Subhasish Basak
email: subhasish.basak@centralesupelec.fr

Julien Bect

Emmanuel Vazquez

Integration of bounded monotone functions: Revisiting the nonsequential case, with a focus on unbiased Monte Carlo (randomized) methods

Keywords: Intégration monotone, Monte Carlo, échantillonnage stratifié, échantillonnage hypercube latin, variable de contrôle, complexité Monotone integration, Monte Carlo, stratified sampling, Latin hypercube sampling, control variate, information-based complexity

Dans cet article, nous revisitons le problème de l'intégration numérique d'une fonction monotone bornée, en nous concentrant sur la classe des méthodes de Monte Carlo non séquentielles. Nous établissons dans un premier une borne inférieure pour l'erreur maximale dans L p d'un algorithme non séquentiel, qui généralise pour p > 1 un théorème de Novak. Nous étudions ensuite, dans le cas p = 2, l'erreur maximale de deux méthodes sans biais-une méthode fondée sur l'utilisation d'une variable de contrôle, et la méthode de l'échantillonnage stratifié.

Introduction

We address in this article the problem of constructing a numerical approximation of the expectation E(g(Y )) = g(y) P Y (dy), where Y is a real random variable with known distribution P Y , and g is a real function that is bounded and monotone. Such a problem occurs naturally in applications where one is interested in computing a risk using a model that provides an increasing conditional risk g(Y ) with respect to some random variable Y . This situation occurs for instance in the field of food safety, with Y a dose of pathogen and g(Y ) the corresponding probability of food-borne illness (see, e.g., [START_REF] Perrin | Quantitative risk assessment of haemolytic and uremic syndrome linked to O157:H7 and non-O157:H7 shiga-toxin producing escherichia coli strains in raw milk soft cheeses[END_REF].

1 Assuming that the cumulative distribution function of Y is continuous, the problem reduces after change of variable and scaling to the computation of S(f ) = E(f (X)) = 1 0 f (x) dx, where X is uniformly distributed on [0, 1] and f belongs to the class F of all non-decreasing functions defined on [0, 1] and taking values in [0,1]. We work in this article in a fixed sample-size setting, where the number n of evaluations of f to be performed is chosen beforehand.

This problem was first studied by [START_REF] Kiefer | Optimum sequential search and approximation methods under minimum regularity assumptions[END_REF], who proved that considering regularlyspaced evaluations at x i = i/(n + 1), 1 ≤ i ≤ n, and then using the trapezoidal integration rule assuming f (0) = 0 and f (1) = 1, is optimal in the worst-case sense among all deterministic-possibly sequential-methods, with maximal error 1/(2(n + 1)). [START_REF] Novak | Quadrature formulas for monotone functions[END_REF] later studied Monte Carlo (a.k.a. randomized) methods, and established that sequential methods are better in this setting than nonsequential ones, with a minimax rate of n -3/2 over F for the L 1 error. Novak's proof relies on the construction of a particular two-stage algorithm, using stratified sampling in the second stage.

This article revisits the nonsequential setting with a focus on unbiased Monte Carlo methods, which are a key building block for the construction of good (rate-optimal) sequential methods-as can be learned from the proof of Theorem 3 in Novak's article. Section 2 derives a lower bound for the maximal L p -error of nonsequential methods, for any p ≥ 1, which is a generalization of a result by [START_REF] Novak | Quadrature formulas for monotone functions[END_REF] concerning the L 1 error. Sections 3 and 4 then study the maximal L 2 error (variance) of two simple unbiased methods, based respectively on the control variate technique and on stratification. Section 5 concludes the article with a discussion.

A lower bound for the maximal L p error

A nonsequential (also called non-adaptive) Monte Carlo method first evaluates the function at n random points X 1 , . . . , X n in [0, 1], and then approximates the integral S(f ) using an estimator Ŝn

(f ) = ϕ (X 1 , f (X 1 ), . . . , X n , f (X n )) , ( 1
)
where ϕ : [0, 1] 2n → R is a measurable function. A nonsequential method is thus defined by two ingredients: the distribution of (X 1 , . . . , X n ) and the function ϕ. The worst-case L p error of such a method over the class F is

e p ( Ŝn ) = sup f ∈F E S(f ) -Ŝn (f ) p 1/p . ( 2
)
Remark 1. The class of nonsequential Monte Carlo methods as usually defined in the literature also allows Ŝn to be randomized (i.e., allows ϕ to be a random function). We have not considered randomized estimators in our definition, however, since Rao-Blackwell's theorem implies that they do not help in this setting, for any convex loss function. [START_REF] Novak | Quadrature formulas for monotone functions[END_REF] proved that for any nonsequential Monte Carlo method with sample size n, the maximal L 1 error e 1 ( Ŝn ) is greater or equal to 1/(8n). We generalize this result to the case of the L p error.

Theorem 2.1. For any nonsequential Monte Carlo methods with sample size n, e p ( Ŝn ) ≥ 1 2

2+1/p 1 n .
Observe that Novak's lower bound is recovered for p = 1. Using Theorem 2.1 with p = 2 we can deduce of lower bound for the variance of unbiased nonsequential methods.

Corollary 2.2. For any unbiased nonsequential Monte Carlo method with sample size n,

sup f ∈F var Ŝn (f ) ≥ 1 32n 2 .
Proof of Theorem 2.1. Consider a nonsequential Monte Carlo methods with evaluation points X 1 , . . . , X n and estimator Ŝn . Divide the interval [0, 1] into 2n equal subintervals of length 1/(2n): then at least one of the subintervals, call it I, will contain no evaluation point with probability at least 1/2. Now construct two functions f 1 , f 2 ∈ F that are both equal to zero on the left of I, equal to one on the right, and such that f 1 = 1 and f 2 = 0 on I. Then S(f 1 ) -S(f 2 ) = 1/(2n), and Ŝn (f 1 ) = Ŝn (f 2 ) on the event A = {{X 1 , X 2 , . . . , X n } ∩ I = ∅}, since f 1 and f 2 coincide outside of I. It follows that

e p ( Ŝn ) p ≥ sup f ∈{f 1 ,f 2 } E |S(f ) -Ŝn (f )| p ≥ 1 2 2 j=1 E |S(f j ) -Ŝn (f j )| p ≥ 1 2 2 j=1 E(|S(f j ) -Ŝn (f j )| p • 1 A ) = 1 2 2 j=1 E(|S(f j ) -T | p • 1 A ),
where T denotes the common value of Ŝn (f 1 ) and Ŝn (f 2 ) on A. We conclude that

e p ( Ŝn ) p ≥ 1 2 |I| p 2 p-1 P (A) ≥ 1 2 2p+1 • 1 n p ,
using the fact that, for any a, b, x ∈ R and p ≥ 1, |a -

x| p + |b -x| p ≥ |a -b| p /2 p-1 .

Uniform i.i.d. sampling

The simple Monte Carlo method is the most common example of a nonsequential method: the evaluation points X 1 , . . . , X n are drawn independently, uniformly in [0, 1], and then the integral is estimated by ŜMC

n (f ) = 1 n n i=1 f (X i ).
The estimator is clearly unbiased, and it follows from Popoviciu's inequality-i.e., var(Z) ≤ 1/4 for any random variable Z taking values in [0, 1]-that

e 2 ŜMC n 2 = max f ∈F var ŜMC n (f ) = 1 4n .
The maximal error is attained when f is a unit step function jumping at x 0 = 1/2. It turns out that a smaller error can be achieved, for the same (uniform i.i.d.) sampling scheme, using the control variate technique. More specifically, we consider the control variate f(X i ) = X i and set

Ŝcv n (f ) = 1 n n i=1 f (X i ) -f (X i ) + 1 2 .
Theorem 3.1. The estimator Ŝcv n (f ) is unbiased, and satisfies

e 2 Ŝcv n 2 = max f ∈F var Ŝcv n (f ) = 1 12n
.

The maximal error is attained for any unit step function.

Proof. The estimator is unbiased since E( f (X i )) = 1/2, and therefore the mean-squared error is equal to var( Ŝcv n (f )) = 1 n var(f (X) -X). For a unit step function f = 1 [x 0 ,1] with a jump at x 0 ∈ [0, 1], the random variable f (X) -X is uniformly distributed over [-x 0 , 1x 0 ], which yields var( Ŝcv n (f )) = 1/(12n) as claimed. It remains to show that var(f (X) -X) ≤ 1 12 for all f ∈ F . Let F m ⊂ F denote the class of all non-decreasing staircase functions of the form

f = m k=1 α k • 1 ( k-1 m , k m ] , with 0 ≤ α 1 ≤ α 2 ≤ . . . ≤ α m ≤ 1. For any f ∈ F , consider the piecewise-constant approximation f m ∈ F m defined by averaging f over each subinterval of length 1/m. Then, E(f m (X)) = E(f (X)) and |var (f m (X) -X) -var (f (X) -X)| ≤ 1 m . Thus, sup f ∈F var (f (X) -X) = lim m→∞ sup f ∈Fm var (f (X) -X) . ( 3
) Let us now show that var (f (X) -X) is maximized over F m when f is a unit step function. Pick any f = m k=1 α k •1 ( k-1 m , k m ] ∈ F m . Set α 0 = 0 and α m+1 = 1. If f is not a unit step function, then there exist k 1 , k 2 ∈ {1, . . . , m} such that k 1 ≤ k 2 and α k 1 -1 < α k 1 = . . . = α k 2 < α k 2 +1 . Denote by f u ∈ F m the function obtained by changing the common value of α k 1 , . . . , α k 2 in f to u ∈ [α k 1 -1 , α k 2 +1 ]. The variance var(f u (X) -X) is a convex function of u, since it can be expanded as au 2 + bu + c with a = k 2 -k 1 +1 m (1 -k 2 -k 1 +1 m ) > 0. Consequently, we have var(f u (X) -X) > var(f (X) -X) at one of the two endpoints of [α k 1 -1 , α k 2 +1
]. Note that the corresponding staircase function f u has one fewer step than f . Iterating as necessary, we conclude that for any f ∈ F m there exists a unit step function g ∈ F m such that var (f (X) -X) ≤ var (g(X) -X) = 1 12 . Therefore sup f ∈Fm var (f (X) -X) = 1 12 , which completes the proof.

Stratified sampling

Consider now a stratified sampling estimator with K strata:

Ŝstr n (f ) = K k=1 w k • 1 n k n k i=1 f (X k,i ) , (4)
where the k-th stratum is

I k = [x k-1 , x k ], 0 = x 0 < x 1 < • • • < x K-1 < x K = 1, the weight w k = |x k-1 -x k |
is the length of the k-th stratum, the allocation scheme (n 1 , . . . , n K ) is such that n k ≥ 0 for all k and k n k = n, and the random variables X k,i are independent, with the X k,i s uniformly distributed in I k . Note that the sampling points are no longer identically distributed here. The estimator Ŝstr n (f ) is unbiased, with variance var Ŝstr

n (f ) = K k=1 w 2 k n k var (f (X k,1 )) . ( 5
)
Theorem 4.1. For any K ≤ n, any choice of strata and any allocation scheme, the stratified sampling estimator (4) satisfies

e 2 Ŝstr n 2 = max f ∈F var Ŝstr n (f ) = 1 4 max k w 2 k n k , ( 6
)
The maximal error is attained for a unit step function with a jump at the middle of I k * , where k * ∈ argmax w 2 k /n k . The minimal value of the maximal error (6) is 1 4n 2 , and is obtained with K = n strata of equal lengths (w k = n -1 and n k = 1 for all k).

The optimal stratified sampling method can be seen as a one-dimensional special case of the Latin Hypercube Sampling (LHS) method [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. (On a related note, [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF] prove that, in any dimension, the LHS method is preferable to the simple Monte Carlo method if the function is monotone in each of its arguments.)

Proof. For all K ∈ N * , let ∆ K = (∆ 1 , . . . , ∆ K ) ∈ R K + | K k=1 ∆ k ≤ 1 .
For a given stratified sampling method with K strata, for all ∆ ∈ ∆ K , define

F ∆ = f ∈ F ∀k ∈ {1, . . . , K}, f (x k ) -f (x k-1 ) = ∆ k .
Then it follows from (5) and Popoviciu's inequality that max

f ∈F ∆ var Ŝstr n (f ) = 1 4 K k=1 w 2 k ∆ 2 k n k , ( 7
)
where the maximum is attained for a non-decreasing staircase function with jumps of height ∆ k at the middle of the strata. Note that K k=1 ∆ 2 k ≤ K k=1 ∆ k ≤ 1. Therefore, the right-hand side of ( 7) is upper-bounded by 1 4 max k w 2 k /n k , which is indeed the value of the variance (5) when f is a unit step function with a jump at the middle of the stratum where w 2 k /n k is the largest. In order to prove the second part of the claim, observe that any stratum with n k ≥ 2 can be further divided into n k sub-strata of equal lengths without increasing the upper bound. Considering then the case where K = n and n k = 1 for all k, the upper bound reduces to 1 4 max k w 2 k , which is minimal when

w 1 = • • • = w n = n -1 since k w k = 1.

Discussion

The stratified sampling (LHS) method provides the best-known variance upper bound over the class F for an unbiased nonsequential method as soon as n ≥ 3, but is outperformed by the control variate method of Section 3 when n ≤ 2. We do not know at the moment if these results are optimal in the class of unbiased nonsequential methods. (The ratio between the best variance upper bound and the lower bound of Corollary 2.2 is 8 3 ≈ 2.67 for n = 1, 16 3 ≈ 5.33 for n = 2 and 8 for n ≥ 3.) Relaxing the unbiasedness requirement, it turns out that both methods are outperformed for all n by the (deterministic) trapezoidal method discussed in the introduction, which has a worst-case squared error of 1/(4(n + 1) 2 ). The ratio of worst-case meansquared errors, however, is never very large-at most 16 9 ≈ 1.78-and goes to 1 when n goes to infinity.

Directions for future work include constructing better sequential methods using the findings of this article, and extending our results to mutivariate integration with partial monotonicity (see, e.g., McKay et al., 1979, Section 2.1).

This work is part of the ArtiSaneFood project (ANR-18-PRIM-0015), which is part of the PRIMA program supported by the European Union.


domain: chim phys

text:
Walaa Mohammad

K David Wegner

Clothilde Comby-Zerbino

Vanessa Trouillet

Marina Paris Ogayar

Jean-Luc Coll

Riccardo Marin

Daniel Jaque

Ute Resch-Genger

Rodolphe Antoine

K David Wegner

Ute Resch- Genger

Xavier Le Guével

Enhanced brightness of ultra-small gold nanoparticles in the second biological window through thiol ligand shell control

Keywords: gold nanoparticles, SWIR, luminescence, ligand shell control, thiol ligand

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

Introduction

In vivo optical imaging has progressed in the past decade to detect non-invasively earlystage diseases and monitor physiological changes in real time in different organs like the brain or heart 1,2 . A successful strategy is to perform optical imaging in the second infrared region between 1000 and 1700 nm, which is referred to as NIR-II (or shortwave infrared, SWIR) 3 .

This optical window provides a high spatial resolution -down to a few millimeters -due to reduced scattering and autofluorescence of biological tissue compared to the visible (400-700 nm) and the first near infrared region (NIR-I) between 700 and 900 nm. Following the seminal work of the research groups of Dai and Bawendi 4 , a myriad of new NIR-II emitting reporters and probes made from different classes of emitters have been designed. These include organic polymethine, xanthene, and bodipy dyes as well as AIEgens [START_REF] Yang | [END_REF]6 , carbon based nanomaterials like carbon nanotubes and inorganic nanoparticles like semiconductor quantum dots, lanthanide nanoparticles, and ultra-small gold nanoparticles (us-AuNPs) [7][8][9] have been designed. NIR-II probes have been meanwhile employed for the diagnostic of several pathologies [START_REF] Du | Near Infrared-Emitting Nanoparticles for Biomedical Applications[END_REF]8,10,11 and especially in oncology including the imaging of vascular and lymphatic systems and the tumor resection by optical guided surgery [12][13][14][15] .

The requirements for NIR-II probes are similar to that for visible and in the NIR-I optical probes : excitability in the infrared for a large penetration depth in living tissue, a high brightness ( i.e., the product of the photoluminescence quantum yield (PLQY) and the molar absorption coefficient or cross section at the excitation wavelength), and a high photostability 10,11 . For in vivo applications, these emitters should be biocompatible and waterdispersible, with a high colloidal stability at physiological conditions such as pH or ionic strength. Moreover, it should be possible to functionalize them with molecules for targeting and/or delivery purposes.

Compared to other inorganic NIR-II probes, luminescent us-AuNPs, also often referred to as gold nanoclusters, have several advantages for photoluminescence (PL) bioimaging, like tunable PL in the 1000-1300 nm range [16][17][18][19] , PL lifetimes in the microsecond range 16 , and PLQY of 0.1 to 6.7% in aqueous solution 14,18,20 . Usually, such us-AuNPs are composed of a metal core of a few tens to hundreds of atoms protected/covered by (bio)organic ligands [21][22][23] .

Therefore, the pharmacokinetic of these probes can be tailored by the chosen surface ligands 24 . Thanks to their ultra-small size, which is usually less than 2 nm and way below the kidney filtration cut-off 25 , these us-AuNPs exhibit a high renal clearance. This prevents the long-term accumulation in the reticuloendothelial system (RES) leading to severe toxicity 26 .

Different strategies to tune the emission properties of NIR-II emissive us-Au-NPs have been explored. A strong emission in the NIR-II 18 can be achieved by controlling the size and the physicochemical properties of the ligand shell, as demonstrated with multishell zwitterionic us-AuNPs 27 . Alternatively, us-AuNPs can be encapsulated in polymers or rigid, inorganic nanostructures 28,29 . The observed enhanced PL is ascribed to the rigidification of the structure of the us-AuNPs or to an increase in ligand shell size. The former reduces the energy loss caused by intramolecular rotations and vibrations, thus increasing the PL intensity of the nanoclusters and therefore their PLQY. Also possible PL quenching by water molecules penetrating into the ligand shell can be significantly reduced with the multi-shell ligand and encapsulation approaches 30 . Surface shell protection can also be obtained with bulky counter-ions or protein templating of us-AuNPs 31,32 . In the latter case, a change in the gold-thiolate interface could lead to both a PL enhancement and a red-shift in PL 32 . Control of the spatial organization of the organic shell protecting the metal core could be also contribute to enhance the NIR absorption and shift the PL emission to longer wavelength 18,33 .

Engineering the thiol ligand shell by collocation of different charged ligands leading to surface charge anisotropy can boost the NIR absorption of Au NPs 34 . A key parameter can be here the tailoring of the thickness of the thiol ligand shell during particle synthesis selecting the nature and strength of the reducing agent that is involved in the formation of the gold thiolate complexes, the nucleation of the gold core, and the formation of disulfide bonds between ligands.

In this study, we prepared us-AuNPs stabilized by monothiol and short dithiol molecules that have shown promising optical properties in earlier studies 18,35 for in vivo applications 14 . Such us-AuNPs are typically obtained by combining metal salt, ligands, and a reducing agent, stirred at room temperature. However, the reducing agent plays a crucial role as it directly reduces Au ions to form the atomically precise metal core in the presence of the ligands. It also controls the growth of the core and the formation of the ligand shell in the resulting us-AuNPs. By adjusting the concentration of the reducing agent, it was possible to fine-tune the physicochemical properties of the co-ligand shell. This resulted in more than a 60-fold enhancement of the absorption cross-section at approximately 990 nm while maintaining a small metal core size below 2 nm. Moreover, we demonstrated the advantages of these us-AuNPs with mixed ligand shells for imaging applications. We successfully detected NIR-II signals from these emitters at a penetration depth of up to 6 mm, exhibiting high sensitivity and a high signal-to-noise ratio even at greater depths in intralipids when excited at 980 nm.

Experimental section

Synthesis :

All chemical products were purchased from Sigma-Aldrich (France) and syntheses were conducted using Ultra MiliQ water (16 MΩ).

Bi-thiol-protected us-AuNPs, AuMHA/HDT, were synthesized by wet chemistry in alkaline condition. Briefly, 250 L HAuCl 4 solution (20 mM) is added to 2.4 mL of water followed by a slow addition of a mixture of 6-mercaptohexanoic acid (MHA,1.25 mL, 5 mM) and hexa(ethyleneglycol) dithiol (HDT, 0.75 mL, 5 mM) under mild stirring (350 rpm) keeping a molar ratio Au:MHA:HDT = 1:1.4:0.6. 250 µL of NaOH (1 M) was then added dropwise to adjust the pH to 10. Afterwards, the freshly prepared reducing agent, NaBH 4 (20 mM), was added dropwise using different amounts (0.5, 2, 4, 6, 10, 12.5, and 16 mol corresponding to molar ratios Au:NaBH 4 of 1:0.1 to 1:3.2) to prepare a series of us-AuNPs that were stirred for16 hours and 30 minutes at room temperature. The reaction solution was washed several times using an Amicon centrifuge filter (Milipore) 3 kDa to remove unreacted species. Then the pH was adjusted to 7 and the NPs were kept at room temperature in the dark.

The monothiol sample AuMHA was prepared using the molar ratio Au:MHA = 1: 2 and 2 mol of NaBH 4 as the amount of reducing agent did not influence the optical properties (absorbance, fluorescence, PLQY) of this compound. The reaction solution was stirred for 3 hours, purified and stored using the same protocol as described above.

Instrumentations and methods:

The sizes of the metal cores were determined by high resolution transmission electron microscopy JEOL2010 (HR-TEM) using a monochromated microscope working at 200 kV.

Prior to imaging, the us-AuNPs were dispersed on copper grids covered with a carbon film. X-ray photoelectron spectroscopy (XPS) investigation was performed in a K-Alpha+ spectrometer (ThermoFisher Scientific) using a microfocused, monochromated Al K X-ray source (400 m spot size). The Thermo Avantage software was used for data acquisition and processing. The K-Alpha+ charge compensation system was employed during analysis, using electrons of 8 eV energy, and low-energy argon ions to prevent any localized charge build-up.

The spectra were fitted with one or more Voigt profiles (BE uncertainty: ±0.2 eV) and Scofield sensitivity factors were applied for quantification 36 . All spectra were referenced to the C 1s peak at 285.0 eV binding energy (C-C, C-H) and controlled by means of the wellknown photoelectron peaks of Cu, Ag, and Au, respectively. Measurements were performed on three different locations of dried samples.

Evolution of the us-AuNPs size was determined by positive ion mode nanoelectrospray ionization-variable temperature (nanoESI-VT, customized source) on a commercial quadrupole time-of-flight (micro-qTOFII, Bruker-Daltonics, Bremen, Germany, mass resolution 10,000). The temperature of the solution was set at 17 °C. The samples were prepared by diluting nanoclusters in water, to a final concentration of 0.25-0.5 mg•mL -1 in water. External calibration was carried out with a set of synthetic peptides.

The hydrodynamic diameter and the zeta potential of the samples dispersed in phosphate buffer (PBS, pH 7.4) were measured in duplicate on a Zetasizer instrument from Malvern.

The absorption spectra of diluted us-AuNPs samples were recorded on Perkin Elmer Lambda 1050 UV-vis spectrophotometer between 350 and 1300 nm. Steady-state photoluminescence spectra were measured from 700 -1600 nm with a calibrated FSP 920 (Edinburgh Instruments, Edinburgh, United Kingdom) spectrofluorometer equipped with a nitrogen-cooled PMT R5509P.

Measurements of the relative PLQYs ( f,x ) were performed using the reference dye IR125 dissolved in dimethylsulfoxide (DMSO). The PLQY of this dye was previously determined absolutely to  f,st = 0.23) 37 . The relative PLQYs were calculated according to the formula of Demas and Crosby 38 , see equation below.

The subscripts x, st, and ex denote sample, standard, and excitation respectively.

is the absorption factor, the integrated spectral fluorescence photon flux, and the refractive index of the solvents used (DMSO for IR125; water for us-AuNPs).

All spectroscopic measurements were done in a 1 cm quartz cuvettes from Hellma GmbH at room temperature using air-saturated solutions.

Lifetime measurements were performed using a thermoelectric cooled NIR Photomultiplier tube (Hamamatsu H10330C-45) connected to a Time-Correlated Single Photon Counting Timeharp 260 system from PicoQuant equipped with two picosecond pulsed diode lasers EPL-640 and EPL-450 (Edinburgh Instruments) with 634.3 nm excitation and 82.4 ps pulse width and 450.8 nm excitation with 79.1 ps pulse width, respectively. An 850 nm long-pass filter (FELH0850 Thorlabs) was used for the 634 nm excitation and a 500 nm long-pass filter (FELH0500 Thorlabs) for the 450 nm excitation. The resulting broadband emission bands were monitored for lifetime analysis using a photomultiplier connected to a time-correlated single photon counting system. All measurements were performed at a fixed temperature of 20 ºC using an air-cooled qpod 3e from Quantum Northwest.

NIR-II imaging was performed using a Princeton camera 640ST (900-1700 nm) coupled with a laser excitation source at = 980 nm (100 mW/cm 2 ). We used short-pass excitation filter at 1000 nm (Thorlabs) and long pass filters on the NIR-II camera at 1064 nm (Semrock), and at 1250 nm (Thorlabs). A 50 mm lens with 1.4 aperture (Navitar) was used to focus on the samples. Capillaries containing us-AuNPs solution were soaked in intralipid solution (1%vol) at different depths and put in front of the camera before acquisition and analyses were performed using FIJI software.

Results and discussion

1-Physicochemical characterizations

The us-AuNPs were synthesized through a two-step process using wet chemistry. This process involves forming gold complexes with both monothiol molecules MHA and dithiol molecules HDT. Subsequently, the us-AuNPs are formed by utilizing NaBH 4 as a reducing agent (Figure 1). The amount of reducing agent used during the synthesis was varied from 0.5 to 16.0 µmol, corresponding to a molar ratio of gold to the reducing agent NaBH 4 ranging from 1:0.1 to 1:3.2. The synthesized us-AuNPs were characterized by TEM, and the size distribution of their cores and interparticle distances were evaluated by analyzing at least 200 NPs (Figure 2, Figures S1-S7). The analysis revealed a consistent size distribution of the gold cores around 1.5±0.3 nm, independently of the amount of NaBH 4 used. This core size was also similar to that of the us-AuNPs prepared using only MHA and 2.0 µmol of the reducing agent (Figure S8, Table 1). The TEM images indicated no aggregation of the us-AuNPs, and the interparticle distances ranged between 2.9 and 4.4 nm for all samples.

Elemental analysis indicated that the gold content in all samples reached 45%wt of the total weight of the NPs (Figure 3a). This value aligns with many other reports in the literature 39,40 and indicates that the overall reaction yield is not affected by the amount of reducing agent used during the synthesis. Only the highest amount of the reducing agent led to a decrease to 26%wt of Au in the AuMHA/HDT sample. This could be attributed to a lower reaction yield, where unreacted Au was removed during the purification step (Figure 3a).

Mass spectrometry using electrospray ionization (ESI) can provide information about the total molecular weight (MW) of the samples, and we observed an increase in the m/z distributions (Figure S9a), corresponding to a quasi-linear increase in the average MW from 8.7 kDa using 0.5 µmol of NaBH 4 to 13.1 kDa for 16.0 µmol of NaBH 4 (Figure S9b). Since a consistent size distribution of the gold core around 1.5±0.3 nm was observed regardless of the amount of NaBH 4 , the mass spectrometry data suggest that the increase in molecular weight for us-AuNPs by increasing the amount of the reducing agent is due to an increase in the number of ligands constituting the organic shell that protects the gold core (Figure 3b, Figure S9, Table 1).

X-Ray photoelectron spectroscopy (XPS) measurements were performed to obtain information on the oxidation state of gold and the nature of the ligand binding to the gold surface (Figures 3c,S10). Due to the presence of only one doublet Au 4f in the spectrum independently of the presence of the co-ligand HDT or the amount of NaBH 4 used during the synthesis, we can affirm that no significant difference in the oxidation of gold could be observed. A binding energy between Au(0) at 84.0 eV and Au(I) at 85.5 eV is commonly reported for us-AuNPs 39 . We quantified the amount of HDT by evaluating the intensity of the carbon peak at 286.5 eV, which is attributed to the C-O species present only in the HDT ligand (Figure 3c). This estimation was validated by the significant difference observed when comparing AuMHA/HDT NPs to AuMHA NPs, where the HDT ligand is not present. We observed a linear increase in HDT molecules when increasing NaBH 4 to 6.0 µmol, suggesting that NaBH 4 promotes the binding of HDT to the gold surface. Above 6.0 µmol, no further correlation was observed and relative error is large, and we hypothesized that this may be due to the thicker organic shell and the difficulty of providing additional HDT to the gold core, possibly due to steric hindrance and the potential rearrangement of the two ligands in the organic shell. Notably, above 6.0 µmol, the increase in average MW is significantly reduced, confirming this saturation effect in the evolution of the ligand shell observed by XPS based on the concentration of carbon at 286 eV (compare Figures 3c andS9b). It was also recently reported that a high amount of the reducing agent can promote the formation of disulfide bonds 41,42 , which in our case may lead to the crosslinking of HDT molecules, thereby weakening the thiol interaction with the gold core surface.

The zeta potential of AuMHA/HDT dispersed in water was measured to be approximately -28±3 mV (see Table 1), confirming the negatively charged surface of these samples due to the presence of carboxyl groups from MHA. Only with the highest amount of NaBH 4 did we observe a less negative zeta potential (-18±2 mV). This could be attributed to a different distribution of HDT within the ligand shell, which is more neutral than MHA at pH 7.

Therefore, the physicochemical characterizations conducted via elemental analyses and mass spectrometry suggest that increasing the amount of the reducing agent leads to a thicker organic shell that protects the gold core and reduces the surface charge. This thicker organic shell may be accompanied by a restructuring of the co-ligand shell, as indicated by the direct interaction (HDT vs MHA) with the gold surface.

2-Optical characterizations

The variation in the amount of sodium borohydride, which initiates the growth of the us-AuNPs, significantly influences the optical properties of the resulting us-AuNPs. These effects were reproducible across all series of samples (Figure S11). As shown in Figure 4a, increasing the amount of NaBH 4 from 0.5 to 16.0 µmol led to an increase in the intensity of the absorption band, which was approximately centered at 990 nm. To evaluate the absorption profile, we divided the absorption spectrum into three optical windows: 650-720 nm (I), 720-920 nm (II), and 920-1250 nm (III). The evolution of the area under each spectral window was plotted as a function of the NaBH 4 concentration after normalizing the spectra and applying deconvolution (see Figure 4b,S12,S13). The first band (I), centered at 670 nm, was not affected by the amount of NaBH 4 . Band II exhibited a rapid and strong increase, reaching up to a 4-fold enhancement at low NaBH 4 concentrations, followed by a linear decrease. A significant 60-fold enhancement of band III, centered at 990 nm, was observed, reaching its maximum with the use of 12.5 µmol of NaBH 4 . The molar extinction coefficient determined at 990 nm, using the average molecular weight of each sample from the ESI-MS data, showed a consistent increase from 0.1x10 3 M -1 cm -1 to 1.69x10 4 M -1 cm -1 (Table 2). Changes in the ratio of the ligands MHA:HDT did not affect the impact of NaBH 4 content on the optical properties (Figure S14). Additionally, we can exclude the influence of the counterion of the reducing agent by using tetramethylammonium borohydride (Figure S15). Only acidic pH below 4 led to a fall of the band III of these samples (Figure S16).

TEM data confirmed also that this absorption band (III) was not related to a plasmon band from large AuNPs (size > 5 nm) or anisotropic NPs (Figures S1-S7). Furthermore, interparticle distances were greater than 3 nm, which rules out electron coupling between nanoparticles that could result in NIR absorption 18 .

Subsequently, a PL study was conducted on AuMHA/HDT dispersed in PBS. All samples exhibited a broad PL emission ranging from 800 to 1400 nm, with three main bands centered at 850 nm, 1050 nm, and 1250 nm. These bands are commonly observed in various types of NIR-II-emitting us-AuNPs 16,35,43 . Here, we present the normalized spectra to highlight the influence of excitation wavelengths on the PL emission spectra. The PL emission bands exhibited a dependence on the excitation wavelength in terms of intensity, shape, and spectral position. These effects were also influenced by the amount of NaBH 4 used in the synthesis of us-AuNPs. Under short excitation wavelengths (λ exc. 450 nm), we observed a shift of the first PL band from 850 to 930 nm and the second band from 1030 nm to 1100 nm with an increase in the NaBH 4 amount (see Figure 5a). The third band centered at 1250 nm exhibited a linear increase in PL intensity. A similar trend, though less pronounced, was observed with excitation at 780 nm (Figure 5b). When excited at 980 nm, corresponding to band III in the absorption spectra, we observed a slight shift of the PL band maximum from 1060 nm to 1080 nm, accompanied by a shift of the PL tail towards longer wavelengths, with an increasing NaBH 4 amount from 2.0 µmol to 16.0 µmol of NaBH 4 (Figure 5c). PL excitation spectra of AuMHA/HDT samples recorded at λ em. 925 nm exhibited a single band at 830 nm (Figure 6a), while selecting the PL emission at λ em. 1050 nm revealed two bands, one at 780 nm and another at 990 nm (Figure 6b). The band at 780 nm exhibited a consistent decrease with increasing NaBH 4 concentration from 4 µmol to 16 µmol. The PLQY of the different AuMHA/HDT samples was determined in water relative to the reference dye IR125 dissolved in DMSO (with a PLQY of 23%). The PLQY values reached a maximum of 2.8% using 2 µmol of NaBH 4 , which is 10 times higher than that observed for the sample obtained with the monothiol MHA (Table 2).

While changing the pH from 12 to 6 did not result in significant changes in the absorbance spectra (Figure S16), we observed higher PL emission at basic pH (Figures S17).

This pH effect has also been reported for various thiol-stabilized gold nanoclusters, attributed to the efficient energy transfer between gold and the thiolate ligand under alkaline conditions 44,45 .

PL lifetime measurements were performed for three AuMHA/HDT samples prepared with different amounts of NaBH 4 : 2.0 µmol, 6.0 µmol, and 16.0 µmol using 450 nm and 634 nm as excitation wavelengths and collecting the photons in a broad NIR-II emission window (1050-1700 nm). All decay curves could be fitted with three components  1 (1 ns), At this stage, we can only speculate about how a thicker organic shell could enhance NIR absorption. One possible explanation is an increase in the rigidity of the ligand shell, as previous studies have shown that this can significantly alter optical properties 32,34,39,44 .

Another hypothesis is that the change in the surface shell could induce variations in the charge states of the gold core. In fact, Zhu et al. discovered that the absorption spectra in the NIR region are altered when the charge state of Au 25 changes from -1 to 0, likely due to a slight distortion of the cluster structures 46 .

The PL properties of the us-AuNPs exhibit a similar trend to that observed in other luminescent ultra-small NPs 47 . This trend involves a shift in PL towards longer wavelengths through the rigidification of the surface, which limits the access of water molecules to the gold surface and reduces non-radiative processes 35 . These findings are supported by the decrease in PL lifetime, indicating more efficient energy transfer to water molecule.

Furthermore, the PL emission of these emitters is dependent on the excitation wavelength, confirming the occurrence of multiple energy transfers within nanoparticles of potentially different sizes. This is illustrated by the attenuation of a specific excitation band at 830 nm.

We observed PL excitation band at 780 nm and 980 nm, which clearly indicate optical transitions at these wavelengths and contribute to the PL of these emitters. We also confirm the absorption at 980-995 nm corresponds to an excited state with PL emission exhibiting a small Stokes shift of approximately 60 nm (emission centered at 1050 nm). This suggests that these transition energies likely occur within the core and involve aurophilic interactions.

Conversely, shorter excitations at 450 nm involve both energy transfer from the core and from the surface, as reported in the literature 48 .

3-NIR-II imaging in intralipids

To demonstrate the potential of these new NIR-II emitters for bioimaging, we filled capillaries with different samples of AuMHA/HDT soaked in an intralipid solution (1%) 49 at various depths and measured the PL intensity that mimics living tissue using a 980 nm excitation. Two sub-NIR-II windows were selected using a long-pass filter at 1064 nm and 1250 nm to evaluate the sensitivity of detection, as it is known to be optimal at longer wavelengths 49 . Images (Figure 8) were obtained, and the PL signal for all samples could be detected down to a depth of 6 mm in intralipids. The sensitivity of detection was better when selecting PL emission above 1250 nm (Figure 9), where the PL signal was still 10 times higher than the background for AuMHA/HDT samples prepared with a high NaBH 4 content.

These samples exhibited the highest absorption at 980 nm, which was significantly above the detection limit (threshold SNR=3). Despite a shorter PL emission range using 980 nm instead of 780 nm excitation wavelength, the strong absorption at 980 nm still provided a high number of photons, with a reasonable PLQY ranging from 0.41 to 2.85%. This enables a sensitive detection of PL in the capillaries down to 6 mm in intralipids.

Conclusion and Outlook

In this work, we demonstrated the ability to design ultra-small gold nanoparticles (us-Au NPs) with a size smaller than 2 nm, which exhibit efficient absorption in the infrared region around 1000 nm. Physicochemical characterizations, including elemental analyses and mass spectrometry, were performed. These characterizations suggest that an increase in the amount of the reducing agent leads to the formation of a thicker organic shell, which serves to protect the gold core while maintaining a relatively constant diameter and reducing the negative charge on the surface. By conducting comprehensive characterizations of these us-Au NPs, we can better understand the structural factors that contribute to their optical properties. It is possible that the arrangement of the organic shell induces distortions and/or changes in the charge state of the gold core, which may be responsible for the significant increase in NIR absorption.

Importantly, these us-AuNPs are dispersible in water and exhibit efficient photoluminescence emission in the NIR-II window, specifically between 1000 and 1400 nm. When excited at 980 nm, PL signals from these particles can be detected through a 6 mm thick intralipid solution that mimics biological tissue. This discovery opens up new avenues for utilizing such ultrasmall probes in bioimaging applications. However, further studies are required to fully understand the nature of this near-infrared absorption and its correlation with the ligand shell.

   2 (40 to 110 ns), and  3 (200 to 750 ns) (Figure S18). Results depicted in Figure 7 indicated a decrease of the PL lifetime with the increase of NaBH 4 concentration at higher excitation wavelengths. This is traduced by the fall of the average PL lifetime = 714 ns (2 mol NaBH 4 ), 501 ns (6 mol NaBH 4 ), 233 ns (16 mol NaBH 4 ) using 450.8 nm excitation wavelength and also using 634.3 nm excitation wavelength: = 660 ns (2 mol NaBH 4 ), 442 ns (6 mol NaBH 4 ), 216 ns (16 mol NaBH 4 ).

Figure 2 .

 2 Figure 2. TEM images and core size analysis of AuMHA/HDT in water prepared with 0.5(a), 2(b), 4(c), 6(d), 10(e), 12.5 (f), and 16 (g) µmol of NaBH 4 .

Figure 3 .

 3 Figure 3. a. Gold concentration in AuMHA/HDT as a function of NaBH 4 content during the synthesis determined by elemental analysis. b. ESI-MS spectra for AuMHA/HDT prepared with different amount of NaBH 4 in positive mode except for AuMHA sample in negative mode. c. Relative HDT concentration on us-AuNPs surface estimated by XPS measurement.

Figure 4 .

 4 Figure 4. a. Absorption spectra of AuMHA/HDT prepared using different concentrations of NaBH 4 . b. Evolution of the area of the absorption bands in the region 650-720 nm (I), 720-920nm (II), and 920-1250 nm (III) on normalized and deconvoluted spectra (see SI).

Figure 5 .

 5 Figure 5. a. PL emission spectra of AuMHA/HDT samples dispersed in PBS buffer (pH 7) at ƛ exc. = 450 nm (a), ƛ exc. = 780 nm (b), and ƛ exc. = 980 nm (c).

Figure 6 .

 6 Figure 6. PL excitation spectra of AuMHA/HDT samples dispersed in PBS buffer (pH 7) measured at ƛ em. = 925 nm (a), and ƛ em. = 1050 nm (b).

Figure 7 .

 7 Figure 7. a. PL decay curves at 20 C for AuMHA/HDT samples dispersed in PBS (pH 7.4) using 450.8 nm excitation (a) and 634.3 nm excitation (b).

Figure 8 .

 8 Figure 8. NIR-II imaging of capillaries filled with AuMHA/HDT samples at different depth in intralipid solution (1%) using long pass filter at 1064 nm and 1250 nm.

Figure 9 .

 9 Figure 9. Sensitivity of detection of AuMHA/HDT samples in capillaries by NIR-II imaging at different depth in intralipid solution (1%). Signal-to-noise ratio (SNR) threshold for limit of detection is fixed at 3.

Acknowledgement

XLG would like to thank "ligue contre le cancer" (R22004CC), ANR SIREN (ANR-20-CE92-0039-01) for their financial support. This work was financed by the Spanish Ministerio de Ciencia e Innovación under project NANONERV PID2019-106211RB-I00. R.M. is grateful to the Spanish Ministerio de Ciencia e Innovación for support to research through a Ramón y Cajal Fellowship (RYC2021-032913-I). WM and XLG would like to thank GDR Or-Nano for their financial support. URG acknowledges financial support by the DFG-ANR grant SIREN (DFG RE 1203/38-1).

All authors declare no conflict of interests.

Author contribution section

WM conducted the synthesis and the imaging experiments. WM, DW, CZZ, VT and MPO performed spectroscopic characterizations and analyses. XLG and RA participate to the conceptualization of this work. All the authors discussed the results and wrote the article.


S Basak

J Bect

L Guillier §

F Tenenhaus-Aziza

E Vazquez

† Phd

Bayesian multi-objective optimization for stochastic simulators

We consider the problem of multi-objective optimization for stochastic simulators, where each evaluation of the objectives f = (f 1 , f 2 , . . . , f q ) at a given point x in a discrete finite search domain X yields random responses Z i = f i (x) + ϵ i , 1 ≤ i ≤ q, with possibly heteroscedastic noise ϵ i . The best possible trade-off between the conflicting objectives is determined by the Pareto domination rule : f (x) ≺ f (x ′ ) (x Pareto dominates x ′ ), when f i (x) ≤ f i (x ′ ) for all i, with at least one of the inequalities being strict. The goal is to estimate the set of Pareto optimal solutions P = {x ∈ X : ∄x ′ ∈ X, f (x ′ ) ≺ f (x)}, from a limited number of evaluations of the objective functions.

In this work, we propose a modified version of the PALS (Pareto Active Learning for Stochastic simulators) algorithm [START_REF] Barracosa | Extension of the pareto active learning method to multi-objective optimization for stochastic simulators[END_REF] called PALS-II, that significantly improves the optimization performance in a high noise setting and we perform a numerical benchmark (see Figure 1) comparing its performance with several other competing approaches. The idea is to construct an efficient, inexpensive and easy to implement algorithm, unlike other Bayesian approaches in the literature based on computationally-intensive criteria for point selection. The metrics proposed to assess the performance are the volume of the symmetric difference, which assesses the global error on Pareto front estimates (in the objective space), and the misclassification rate, which measures the global error on the Pareto set estimates (in the input space). PALS builds a Gaussian Process surrogate model over the computationally expensive stochastic simulator and iteratively classifies each point in X as either potentially Pareto optimal (P n ), non Pareto optimal (N n ) or unclassified (U n ), using confidence hyper-rectangles of the form

{z ∈ R q : µ n (x) -β 1/2 σ n (x) ≺ z ≺ µ n (x) + β 1/2 σ n (x)},
where β is a parameter controlling the size of the confidence region, and where µ n (x) and σ 2 n (x) stand for the GP posterior mean and variance at the n-th iteration. The point with the highest uncertainty (maximum rectangle diameter) is evaluated at the next iteration. PALS suffers from one major drawback: the tuning of the parameter β can be difficult. It is not directly related to a probabilistic interpretation in terms of Pareto domination and it affects the performance in a significant way.

To address this drawback, we propose a new rectangle-free classification rule that takes into account the global uncertainty and the correlation between the points. At the n-th iteration,we simulate sample paths from the GP, conditional on the observed evaluations. The conditional sample paths are used to obtain empirical estimates of the probabilities of being Pareto optimal for each point in X. These probabilities are used to classify points in P n , N n and U n . The rule for selecting the next evaluation point remains the same as PALS. We carry out a numerical comparison of PALS-II with other algorithms from the literature of Bayesian ranking and selection where the main idea is to allocate the simulation budget to the input design points such that the Pareto set can be estimated with high confidence. In particular, we compare PALS-II to MOCBA (Multi-objective Optimal Computing Budget Allocation [START_REF] Lee | Finding the non-dominated pareto set for multi-objective simulation models[END_REF]) that perform allocation by minimizing the asymptotic upper bounds of misclassification error of the design points.

Figure 1 :

 1 Figure 1: Figure comparing misclassification rate (mcl) and volume of symmetric difference (svd) averaged over 200 repetitions for PALS, PALS-II and a baseline random search approach, for a test problem over a grid of size 21 × 21.

Application case: In the context of the project ArtiSaneFood, we aim to find optimal process intervention parameters in cheese production, which translates into a multi-objective optimization problem of a stochastic simulator [START_REF] Basak | Bayesian multi-objective optimization for quantitative risk assessment in microbiology[END_REF], based on a Quantitative Risk Assessment model.

Implementation details: PALS-II is coded in Python, using the open source packages GPmp (Gaussian process micro package) and GPmp-contrib. Our code will be published as a part of these packages, available at https://github.com/gpmp-dev.

Short biography (PhD student)

Subhasish Basak is a third-year doctoral student at the Laboratoire des Signaux et Systèmes (L2S), Université Paris-Saclay and at Agence nationale de sécurité sanitaire (ANSES). His thesis is a part of the project ArtiSaneFood project (grant number : ANR-18-PRIM-0015) which is part of the PRIMA program supported by the European Union. This project aims to establish methodological recommendations to cheese producers in France, in order to reduce the risk of Haemolytic Uremic Syndrome (HUS) in children, from the consumption of raw-milk soft cheese.