G Mirivel
email: mirivel@ensm-douai.fr

V Riffault
email: riffault@ensm-douai.fr

J C Galloo
email: galloo@ensm-douai.fr

DEVELOPMENT AND VALIDATION OF A LC/APCI/TOF-MS METHOD TO QUANTIFY NITRATED AND OXYGENATED PAHS IN ATMOSPHERIC AEROSOLS

Keywords: nitrated polycyclic aromatic hydrocarbons oxygenated polycyclic aromatic hydrocarbons, particulate organic matter, ultra-performance

A new method for the analysis of 12 nitrated polycyclic aromatic hydrocarbons (NPAHs) and 9 oxygenated polycyclic aromatic hydrocarbons (OPAHs) has been developed using reversed-phase liquid chromatography (LC) coupled to mass spectrometry (MS). Validation has been carried out on NIST standard reference materials and compared with certified data. Results are briefly discussed.

INTRODUCTION

Atmospheric aerosols are complex matrices composed of different chemical species potentially harmful to human health. Polycyclic aromatic hydrocarbons (PAHs) are wellknown carcinogenic and/or mutagenic compounds, emitted by incomplete combustion (e.g. [START_REF] Baek | A review of atmospheric polycyclic aromatic hydrocarbons: sources, fate and behaviour[END_REF], [START_REF] Straif | Carcinogenicity of polycyclic aromatic hydrocarbons[END_REF]). Once in the atmosphere, they redistribute between both the particulate and the gas phases.

PAHs derivatives, such as NPAHs and OPAHs, seem to be even more toxic than their related parent PAHs and are also found in airborne particulate matter [START_REF] Allen | Measurement of oxygenated polycyclic aromatic hydrocarbons associated with a sizesegregated urban aerosol[END_REF][START_REF] Bamford | Nitro-polycyclic aromatic hydrocarbon concentrations and sources in urban and suburban atmospheres of the mid-Atlantic region[END_REF]. They can be directly emitted by combustion processes or formed in the atmosphere by both gas and heterogeneous reactions of PAHs induced by atmospheric oxidants (OH, NO 3 ; NO 2 , O 3 ). Little is known about their behavior and their typical concentrations in airborne particulate matter, as their analysis generally requires several purification and preconcentration steps.

NPAHs and OPAHs have already been analyzed by gas chromatography/mass spectrometry methods (e.g. [START_REF] Koeber | Determination of benzo[a]pyrene diones in air particulate matter with liquid chromatography mass spectrometry[END_REF]). But this technique involves time-consuming procedures like sample purification and derivatization steps. LC generally requires less sample clean-up, and derivatization steps are usually not necessary. For some of these compounds LC coupled to mass spectrometry with atmospheric pressure ionization (LC/APCI/MS) has already been applied successfully [START_REF] Albinet | Simultaneous analysis of oxygenated and nitrated polycyclic aromatic hydrocarbons on standard reference material 1649a (urban dust) and on natural ambient air samples by gas chromatography-mass spectrometry with negative ion chemical ionisation[END_REF][START_REF] Lintelmann | Determination of oxygenated polycyclic aromatic hydrocarbons in particulate matter using high-performance liquid chromatography-tandem mass spectrometry[END_REF][START_REF] Schauer | Analysis of nitrated polycyclic aromatic hydrocarbons by liquid chromatography with fluorescence and mass spectrometry detection: air particulate matter, soot, and reaction products studies[END_REF][START_REF] Delhomme | Determination of oxygenated polycyclic aromatic hydrocarbons in atmospheric aerosol samples by liquid chromatographytandem mass spectrometry[END_REF]. To the best of our knowledge, few studies [START_REF] Schauer | Analysis of nitrated polycyclic aromatic hydrocarbons by liquid chromatography with fluorescence and mass spectrometry detection: air particulate matter, soot, and reaction products studies[END_REF] have used ToF-MS to characterize and quantify PAHs derivatives in particulate matter. One of its main advantages is its accurate mass measurements which can prove very useful when analyzing air samples since it can lessen the matrix interferences from coeluting impurities and it allows unequivocal identification of target compounds from complex matrices, as well as the possibility of quantification at low-level concentrations.

In this work, a new method for the simultaneous detection and quantification of 21 NPAHs and OPAHs (Figure 1) using LC/MS is reported.

EXPERIMENTAL

Materials

ULC-MS solvents (CH

3 CN, CH 3 OH, H 2 O, CH 2 O 2 )
were purchased from Biosolve (Valkenswaard, the Netherlands). Reference NPAHs and OPAHs were obtained from Aldrich (St-Quentin Fallavier, France) and Carlo Erba SDS (Peypin, France). If standard substances at different qualities were available, the highest purity was chosen. Leucine Enkephalin was purchased from Aldrich.

Stock solutions of NPAHs and OPAHs (0.02-10 mg.L -1 ) were prepared in methanol. The stock solutions were further diluted between 10 and 40,000 times to give a series of standards.

Standard reference samples of Diesel Particulate Matter (SRM 1650) and Urban Dust (SRM 1649) from the National Institute of Standards and Technology (NIST, Gaithersburg, MD, USA) were used.

Extraction

Extractions of NPAHs and OPAHs were carried out by a pressurized solvent extractor (ASE 200, Dionex SA, Voisins-le-Bretonneux, France) with cells of 11 mL. Methanol was used as the extraction solvent. A cellulose filter (∅ = 1.923 cm, 049458, D28, Dionex SA) was inserted at the bottom of the cell. Then particles were weighted on a quartz fiber filter (∅ = 47 mm, PALLFLEX, VWR, Fontenay-sous-Bois, France) which was folded to be introduced into the cell. Finally, Fontainebleau sand (VWR) was added to fill up the cell. Extraction parameters were as follows: 120°C extraction temperature, 2,000 psi nitrogen pressure, 8-min static time, 60% flush, 40 s purge.

The extracts were concentrated by evaporation under a nitrogen stream (TurboVap, Zymark Corp., Hopkinton, MA, USA) at 60°C down to a volume of 1 mL.

Instrumentation

Analyses of the extracts were performed by a Waters Acquity UPLC ® pump coupled to a PDA detector and a LCT Premier XE Time-of-Flight mass spectrometer (Waters, Millford, MA, USA). The MS detector was connected to the LC system through an atmospheric pressure chemical ionization (APCI) source running in negative mode. Instrument control, data acquisition and analysis were performed with MassLynx version 4.1.

The LCT Premier XE Time-of-Flight mass spectrometer was coupled to the UPLC ® system with an atmospheric chemical ionization (APCI) interface, which consists of a heated nebulizer probe and an atmospheric pressure source equipped with a corona discharge pin. Accurate mass spectra for negative ions were recorded across the range from m/z 50-1000 each 0.2 s. Nitrogen was used as the drying gas. The instrument was tuned to provide a resolution of 12,000 FWHM. The mass axis was calibrated using a sodium formate solution over the m/z 50-1000 range. A lock-mass of leucine enkephalin ([M-H] -= 555.2645 for the negative ion mode) at a concentration of 2.0 ng µL -1 in 50:50 acetonitrile:water containing 0.1% formic acid was infused at a flow rate of 0.05 mL min -1 via a lock spray ionization source with the help of a second HPLC pump (Waters Reagent Manager). Data were collected in the centroid mode and the lock-mass data were averaged over 50 scans for correction.

RESULTS AND DISCUSSION

Extraction procedure

The applicability of ASE for the extraction of PAHs from aerosol samples has been evaluated and optimized in a parallel study where it provides good repeatability and recoveries, as well as low time and solvent consumption. Similarly to the work carried out in the case of PAHs, blank filters (n=5) were spiked with a known amount of NPAHs and OPAHs and were extracted keeping parameters the same as for PAHs extraction. Methanol, also the analysis solvent, was used for extraction. Possible evaporation losses from the concentration step were investigated and found to be less than 10% for almost all the standard compounds, except for 2-fluorenecarboxaldehyde and 9,10anthraquinone.

Suitable chromatographic conditions

A good chromatographic separation of the analytes (Figure 2) needed to be achieved, in order to differentiate isomeric compounds such as nitrofluoranthenes in the mass spectrometer. The chromatographic separation was performed on an Acquity UPLC ® BEH C18 column (100 mm × 2.1 mm i.d., 1.7 µ m, Waters). Different elution conditions -with water, methanol and acetonitrile -and different concentrations of acid in the mobile phase have been tested. The addition of an acid to the mobile phase was required to enhance the sensitivity of the APCI. The two mobile phases were -phase A: acidified water (10 mM formic acid); phase B: methanol. A linear gradient was programmed -0 min: 20% B; 20 min: 90% B; 27 min: 90% B; 30 min: 20% B. The flow rate was 0.2 mL min -1 . The column was held at 30°C. The injection volume was 5 µL. The Acquity UPLC ® HSS T3 column (100 mm × 2.1 mm i.d., 1.8 µm, Waters) was also tested without improving the separation. UV chromatograms were monitored across the range 190-500 nm with a resolution of 1.2 nm.

LC/APCI/ToF-MS parameter optimization

The main instrumental parameters (corona current, cone voltage, probe gas flow and temperature, and aperture 1 voltage) were optimized to provide the best possible sensitivity within the recommended range of values (cf. Table 1).

Optimizations were carried out by chromatographic separation of a standard solution and the signal area obtained for each compound was plotted as a function of the settings value. Parameters were sequentially optimized, using the optimized values in the following optimization steps. For all compounds, the major peak in the mass spectra was either the [M] -or the [M-H] -molecule and for each parameter, a compromise providing the optimal sensitivity for all compounds was chosen.

Interestingly, in the higher range of aperture 1 voltages the fragmentation of the analytes is increased and allows the use of one or two specific fragments for confirmation purposes.

Table 1. Optimized parameter values of the APCI interface

Accurate mass measurements

Checking the accuracy of mass measurements was carried out using the following procedure. The m/z of the analytes (using a mass interval of 0.01 Da) was extracted from the total ion chromatogram (TIC), to obtain an extracted ion chromatogram. The accurate masses of the molecular ions [M] -were used for both identification and quantification purposes in all cases, except for 2-fluorenecarboxaldehyde, 5-nitroacenaphthene and 2-nitrofluorene, which showed deprotonated ions. Accurate mass data of the molecular ions were then processed through the software, which provided a list of molecular formula assignments based on the accurate mass and the isotopic intensity pattern (i-fit). For each compound, the best elemental formula corresponded to the expected molecule. Accuracy of mass measurements for a standard solution of the 21 PAH derivatives is shown in Table 2. Errors were less than 5 ppm in most cases with an average value of about 1 ppm. The widely accepted accuracy threshold for confirmation of elemental compositions has been established at 5 ppm. The corresponding mass spectrum together with the characteristic retention time and UV spectrum provided unique elemental composition assignment.

The effect of matrix complexity from standard reference samples and filters was evaluated and no significant differences were observed in the accuracy compared to those prepared with pure solvents, keeping the relative error below 5 ppm. Therefore the method can be considered as specific and selective.

Method validation

Sensitivity, accuracy, precision and linearity of the method were investigated for validation.

Instrumental limits of detection (LOD) were calculated for all standard compounds and were defined as three times the signal-to-noise ratio (S/N=3). LOD ranged from 10 to 295 pg for NPAHs, and from 45 to 1185 pg for OPAHs. Values of instrumental LODs obtained for NPAHs were in the same order of magnitude as those obtained by [START_REF] Schauer | Analysis of nitrated polycyclic aromatic hydrocarbons by liquid chromatography with fluorescence and mass spectrometry detection: air particulate matter, soot, and reaction products studies[END_REF]. Those obtained for OPAHs were less sensitive than those obtained by [START_REF] Delhomme | Determination of oxygenated polycyclic aromatic hydrocarbons in atmospheric aerosol samples by liquid chromatographytandem mass spectrometry[END_REF]. However in this latter work an LC-APCI-MS-MS was used which produced some selective fragmentation patterns leading to an increase of selectivity and sensitivity. Quantitation limits (LOQ) were considered as the analyte concentration that gives S/N = 10.

Accuracy was estimated with the determination of recovery factors from the extraction of five blank filters spiked with a standard mixture. The extraction method presented here gives acceptable recovery values (> 80%) for almost all the standard compounds. However the obtained value for chrysene-5,6-dione and phenanthrene-9,10-dione were as low as 39% and 41%, respectively, but with estimated concentrations comparable to the detection limits, where small area variations strongly affect the measured concentrations. Recovery corrections were applied to the NPAHs and OPAHs measurement values.

The repeatability of the response was by injecting the same standard solution seven times in one day. The reproducibility of the response was carried out for three successive days using the same solution. From the results obtained, the developed method was found to be precise (with run-to-run instrumental RSD < 7% and day-to-day RSD < 13%). In addition, RSD values of retention time (R t ) were less than 0.1% for repeatability and reproducibility.

Calibration plots were drawn from six different concentration levels up to 3 orders of magnitude, using standard solutions ranging from 0.02 to 10 mg L -1 . Excellent linearity was observed for all compounds with determination coefficients R² > 0.992 in most cases. Due to some disadvantages and limitations, such as the narrow dynamic range, LC/ToF-MS has been applied mainly for confirmatory analyses rather than for quantification in the past few years. A Dynamic Range Enhancement (DRE) is used in this work providing thus an enhanced linear dynamic range and this makes possible its successful applicability to NPAHs and OPAHs analyses.

OPAHs and NPAHs concentrations in air particulate reference samples

The developed method was applied to the analysis of two NIST standard reference samples where some PAH derivatives can be found: Diesel Particulate Matter (SRM 1650) and Urban Dust (SRM 1649) (approximately 10 and 50 mg, respectively, one replicate for each). Tables 3 and4 show our measurements together with those reported in previous studies. There is a generally good agreement for OPAHs in the literature values obtained by GC/MS analysis of SRM 1649. Most NPAHs are below our detection limits for this reference sample. Regarding SRM 1650 this study is to the best of our knowledge only the second one to report OPAHs concentrations. The high value found for 1nitrofluoranthene might indicate that the analysis is not specific enough for this compound and might include 2nitrofluoranthene measurements as well. Despite sufficiently low LODs, 7-nitrobenz[a]anthracene and 6-nitrobenz[a]pyrene have not been detected in SRM 1650. These discrepancies might be explained by an ineffective extraction from particulate material, even if recoveries for these two compounds in spiked blank filters were excellent (>95%). The measurement values obtained for 9nitroanthracene and 1-nitropyrene were in good agreement with NIST certified values (113 and 78% recovery, respectively).

CONCLUSION

In this work, a new method for the simultaneous detection and quantification of 21 NPAHs and OPAHs by LC/APCI/ToF-MS has been developed. The resolving power, accurate mass measurement capability and full spectral sensitivity allowed for the determination of the target compounds in complex matrices. So far, the analytical procedure has been applied to NIST SRM1649 and SRM1650, giving results in good agreement with the few data available in the literature and leading to the first determination of NPAHs and OPAHs in the Diesel particulate. This method will be used to investigate NPAHs and OPAHs concentrations in ambient air samples.

Fig. 1 .

 1 Fig. 1. Structures of the 12 NPAHs and 9 OPAHs studied The superscript number refers to the elution order in Fig.2 chromatogram

Fig. 2 .

 2 Fig.2. LC/ToF-MS ion chromatograms of a standard solution of NPAHs and OPAHs with APCI in negative mode

Table 2 . LC/APCI/ToF-MS accurate mass measurements for a standard solution of 21 NPAHs and OPAHs in MeOH

 2

	Error

Table 3 . OPAHs and NPAHs concentrations (ng g -1 ) in SRM 1649 (urban dust)

 3

		This work a	NIST	Albinet et al. [6] e	Fernandez & Bayona [10] d	Durant et al. [11] f	Cho et al. [12] g	Chiu & Miles [13] h	Bamford et al. [14] h	Crimmins & Baker [15] h	Oda et al. [17]
	9,10-Anthraquinone	3046 (152)	-	2238 (363)	220 (40)	2700 (120)	2030 (192)	-	-	-	1600
	Phenanthrene-9,10-dione	5414 (271)	-

ACKNOWLEDGEMENTS

Our laboratory participates in the Research Institute of Industrial Environment (IRENI) which is financed by the Communauté Urbaine de Dunkerque, the Nord-Pas de Calais Regional Council, the French Ministry of Education and Research, and European funds (FEDER).


D Efimov

E Fridman

On ISS with respect to average value of disturbances: a time-delay approach

For input-affine nonlinear dynamical systems, an input-to-state stability analysis with respect to (weighted) average values of exogenous perturbations is proposed. The time-delay method is used to represent the system in a suitable form for investigation: a kind of neutral-type differential equation. The introduced approach allows the asymptotic gains with respect to zero-mean periodic signals to be evaluated for nonlinear systems (an analogue of Bode magnitude plot), as well as for integral input-to-state stable systems with periodic inputs. The results are illustrated on the class of homogeneous systems.

I. INTRODUCTION

Analysis of robustness of stability properties for dynamical systems with respect to external disturbances, measurement noises, variations of parameters, unmodeled dynamics, delays, etc., is one of the central issues in the control and estimation theory [START_REF] Khalil | Nonlinear Systems[END_REF]. For linear systems all these notions have been wellstudied through H ∞ norms or L 2 gains in 60s of the previous century. For nonlinear systems there are several approaches for robust stability analysis, and one of the most popular frameworks is built around the input-to-state stability (ISS) concept [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]. The advantage of this theory is that it has a list of properties describing different stability characteristics, as well as efficient necessary and sufficient conditions to check these properties in applications (existence of proper Lyapunov functions are usually verified). This framework allows one to make qualitative and quantitative analysis of behavior of the system trajectories with respect to the L ∞ norm of the input (or an integral of the norm).

In some cases, the analysis should be performed not with respect to the maximal amplitude of the input, but evaluating the bounds on trajectories as a function of the frequency of the disturbance, or its average value on an interval of time (see the respective extensions of the ISS framework [START_REF] Angeli | Power characterizations of input-to-state stability and integral input-to-state stability[END_REF], [START_REF] Grø Tli | Robustness of ISS systems to inputs with limited moving average: Application to spacecraft formations[END_REF]). This is often the case in vibration analysis, or when the external inputs are (almost) periodic functions of time. In such a scenario, the time-averaging approach was found to be very efficient [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-Linear Oscillations[END_REF], and later its different extensions have been proposed [START_REF] Kokotovic | Singular perturbations in systems and control[END_REF], [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF], [START_REF] Mazenc | Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems[END_REF], also in conjunction with ISS [START_REF] Nesic | A note on input-to-state stability and averaging of systems with inputs[END_REF], [START_REF] Teel | A unified framework for input-tostate stability in systems with two time scales[END_REF] (where the stability property of the system is substantiated from the characteristics of the averaged dynamics). Recent works [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], [START_REF] Zhang | L2-gain analysis via time-delay approach to periodic averaging with stochastic extension[END_REF], which are based on the time-delay approach, introduced the idea of a transformation of coordinates, which allowed quantitative estimates to be obtained on the averaging period.

The objective of this work is to use the latter results to propose a tool for studying ISS property of nonlinear systems with respect to an averaged value of the input, which can also be transformed in a frequency gain analysis (similar to the amplitude Bode plot for linear systems or its counterpart given for nonlinear convergent dynamics in [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF]). Applying the transformation from [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], [START_REF] Zhang | L2-gain analysis via time-delay approach to periodic averaging with stochastic extension[END_REF], it will be shown that a nonlinear affine in the inputs system can be transformed to the neutral-type delayed dynamics dependent on the average values of the inputs. The obtained theoretical findings are illustrated on a class of homogeneous systems.

For homogeneous dynamical systems, their local and global behaviors coincide, and for stability/instability analysis, Lyapunov or Chetaev function of a homogeneous system can also be chosen homogeneous [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]. In addition, if a system is homogeneous (considering disturbance as an auxiliary variable) and asymptotically stable without disturbances, then it is ISS [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. These facts make homogeneous systems suitable for analysis and modeling of complex nonlinear compartments, disposing many efficient and helping design procedures. The averaging method has been already used with homogeneous systems in [START_REF] Peuteman | Averaging results for homogeneous differential equations[END_REF], [START_REF] Peuteman | Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying[END_REF], [START_REF] Aleksandrov | Stability analysis for a class of nonlinear nonstationary systems via averaging[END_REF], [START_REF] Aleksandrov | Rigid body stabilization under time-varying perturbations with zero mean values[END_REF], where stability of a system was related with the same property of its averaged dynamics.

The outline of this work is as follows. The preliminary definitions and the homogeneity framework are given in Section II. A method for analysis of robust stability with respect to an average value of perturbations is presented in Section III. Application of the developed theory to homogeneous dynamical systems is considered in Section IV.

II. PRELIMINARIES

A. Notation

The real and integer numbers are denoted by R and Z, respectively, R + = {s ∈ R : s ≥ 0} and Z + = R + ∩ Z. Euclidean norm for a vector x ∈ R n is denoted by ∥x∥ (for a matrix A ∈ R n×m , ∥A∥ corresponds to the induced norm), and the absolute value of s ∈ R by |s|. We denote by

C n [a,b] , -∞ < a < b < +∞ the Banach space of continuous functions ϕ : [a, b] → R n with the uniform norm |ϕ| C = sup a≤ς≤b ∥ϕ(ς)∥ (similarly for C n [a,b) ), for x : [-T, +∞) → R n with T > 0 we denote by x t ∈ C n [-T,0] , t ∈ R + its respective restriction x t (s) = x(t + s) for s ∈ [-T, 0]. For a Lebesgue mea- surable function of time d : [a, b) → R m define the norm ∥d∥ [a,b) = ess sup t∈[a,b) ∥d(t)∥, then ∥d∥ ∞ = ∥d∥ [0,+∞) and the space of d with ∥d∥ [a,b) < +∞ (∥d∥ ∞ < +∞) we further denote as L m [a,b] (L m ∞ ).
For a continuously differentiable function V : R n → R + , its directional derivative along a given vector v ∈ R n at a point x ∈ R n is defined as follows:

DV (x)v = ∂V (x) ∂x v.
A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R + ×R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is a strictly decreasing to zero for any fixed r>0.

A sequence of integers, 1, 2, . . . , n, is further denoted by 1, n.

B. Input-to-state stability

Consider a dynamical system:

ẋ(t) = F (x(t), u(t)), t ≥ 0, (1)
where x(t) ∈ R n is the state and u(t) ∈ R m is the input, u ∈ L m ∞ ; F : R n+m → R n is a continuous function, and it ensures forward existence and uniqueness of solutions of the system at least locally in time (e.g., it is locally Lipschitz continuous out the origin), F (0, 0) = 0. For any x 0 ∈ R n and u ∈ L m ∞ the respective solution is denoted by x(t, x 0 , u) with x(0, x 0 , u) = x 0 .

The definitions and results given in this subsection follow [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF].

Definition 1. The system (1) is called practically ISS, if there exist β ∈ KL, γ ∈ K and q ≥ 0 such that ∥x(t, x 0 , u)∥ ≤ max{β(∥x 0 ∥, t), γ(∥u∥ [0,t) ), q}, ∀t ≥ 0 for all x 0 ∈ R n and u ∈ L m ∞ . If q = 0 then (1) is called ISS.

The function γ from Definition 1 quantifies an asymptotic gain of (1): lim sup t→+∞ ∥x(t, x 0 , u)∥ ≤ max{γ(∥u∥ [0,t) ), q} satisfying for all x 0 ∈ R n and u ∈ L m ∞ . Another property, followed by (practical) ISS, is (practical) global stability:

there exist ϑ x , ϑ u ∈ K ∞ such that ∥x(t, x 0 , u)∥ ≤ max{ϑ x (∥x 0 ∥), ϑ u (∥u∥ [0,t) ), q}, ∀t ≥ 0 for all x 0 ∈ R n and u ∈ L m ∞ (obviously, a possible choice is ϑ x (s) = β(s, 0) and ϑ u (s) = γ(s)).
If u ≡ 0, then from ISS we recover the conventional global asymptotic stability at the origin. Definition 2. A continuously differentiable function V : R n → R + is called practical ISS-LF for the system (1) if there exist α 1 , α 2 ∈ K ∞ , η, ϱ ∈ K and r ≥ 0 such that for all x ∈ R n and all u ∈ R m :

α 1 (∥x∥) ≤ V (x) ≤ α 2 (∥x∥), V (x) ≥ max{ϱ(∥u∥), r} =⇒ DV (x)F (x, u) ≤ -η(∥x∥). If r = 0, then such a function V is called ISS-LF.
Theorem 1. The system (1) is (practically) ISS if and only if it admits a (practical) ISS-LF.

C. Homogeneity

For any r i > 0, i ∈ 1, n and λ > 0, define the vector of weights r = [r 1 , ..., r n ] and the dilation matrix Λ r (λ) = diag{λ ri } n i=1 ; r min = min i∈1,n r i and r max = max i∈1,n r i . Definition 3. [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF] The function h : R n → R is called r-homogeneous, if for any x ∈ R n the relation

h(Λ r (λ)x) = λ ν h(x)
holds for some ν ∈ R and all λ > 0.

The vector field f : R n → R n is called r-homogeneous, if for any x ∈ R n the relation

f (Λ r (λ)x) = λ ν Λ r (λ)f (x)
holds for some ν ≥ -r min and all λ > 0.

In both cases, the constant ν is called the degree of homogeneity.

A dynamical system

ẋ(t) = f (x(t)), t ≥ 0 (2)
is called r-homogeneous of degree ν if this property is satisfied for f in the sense of Definition 3.

For any x ∈ R n and ϖ ≥ r max , a homogeneous norm can be defined as follows

∥x∥ r = n i=1 |x i | ϖ/ri 1/ϖ .
For all x ∈ R n , its Euclidean norm ∥x∥ is related with the homogeneous one:

σ r (∥x∥ r ) ≤ ∥x∥ ≤ σr (∥x∥ r ) (3)
for some σ r , σr ∈ K ∞ [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]. In the following, due to this "equivalence", stability analysis with respect to the norm ∥x∥ can be substituted with analysis for the norm ∥x∥ r .

The homogeneous norm has an important property: it is rhomogeneous of degree 1, that is ∥Λ r (λ)x∥ r = λ∥x∥ r for all x ∈ R n and λ > 0.

Homogeneous systems possess certain robustness with respect to external disturbances.

Theorem 2. [15] Let F (Λ r (λ)x, Λ r (λ)u) = λ ν Λ r (λ)F (x, u) for all x ∈ R n , u ∈ R m
and all λ > 0 with the weights r = [r 1 , . . . , r n ] > 0, r = [r 1 , . . . , rm ] > 0 and a degree ν ≥ -r min . Assume that the system (1) is asymptotically stable for u = 0, then (1) is ISS.

III. MAIN RESULT

Consider an affine in the inputs nonlinear system:

ẋ(t) = f (x(t)) + g(x(t))u(t), t ≥ 0, (4)
where

x(t) ∈ R n is the state, u(t) ∈ R m is the input, u ∈ L m ∞ ; f : R n → R n
and g : R n → R n×m are locally Lipschitz continuous functions for all x ∈ R n \ {0}, ensuring forward existence and uniqueness of the system solutions at least locally in time, f (0) = 0.

Our goal is to derive ISS-like estimates for (4) with respect to average or weighted average values of the input u, and for this purpose we define the signals of interest:

δ 1 (t) = 1 T t t-T (s -t + T )u(s)ds, δ 2 (t) = 1 T t t-T u(s)ds, δ1 (t) = 1 T t t-T (s -t + T )∥u(s)∥ds, δ2 (t) = 1 T t t-T ∥u(s)∥ds,
where T > 0 is the window of averaging (a parameter for our analysis). These quantities are well-defined for t ≥ T , while for t ∈ [0, T ) in the sequel we will assume that u(t -T ) = 0, then all these averaged variables are defined for all t ≥ 0. Remark 1. Note that δ 1 , δ 2 , δ1 and δ2 are continuous functions of time t ≥ 0, and the upper bounds of these averaged values are related to the respective norms of u and the period T :

∥δ 1 (t)∥ ≤ ∥ δ1 (t)∥ ≤ T 2 ∥u∥ ∞ , ∥δ 2 (t)∥ ≤ ∥ δ2 (t)∥ ≤ ∥u∥ ∞ .
A. Time-delay approach analysis Similar to [START_REF] Fridman | Averaging of linear systems with almost periodic coefficients: A time-delay approach[END_REF], [START_REF] Zhang | L2-gain analysis via time-delay approach to periodic averaging with stochastic extension[END_REF], [START_REF] Yang | A robust time-delay approach to continuoustime extremum seeking for multi-variable static map[END_REF], let us define a transformation of state variables:

x(t) = 1 T t t-T (s -t + T )g(x(s))u(s)ds + z(t), (5)
x(s) = x(0), u(s) = 0, ∀s < 0,
where z(t) ∈ R n is a new auxiliary residual variable defined for all t ≥ 0. Differentiating ( 5) we obtain:

ż(t) = f (x(t)) + 1 T t t-T g(x(s))u(s)ds (6) = f (z(t) + d 1 (t)) + d 2 (t),
where

d 1 (t) = - 1 T t t-T (s -t + T )g(x(s))u(s)ds, (7)
d 2 (t) = 1 T t t-T g(x(s))u(s)ds.
The right-hand side of (5) depends on the past values of x(t) (i.e., x(s) for s ∈ [t -T, t)). Thus, (5) can be written as

x(t) -G(x t , u t ) = z(t) for G(x t , u t ) = 1 T t t-T (s -t + T )g(x(s))u(s)ds, where x t ∈ C n [-T,0] with x t (s) = x(t + s) and u t (s) = u(t + s) for s ∈ [-T, 0].
It can be considered as a difference equation with respect to x(t) with disturbances u and z, which is ISS in z(t) and δ1 (t) under introduced restrictions, as it is demonstrated later. To this end note that for any x(0) ∈ R n and u ∈ L m ∞ in (5), z(t) exists on maximal time interval T = [0, t ′ ) for some t ′ > 0 and z(0) = x(0). By continuity of the solutions x(t) of (4), z(t) is a continuous function of time t ∈ T by construction.

The starting point for our analysis is ISS of (6) with respect to the auxiliary inputs d 1 and d 2 :

Assumption 1. There exists a continuously differentiable function V : R n → R + such that

α 1 (∥z∥) ≤ V (z) ≤ α 2 (∥z∥), V (z) ≥ max{σ 1 (∥d 1 ∥), σ 2 (∥d 2 ∥)} ⇒ DV (z) (f (z + d 1 ) + d 2 ) ≤ -α(∥z∥), for all z, d 1 , d 2 ∈ R n , with some α 1 , α 2 , α ∈ K ∞ and σ 1 , σ 2 ∈ K.
These conditions imply that V is an ISS-LF for the system (6) with disturbances d 1 , d 2 ∈ L n ∞ and, hence, due to Theorem 1 this system is ISS with respect to these perturbations.

Since in [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF] there is additionally a nonlinear gain g multiplying the input u, supplementary constraints have to be imposed. Note that by continuity of g, its norm can be bounded by a growing function of ∥x∥, hence, there exist ρ ∈ K and ρ 0 ≥ 0 such that

∥g(x)∥ ≤ ρ(∥x∥) + ρ 0 , ∀x ∈ R n . ( 8
)
Assumption 2. There exist θ δ , θ z ∈ K ∞ such that

θ z (s) + (ρ(s) + ρ 0 )θ δ (s) < s, ∀s > 0. Assumption 3. There exist γ V ∈ K, γ 1 , γ 2 ∈ K ∞ such that ξ ≥ γ V (ζ) ⇒ σ i (γ i (ξ)(ρ(ζ) + ρ 0 )) ≤ ξ, i = 1, 2, ∀ξ, ζ ∈ R + ,
where σ 1 , σ 2 ∈ K are given in Assumption 1.

In these assumptions, some functions from classes K and K ∞ are introduced, whose existence guarantees ISS of ( 4) with respect to averaged signals δ1 and δ2 , as it will be shown below (assumptions 1 and 3 imply ISS of (4) with respect to the input

u if γ V (s) ≥ α 1 (s) for all s ∈ R + ). Lemma 1. Let Assumption 2 hold and given x(0) ∈ R n , z ∈ L n ∞ ∩ C n [0,+∞) , u ∈ L m ∞ the relation (5)
be satisfied for all t ∈ T . Then T = R + and there exists β 1 ∈ KL such that [START_REF] Mazenc | Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems[END_REF] an estimate follows from (5) for all t ∈ T :

∥x(t)∥ ≤ max{β 1 (∥x(0)∥, t), θ -1 z (∥z∥ [0,t) ), (9)
θ -1 δ (∥ δ1 ∥ [0,t) )} for all t ≥ 0, all x(0) ∈ R n , z ∈ L n ∞ ∩ C n [0,+∞) and u ∈ L m ∞ . Proof. Using
∥x(t)∥ ≤ 1 T t t-T (s -t + T )∥g(x(s))u(s)∥ds + ∥z(t)∥ ≤ (ρ(|x t | C ) + ρ 0 ) δ1 (t) + ∥z(t)∥.
Under Assumption 2, this estimate leads to the following relation:

∥x(t)∥ ≥ max{θ -1 z (∥z(t)∥), θ -1 δ ( δ1 (t))} ⇒ ∥x(t)∥ ≤ θ z (|x t | C ) + (ρ(|x t | C ) + ρ 0 ) θ δ (|x t | C ) = µ(|x t | C ) with µ ∈ K ∞ . The contraction property, µ(|x t | C ) < |x t | C for all |x t | C ̸ = 0, is introduced in Assumption 2,
and it implies convergence of x(t) while the latter implication holds. Indeed, assume that T = T z ∪ T x , where

t ∈ T z : ∥x(t)∥ ≤ max{θ -1 z (∥z∥ ∞ ), θ -1 δ (∥ δ1 ∥ ∞ )}, t ∈ T x : ∥x(t)∥ ≥ max{θ -1 z (∥z∥ ∞ ), θ -1 δ (∥ δ1 ∥ ∞ )},
then due to continuity of x(t) there exists a sequence of nonempty intervals such that T x = j∈Z+ [t j s , t j e ] (all isolated points can be placed in T z ). Denote τ t = min{arg max s∈[-T,0] ∥x(t + s)∥} (we use min{•} to avoid non-uniqueness of arg max operation), i.e., |x t | C = ∥x(τ t )∥, and for some j ∈ Z + consider t ∈ [t j s , t j e ], then

∥x(t)∥ ≤ µ(∥x(τ t )∥) ≤ µ • µ(|x τt | C ) ≤ µ • µ(∥x(τ τt )∥) ≤ • • • ≤ µ ceil t-t j s T • • • • • (|x t j s | C )
, where ceil (s) is the function of truncation to the smallest integer bigger than s ∈ R. Hence, there is

β 1 ∈ KL such that ∥x(t)∥ ≤ β 1 (|x t j s | C , t -t j s ) for all t ∈ [t j
s , t j e ] and any j ∈ Z + . Note that β 1 (s, 0) = s by the construction, then for the value of |x t j s | C two options are possible: either t j s = 0 and |x

t j s | C = |x 0 | C = ∥x(0)∥, or t j s > 0 and |x t j s | C = max{θ -1 z (∥z∥ ∞ ), θ -1 δ (∥ δ1 ∥ ∞ )
} by continuity. Therefore, the following estimate is satisfied for all t ∈ T :

∥x(t)∥ ≤ max{β 1 (∥x(0)∥, t), θ -1 z (∥z∥ ∞ ), θ -1 δ (∥ δ1 ∥ ∞ )}, implying that T = R + , which finishes the proof.
In the framework of the trajectory based approach [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], a similar ISS result was obtained for µ(s) = µ 0 s with µ 0 ∈ (0, 1) and the exponential convergence, then Lemma 1 can be seen as an extension of that method to the case of asymptotic convergence. Now let us investigate the behavior of z(t) in [START_REF] Kokotovic | Singular perturbations in systems and control[END_REF]. Under Assumption 1, for d 1 and d 2 defined by [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF], considering V (z) as an ISS-LF candidate for (6), the following relation is valid:

V (z(t)) ≥ max{σ 1 (∥d 1 (t)∥), σ 2 (∥d 2 (t)∥)} ⇒ V (t) = DV (z(t)) (f (z(t) + d 1 (t)) + d 2 (t)) ≤ -α(∥z(t)∥), Recalling (8), ∥d 1 (t)∥ ≤ (ρ(|x t | C ) + ρ 0 ) δ1 (t), ∥d 2 (t)∥ ≤ (ρ(|x t | C ) + ρ 0 ) δ2 (t).
In addition, under Assumption 3 the following implications are satisfied:

V (z(t)) ≥ max{γ V (|x t | C ), γ -1 1 ( δ1 (t)), γ -1 2 ( δ2 (t))} ⇒ V (z(t)) ≥ max{σ 1 (ρ(|x t | C ) + ρ 0 ) δ1 (t) , σ 2 (ρ(|x t | C ) + ρ 0 ) δ2 (t) } ⇒ V (z(t)) ≥ max{σ 1 (∥d 1 (t)∥), σ 2 (∥d 2 (t)∥)},
which implies due to Definition 2 that V is an ISS-LF for [START_REF] Kokotovic | Singular perturbations in systems and control[END_REF] with respect to the inputs |x t | C , δ1 (t) and δ2 (t):

V (z) ≥ max{γ V (|x t | C ), γ -1 1 ( δ1 ), γ -1 2 ( δ2 )} ⇒ V ≤ -α(∥z∥) for all z ∈ R n , δ1 , δ2 ∈ R + and x t ∈ C n [-T,0]
. Therefore, according to Theorem 1, the following estimate is valid:

∥z(t)∥ ≤ α -1 1 • max{β 2 (∥z(0)∥, t), γ V (∥x∥ [0,t) ), (10)
γ -1 1 (∥ δ1 ∥ [0,t) ), γ -1 2 (∥ δ2 ∥ [0,t)
)} for all t ≥ 0 and some β 2 ∈ KL (note that again β 2 (s, 0) = s).

We are in position to formulate the main result. Theorem 3. Let Assumptions 1-3 be satisfied for (4), then there exists

β 3 ∈ KL such that ∥x(t)∥ ≤ max{β 3 (∥x(0)∥, t), θ -1 δ (∥ δ1 ∥ [0,t) ), θ -1 z • α -1 1 • γ -1 1 (∥ δ1 ∥ [0,t) ), θ -1 z • α -1 1 • γ -1 2 (∥ δ2 ∥ [0,t) )} for all t ≥ 0, all x(0) ∈ R n and u ∈ L m ∞ provided that θ -1 z • α -1 1 • γ V (s) < s, ∀s > 0, meaning that (4) is ISS with respect to δ1 , δ2 ∈ L 1
∞ that are averages of |u|.

Proof. Since all conditions of Lemma 1 are verified, the estimate ( 9) is valid for t ∈ T ⊆ R + . Above, it has been shown that the estimate ( 10) is also true for all t ∈ T . Using the standard small-gain arguments [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF], [START_REF] Dashkovskiy | An ISS small gain theorem for general networks[END_REF], first, the global stability of the feedback interconnection ( 5), ( 6) can be proven, i.e., there exist functions ψ i ∈ K ∞ for i = 0, 1, 2 such that max{∥x(t)∥, ∥z(t)∥} ≤ max{ψ 0 • max{∥x(0)∥, ∥z(0)∥},

ψ 1 (∥ δ1 ∥ ∞ ), ψ 2 (∥ δ2 ∥ ∞ )}, ∀t ≥ 0,
for all x(0), z(0) ∈ R n and u ∈ L m ∞ , implying also that T = R + . Under the observation that ∥z(0)∥ ≤ (ρ(∥x(0)∥) + ρ 0 ) ∥ δ1 ∥ ∞ + ∥x(0)∥, the latter estimate can be rewritten as

max{∥x(t)∥, ∥z(t)∥} ≤ max{ ψ0 (∥x(0)∥), ψ1 (∥ δ1 ∥ ∞ ), ψ 2 (∥ δ2 ∥ ∞ )}
for all t ≥ 0, x(0) ∈ R n and u ∈ L m ∞ , for suitably modified ψ0 , ψ1 ∈ K ∞ . Second, the asymptotic gain function can be derived (we are further interested only in the behavior of x(t) due to its physical meaning in (4)):

lim sup t→+∞ ∥x(t)∥ ≤ max{θ -1 δ (∥ δ1 ∥ ∞ ), θ -1 z • α -1 1 • max{γ -1 1 (∥ δ1 ∥ ∞ ), γ -1 2 (∥ δ2 ∥ ∞ )}}.
If the system (4) admits both properties, global stability and asymptotic gain, then it is ISS [26, Lemma 2.7].

Remark 2. All assumptions, 1-3, as well as the small-gain condition in Theorem 3, are formulated globally (i.e., for all x ∈ R n and s > 0). Restricting validity of the introduced inequalities it is possible to get the respective local stability results.

In the conditions of Theorem 3, the system (4) is not obligatory ISS in u, while the power ISS or the moving average ISS properties from [START_REF] Angeli | Power characterizations of input-to-state stability and integral input-to-state stability[END_REF] are equivalent to ISS under Assumption 1. In [START_REF] Grø Tli | Robustness of ISS systems to inputs with limited moving average: Application to spacecraft formations[END_REF], ISS of (4) with respect to δ2 was established starting from ISS property in u.

Note that rewriting (6) in the coordinates x(t),

d dt x(t) - 1 T t t-T (s -t + T )g(x(s))u(s)ds = f (x(t)) + 1 T t t-T g(x(s))u(s)ds,
we get a special neutral-type time-delay system. Then the main result of this work can be interpreted as an extension of Lyapunov-Razumikhin ISS result of [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF] to this neutral-type system.

The results of Theorem 3 can be easily applied if the function g(x) is just bounded, then ρ ∈ K from (8) admits a restriction:

ρ(s) ≤ ρ 1 < +∞, ∀s > 0,
for some ρ 1 > 0, and we can take θ δ (s) = θz(s) ρ1+ρ0 with θ z (s)

< s 2 , γ i (s) = σ -1 i (s)
ρ1+ρ0 with i = 1, 2 and γ V (s) < α 1 • θ z (s). In addition, if g is a constant function, then a more constructive result can be derived given below.

B. A static input gain

A constant input gain, g(x) = B ∈ R n×m , represents an interesting practical case, where rather technical Assumptions 2, 3 and the small-gain condition can be skipped (they characterize the properties of g as a function of x): Theorem 4. Let Assumption 1 be satisfied for (4) with g(x) ≡ B, then there exists β 4 ∈ KL such that

∥x(t)∥ ≤ max{β 4 (∥x(0)∥, t), α -1 1 • σ 1 (∥B∥∥δ 1 ∥ [0,t) ), α -1 1 • σ 2 (∥B∥∥δ 2 ∥ [0,t) )} + ∥B∥∥δ 1 ∥ [0,t) for all t ≥ 0, all x(0) ∈ R n and u ∈ L m ∞ , meaning that (4) is ISS with respect to δ 1 , δ 2 ∈ L m ∞ ∩ C m [0,+∞) that are averages of u.
Proof. In such a case the equations ( 5) and ( 6) take the form:

x(t) = Bδ 1 (t) + z(t) and ż(t) = f (z(t) -Bδ 1 (t)) + Bδ 2 (t),
respectively. Hence, from the first equation, which is static here, we immediately get:

∥x(t)∥ ≤ ∥B∥∥δ 1 (t)∥ + ∥z(t)∥
for all t ≥ 0, while applying Assumption 1 to the dynamics of z, its ISS property can be substantiated, i.e., there exists β4 ∈ R n such that

∥z(t)∥ ≤ α -1 1 • max{ β4 (∥z(0)∥, t), σ 1 (∥B∥∥δ 1 ∥ [0,t) ), σ 2 (∥B∥∥δ 2 ∥ [0,t) )}, ∀t ≥ 0, for all z(0) ∈ R n and δ 1 , δ 2 ∈ L m ∞ .
Combining both inequalities, due to ∥z(0)∥ = ∥x(0)∥ we obtain:

∥x(t)∥ ≤ ∥B∥∥δ 1 ∥ [0,t) + α -1 1 • max{ β4 (∥x(0)∥, t), σ 1 (∥B∥∥δ 1 ∥ [0,t) ), σ 2 (∥B∥∥δ 2 ∥ [0,t) )}, ∀t ≥ 0, for all x(0) ∈ R n and δ 1 , δ 2 ∈ L m
∞ , leading to the required estimate.

Remark 3. Note that we may have δ 2 ∈ L m ∞ while u / ∈ L m ∞ . For a T -periodic time signal u with zero mean value (in such a case δ 2 (t) = 0 for all t ≥ 0), the following corollary is a direct consequence of Theorem 4 and Remark 1:

Corollary 1. Let Assumption 1 be satisfied for d 2 = 0 and g(x) ≡ B in (4). Then for all x(0) ∈ R n and all T -periodic signals u ∈ L m ∞ with zero mean value the following holds:

lim sup t→+∞ ∥x(t)∥ ≤ α -1 1 • σ 1 (∥B∥ T 2 ∥u∥ ∞ ) + ∥B∥ T 2 ∥u∥ ∞ .
Remark 4. If the signal u is T -periodic with non-zero average, then practical asymptotic gain can be obtained. Local versions of the above results follow the respective restrictions in Assumption 1 (e.g., if V is a local ISS-LF).

Defining the frequency ω = 2π T , this result allows us to introduce an analogue of Bode magnitude plot for the nonlinear system (4) with g(x) ≡ B:

H(ω) = 20 log α -1 1 • σ 1 (∥B∥ π ω ∥u∥ ∞ ) + ∥B∥ π ω ∥u∥ ∞ ,
which indicates that for a constant input amplitude, increasing the frequency improves the filtering abilities for the class of nonlinear systems satisfying the conditions of Corollary 1. Comparing with [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF] (see also [START_REF] Margaliot | Contraction after small transients[END_REF]), where a frequency response of convergent systems is studied, and to find the response it is suggested either to solve a PDE or to make simulations, in the present work an upper estimate H(ω) of the response is obtained dependent on the functions α 1 and σ 1 defining an ISS-LF of ( 6), which are easier to derive.

Example 1. In [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF], the frequency response of

ẋ1 (t) = -x 1 (t) + x 2 2 (t), ẋ2 (t) = -x 2 (t) + u(t), was analytically derived, x(t) = [x 1 (t) x 2 (t)] ⊤ ∈ R 2 . To apply Corollary 1, consider in Assumption 1 a Lyapunov function V (x) = 1 2 x 2 1 + κ 2 x 2 2 + ℓ 4 x 4 2
for some κ > 0 and ℓ > 0, then an estimate

DV (x)f (x + d 1 ) ≤ -(1 -1.5ϵ)x 2 1 - κ 2 x 2 2 -ℓ 4 -ϵ -1 x 4 2 +0.5ϵ -1 (2d 2 12 +|d 11 |) 2 + κ 2 d 2 12 + ℓ 4 d 4 12 for d 1 = [d 11 d 12 ] ⊤ ∈ R 2 and some ϵ ∈ (0, 2
3 ) can be obtained, which for ℓ > 4ϵ -1 gives

α 1 (s) = min{1, κ} 2 s 2 , σ 1 (s) = 2 ℓ 4 + 4 ϵ s 4 + max{ϵ -1 , κ 2 }s 2 min{1, 2 -3ϵ, 1 -4 ℓϵ }
.

An interesting feature of the obtained result is that in (4), in order to have a well-defined asymptotic gain with respect to an average value of the input u, according to Assumption 1, the system should be ISS with respect to a "measurement noise" signal d 1 . However, the noise sensitivity of a system usually has other origins and mechanisms than the robustness in the exogenous state disturbances leading to insightful results, which is demonstrated in the next example.

Example 2. Consider a system ẋ(t) = -k|x(t)| κ sign(x(t)) + sin(ωt), where x(t) ∈ R and k, κ > 0 are parameters. Assumption 1 is satisfied for where ϵ 1 ∈ (0, 1), ϵ 2 , ϵ 3 > 0 and k(1

V (x) = x 2 with α 1 (s) = α 2 (s) = s 2 , σ 1 (s) = ϵ -2 1 s 2 , σ 2 (s) = ϵ -2 κ 2 s 2 κ , α(s) = 2ϵ 3 s 1+κ ,
-ϵ 1 ) κ ≥ ϵ 2 + ϵ 3 . Therefore, H(ω) = 20 log (1 + ϵ -1 1 )
π ω and optimizing the value of ϵ 1 we can get a lower bound H(ω) = 20 log 2 π ω independent in k and κ, which may indicate that the obtained bound is qualitative representing the worst-case estimate. To evaluate its conservativeness, we simulate this system on the time interval [0, 500] using the explicit Euler method with the time step h = 0.005, for different values of ω, κ, k and calculating the map

a k (ω, κ) = log max 400≤t≤500 |x(t)|
we obtain the results given in Fig. 1 for k = 0.01, 1, 100. These results surprisingly demonstrate indeed a weak dependence on κ, and also a limited sensitivity for the decay in k, while increasing ω leads to lower values of a k confirming the theoretical findings.

C. Integral ISS

Another interesting consequence of Corollary 1 is that if the system (4) is integral ISS (iISS) [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF], then it has an asymptotic gain with respect to η t 0 γ(∥u(s)∥)ds for some η, γ ∈ K ∞ (and not with respect to γ(∥u∥ ∞ ) as in the ISS case), and it is easy to observe that t 0 γ(∥u(s)∥)ds for any γ ∈ K ∞ and a bounded non-zero periodic u is an unbounded signal in t. Therefore, iISS property, formally, does not provide a reliable conclusion about the boundedness of solutions of (4) in such a setting. However, if the conditions of Corollary 1 are verified, it could be the case that the system is still ISS with respect to d 1 (Assumption 1 is satisfied for d 2 = 0), then the theory proposed in this work allows to guarantee the boundedness with respect to periodic signals for iISS systems.

Example 3. For an illustration consider a scalar system ẋ(t) = -kφ(x(t)) + sin(ωt), where x(t) ∈ R, k > 0 is a parameter, and φ : R → R is a bounded continuous function such that φ(s)s > 0 for all s ∈ R \ {0}. Due to boundedness of φ this system is only iISS with respect to an additive input. However, Assumption 1 is satisfied for d 2 = 0 with V (x) = x 2 and

α 1 (s) = α 2 (s) = s 2 , σ 1 (s) = ϵ -2 1 s 2 , α(s) = 2ksφ ϵ1 (s),
where ϵ 1 ∈ (0, 1) is a parameter, and

φ ϵ1 (x) =      min s∈[(1-ϵ1)x,(1+ϵ1)x] f (s) if x > 0 0 if x = 0 max s∈[(1+ϵ1)x,(1-ϵ1)x] f (s) if x < 0
is a continuous function verifying the passivity condition φ ϵ1 (s)s > 0 for all s ∈ R \ {0}. Therefore, again

H(ω) = 20 log (1 + ϵ -1 1 )
π ω and optimizing the value of ϵ 1 we can get a lower bound H(ω) = 20 log 2 π ω .

IV. APPLICATION TO HOMOGENEOUS SYSTEMS

Let us consider a subclass of ( 4) with homogeneous functions f and g, which allows us to replace Assumptions 1-3 by the following one (denote g(x) = [g 1 (x) . . . g m (x)], where g i (x) is a column of g for i ∈ 1, m, then g i,j (x) is the element of g i for j ∈ 1, n): Assumption 4. The vector field f and functions g i,j for i ∈ 1, m, j ∈ 1, n are r-homogeneous of degrees ν ≥ -r min and ς i > 0, respectively, and the system (4) is asymptotically stable for u ≡ 0.

Following [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], under Assumption 4, for the system (4) there is a twice continuously differentiable and r-homogeneous of degree µ > r max Lyapunov function V : R n → R + , which implies fulfillment of the following inequalities:

c 1 ∥x∥ µ r ≤ V (x) ≤ c 2 ∥x∥ µ r , DV (x)f (x) ≤ -c 3 ∥x∥ µ+ν r , sup ∥x∥r=1 ∂V (x) ∂x ≤ c 4
for all x ∈ R n and some c i > 0, i = 1, . . . , 4. Moreover, following [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] and using (3) there exist ω 1 > 0 and ω 2 > 0 (dependent only on the properties of f and c 3 , c 4 ) such that

∥x∥ r ≥ max{ω 1 ∥d 1 ∥ r , ω 2 ∥d 2 ∥ r } ⇒ DV (x)(f (x + d 1 ) + d 2 ) ≤ - c 3 2 ∥x∥ µ+ν r for all x, d 1 , d 2 ∈ R n (see Theorem 2), r = r + ν. Hence, Assumption 1 is verified for this V with α(s) = c3 2 s µ+ν , σ i (s) = c 2 ω µ i s µ for i = 1, 2,
where the homogeneous norm is used instead of the standard one (recall that these norms are equivalent due to (3)), then |x t | C = sup -T ≤ς≤0 ∥x(t + ς)∥ r in this section. As a consequence, to verify Assumptions 2-3 we will use also the homogeneous norm, then to simplify the computations let m = 1 and ς = ς 1 . From Assumption 4 we get that ∥g(x)∥ ≤ c 5 ∥x∥ ς r for all x ∈ R n and some c 5 > 0, therefore, ρ(s) = c 5 σ -ς r (s) and ρ 0 = 0 in (8). Next, let us repeat the steps made in the previous section in order to replace the conditions in Assumptions 2 and 3. An estimate follows from ( 5) for all t ∈ T :

∥x(t)∥ ≤ 1 T t t-T (s -t + T )∥g(x(s))u(s)∥ds + ∥z(t)∥ ≤ 1 T t t-T (s -t + T )∥u(s)∥∥g(x(s))∥ds + ∥z(t)∥ ≤ c 5 |x t | ς C δ1 (t) + ∥z(t)∥ for x(0) ∈ R n , z ∈ L n ∞ and u ∈ L m ∞ .
This estimate leads to the following relation:

∥x(t)∥ r ≥ max{θ -1 z (∥z(t)∥), θ -1 δ ( δ1 (t))} ⇒ ∥x(t)∥ ≤ θ z (|x t | C ) + c 5 |x t | ς C θ δ (|x t | C ) ⇒ ∥x(t)∥ r ≤ σ -1 r (θ z (|x t | C ) + c 5 |x t | ς C θ δ (|x t | C )) < |x t | C , which is valid provided that σ -1 r (θ z (s) + c 5 s ς θ δ (s)) < s, ∀s > 0. ( 11
)
Repeating the same computations as before, the following inequality is satisfied for all t ≥ 0, all x(0

) ∈ R n , z ∈ L n ∞ and u ∈ L m ∞ : ∥x(t)∥ r ≤ max{β 1 (∥x(0)∥ r , t), θ -1 z (∥z∥ ∞ ), θ -1 δ (∥ δ1 ∥ ∞ )}
for some β 1 ∈ KL. Recalling the definition of d 1 and d 2 in (7), we obtain

σ r (∥d 1 (t)∥ r ) ≤ c 5 |x t | ς C δ1 (t), σ r (∥d 2 (t)∥ r ) ≤ c 5 |x t | ς C δ2 (t).
Next, the following implications should be satisfied:

V (z(t)) ≥ max{γ V (|x t | C ), γ -1 1 ( δ1 (t)), γ -1 2 ( δ2 (t))} ⇒ V (z(t)) ≥ max{c 2 ω µ 1 σ -µ r c 5 |x t | ς C δ1 (t) , c 2 ω µ 2 σ -µ r c 5 |x t | ς C δ2 (t) } ⇒ V (z(t)) ≥ max{c 2 ω µ 1 ∥d 1 (t)∥ µ r , c 2 ω µ 2 ∥d 2 (t)∥ µ r },
leading to the property:

ξ ≥ γ V (ζ) ⇒ c 2 ω µ 1 σ -µ r (c 5 ζ ς γ 1 (ξ)) ≤ ξ, (12)
c 2 ω µ 2 σ -µ r (c 5 ζ ς γ 2 (ξ)) ≤ ξ, ∀ξ, ζ ∈ R + .
Therefore, according to Theorem 1, the following estimate is valid for all t ≥ 0:

∥z(t)∥ r ≤ c -1/µ 1 max{β 2 (∥z(0)∥ r , t), γ V ( sup s∈[0,t) ∥x(s)∥ r ), γ -1 1 (∥ δ1 ∥ [0,t) ), γ -1 2 (∥ δ2 ∥ [0,t) )} 1/µ , for all z(0) ∈ R n , x ∈ L n ∞ and u ∈ L m ∞
, and a suitably defined β 2 ∈ KL. Finally, the small-gain condition of Theorem 3 takes the form: Recall from [START_REF] Efimov | Homogeneous Lyapunov functions: From converse design to numerical implementation[END_REF] that for r max ≤ 1 and ϖ = 1 (this property can be always provided by scaling the weights in r) the following choice is possible Let us perform verification of the conditions ( 11)-( 13) (Assumptions 2-3 and the small-gain condition) for s > 1. After straightforward computations (applying Jensen's inequality) we get that ( 11) is satisfied for θ z (s) = c 6 s rmin and θ δ (s) = c 6 s rmin-ς with some sufficiently small c 6 > 0 under constraint ς < r min ; the condition ( 12) is valid for γ V (s) = c 7 s otherwise; finally the small-gain relation [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF] can be checked for s > 1 if ς ≤ rmin 2rmax min{r min , r min + ν}. Since r max = 1 is a possible/maximal choice, the restriction on the degree of g is ς ≤ rmin 2 min{r min , r min + ν}, and the following result can be deduced from Theorem 3: Corollary 2. For the case m = 1, the system (4) under Assumption 4 with ς ≤ rmin 2 min{r min , r min +ν} is practically ISS with respect to the inputs δ1 , δ2 ∈ L 1 ∞ . Example 4. Consider a system ẋ(t) = -k|x(t)| κ sign(x(t)) + |x(t)| ς sign(x(t))u(t), where x(t) ∈ R and k, κ, ς > 0 are parameters. Assumption 4 is satisfied for r = 1 and ν = κ -1, and according to the established result, ς ≤ κ 2 . Obviously, if κ > ς then the system is ISS with respect to u ∈ L 1 ∞ and for u(t) = A sin( 2π T t), for some A > 0 and T > 0, this system is practically ISS with respect to δ1 (t) ≤ AT 2 and δ2 (t) = 2A π . The results of simulation for k = 1, κ = 0.25, ς = 1/3 and A = 5 are presented in Fig. 2, where blue, red and green curves correspond to T = 0.1, 1, 10, respectively. A kind of finitetime convergence to the origin is observed since the term g(x(t))u(t) acts as a stabilizer for the instants of time when u(t) is negative.

θ -1 z • σr c -1/µ 1 γ 1/µ V (s) < s, ∀s > 0. (13)

V. CONCLUSIONS

Using the time-delay approach, the conditions of ISS are investigated for nonlinear affine in the input systems with respect to an average value of the disturbances. For periodic signals and static input gains, a kind of Bode amplitude plot is derived for this class of nonlinear dynamics. For integral ISS systems, the gains are obtained for periodic inputs implying boundedness of solutions. For the case of homogeneous systems, it is shown that stability can be preserved with respect to an average value of the input even in the case when degree of the perturbation gain is higher than the degree of the nominal dynamics. The results are illustrated by simple examples. Relaxing conservatism of the imposed assumptions can be considered as a direction for further research.

Figure 1 .

 1 Figure 1. Plot of a k for different value of k

Figure 2 .

 2 Figure 2. An illustration for Corollary 2

1 ,

 1 σr (s) = √ n s rmin s ≤ 1 s rmax s > 1 .

2ςµ r min and γ 1

 1 (s) = c 8 s r min 2µ , γ 2 (s) = c 8 s r min +2ν 2µ for some c 7 > 0 and c 8 > 0 if ν ≥ 0, and γ V (s) = c 7 s 2ςµ r min +ν , γ 1 (s) = c 8 s r min 2µ , γ 2 (s) = c 8 s r min +ν 2µ

This work was supported by Israel Science Foundation (grant no. 673/19) and by Chana and Heinrich Manderman Chair at Tel Aviv University.


Alba Tor-Roca

Alex Sánchez-Pla

Aniko Korosi

Mercè Pallàs

Paul J Lucassen

Pol Castellano-Escuder

Ludwig Aigner

Raúl González-Domínguez

Claudine Manach

Francisco Carmona

Esteban Vegas

Catherine Helmer

Catherine Feart

Sophie Lefèvre-Arbogast

Jeanne Neuffer

Hyunah Lee

Sandrine Thuret

Cristina Andres-Lacueva

Cécilia Samieri

Mireia Urpi-Sarda

A Mediterranean Diet-Based Metabolomic Score and Cognitive Decline in Older Adults: A Case-Control Analysis Nested within the Three-City Cohort Study

Keywords: aging, Alzheimer's disease, cognitive dysfunction, cognitive impairment, metabolite

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

Introduction

Dementia is a rapidly growing public health problem affecting around 50 million people worldwide. Every year, there are nearly 10 million new cases, and this figure is projected to triple by 2050 due to the global aging population and its anticipated impact and costs for society are huge. [START_REF]World Health Organization, Risk Reduction of Cognitive Decline and Dementia, WHO Guidelines[END_REF] Dementia is characterized by progressive cognitive decline (CD), a major cause of disability and dependency among older people.

While there is currently no treatment for dementia, certain lifestyle factors have been associated with a delay in the age-at-onset or with a slowing down of disease progression. In fact, a healthy diet is thought to have great preventive potential for CD, both directly and through its role in reducing other risk factors (such as hypertension and type 2 diabetes). [START_REF] Samieri | [END_REF] Healthy dietary patterns have indeed been associated with a lower risk of dementia [3][4][5] and better cognitive performance. [6] Also, several observational studies have concluded that a high adherence to in particular the Mediterranean diet (MD) is associated with a decreased risk of mild cognitive impairment and Alzheimer's disease (AD), [7] and with better episodic memory and global cognition. [8] Two other related dietary patterns also associated with better cognitive performance are the Dietary Approaches to Stop Hypertension (DASH) [9] and the Mediterranean-DASH Intervention for Neurodegenerative Delay (MIND) diets. [10,11] However, evidence of the associations between dietary patterns and cognitive function is still inconclusive. [12] For instance, a meta-analysis of randomized controlled trials investigating MD effects on cognition, or on brain morphology and function, concluded that the associations were mostly non-significant, with small effect sizes. [13] Gauci et al. also found conflicting results in studies investigating the effects of the DASH and MIND diets on neurocognitive function. [14] Some of these inconsistencies may be attributable to the fact that current epidemiological data on dietary patterns mostly rely on self-reported questionnaires, which may misreport the actual food consumption and thereby underestimate the inter-and intraindividual variability in bioavailability of bioactive food components that could arise from numerous factors (e.g., sex, age, genetic background, chronic diseases, liver function, and gut microbiota). [15] To overcome these limitations, the use of a panel of dietary biomarkers would constitute an alternative (or complement) to provide a more specific assessment of dietary exposure and to better evaluate the association between dietary exposure and health outcomes. [16] Prior studies have indeed identified protective associations between urinary total polyphenols, individual plant-derived metabolites -including phenolic acids such as 3-hydroxyhippuric acid and 3-hydroxyphenylacetic sulfate, urinary urolithin derivatives, lutein, ergothioneine, and equol -and CD, as well as negative associations with metabolites related to generally considered unhealthy dietary components, such as saccharin and propionic acid. [17][18][19][20][21][22][23] Yet, studies using combinations of diverse biomarkers to monitor compliance with specific dietary patterns are scarce.

Here, we developed an MD-based metabolomic score and investigated its association with CD in two sample sets from the centers of Bordeaux and Dijon, as of the Three-City Cohort (3C Study).

Experimental Section

Study Design and Population

This study conducted a nested case-control study using data from the 3C Study, a population-based cohort on dementia that included older persons (≥65 years) from three French cities. [24] A detailed description was provided elsewhere. [17][18][19][24][25][26][27][28][29] Briefly, sociodemographic and lifestyle characteristics, medical information, neuropsychological testing, blood pressure, anthropometric measurements, and fasting blood samples were collected at baseline from 1999 to 2000, and follow-up visits were then scheduled every 2-3 years for repeated health and neuropsychological assessment. [24] The study was performed in accordance with the principles contained in the Declaration of Helsinki. The Consultative Committee for the Protection of Persons participating in Biomedical Research at Kremlin-Bicêtre University Hospital (Paris, France) approved the 3C Study protocol (IRB approval number 9928, Kremlin Bicêtre, June 10, 1999), and all participants provided written consent.

In this nested case-control study, two sample sets were built among participants from the centers of Bordeaux and Dijon of the 3C Study, with Bordeaux as the discovery set and Dijon as the set used for external validation. [17,18] From the entire Bordeaux and Dijon cohorts, eligible participants were selected for the present study if they were not diagnosed with dementia at baseline, had available serum samples, and had at least one repeated cognitive evaluation over the subsequent 12 years.

To build the case-control samples on CD, the study defined a composite score of global cognition at each follow-up visit as the average Z-scores of five neuropsychological tests (Mini-Mental State Examination [MMSE], Benton Visual Retention Test, Isaac's Set Test, Trail-Making Test part A, and Trail-Making Test part B). [17,18] Individual slopes of cognitive change were then estimated using linear mixed models, as previously detailed. [18] Cases were defined as the participants with the worst CD slopes. Then, it matched each case to a control (i.e., a participant with a CD slope better than the population median) based on age at baseline, sex, and education level. Overall, 209 cases were individually matched to a control in Bordeaux, leading to a discovery sample of 418 subjects. In Dijon, 211 cases were successfully matched to 211 controls, leading to a validation sample of 422 subjects.

Targeted Metabolomics Analysis of Serum Samples

The study conducted a targeted metabolomics analysis of serum samples using a multi-metabolite platform for the simultaneous detection and quantification of food-related metabolites, microbiota derivatives, and endogenous metabolites following the methodology detailed in previous publications. [17,30] The samples were randomly distributed for metabolomics analysis utilizing an Infinity UHPLC system (Agilent, Santa Clara, CA, USA) coupled to a QTRAP 6500 mass spectrometer equipped with an Ion Drive Turbo V ion source (Sciex, Framingham, MA, USA) using the operating conditions described elsewhere. [17]

Mediterranean Diet Metabolomic Score

In line with the Mediterranean diet score (MDS) defined by Trichopoulou et al. [31] the MD metabolomic score (MDMS) was developed de novo using a 14-point linear scale that incorporated two putative dietary metabolomic biomarkers of seven key components of the MD (vegetables, legumes, fruits, cereals, dairy, fats, and fish) (Table 1 and Table S1, Supporting Information). [31,32] The metabolites for the MDMS were selected based on three criteria: a) metabolites reported as putative biomarkers of key MD food groups, [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] b) metabolites available in the serum metabolomic panel described previously, [17,30,48] and c) metabolites with described health-related proprieties (Table S1, Supporting Information). Detailed information about the eligibility criteria of the metabolites for the MDMS is shown in Table S1 (Supporting Information). In this MDMS, the meat item from the MDS was excluded as it was not possible to establish a set of metabolites that can reliably discriminate between meat in-take, products of human muscle catabolism, and other proteinrich foods (e.g., fish and seafood), as well as between different meat sources (e.g., red meat vs. poultry). [44] This study did not include an alcohol-related item either since low levels of alcohol have shown slight reductions in health-related harm (i.e., ischemic heart disease and diabetes in very specific populations) but an increased risk of other health-related conditions, including cirrhosis, infectious diseases, early dementia, and cancer. [49,50] We thus worked with n = 14 serum biomarker components. We decided to group four categories of foods (vegetables, fruits, legumes, and cereals) as a plant-based food group since some of the biomarkers described could be ubiquitously present among all these food groups. Enterolactone (EL), hippuric acid (HA), proline betaine; trigonelline; 3,8-dihydroxyurolithin glucuronide (UroA-G), 3,4-dihydroxybenzoic acid (3,4-DHBA), 3′,4′-dihydroxyphenylacetic acid sulfate (3,4-DHPAA-S), and 5-(3′,4′-dihydroxyphenyl)-𝛾-valerolactone sulfate (3,4-DHPV-S) represented the plant-based food group. Docosahexaenoic acid (DHA) and eicosapentaenoic acid (EPA) depicted the fish group, while pentadecanoic acid and margaric acid represented the dairy products group. Finally, the dietary intake of MUFAs and saturated fatty acids (SFAs) were depicted by oleic acid and palmitic acid, respectively (Tables S1 andS2, Supporting Information).

Following the same design used by Trichopoulou et al. [31] the study assigned a value of 0 or 1 to each dietary biomarker with the use of the sex-specific median as the cutoff (Table S3, Supporting Information). For dietary biomarkers of beneficial components (vegetables, legumes, fruits, cereals, and fish), subjects whose circulating level was below the median were assigned a value of 0, and individuals whose level was at or above the median were assigned a value of 1. According to the MDS defined by Trichopoulou et al., [31,32] dairy products were presumed to be detrimental. For this reason, the scoring for dietary biomarkers of dairy products was inverted. [31,32] Finally, for the relation between MUFAs and SFAs, the study assigned a value of 1 to those individuals whose oleic acid concentration was at or above the median and a value of 0 to those whose concentrations were below it. The scoring was inverted for palmitic acid concentrations. Oleic acid and palmitic acid are the most abundant dietary and plasma FAs. [47,51] Moreover, oleic acid is a biomarker of olive oil intake, the main source of MUFAs in the MD context. [31,52] Thus, the total MDMS ranged from 0 (minimal adherence) to 14 (maximal adherence). The MDMS was secondarily categorized, using cutoff values based on the sex-and cohort-specific medians, as low (<7 points) and high (≥7 points) adherence for both sexes and cohorts.

Dietary Data

Details of dietary habits assessment were described elsewhere. [53,54] Briefly, the study used a comprehensive food frequency questionnaire (FFQ) [53] and a 24 h dietary recall (24-HDR) [54] administered in 3C Bordeaux at the first follow-up visit in the period 2001-2002 (2-3 years after baseline when blood was drawn). For foods from the FFQ, data were available for 351 Bordeaux sample subjects, and for MUFAs and SFAs intake from the 24-HDR, data were available for 359 subjects. An MDS based on the FFQ and the 24-HDR was available for 349 subjects from Bordeaux (Table S4, Supporting Information).

Statistical Analysis

Baseline characteristics of the participants were expressed as means and standard deviations (SDs) for variables with normal distribution, medians, and interquartile ranges (IQRs) for variables with asymmetric distribution, and percentages for categorical variables. Baseline characteristic comparisons were tested using conditional logistic regression models. Correlation analyses were performed to examine the relation between MDMS and MDS, and between food groups and dietary biomarkers of MD key food groups in the Bordeaux cohort.

To study the association between MDMS and CD, the study estimated odds ratios (ORs) and their 95% confidence interval (CI) using logistic regression models conditioned on matching variables (i.e., age at baseline, sex, and education level) (Model 1). For multivariable-adjusted analyses, the study considered potentially modifiable risk factors for dementia (i.e., BMI, diabetes, alcohol intake, smoking status, hypertension, depression, and hypercholesterolemia) [55,56] (Model 2) as well as genetic factors (i.e., apolipoprotein E-ɛ4 genotype [ApoE-ɛ4]) [56] and the total number of medications regularly consumed [17] (Model 3). The goodness of fit of the proposed models was evaluated using the likelihood-ratio chi-square test of the full model versus the null model, where p < 0.05 was considered to be a good fit.

Effect modification by age at baseline, sex, education level, BMI, diabetes, alcohol intake, smoking status, hypertension, depression, hypercholesterolemia, the total number of medications regularly consumed, and ApoE-ɛ4 was assessed by adding product terms in the fully adjusted regression models. Bonferroni cor-rection was utilized for multiple testing correction (p < 0.05/12, statistical threshold at 𝛼 = 0.004).

For covariates, missing values comprised <2% of the samples and the reference category was assigned to missing data (for categorical variables) or the median value (for continuous variables). All statistical analyses were performed using the SPSS software package (SPSS 25.0, IBM, NY) and the significance level was a two-tailed p < 0.05.

Results

Characteristics of the Study Populations

Clinical and demographic characteristics of the study populations at baseline are shown in Table 1. The participants were matched for age, sex, and education level within the two samples, and these characteristics were similar between the discovery (i.e., Bordeaux) and validation (i.e., Dijon) sample sets. As expected, we observed lower scores for MMSE tests assessed in this study among subjects with CD. Medication use was significantly higher in cases, as well as the prevalence of diabetes. In the Bordeaux cohort, the presence of ApoE-ɛ4 was higher among cases than among controls, although the percentage was similar in the Dijon set. Cases from the Dijon cohort also showed a significantly higher percentage of participants with depression compared to controls (Table 1). However, in the Bordeaux set, cases and controls had a similar percentage of subjects with depression (Table 1).

Mediterranean Diet Metabolomic Score

The correlations among FFQ dietary components of MD key food groups and concentrations of serum biomarkers in the Bordeaux set are presented in Table S5 (Supporting Information). Nearly all MD key food groups had a positive and significant correlation with their serum metabolites. The strongest correlation was between the intake of fish and seafood and the sum of EPA and DHA (Table S5, Supporting Information). The plant-based food group (i.e., total intake of vegetables, fruits, legumes, and cereals) and the sum of their selected biomarkers were also positively correlated (Table S5, Supporting Information). Dairy intake and serum pentadecanoic acid and margaric acid were not correlated. MDS was also positively correlated with MDMS (r = 0.135, p = 0.012) in the Bordeaux cohort (n = 349).

MDMS and CD

The associations between MDMS and CD are shown in Figure 1. In the discovery set, CD was not associated with MDMS in the basic model (Figure 1A, Model 1). This association was unchanged after further adjustment for CD risk factors (i.e., BMI, diabetes, alcohol intake, smoking status, hypertension, depression, and hypercholesterolemia) (Figure 1A, Model 2). However, after additionally adjusting for the total number of medications regularly consumed and for ApoE-ɛ4 genotype (Model 3), subjects with a higher MDMS were less likely to experience CD (OR [95% CI] OR (95% CI). Model 1 is unadjusted. Model 2 is adjusted for BMI, diabetes, alcohol intake, hypertension, smoking status, depression, and hypercholesterolemia. Model 3 is further adjusted for ApoE-ɛ4 and number of medications. The CD variable was built defining the composite score of global cognition at each follow-up visit. [17,18] Individual slopes of cognitive change were then evaluated using linear mixed models, as detailed in the previous publication. [18] Cases were defined as the participants with the CD worst slopes. ApoE-𝜖4, apolipoprotein E-ɛ4 genotype; 3C, Three-City; CD, cognitive decline; CI, confidence interval; MDMS, Mediterranean diet metabolomic score; OR, odds ratio. = 0.90 (0.80-1.00); p = 0.048, MDMS as continuous variable) (Figure 1A, Model 3). This was consistent with the results observed in Dijon, with a trend toward significance in this validation sample (OR [95% CI] = 0.91 [0.83-1.01]; p = 0.084 for Model 3 and MDMS as a continuous variable) (Figure 1B, Model 3). The omnibus test suggested a significant improvement in fit of the models relative to the null (𝜒 2 (10) = 34.292; p ≤ 0.001 and 𝜒 2 (10) = 37.670; p ≤ 0.001 in Bordeaux and Dijon, respectively, for Model 3, MDMS as continuous variable).

Stratified analyses were performed to assess the association between MDMS and the risk of CD in various subgroups (Table S6, Supporting Information). We noticed a close to significant interaction between MDMS and hypertension in the Bordeaux cohort (no hypertension, adjusted OR [95% CI] = 1.19 [0.95-1.50], compared with hypertension, adjusted OR [95% CI] = 0.83 [0.66-1.05]; p for interaction = 0.006). None of the other variables, including age at baseline, sex, education level, BMI, diabetes, alcohol intake, smoking status, hypercholesterolemia, depression, the total number of medications regularly consumed, and ApoE-ɛ4, significantly modified the association between MDMS and the risk of CD in the Bordeaux cohort. No interaction between MDMS and the abovementioned variables was found in the Di-jon cohort (all p for interactions > 0.004 for multiple testing correction) (Table S6, Supporting Information).

Discussion

Leveraging targeted large-scale metabolomics data from the 3C Study, we here developed a Mediterranean diet metabolomic score (MDMS) based on serum biomarkers of MD key food groups. Our findings show that a higher MDMS is prospectively associated with a lower risk of CD in older adults free of dementia at baseline (median follow-up 12 years).

These results are in line with some prior prospective studies, where a high adherence to the MD, as assessed using dietary questionnaires, has been associated with a slower decline in global cognition in older adults. [57,58] However, despite the efforts made to reduce the heterogeneity of the scoring systems used for assessing adherence to the MD, it is still challenging to elucidate why current literature still reports mixed results for cognitive function and the MD in older adults. [57,59] For instance, in the 3C Study, Féart et al. suggested that a higher adherence to the MD, using the MDS, was associated with a slower CD, as based on MMSE score, but not with the risk of incident dementia or AD. [32] In another large-scale prospective study including 16 058 women from the Nurses' Health Study aged ≥70 years, authors found that an alternate MDS was related to moderately better cognition overall, but not to change in cognition over four repeated measures after 6 years of follow-up. [60] By contrast, in the EPIC-Spain Dementia Cohort Study, recent results showed that MD appeared to be protective and lowered the incidence of dementia. [61] Also, Scarmeas et al. reported that a higher MD adherence was associated with a reduced risk of AD. [62] One source of these biases could be linked to the modality of assessing dietary intake, as most studies used dietary records, which lack accuracy and are prone to random bias and systematic errors. As a matter of fact and as potentially expected, the MDMS was weakly correlated with the MDS, as were most of the MD key food groups with their metabolites. The MDMS was created a priori based on the current state of the art on food intake biomarkers [63,64] and aiming at taking a step forward by surrogating traditional indicators of dietary patterns intake (i.e., FFQ and 24-HDR derived scores) and thus their inherent prone-tosubjectivity and lack of accuracy. [65][66][67] For instance, dairy intake and serum pentadecanoic and margaric acids were not correlated in this cohort. This may be attributable to the heterogeneity of fat content through different dairy products in the marketplace or within a specific product resulting from differences in processing (e.g., skimming, among others), as well as in the quality of the milk used as raw material (e.g., cattle breed and feeding, among others). [45] This particular case might serve as a revealing example of the need of better dietary definition, since in the MDS defined by Trichopoulou et al. the dairy products are presumed to be detrimental on account of their fat content. [31] The weakness of correlation between the other food groups and their dietary biomarkers could be attributed to i) the variability of phytochemical or fatty acid content through a food group or a specific product resulting from differences in harvesting or between species (e.g., fatty fish have higher levels of long chain n-3 PUFA than lean fish and seafood), [START_REF]Dietary Guidelines for Americans[END_REF] among others; ii) the presence of the selected metabolites in other food sources not included in the MD key food groups (e.g., coffee, tea, and cocoa products) [START_REF] Wishart | [END_REF] ; and iii) the presence of other metabolites that could contribute to the real content of an MD key food group (i.e., MUFA and SFA).

In a move to obtain more revealing information from dietary intake data and to overcome the above-mentioned problems, food biomarkers could be valuable in this respect as they can capture not only food composition but also their bioavailability. In this sense, several works have highlighted promising results when evaluating single food biomarkers in relation to CD. Rabassa et al. showed in their 3-year follow-up study that higher levels of urinary total polyphenols, a biomarker of fruit and vegetable intake, were associated with a lower risk of CD in older adults. [21] Interestingly, in the 3C Study, distinct associations were found between several serum food-derived metabolites and CD. [17][18][19] Yet, in another move aimed at obtaining more meaningful evidence, nutritional researchers have begun examining biomarkers as measures of dietary patterns rather than of single nutrients or dietary components. Unlike individual compounds, dietary patterns better reflect a person's habitual diet and provide a more integrated measure of the synergistic and accumulative effects of a combination of dietary factors.

To date, few studies have captured metabolomic signatures related to the MD through predictive and discriminant analysis, and some have studied their association with health outcomes, such as cardiovascular diseases and its risk factors. [70][71][72][73][74][75] To the best of our knowledge, our current work is the first study to have developed an a priori dietary metabolomic score based on serum biomarkers of MD food groups and to evaluate it in association with CD. Specifically, the MDMS comprises a set of serum phytochemical and fatty acid metabolites that reflect the individual bioavailability. Moreover, some of them have been recognized as exposure markers of the key MD food groups to which, in turn, the protective effects on health of the MD have been attributed to (Supporting Information). [76] Validation is important but has rarely been carried out in previous food metabolomics studies, likely because it is challenging in research on nutrimetabolomics and cognitive impairment as there is an intrinsic and large heterogeneity across studies, and profound intervariability arising from factors like gut microbiota and/or health status, [15,77] and the complexity of outcomes based on long-term trajectories such as CD and different types of dementia. For example, despite belonging to the same cohort study, the features of our validation sample differed somewhat from the discovery sample (e.g., participants were older, showed a higher percentage of depression and ApoE-ɛ4 carriers, and had more cardiovascular risk factors at baseline). This could explain the weaker association we found in the Dijon (validation) sample compared to the discovery one, even though both sample sets were positively related to a greater MDMS, indicating a protective effect.

The major strengths of our study are the population-based and prospective design with repeated cognitive assessment over up to 12 years, a rigorous case-control sampling, and the application of a powerful targeted metabolomics approach to serum samples collected at baseline, i.e., at an age prior to the appearance of any dementia symptoms, which reduces the possibility of reverse causation. In addition, the use of two separate nested case-control sample sets noticeably enhances the reliability of our findings.

However, also several limitations warrant further discussion. First, serum samples for metabolomics analysis were only available at baseline, and thus we could not examine prior exposures or changes during the follow-up. An approach based on multiple sampling would have been of great interest to investigate the trajectory of CD. Moreover, although we adjusted the model by several potential confounders including ApoE-ɛ4, residual confounding factors cannot be dismissed. Another limitation is that the MDMS scoring system is based on sex-specific median values across metabolite concentrations of the studied populations. Therefore, the score may not be related to a healthy level of concentrations per se. Although we evaluated the reproducibility of the results in a separate nested case-control sample set, this metabolomic score deserves to be validated also in other cohorts and geographic areas, and its association with other chronic diseases should be examined.

In summary, a greater adherence to the MD assessed by a novel MDMS was associated with a lower risk of CD in older adults during a 12-year follow-up. The development of dietary metabolic scores based on dietary patterns may help further refine dietary assessment measures and will hopefully contribute to a better understanding of the biological mechanisms via which diet impacts cognitive health in the aging population. AC19/00096), CIBERFES funded by the Instituto de Salud Carlos III and co-funded by the European Regional Development Fund "A way to make Europe", the Generalitat de Catalunya

  Authors. Molecular Nutrition & Food Research published by Wiley-VCH GmbH

Figure 1 .

 1 Figure 1. Multivariate conditional logistic regression models for association between MDMS (continuous and categorical) and CD in the 3C discovery (Bordeaux, n = 418) A) and validation (Dijon, n = 422) B) cohorts.OR (95% CI). Model 1 is unadjusted. Model 2 is adjusted for BMI, diabetes, alcohol intake, hypertension, smoking status, depression, and hypercholesterolemia. Model 3 is further adjusted for ApoE-ɛ4 and number of medications. The CD variable was built defining the composite score of global cognition at each follow-up visit.[17,18] Individual slopes of cognitive change were then evaluated using linear mixed models, as detailed in the previous publication.[18] Cases were defined as the participants with the CD worst slopes. ApoE-𝜖4, apolipoprotein E-ɛ4 genotype; 3C, Three-City; CD, cognitive decline; CI, confidence interval; MDMS, Mediterranean diet metabolomic score; OR, odds ratio.

  's Agency AGAUR (2021SGR00687), the MCIN/AEI/10.13039/501100011033 (PID2020-114921RB-C21), and the INSA-UB institute recognized as a Maria de Maeztu Unit of Excellence grant (CEX2021-001234-M) funded by MICINN/AEI/FEDER, EU. The Three-City Study is conducted under a partnership agreement between the Institut National de la Santé et de la Recherche Médicale (INSERM), the Institut de Santé Publique et Développement of the Victor Segalen Bordeaux 2 University, and Sanofi-Aventis. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by the Caisse Nationale de l'Assurance Maladie des Travailleurs Salariés, Direction Générale de la Santé, Mutuelle Générale de l'Education Nationale, Institut de la Longévité, Regional Governments of Aquitaine and Bourgogne, Fondation de France, Ministry of Research-INSERM Program "Cohortes et collections de données biologiques", the French National Research Agency COGINUT ANR-06-PNRA-005, the Fondation Plan Alzheimer (FCS 2009-2012), and the Caisse Nationale pour la Solidarité et l'Autonomie (CNSA). A.T.-R. is grateful for the FPU 2019 contract from the Ministry of Science, Innovation and Universities (FPU19/06044), R.G.-D. for the "Juan de la Cierva" program from Mol. Nutr. Food Res. 2023, 2300271 2300271 (6 of 9) © 2023 The Authors. Molecular Nutrition & Food Research published by Wiley-VCH GmbH

Table 1 .

 1 Clinical and demographic characteristics of the discovery (Bordeaux, n = 418) and validation (Dijon, n = 422) case-control samples.

			Bordeaux cohort			Dijon cohort
		Controls (n = 209)	Cases (n = 209)	p a)	Controls (n = 211)	Cases (n = 211)	p a)
	Age [years]	75.7 (4.2)	75.9 (4.5)	-	76.2 (4.7)	76.5 (5.2)	-
	Sex, female (n [%])	138 (66)	138 (66)	-	133 (63)	133 (63)	-
	Educational level, ≥ secondary school (n [%])	60 (28.7)	61 (29.2)	-	60 (28.4)	60 (28.4)	-
	BMI [kg m -2 ]	25.8 [4.3]	26.4 [5.6]	0.12	24.9 [4.4]	25.4 [6]	0.099
	Number of medications regularly consumed	4.1 (2.4)	4.9 (2.7)	0.003	4.0 [3.0]	5.0 [4.0]	< 0.001
	MMSE (points)	28.0 [2.0]	28.0 [3.0]	< 0.001	29.0 [1.0]	26.0 [4.0]	< 0.001
	Diabetes, yes (n [%])	12 (5.7)	27 (12.9)	0.017	12 (5.7)	27 (12.8)	0.017
	ApoE-𝜖4, carrier (n [%])	25 (12.0)	54 (25.8)	0.001	44 (20.9)	56 (26.5)	0.187
	Alcohol intake [g d -1 ]	9.6 [19.2]	9.6 [17.8]	0.263	9.6 [16.5]	9.6 [17.8]	0.822
	Current smoking, yes (n [%])	9 (4.3)	10 (4.8)	0.695	4 (1.9)	13 (6.2)	0.150
	Depression, yes (n [%])	15 (7.2)	20 (9.6)	0.386	25 (11.8)	47 (22.3)	0.005
	Hypercholesterolemia, yes (n [%])	128 (61.2)	127 (60.8)	0.922	119 (56.4)	126 (59.7)	0.482
	Hypertension, yes (n [%])	159 (76.1)	164 (78.5)	0.564	175 (82.9)	177 (83.9)	0.789
	MDS (points) b)	4.3 (1.6)	4.3 (1.7)	0.404	-	-	-
	MDMS (points)	6.7 (2.1)	6.4 (2.2)	0.168	6.8 (2.0)	6.5 (2.0)	0.067
	Values are mean (SD), median [IQR], or number of subjects with the characteristic (%). ApoE-𝜖4, apolipoprotein E-ɛ4 genotype; IQR, interquartile range; MDMS, Mediterranean

diet metabolomic score; MDS, Mediterranean diet score; MMSE, Mini-Mental State Examination; SD, standard deviation. a) Estimated using conditional logistic regression models; b) Data available from 349 subjects from the Bordeaux cohort.

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Acknowledgements

This work was accomplished as part of the D-CogPlast project ("Identification of dietary modulators of cognitive ageing and brain plasticity and proof of concept of efficacy for preventing/reversing cognitive decline") supported within the European Joint Programming Initiative "A Healthy Diet for a Healthy Life" (JPI HDHL, http://www.healthydietforhealthylife. eu/), granted by MINECO (Spain, PCIN-2015-229), ANR (France, ANR-15-HDHL-0002-05), and the Medical Research Council UK (UK, MR/N030087/1). This work also received funding from the JPI-HDHL ERA-Net Cofund on INtesTInal MICrobiomics (ERA-HDHL INTIMIC, www.mnf-journal.com MINECO (FJCI-2015-26590), C.A.-L. for the ICREA Academia Award 2018 from the Generalitat de Catalunya, and M.U.-S. for the I3 Program of Ministry of Science and Innovation. P.J.L. is supported by Alzheimer Netherland, ZonMW MODEM, and the Center for Urban Mental Health. A.K. is supported by JPND, Alzheimer Nederland and ZonMW MODEM. J.N. is supported by the grant "SilverBrainFood" within the framework of the "Future Investment Program" (Programme d'Investissements d'Avenir PIA3), "Competitiveness cluster structuring projects" (Projets structurants des pôles de compétitivité, PSPC), operated by BPI France.

Data Availability Statement

Data used in the manuscript will be made available by the corresponding author upon request pending application and approval.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Conflict of Interest

The authors declare no conflict of interest.

Author Contributions