the limit. Therefore it could be expected that these utensils continued to be no compliant after several exposures. It has been reported elsewhere that after simulating 1-2 years of household use, the samples continued to release PAA (Mortenseen 2005).

Conclusions

A simple LC-MS/MS method able to analyze directly (without clean up) a food simulant has been developed and the low validation level achieved guarantees the method is suitable for routine analysis to check for compliance with Directive 2002/72/EC. Moreover several unknown compounds have been detected that could be related to PAAs, so further work is needed in this field. Most notifications related to cooking utensils concern products originating from third countries. The proposed method has proved to be adequate for testing compliance with European Legislation. Though the first RASFF notification concerning PAAs migration from cooking utensils was in 2004, this is not a resolved issue, as it is demonstrated by the fact that in the following years the number of alerts did not show a decreasing tendency; 22 notifications in 2005;

  122.8>105.8 which corresponds to [M+H-NH 3 ] + ion; for ANL was 93.8>50.7 i.e. [M+H-C 2 H 5 N] + due to the aromatic ring rupture; for 4,4'-MDA 199.2>76.8 (complete breaking down of the molecule arising the aromatic rings) and for 3,3'-DMB 213.2>181.1, that is [M+H-CH 3 -NH 3 ] + . In all cases higher collision voltages were required than those needed for the quantification transitions.

  p<0.001) and the lack of fit was not significant (p>0.05) for all amines in all the curves, what demonstrates the validity of the lineal model at the working range. Besides, the correlation coefficients of the calibration curves were always greater than 0.994 and the linearity values, calculated according to the equation given under section 2.6 Validation experiments were above 96%.

  have obtained lower DLs (five to ten times below) employing GC-SPE derivatization or LC-UV-SPE (Brauer and Funke 2002), but these techniques are costly and time consuming. Some authors have, nevertheless, pointed out (Morteresen 2005) that once the individual determination of PAAs has become an easy task, expressing the DLs as ANL equivalents is inappropriate due to the differences in the molecular weight of the amines. The fourth amendment /72/EC, Directive 2007/19/EC, (EC 2007) when setting restrictions for PAAs does not refer anymore to aniline equivalents.

  were: 0.1 for 2,4-TDA, 0.3 for ANL and -0.3 for 4,4-MDA. These z-scores values confirm the method suitability for the proposed objectives.Regarding selectivity, no relevant interferences were detected in the analysis of blank migration solutions (acetic acid 3%) obtained from nylon cooking utensils. Representative chromatograms of PAAs standards and a blank migration solution are given in figure2. The stability of PAAs under two possible migration conditions for nylon utensils was studied for the eight PAAs included in the

  A background spectrum of air was recorded each time. The results obtained were compared with those included in a database of commercial libraries (Biorad ® ). The results obtained

  30 in 2006; 23 in 2007 28 in 2008 and 15 notifications during the firsts weeks of 2009.

  2000. Recent trends in inter-laboratory precision at ppb and sub-ppb concentrations in relation to fitness for purpose criteria in proficiency testing.

Figure 1 :

 1 Figure 1: Daughter scan of 1,5 DAN (159 m/z).

Figure 2 :

 2 Figure 2: Chromatogram of 3 rd migration extract of sample 16, A: spiked with the PAAs mix (10 µg L -1 ) showing the quantification traces used; B: without spiking.

Figure 3 :

 3 Figure 3: Spectrum of sample 15 (black line) compared with the reference spectrum of Polyamide

Figure 3 :

 3 Figure 3: Comparison of the three migrations experiments.

Figure 1 .

 1 Figure 1.: Daughter scan of 1,5-DAN (159 m/z).

Figure 2 :

 2 Figure 2: Chromatogram of 3 rd migration extract of sample 16, A: spiked with the PAAs mix (10 µg L -1 ) showing the quantification traces used; B: without spiking.

Figure 3 :

 3 Figure 3: Spectrum of sample 15 (black line) compared with the reference spectrum of Polyamide

Table 1 :

 1 Experimental data for quantification purposes.

Table 2 :

 2 Description of the analyzed cooking utensils.

Table 3 :

 3 Number of analysis for estimation of trueness and precision

Table 3 :

 3 Detection limits.

Table 4 :

 4 Precision expressed as internal reproducibility (RSD R %) of the method in migration solutions from nylon cooking utensils.

Table 6 .

 6 Recoveries (%) of PAAs in migration solutions from nylon cooking utensils.

Table 5 :

 5 Results of migration experiments. The data showed correspond to the 3 rd exposure. All results are expressed as the sum of all compounds identified.

Table 1 :

 1 Experimental data for quantification purposes.

	Name	Abbreviation CAS Nº	Structure	Quantification MRM traces	Cone (V)	Collision (eV)
	m-phenylenediamine	m-PDA	108-45-2		108.7>91.7	20	15
	2,6-toluenediamine	2,6-TDA	823-40-5		122.8>107.8	20	18
	2,4-toluenediamine	2,4-TDA	95-80-7		122.8>107.8	20	18
	1,5-diaminonaphthalene	1,5-DAN	2243-62-1		158.8>142.8	25	20
	Aniline	ANL	62-53-3		93.8>76.8	20	18
	4,4'-diaminodiphenylether	4,4'-DPE	101-80-4		201.1>107.8	20	20
	4,4'-methylenedianiline	4,4'-MDA	101-77-9		199.2>105.9	25	25
	3,3'-dimethylbenzidine	3,3'-DMB	119-93-7		213.2>196.1	25	25

Table 3 :

 3 Number of analysis for estimation of trueness and precision

	PAAs	2 ng mL -1 5 ng mL -1 10 ng mL -1 20 ng mL -1
	m-PDA	15	16	20	9
	2,6-TDA	15	16	20	10
	2,4-TDA	15	13	20	10
	1,5 DAN	14	16	20	10
	ANL	8	16	16	10
	4,4`-DPE	15	16	22	10
	4,4`-MDA	15	16	12	10
	3,3`-DMB	9	16	22	10

Table 3 :

 3 Detection limits.

		µg L -1		µg dm -2
	PAAs	Standard	Standard ratio	Lowest ratio	Highest ratio
		solution	6 dm 2 kg -1	2.62 dm 2 kg -1	(10.28 dm 2 kg -1 )
	m-PDA	0.5	0.083	0.191	0.049
	2,6-TDA	0.5	0.083	0.191	0.049
	2,4-TDA	0.5	0.083	0.191	0.049
	1,5 DAN	1.0	0.167	0.382	0.097
	ANL	1.0	0.167	0.382	0.097
	4,4`-DPE	0.5	0.083	0.191	0.049
	4,4`-MDA	0.5	0.083	0.191	0.049
	3,3`-DMB	0.5	0.083	0.191	0.049
	TOTAL*	4.5	0.750	1.718	0.438
	*Total express the sum of all amines showed except m-PDA (i.e. it has its own SML)

Table 4 :

 4 Precision expressed as internal reproducibility (RSD R %) of the method in migration solutions from nylon cooking utensils.

	PAAs	2 ng mL -1 5 ng mL -1 10 ng mL -1 20 ng mL -1
	m-PDA	12.3	7.9	9.4	9.3
	2,6-TDA	13.6	9.7	11.9	10.3
	2,4-TDA	21.4	11.3	16.5	9.7
	1,5 DAN	18.4	9.0	11.6	12.1
	ANL	9.8	20.0	11.3	20.3
	4,4`-DPE	12.2	6.8	10.3	5.6
	4,4`-MDA	8.4	7.4	12.8	9.9
	3,3`-DMB	15.2	8.9	13.1	13.1

Table 6 .

 6 Recoveries (%) of PAAs in migration solutions from nylon cooking utensils.

	PAAs	2 ng mL -1	5 ng mL -1 10 ng mL -1 20 ng mL -1 Global mean
	m-PDA	96.9	100.0	99.4	97.2	98.4
	2,6-TDA	93.9	96.9	92.6	97.4	95.2
	2,4-TDA	91.9	95.9	89.6	94.9	93.1
	1,5 DAN	99.1	94.5	88.9	92.2	93.7
	ANL	80.0	90.4	100.1	85.4	89.0
	4,4`-DPE	104.0	101.0	97.0	95.1	99.3
	4,4`-MDA	101.4	101.1	99.3	99.6	100.4
	3,3`-DMB	99.5	99.9	96.6	93.4	97.4

Table 5 :

 5 Results of migration experiments. The data showed correspond to the 3 rd exposure. All results are expressed as the sum of all compounds identified.

	Sample	Utensil	Contact surface (dm 2 )	Ratio (dm 2 L -1 )	MIGRATION (µg dm -2 )	PAAs identified
	1	Slotted turner	1.3	3.75
	2	Slotted turner	1.1	2.62	909.21	4 4´-MDA & ANL
	3	Soup Spoon	2.4	5.49	559.25	4 4´-MDA & ANL
	4	Ladle	1.5	3.36	230.44	4 4´-MDA & ANL
	5	Slotted Spatula	1.6	4.26	74.30	4 4´-MDA
	6	Ladle	1.2	2.98	1.97 2.0	ANL
	7	Ladle	1.5	3.28
	8	Slotted turner	1.5	3.98
	9	Sauce spoon	1.5	4.32	107.01	4 4´-MDA & ANL
	10	Soup Spoon	1.9	3.71
	11	Ladle	1.5	3.44	2470.34	4,4´-MDA & ANL
	12	Ladle	1.8	3.53
	13	Serving Tong	0.9	6.42
	14	Skimmer	2.1	5.55
	15	Ladle	1.4	3.17
	16	Slotted turner	1.3	3.50
	17	Slotted turner	1.1	2.68	751.24	4 4´-MDA & ANL
	18	Soup Spoon	2.4	5.25	926.69	4 4´-MDA & ANL
	19	Ladle	1.4	4.33	315.73	4 4´-MDA & ANL
	21	Ladle	1.6	3.55	7.659	ANL & 4,4´-MDA
	22	Soup Spoon	2.7	7.10	140.70	4 4´-MDA & ANL
	24	Soup Spoon	2.6	8.98
	25	Ladle	2.1	7.26
	26	Ladle	2.1	7.14
	27	Fork	0.8	3.28	68.13	ANL
	28	Spoon	0.9	3.46	105.38	ANL
	29	Slotted turner	0.9	3.92	80.90	ANL
	30	Ladle	1.6	4.79
	31	Ladle	1.6	5.66
	32	Ladle	1.5	5.67
	33	Soup Spoon	2.6	6.93
	34	Sauce spoon	1.4	8.72
	35	Ladle	1.4	3.89
	36	Sauce spoon	1.4	10.28	329.20	4,4'-MDA & ANL
	37	Serving Fork	0.7	4.70	369.94	4,4'-MDA & ANL
	38	Spatula	2.1	6.38	798.03	4,4'-MDA & ANL
	39	Skimmer	2.4	4.11	179.03	4,4'-MDA & ANL
	40	Slotted turner	1.1	2.90	860.81	4,4'-MDA & ANL
	41	Ladle	1.3	4.70	562.28	4,4'-MDA & ANL

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgements

The content of this paper are the responsibility of the authors alone and should not be taken to represent the opinions of the supporting organizations. Authors are grateful to the ''Angeles Alvariño" Program financed by "Consellería de Innovación e Industria, Xunta de Galicia" for the postdoctoral contract of R. Sendón. The authors are grateful to Ms. Carmen Tejedor Román and Ms. Libertad García Martín for their excellent technical assistance.


Frédéric Mazenc
email: frederic.mazenc@lss.supelec.fr

Christophe Prieur
email: christophe.prieur@gipsa-lab.grenoble-inp.fr

Strict Lyapunov functionals for nonlinear parabolic partial differential equations

Keywords: Strictification, Lyapunov functional, partial differential equation

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functionals are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functionals for the nonlinear source term. The strict Lyapunov functionals are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov functional makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

INTRODUCTION

Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). The technics are useful for the stability analysis of systems of many different families (although other approaches can be used too, especially when parabolic PDEs are studied; see in particular the contributions [START_REF] Matignon | Asymptotic stability of linear conservative systems when coupled with diffusive systems[END_REF]; [START_REF] Iftime | Optimal control of switched distributed parameter systems with spatially scheduled actuators[END_REF]; [START_REF] Drame | Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model[END_REF]).

Amongst the remarkable results for PDEs which extensively use Lyapunov functionals, it is worth mentioning the following. In [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] a Lyapunov functional is used to establish the existence of a global solution to the celebrated heat equation. In [START_REF] Krstic | Adaptive boundary control for unstable parabolic PDEs-part I: Lyapunov design[END_REF], Lyapunov functionals are designed for the heat equation with unknown destabilizing parameters (see also Smyshlyaev and Krstic (2007a,b) for further results on the design of output stabilizers). Lyapunov functionals have been also used to establish controllability results for semilinear heat equations. For example in [START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF] the computation of a Lyapunov functional, in combination with the quasi-static deformation method, is a key ingredient of the proof of the global controllability of this equation. In all the preceding papers, parabolic PDEs are considered, but Lyapunov functionals can also be useful for other kinds of dynamics. For instance, in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] the stabilization of a linear dynamic equation modeling a rotating beam is achieved through a control design relying on weak Lyapunov functional, i.e. a Lyapunov functional whose derivative along the trajectories of the system which is considered is non-positive (but not necessarily negative definite), and in [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF] the controllability of the wave equation is demonstrated via a Lyapunov func-tional. Besides, the knowledge of Lyapunov functionals can be useful for the stability analysis of nonlinear hyperbolic systems (see the recent work [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]) or even for designing boundary controls which stabilize a system of conservation laws (see [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] To demonstrate asymptotic stability through the knowledge of a weak Lyapunov functional, the celebrated LaSalle invariance principle has to be invoked (see e.g. [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]; [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]; [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]). It requires to demonstrate a precompactness property for the solutions, which may be difficult to prove (and is not even always satisfied, as illustrated by the hyperbolic systems considered in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]). This technical step is not needed when is available a strict Lyapunov functional i.e. a Lyapunov functional whose derivative along the trajectories of a system which is considered is negative definite. Thus designing such a Lyapunov functional is a way to overcome this technical difficulty. This is not the unique motivation for designing strict Lyapunov functionals. From the knowledge of the explicit expression of strict Lyapunov functionals one can estimate the robustness of the stability of a system with respect to the presence of uncertainties and one can analyze the sensitivity of the solutions with respect to external disturbances.

The present paper is devoted to new techniques of constructions of strict Lyapunov functionals for parabolic PDEs. For particular families of PDEs with diffusion and convection terms and specific boundary conditions, we modify weak Lyapunov functionals, which are readily available, to obtain strict Lyapunov functionals given by explicit formulas. The resulting functionals have rather simple explicit expressions. The underlying concept of strictification used in our paper is the same as the one exposed in [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF], (see also [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF], [START_REF] Mazenc | A simplified design for strict Lyapunov functions under Matrosov conditions[END_REF]). However, due to the specificity of PDEs, the techniques of construction that we shall present are by no means a direct application of any constructions available for ordinary differential equations.

In a second part of our work, we design strict Lyapunov functionals to establish robustness properties of Input-to-State (ISS) type for a family of globally asymptotically stable PDEs with disturbances. Although the ISS notion is very popular in the area of the dynamical systems of finite dimension (see e.g. the recent survey [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]) and, for a few years, begins to be used in the domain of the systems with delay (see for instance [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], [START_REF] Pepe | Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems[END_REF], [START_REF] Pepe | On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign[END_REF], [START_REF] Karafyllis | Inputto-output stability for systems described by retarded functional differential equations[END_REF]), the present work is, to the best of our knowledge, the first one which uses it to characterize a robustness property of a PDE.

Our paper is organized as follows. Basic definitions and notations are introduced in Section 2. Constructions of Lyapunov functionals under various sets of assumptions are performed in Section 3. In Section 4 the analysis of the robustness of a family PDEs with uncertainties is carried out by means of the design of a so called ISS Lyapunov functional. An example in Section 5 illustrates the main result of Section 4. Concluding remarks in Section 7 end the work. Due to space limitation, some of the proofs are omitted.

Notation. Throughout the paper, the argument of the functions will be omitted or simplified when no confusion can arise from the context. Given a matrix A, its induced matrix,

Sym(A) = 1 2 (A + A ⊤ ) stands for the symmetric part of A. The norm | • | L 2 (0,L) is defined by: |φ| L 2 (0,L) = L 0 |φ(z)| 2 dz. Finally, we denote C L = C 2 ([0, L], R n ),
the set of all twice-differentiable R n -valued functions defined on a given interval [0, L].

BASIC DEFINITIONS AND NOTIONS

Throughout our work, we will consider partial differential equations of the form

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + ∆(z, t) ∂X ∂z (z, t) +f (X(z, t)) + u(z, t) , (1)
with z ∈ [0, L] and X(., t) ∈ C L for all t ≥ 0, where ∆ is continuous and bounded in norm, f : R n → R n is continuously differentiable and where u is a continuous function (which typically is unknown and represents disturbances).

Let us introduce the notions of weak and strict Lyapunov functionals that we will consider in this paper (see e.g. (Luo et al., 1999, Def. 3.62)). Definition 2.1. Let µ : C L → R be a continuously differentiable function. The functional µ is said to be a weak Lyapunov functional for (1), if there are two functions κ S and κ M of class K ∞ such that, for all functions φ ∈ C L ,

κ S |φ| L 2 (0,L) ≤ µ(φ) ≤ L 0 κ M (|φ(z)|) dz (2)
and, in the absence of u, for all solutions of (1), for all t ≥ 0, dµ(X(., t)) dt ≤ 0 .

The functional µ is said to be a strict Lyapunov functional for (1) if, additionally, in the absences of u, there exits λ 1 > 0 such that, for all solutions of (1), for all t ≥ 0, dµ(X(., t)) dt ≤ -λ 1 µ(X(., t)) .

The functional µ is said to be an ISS Lyapunov functional for (1) if, additionally, there exit λ 1 > 0 and a function λ 2 of class K such that, for all continuous functions u, for all solutions of (1), and for all t ≥ 0,

dµ(X(., t)) dt ≤ -λ 1 µ(X(., t)) + L 0 λ 2 (|u(z, t)|)dz .
Remark 1. 1. For conciseness, we will often use the notation μ instead of dµ(X(.,t)) dt .

2. When a strict Lyapunov functional exists and u is not present, then the value of a strict Lyapunov functional for (1) along the solutions of (1) exponentially decays to zero and therefore each solution X(z, t) satisfies lim t→+∞ |X(., t)| L 2 (0,L) = 0. When in addition, there exists

a function κ L of class K ∞ , such that, for all functions φ ∈ C L , µ(φ) ≤ κ L |φ| L 2 (0,L) ,
(3) then the system (1) is globally asymptotically stable for the topology of the norm L 2 .

3. When the system (1) admits an ISS Lyapunov functional µ, then, one can check through elementary calculations that, for all solutions of (1) and for all instants t ≥ t 0 , the inequality

|X(., t)| L 2 (0,L) ≤ κ -1 S 2e -λ1(t-t0) L 0 κ M (|X(z, t 0 )|)dz + κ -1 S 2 λ 1 sup ℓ∈[t0,t] L 0 λ 2 (|u(z, ℓ)|)dz
holds. This inequality is the analogue for the PDE (1) of the ISS inequalities for ordinary differential equations.

It gives an estimation with the L 2 norm of the influence of the disturbance u on the solutions of the system (1).

Of course this inequality does not imply that a similar inequality holds when another norm is selected. •

CONSTRUCTIONS OF LYAPUNOV FUNCTIONALS

In this section, we give several constructions of Lyapunov functionals for the system

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + f (X(z, t)) (4)
with z ∈ [0, L], X(z, t) ∈ R n and where f is a nonlinear function of class C 1 .

Weak Lyapunov functional for the system (4)

To prepare the construction of strict Lyapunov functionals of the forthcoming sections, we recall how a weak Lyapunov functional can be constructed for the system (4) under the following assumptions:

Assumption 1. There is a symmetric positive definite matrix Q such that the function

W 1 (Ξ) := - ∂V ∂Ξ (Ξ)f (Ξ) , (5)
with V (Ξ) =1 2 Ξ ⊤ QΞ, is nonnegative. Assumption 2. The boundary conditions are such that, for all t ≥ 0,

either |X(L, t)| ∂X ∂z (L, t) = |X(0, t)| ∂X ∂z (0, t) = 0 , or X(L, t) = X(0, t) and ∂X ∂z (L, t) = ∂X ∂z (0, t) .
(6) Some comments on Assumptions 1 and 2 follow. Remark 2. 1. Assumption 1 is equivalent to claiming that V is a weak Lyapunov function for the ordinary differential equation Ξ = f (Ξ) (7) with Ξ ∈ R n . Therefore it implies that this system is globally stable.

2. Assumption 2 is satisfied in particular if the Dirichlet or Neumann conditions or the periodic conditions, i.e. X(0, t) = X(L, t) and ∂X ∂z (0, t) = ∂X ∂z (L, t) for all t (see [START_REF] Chen | Convergence, asymptotic periodicity, and finite-point blow-up in onedimensional semilinear heat equations[END_REF]), are satisfied.

3. Since Q is positive definite, there exist two positive real values q 1 and q 2 such that, for all Ξ ∈ R n ,

q 1 |Ξ| 2 ≤ V (Ξ) ≤ q 2 |Ξ| 2 .
(8) The constants q 1 and q 2 will be used in the constructions of strict Lyapunov functionals we shall perform later. •

The construction we perform below is given for instance in [START_REF] Krstic | Adaptive boundary control for unstable parabolic PDEs-part I: Lyapunov design[END_REF]; [START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF]. Lemma 3.1. Under Assumptions 1 and 2, the functional

U (φ) = L 0 V (φ(z))dz (9)
is a weak Lyapunov functional whose derivative along the solutions of (4) satisfies

U = - L 0 ∂X ∂z (z, t) ⊤ Q ∂X ∂z (z, t)dz - L 0 W 1 (X(z, t))dz . (10)
Proof. The proof is omitted.

Strict Lyapunov functional for the system (4): first result

In this section, we show that the functional U given in ( 9) is a strict Lyapunov functional for (4) when this system is associated with special families of boundary conditions or when W 1 is larger than a positive definite quadratic function. We state and prove the following result: Theorem 1. Assume that the system (4) satisfies Assumptions 1 and 2 and that one of the following property is satisfied (i) there exists a constant α > 0 such that, for all Ξ in R n , W 1 (Ξ) ≥ α|Ξ|2 , (ii) X(0, t) = 0 for all t ≥ 0, (iii) X(L, t) = 0 for all t ≥ 0.

Then the functional U given in ( 9) is a strict Lyapunov functional for (4).

Proof. Let us assume that the property (i) holds. Then it follows straightforwardly from ( 8) and ( 10) that

U ≤ - α q 2 L 0 V (X(z, t))dz
and U is a strict Lyapunov functional (see Definition 2.1).

We consider now the cases (ii) and (iii).We consider now the cases (ii) and (iii). Let us recall the Poincaré inequality. Lemma 3.2. For any function w, continuously differentiable on [0, 1], and for c = 0 or c = 1,

1 0 |w(z)| 2 dz ≤ 2w 2 (c) + 4 1 0 ∂w ∂z (z) 2 dz . ( 11
)
From this lemma, we deduce that for all L ≥ 0 and c = 0 or L, and for any function w, continuously differentiable on [0, L], the inequality

L 0 |w(z)| 2 dz ≤ 2Lw 2 (c) + 4L 2 L 0 ∂w ∂z (z) 2 dz , (12)
is valid. We deduce that when X(0, t) = 0 for all t ≥ 0 or X(L, t) = 0 for all t ≥ 0 then, for all t ≥ 0, the inequality

L 0 |X(z, t)| 2 dz ≤ 4L 2 q 1 L 0 ∂X ∂z (z, t) ⊤ Q ∂X ∂z (z, t)dz
where q 1 is the constant in (8), is satisfied. Combining this inequality with (10) yields

U ≤ - q 1 2L 2 L 0 |X(z, t)| 2 dz .
Using (8) again, we can conclude that U is a strict Lyapunov functional for the system (4). •

Strict Lyapunov functional for the system (4): second result

One can check easily that Assumptions 1 and 2 alone do not ensure that the system (4) admits the zero solution as an asymptotically stable solution. 1 Therefore an extra assumption must be introduced to guarantee that a strict Lyapunov functional exists. In Section 3.2 we have exhibited simple conditions which ensure that U is a strict Lyapunov functional. In this section, we introduce a new assumption, less restrictive than the condition (i) of Theorem 1, which ensures that a strict Lyapunov functional different from U can be constructed.

Assumption 3. There exist a nonnegative function M : R n → R of class C 2 , and a continuous function

W 2 : R n → R such that M (0) = 0 , ∂M ∂Ξ (0) = 0 , ( 13
) ∂M ∂Ξ (Ξ)f (Ξ) ≤ -W 2 (Ξ) , ∀Ξ ∈ R n , (14)
∂ 2 M ∂Ξ 2 (Ξ) ≤ q 1 2 , ∀Ξ ∈ R n , (15)
and there exists a constant q 3 > 0 such that W 1 + W 2 is positive definite and

W 1 (Ξ) + W 2 (Ξ) ≥ q 3 |Ξ| 2 , ∀Ξ ∈ R n : |Ξ| ≤ 1 (16)
where W 1 is the function defined in (5).

We are ready to state and prove the following result: Theorem 2. Under Assumptions 1 to 3, there exists a function k of class K ∞ , of class C 2 such that k ′ is positive, k ′′ is nonnegative and the functional

U (φ) = L 0 k(V (φ(z)) + M (φ(z)))dz (17)
is a strict Lyapunov functional for (4). Remark 3. Assumption 3 seems to be restrictive. In fact, it can be significantly relaxed. Indeed, if the system Ξ = f (Ξ) ( 18) is locally exponentially stable and satisfies one of Matrosov's conditions which ensure that a strict Lyapunov function can be constructed then one can construct a function M which satisfies Assumption 3. For constructions of strict Lyapunov functions under Matrosov's conditions, the reader is referred to [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF].

•

Proof. Let us consider the functional U defined in ( 17).

Since we impose a priori on k to be of class K ∞ and ( 15) holds, we deduce easily that inequalities of the type (2) are satisfied.

Next, let us evaluate what is the time derivative of U along the solutions of ( 4). With the notation S = V +M , we have

U = L 0 k ′ (S(X(z, t))) ∂S ∂Ξ (X(z, t)) ∂X ∂t (z, t)dz = T 1 (X(., t)) + T 2 (X(., t)) (19)
with

T 1 (φ) = L 0 k ′ (S(φ(z))) ∂S ∂Ξ (φ(z))f (φ(z))dz , T 2 (φ) = L 0 k ′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂ 2 φ ∂z 2 (z)dz . ( 20
) Since ∂S ∂Ξ (Ξ)f (Ξ) = ∂V ∂Ξ (Ξ)f (Ξ) + ∂M ∂Ξ (Ξ)f (Ξ) ,
we deduce from Assumptions 1 and 3 that

T 1 (φ) ≤ - L 0 k ′ (S(φ(z)))[W 1 (φ(z))+W 2 (φ(z))]dz . (21)
Now, we consider T 2 . By integrating by part, we obtain

T 2 (φ) = T 3 (φ) - L 0 ∂H(φ(z)) ∂z ∂φ ∂z (z)dz ( 22
)
with

T 3 (φ) = k ′ (S(φ(L))) ∂S ∂Ξ (φ(L)) ∂φ ∂z (L) -k ′ (S(φ(0))) ∂S ∂Ξ (φ(0)) ∂φ ∂z (0) , (23)
and

H(Ξ) = k ′ (S(Ξ)) ∂S ∂Ξ (Ξ) . ( 24
) Since ∂H(φ(z)) ∂z = k ′′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂φ ∂z (z) ∂S ∂Ξ (φ(z)) +k ′ (S(φ(z))) ∂φ ∂z (z) ⊤ ∂ 2 S ∂Ξ 2 (φ(z)) , (25)
we deduce from ( 22) that

T 2 (φ) = T 3 (φ) -T 4 (φ) -T 5 (φ) (26) with T 4 (φ) = L 0 k ′′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂φ ∂z (z) 2 dz , T 5 (φ) = L 0 k ′ (S(φ(z))) ∂φ ∂z (z) ⊤ ∂ 2 S ∂Ξ 2 (φ(z)) ∂φ ∂z (z)dz .
Since we impose on k to be such that k ′′ is nonnegative, we immediately deduce that

T 2 (φ) ≤ T 3 (φ) -T 5 (φ) . (27) Now, observe that ∂ 2 S ∂Ξ 2 (φ(z)) = Q + ∂ 2 M ∂Ξ 2 (φ(z)) . (28)
This equality, inequalities (8), Assumption 3 and the fact that we impose on k to be such that k ′ > 0 ensure that

T 5 (φ) ≥ 2q 1 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz - q 1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz = 3q 1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz .
It follows that

T 2 (φ) ≤ T 3 (φ) - 3q 1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz . (29)
Hence ( 19), ( 21), and (29) give

U ≤ - L 0 k ′ (S(X(z, t)))W 3 (X(z, t))dz + T 3 (X(., t)) - 3q 1 2 L 0 k ′ (S(X(z, t))) ∂X ∂z (z, t) 2 dz with W 3 = W 1 + W 2 .
Assumption 2 ensures that, for all t ≥ 0, T 3 (X(., t)) = 0. We deduce that U ≤ -

L 0 k ′ (S(X(z, t)))W 3 (X(z, t))dz .
By ( 16) and the inequality S(Ξ) ≥ q 1 |Ξ| 2 , we can construct, through simple but lengthy calculations, a function

k of class C 2 of class K ∞ such that k ′ is positive, k ′′ is nonnegative and k ′ (S(Ξ))W 3 (Ξ) ≥ Ck(S(Ξ)) , ∀Ξ ∈ R n .
where C is a positive constant. Therefore, selecting this function k, we obtain U ≤ -CU (X(., t)) .

It follows that U is a strict Lyapunov functional for (4). •

ISS PROPERTY FOR A FAMILY OF PDES

In the previous section, we have constructed Lyapunov functionals for PDEs without uncertainties and without convection term. In this section, we show how our technique of construction can be used to estimate the impact of uncertainties on the solutions of PDEs with a convection term and uncertainties of the form

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + [D 1 + v(z, t)] ∂X ∂z (z, t) +f (X(z, t)) + u(z, t) (30)
where D 1 is a constant matrix, v is an unknown matrix function and u is an unknown continuous function.

Remark 4. For a linear finite dimensional system ẋ = Ax + Bu (31) where A and B are constant matrices respectively in R n×n and in R n×1 , it is well-known that if the linear system (31) without input u, i.e. ẋ = Ax, is asymptotically stable then bounded inputs result in bounded solutions. However, for nonlinear finite dimensional systems, global asymptotic stability does not imply input/state stability of any sort. See (Sontag, 2007, Section 2.6) for a simple scalar example, which is globally asymptotically stable but which have solutions with a finite time explosion for a suitable constant input. Also for linear infinite dimensional system, asymptotic stability does not imply input-to-state stability. More precisely, we exhibit in Section 6 below an example of linear system which is globally asymptotically stable without any input, but which may have unbounded solutions in the presence of a bounded input.

•

To cope with the presence of a convection term and the uncertainty v, we introduce the following assumption: Assumption 4. Considering the symmetric positive definite matrix Q given by Assumption 1, there exists a nonnegative real number δ such that

|v(z, t)| ≤ δ |Q| , ∀z ∈ [0, L] , t ≥ 0 . (32)
Moreover, the matrix QD 1 is symmetric.

Moreover, we replace Assumption 3 by a more restrictive assumption: Assumption 5. There exists a nonnegative function M : R n → R such that, for all Ξ ∈ R n ,

M (0) = 0 , ∂M ∂Ξ (Ξ)f (Ξ) = -W 2 (Ξ) , (33)
where W 2 is a nonnegative function and there exist c a > 0, c b > 0 and c c > 0 such that, for all Ξ ∈ R n , the inequalities

∂M ∂Ξ (Ξ) ≤ c a |Ξ| , ∂ 2 M ∂Ξ 2 (Ξ) ≤ c b , (34)
|Ξ| 2 ≤ c c [W 1 (Ξ) + W 2 (Ξ)] , (35
) where W 1 is the function defined in (5), are satisfied.

Remark 5. If f is linear and Ξ = f (Ξ) is exponentially stable, then Assumption 5 is satisfied with a positive definite quadratic function as function M .

•

We are ready to state and prove the main result of the section Theorem 3. Assume that the system (30) satisfies Assumptions 1, 4 and 5 and is associated with boundary conditions satisfying X(L, t) = X(0, t) and ∂X ∂z (L, t) = ∂X ∂z (0, t) , ∀t ≥ 0 .

(36) Then the functional

U (φ) = L 0 [KV (φ(z)) + M (φ(z))]dz (37) with K = max 1, 2c b q 1 , 8c c c 2 a (|D 1 | + 1) 2 q 1 (38)
satisfies, along the trajectories of (30),

U ≤ -λ 1 U (X(z, t)) + λ 2 L 0 |u(z, t)| 2 dz (39)
for some positive constants λ 1 , λ 2 , provided that δ in Assumption 4 satisfies

δ ≤ min |Q|, √ q 1 2 √ 2c c K . ( 40
)
Remark 6. Using Assumption 5, one can prove easily that the ISS Lyapunov functional U defined in ( 37) is upper and lower bounded by a positive definite quadratic functional. We deduce easily that (39) leads to an ISS inequality of the type

|X(., t)| L 2 (0,L) ≤ Λ 1 e -λ 1 2 (t-t0) |X(., t 0 )| L 2 (0,L) +Λ 2 sup m∈[t0,t] L 0 |u(z, m)| 2 dz , (41)
where Λ 1 , Λ 2 are positive real numbers. •

Proof. The proof is omitted.

EXAMPLE

In this section, we illustrate Theorem 3 through the nonlinear system

             ∂x 1 ∂t (z, t) = ∂ 2 x 1 ∂z 2 (z, t) - ∂x 1 ∂z (z, t) +x 2 (z, t)[1 + x 1 (z, t) 2 ] + u 1 (z, t) ∂x 2 ∂t (z, t) = ∂ 2 x 2 ∂z 2 (z, t) -x 1 (z, t)[1 + x 1 (z, t) 2 ] -x 2 (z, t)[2 + x 1 (z, t) 2 ] + u 2 (z, t) (42)
with X = (x 1 , x 2 ) ⊤ ∈ R 2 and where u 1 and u 2 are continuous real-valued functions. Equation (42) is a system of two heat equations with a convection term in the first.

Let us check that Theorem 3 applies. One can check readily that Assumption 1, 4 and 5 are satisfied with the positive definite quadratic functions

V (Ξ) = 1 2 [ξ 2 1 + ξ 2 2 ] and M (Ξ) = ξ 2 1 + ξ 2 2 + ξ 1 ξ 2 , and the matrix Q = 1 0 0 1 . Indeed, with f (Ξ) = (ξ 2 [1 + ξ 2 1 ] , -ξ 1 [1 + ξ 2 1 ] -ξ 2 [2 + ξ 2 1 ]) ⊤ , we have, for all Ξ in R 2 , ∂V ∂Ξ (Ξ)f (Ξ) = -ξ 2 2 [2 + ξ 2 1 ] , ∂M ∂Ξ (Ξ)f (Ξ) = -(3 + ξ 2 1 )ξ 2 2 -(1 + ξ 2 1 )ξ 2 1 -ξ 1 ξ 2 (2 + ξ 2 1 ) ≤ -ξ 2 2 - 1 2 + 3 4 ξ 2 1 ξ 2 1 .
Moreover, through elementary calculations, one obtains the following values:

q 1 = 1 2 , |D 1 | = 1, c a = c b = 3, c c =
2 for the constants in (38). Therefore Theorem 3 guarantees that, if (36) is satisfied, then the functional

U (φ) = 1153 L 0 φ 1 (z) 2 + φ 2 (z) 2 dz + L 0 φ 1 (z)φ 2 (z)dz
is an ISS Lyapunov functional for the system (42).

APPENDIX: ILLUSTRATION OF REMARK 4

In this section, we introduce an example of a linear infinite dimensional system which is globally asymptotically stable without any input, but which may have unbounded solutions in presence of a bounded input.

To do that, let us consider the following system composed by an infinite number of scalar ordinary differential equations written by, for each n in N,

Ẋn = - 1 n + 1 X n + u n . ( 43
)
Given an initial condition in l 2 (N), there exists a solution of ( 43) defined for all time t ≥ 0 for u n ≡ 0, for all n in N. Moreover system (43) is globally asymptotic stable, without any input, i.e. when u n ≡ 0, for all n in N.

Let us consider the input satisfying, for each n in N, and for all t ≥ 0, u n (t) = 1 (n + 1) 2 e -1 (n+1) 2 t . Note that, denoting by • l 2 (N) the usual norm in l 2 (N), we have t → (u n (t)) n∈N l 2 (N) is bounded. We even have (u n (t)) n∈N l 2 (N) ≤ π 2 6 e -t → 0, as t → ∞. Moreover the solution of (43) with, for all n in N, X n (0) = 0 is given by, for all t ≥ 0,

X n (t) = 1 n e - 1 (n+1) 2 t -e -1 n+1 t
for n ≥ 1. Therefore X n (t) is not in l 2 (N) for t > 0, and the system (43) is not asymptotically stable. System ( 43) is an example of asymptotic stable systems which is not input-to-state stable, as considered in Remark 4.

CONCLUSIONS

For important families of PDEs, we have shown how weak Lyapunov functionals can be transformed into strict Lyapunov functionals. Robustness properties of the ISS type can be inferred from our constructions. Much remains to be done. Other types of equations can be studied and robustness with respect to other types of disturbances can be proved.

More precisely, we can construct examples of systems (4) which are not asymptotically stable when Assumption 1 is satisfied and Assumption

holds with the Neumann boundary conditions.


W Lamm

Willenbacher

Zielinski Cc

MD Johannes Drach
email: johannes.drach@meduniwien.ac.at

Efficacy of the combination of bortezomib and dexamethasone in systemic AL amyloidosis

Keywords: AL amyloidosis, bortezomib, proteasome inhibition, complete remission

Bortezomib -Dexamethasone (Btz/Dex) is an active regimen in patients with multiple myeloma and has been used in few patients with amyloidosis. Here we report a retrospective evaluation of the efficacy and toxicity of Btz/Dex in 26 patients with AL amyloidosis. 18 patients (69%) received Btz/Dex as first-line treatment. Organs most frequently involved were kidneys (100%) and heart (35%), 5 patients (19%) had > 2 organs involved. The overall response rate was 54% (14 of 26 patients), with 8 patients (31%) achieving a hematologic complete remission (CR). All patients who reached a CR received Btz/Dex as first-line therapy. Median time to response was 7.5 weeks. Improvement in organ function was noticed in 3 patients (12%). Median progression free survival (PFS) and overall survival (OS) was 5.0 and 18.7 months, respectively; in CR patients, however, median PFS and OS have not yet been reached. Toxicities were manageable, with haematological side effects being most common. No grade 3/4 neuropathy was observed. Our results confirm the activity of bortezomib/dexamethasone in patients with AL amyloidosis and suggest that patients achieving a CR have a marked benefit for survival.

Introduction

Primary systemic AL-amyloidosis (AL) is a clonal plasma cell disorder characterized by the deposition of amyloid fibrils in various organs, which eventually leads to organ dysfunction and death [START_REF] Merlini | Molecular mechanisms of amyloidosis[END_REF]. Prognosis of AL is poor, with a median survival of about a year from diagnosis without treatment, but survival may be prolonged by effective treatment [START_REF] Kyle | A trial of three regimens for primary amyloidosis: colchicine alone, melphalan and prednisone, and melphalan, prednisone and colchicine[END_REF]. The pathogenic light-chains are produced by a usually relatively small clonal population of immunoglobulinsecreting, transformed plasma cells. Therefore, the objective of treatment in AL is suppression of the abnormal plasma cell clone, which then leads to a reduction of amyloidogenic light chains. Treatment choices depend on the patient`s age, performance status and the type of organ damage [START_REF] Merlini | Dangerous small B-cell clones[END_REF].

Aggressive therapy consisting of high-dose melphalan followed by autologous stem cell transplantation has been shown to be effective in AL [START_REF] Comenzo | Autologous stem cell transplantation for primary systemic amyloidosis[END_REF], but only a minority of patients is eligible for this treatment modality [START_REF] Sanchorawala | An overview of the use of high-dose melphalan with autologous stem cell transplantation for the treatment of AL amyloidosis[END_REF]. Less intensive treatment approaches include an alkylating agent (usually melphalan) in combination with a corticosteroid [START_REF] Jaccard | Myélome Autogreffe (MAG) and Intergroupe Francophone du Myélome (IFM) Intergroup. (2007) High-dose melphalan versus melphalan plus dexamethasone for AL amyloidosis[END_REF]. Melphalan plus dexamethasone (Dex) can induce a hematologic response in up to 67% of patients with AL; however, median time to response is rather long (4.5 months). Thalidomide combined with Dex appears to be only moderately active in AL (hematologic response rate of less than 50%), but of even greater concern is the considerable toxicity of this regimen in patients with AL [START_REF] Palladini | The combination of thalidomide and intermediate-dose dexamethasone is an effective but toxic treatment for patients with primary amyloidosis (AL)[END_REF]. Therefore, effective treatment strategies are still urgently required for patients with AL, both in newly diagnosed and relapsed disease.

Novel agents with proven activity in multiple myeloma (MM) are investigated in AL. Initial experiences with lenalidomide in AL have been promising, but more data are needed to establish its therapeutic role in this plasma cell disorder [START_REF] Kastritis | Treatment of light chain (AL) amyloidosis with the combination of bortezomib and dexamethasone[END_REF][START_REF] Wechalekar | Efficacy of bortezomib in systemic AL amyloidosis with relapsed/refractory clonal disease[END_REF][START_REF] Dispenzieri | The activity of lenalidomide with or without dexamethasone in patients with primary systemic amyloidosis[END_REF][START_REF] Sanchorawala | Lenalidomide and dexamethasone in the treatment of AL amyloidosis: results of a phase 2 trial[END_REF].

Bortezomib (Btz) is a potent, specific and reversible inhibitor of the 26S proteasome with remarkable activity in multiple myeloma (MM) [START_REF] Richardson | Assessment of Proteasome Inhibition for Extending Remissions (APEX) Investigators (2005) Bortezomib or high-dose dexamethasone for relapsed multiple myeloma[END_REF]. Based upon the sensitivity of malignant plasma cells to Btz and the usually rapid onset of a remission induced by Btz, it has been hypothesized that proteasome inhibition should also be beneficial in AL. Two reports with single-agent Btz have shown promising activity and manageable toxicity in patients with relapsed AL [START_REF] Kastritis | Treatment of light chain (AL) amyloidosis with the combination of bortezomib and dexamethasone[END_REF][START_REF] Wechalekar | Efficacy of bortezomib in systemic AL amyloidosis with relapsed/refractory clonal disease[END_REF]. In MM, the efficacy of Btz can be enhanced by the addition of Dex, which may also be true in AL. We therefore evaluated the efficacy and toxicity of the combination of Btz and Dex in a series of consecutive patients with AL.

Materials and methods

Patients' characteristics

This multicenter retrospective analysis outside the context of a clinical trial was performed on a consecutive cohort of 26 patients (male = 13, female = 13) with biopsy-proven, symptomatic AL amyloidosis. Diagnosis of AL and assessment of amyloid related organ involvement followed recently published consensus guidelines [START_REF] Gertz | Definition of organ involvement and treatment response in immunoglobulin light chain amyloidosis (AL): a consensus opinion from the 10th International Symposium on Amyloid and Amyloidosis, Tours, France, 18-22 April 2004[END_REF]. Baseline characteristics including organ involvement are detailed in Table 1. Median age was 62 years (range 42-83).

Treatment with Btz/Dex was performed between October 2006 and October 2009. Eighteen patients received Btz/Dex as first-line treatment, whereas 8 patients were treated at relapse. In patients with previously treated AL, the median number of prior treatment lines was 2 (range, 1 to 6). The most common prior treatment regimens consisted of melphalan/prednisone (already administered in 50% of pretreated patients), vincristine/adriamycin/dex (VAD; 25%) and DCEP (25%). In 23 of the patients, ECOG performance status was < 2, 3 patients had an ECOG performance status > 3.

Treatment regimen

Btz was administered at a standard dose 1.3mg/m 2 as an intravenous bolus twice weekly for two weeks (days 1, 4, 8 and 11) followed by a 10-day rest period (days 12-21). Dex was administered at a daily dose of 20mg on the days of Btz administration and the day after.

Cycles were repeated every 3 weeks. Patients could receive up to 8 cycles of therapy. All patients received antiviral prophylaxis and proton-pump inhibitor therapy according to the standards of individual institutions. Since this was not a prospective trial, dose modifications of Btz and Dex were performed in agreement with guidelines from standard MM treatment at the discretion of the individual physician. Supportive therapy for side effect management was administered according to clinical requirements.

Patients were informed that treatment with Btz had been demonstrated to be active in MM, and that it was licensed for the treatment of patients with MM, but that experience with Btz in AL was limited thus far. All patients gave written informed consent to treatment.

Assessments

Physical examination, standard laboratory parameters, serum electrophoresis and serum free light chains were obtained on a monthly basis. Performance status was assessed as described by the Eastern Cooperative Oncology Group (ECOG) criteria. Toxicity was recorded according to the National Cancer Instiute Cancer Therapy Evaluation Program, Common Terminology Criteria for Adverse Events (Version 3.0).

Patients were considered to be evaluable for outcome after completion of at least two cycles of Btz/Dex. Response to therapy was defined according to recently published consensus guidelines [START_REF] Gertz | Definition of organ involvement and treatment response in immunoglobulin light chain amyloidosis (AL): a consensus opinion from the 10th International Symposium on Amyloid and Amyloidosis, Tours, France, 18-22 April 2004[END_REF]. Partial response (PR) was defined as a > 50% decrease of the involved monoclonal class; complete remission (CR) was defined as a normalization of the serum free light chain (kappa/lambda) ratio and both light chain classes. Organ response (OR) was only assessed in patients who responded to Btz.

Statistical analyses

Progression free survival (PFS) was calculated from the date of Btz/Dex initiation until the date of progression or death. Overall survival (OS) was calculated from the time of initiation of therapy and death from any cause or last follow-up. Survival curves were plotted using the Kaplan-Meier method, and survival curves were compared by the log-rank test.

Results

Response

Treatment outcome of Btz/Dex in 26 patients with AL is summarized in Table 2. The overall response rate was 54% (14 of 26 patients), with 8 patients achieving a hematologic CR (31%) and 6 additional patients achieving a PR (23%). Of note, only one patient experienced a disease progression. Similar overall response rates were observed among untreated patients (58%, 10 of 18 patients) and previously treated patients (50%, 4 of 8 patients); however, all patients who reached a CR received Btz/Dex as first-line therapy. Median time to response was 7.5 weeks.

Improvement in organ function was noticed in 3 patients (12%); in 20 patients (76%), organ function remained stable, whereas progressive organ failure was noted in 3 patients (12%).

Survival

Median follow-up of surviving patients was 14.6 months. Median PFS and OS were 5.0 months and 18.7 months, respectively (Table 2, Figure 1234). When survival times were analyzed according to the quality of response, it became evident that patients achieving a hematologic CR had longer PFS and OS (medians not yet reached) compared to AL patients who did not achieve a CR (median PFS, 2.6 month; median OS 8.9 months). Toxicity A summary of observed side effects is given in Table 3. The most common grade 3/4 toxicities were thrombocytopenia and hyponatremia. Grade 2 neurotoxicity was observed in 42% of patients. Other grade 1 and 2 toxicities observed at high frequencies included edema (27%), hypotension (23%) and fatigue (23%). Two patients with previously treated AL experienced cardiac side effects (arrhythmias) judged to be related to Btz.

Discussion

The aim of this retrospective multicenter study was to evaluate the effect of Btz/Dex in patients with AL amyloidosis. The documented activity of Btz/Dex in MM and the recently developed hypothesis that amyloidogenic plasma cells may be particularly sensitive to proteasome inhibition [START_REF] Sitia | Bortezomib in the treatment of AL amyloidosis: targeted therapy?[END_REF] provide the background for exploring Btz in patients with AL.

Plasma cells in AL tend to accumulate unfolded toxic light chains, which could further increase the endoplasmic reticulum stress [START_REF] Teixeira | Endoplasmatic reticulum stress associated with extracellular aggregates. Evidence from transthyretin deposition in familial amyloid polyneuropathy[END_REF] eventually leading to apoptosis. Inhibition of the proteasome by Btz exaggerates endoplasmic reticulum stress, further stabilizes proapoptotic factors and thereby enhances induction of apoptosis [START_REF] Sitia | Bortezomib in the treatment of AL amyloidosis: targeted therapy?[END_REF]. Response rates to Btzbased regimens in the order of 50 -94% including a hematologic CR rate of 15% to 43% are in support of this hypothesis [START_REF] Kastritis | Treatment of light chain (AL) amyloidosis with the combination of bortezomib and dexamethasone[END_REF][START_REF] Wechalekar | Efficacy of bortezomib in systemic AL amyloidosis with relapsed/refractory clonal disease[END_REF]. In agreement with these reports, our results with Btz/Dex show a 54% overall response rate, with 31% of patients achieving a CR. Furthermore, the short time to the onset of a remission (median, 7.5 weeks) is an important feature of Btz/Dex treatment for AL. This is markedly different from what has been reported with other regimens, e.g. melphalan plus Dex and thalidomide plus Dex (median time to response, 4.5 months and 3.6 months, respectively) [START_REF] Jaccard | Myélome Autogreffe (MAG) and Intergroupe Francophone du Myélome (IFM) Intergroup. (2007) High-dose melphalan versus melphalan plus dexamethasone for AL amyloidosis[END_REF][START_REF] Palladini | The combination of thalidomide and intermediate-dose dexamethasone is an effective but toxic treatment for patients with primary amyloidosis (AL)[END_REF]. With the known limitations of a retrospective analysis, these results reflect a real-life scenario of AL patients because data were derived from a consecutive series of AL patients treated quite uniformly at major hematology/oncology centres.

A particular finding in our patient series was the association of a hematologic CR with prolonged PFS and OS (Figures 2 and4). Response to treatment was linked to survival of patients with AL in previous studies employing high-dose melphalan and autologous transplantation [START_REF] Comenzo | Autologous stem cell transplantation for primary systemic amyloidosis[END_REF]. This observation can now be extended to the setting of Btz-based therapy in AL. Overall, the importance of achieving a good quality remission is reminiscent of what has been described in MM, where depth of remission is associated with improved outcome [START_REF] Niesvizky | The relationship between quality of response and clinical benefit for patients treated on the bortezomib arm of the international, randomized, phase 3 APEX trial in relapsed multiple myeloma[END_REF][START_REF] Harousseau | Superior outcomes associated with complete response in newly diagnosed multiple myeloma patients treated with non-intensive therapy: analysis of the phase 3 VISTA study of bortezomib plus melphalan-prednisone versus melphalan-prednisone[END_REF]. Kastritis et al. reported for the first time the efficacy of the proteasome inhibitor Btz in patients with AL amyloidosis. The hematologic response rate was observed in 94%, including a 44% CR. The median follow up was 11.2 months [START_REF] Kastritis | Treatment of light chain (AL) amyloidosis with the combination of bortezomib and dexamethasone[END_REF]. Wechalekar et al. conducted a trial in which heavily pretreated patients received Btz with or without Dex. Btz was median the third line therapy (range 1 -6). Seven out of 9 (77%) patients who were given the combination of Btz/Dex responded. The median survival was not reached at 24 months from the end of treatment [START_REF] Wechalekar | Efficacy of bortezomib in systemic AL amyloidosis with relapsed/refractory clonal disease[END_REF]. Due to the multisystemic organ involvement toxicity is a major problem in these patients. Different therapies, e.g. thalidomide or lenalidomide, were not tolerated in patients with AL amyloidosis [START_REF] Palladini | The combination of thalidomide and intermediate-dose dexamethasone is an effective but toxic treatment for patients with primary amyloidosis (AL)[END_REF][START_REF] Dispenzieri | The activity of lenalidomide with or without dexamethasone in patients with primary systemic amyloidosis[END_REF][START_REF] Sanchorawala | An overview of the use of high-dose melphalan with autologous stem cell transplantation for the treatment of AL amyloidosis[END_REF]. In our patient series, toxicity of Btz/Dex treatment was limited and adverse events were manageable with routine clinical care. Most grades 3 + 4 toxicities consisted of thrombocytopenia and hyponatriemia (4%), respectively, and appeared to be lower than in other studies. Pretreated patients with cardiac involvement should be monitored closely because of their high risk of cardiac complications [START_REF] Kastritis | Treatment of light chain (AL) amyloidosis with the combination of bortezomib and dexamethasone[END_REF][START_REF] Kastritis | Bortezomib with or without dexamethasone in primary systemic (light chain) amyloidosis[END_REF].

In conclusion, our results confirm the activity of the combination of bortezomib and dexamethasone in elderly untreated und treated patients with AL amyloidosis. Adverse events are rare and manageable. Patients achieving a hematologic CR derive the greatest benefit from treatment with Btz/Dex which suggests that therapeutic strategies enhancing the CR-rate in AL are warranted for future investigations.

Figures:Figure 1 :

 1 Figures:Figure1: Progression free survival for all patients:

Figure 2 :

 2 Figure 2: Progression free survival CR versus Non-CR:

Figure 3 :

 3 Figure 3: Overall survival for all patients:

Figure 4 :

 4 Figure 4: Overall survival CR versus Non-CR:

Table 1 : Characteristics of 26 patients with AL:

 1

	Number of patients

Table 2 : Response to Therapy:

 2

Table 3 : Major toxicities of treatment with B/D:

 3