Vinh Thong
email: vinhthong.ta@unicaen.fr

Sébastien Bougleux
email: sebastien.bougleux@greyc.ensicaen.fr

Abderrahim Elmoataz
email: abder.elmoataz@greyc.ensicaen.fr

Olivier Lézoray
email: olivier.lezoray@unicaen.fr

Nonlocal Anisotropic Discrete Regularization for Image, Data Filtering and Clustering

Keywords: anisotropic discrete regularization, nonlocal operators, graph p-Laplacian, filtering, clustering

In this paper, we propose a nonlocal anisotropic discrete regularization on graphs of arbitrary topologies as a framework for image, data filtering and clustering. Inspired by recent works on nonlocal regularization and on the TV digital filter, a family of discrete anisotropic functional regularization on graphs is proposed. This regularization is based on the L p -norm of the nonlocal gradient and the discrete p-Laplacian on graphs. It can be viewed as the discrete analogue on graphs of the continuous p-TV anisotropic functionals regularization formulations. After providing definitions and algorithms to resolve such a discrete nonlocal anisotropic regularization, we show its properties for filtering, clustering on different types of data living on different graph topologies (image, data). In particular we investigate the cases of p = 2, p = 1 and p < 1, this latter being very few considered in literature.

Introduction

Image regularization, based on variational and partial differential equations (PDEs) approaches, are one of the most important tools in image process-ing and computer vision. They have been applied to handle a wide variety of problems such as denoising, segmentation, optical flow or object tracking. A complete review regarding the theory and the applications of this area can be found, in [START_REF] Alvarez | Axioms and fundamental equations of image processing[END_REF][START_REF] Aubert | Mathematical Problems in Image Processing[END_REF][START_REF] Chan | Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[END_REF][START_REF] Tsai | Total variation and level set methods in image science[END_REF] and references therein. However, partial and variational methods have some limitations in the functionals used in regularization processes, such as the anisotropic diffusion models [START_REF] Perona | scale-space and edge detection using anisotropic diffusion[END_REF][START_REF] Tschumperlé | Fast anisotropic smoothing of multi-valued images using curvature-preserving PDEś[END_REF], the total variation (TV) models [START_REF] Chan | Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[END_REF][START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] or the active contour models [START_REF] Tsai | Total variation and level set methods in image science[END_REF]. Indeed, these methods are based on derivatives which only consider local features of the data.

Recently, the use of nonlocal interactions to capture the complex structure within images, has received a lot of attention. This methodology has shown to be very effective, and allows more flexibility in the regularization processes. Filters based on nonlocal interactions shown superior abilities for image and video processing such as denoising [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF][START_REF] Kervrann | Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal[END_REF], enhancement, demosaicing [START_REF] Buades | Non local demosaicing[END_REF], texture analysis and synthesis [START_REF] Peyré | Image processing with non-local spectral bases[END_REF]. Kindermann and Osher [START_REF] Kindermann | Deblurring and denoising of images by nonlocal functionals[END_REF] were the first to interpret the nonlocal means filters and neighborhoods filters as nonlocal regularization functionals. Later, Guilboa and Osher [START_REF] Gilboa | Nonlocal linear image regularization and supervised segmentation[END_REF] proposed a nonlocal functional, based on weighted differences, for image regularization and semisupervised segmentation. These works can be regarded as the nonlocal analogues of total variation models for image regularization. Moreover, the use of nonlocal functionals, into regularization and diffusion processes not new. Indeed, it has been widely used to model diffusion processes arising from statistical mechanics [START_REF] Bates | A discrete convolution model for phase transitions[END_REF], or neural activity modelization [START_REF] Viéville | An unbiased implementation of regularization mechanisms[END_REF]. For recent works, see [START_REF] Andreu | A nonlocal p-laplacian evolution equation with neumann boundary conditions[END_REF][START_REF] Bogoya | Neumann boundary conditions for a nonlocal nonlinear diffusion operator. continuous and discrete models[END_REF] and references therein.

Most of the proposed nonlocal regularization functionals have been formulated for image processing, where images are expressed as continuous functions on continuous domains. Then, a continuous energy functional is considered and classically solved by the corresponding continuous Euler-Lagrange equations or its associated flows. Unfortunately, the discretization of these equations is difficult for high dimensional data, or functions defined on this data. It is also the case for irregular domains. Moreover, in many research fields such as geography, data-mining, social sciences, or robotics, data can be represented by graphs. The interactions within the data can be expressed by functionals on graphs, and the related problems can be treated in terms of regularization.

To regularize images or discrete data on irregular domains, another methodology consists in performing complete digitization of continuous models [START_REF] Osher | Digitized PDE for data restoration[END_REF]. The latter approach directly considers discrete variational problems, and therefore works on general discrete data. An example of such a digitization is the TV digital filter, introduced and analyzed by [START_REF] Chan | The digital TV filter and nonlinear denoising[END_REF] in the context of image restoration. This filter can be interpreted as a precise translation of the TV model [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] in discrete settings on unweighted graphs. Inspired by this work, we have recently proposed a general p-TV discrete model on weighted graphs [START_REF] Bougleux | Discrete regularization on weighted graphs for image and mesh filtering[END_REF][START_REF] Elmoataz | Nonlocal discrete regularization an weighted graphs: a framework for image and manifolds processing[END_REF]. A similar approach has also been investigated in the context of semi-supervised learning on graphs [START_REF] Zhou | Discrete Regularization, Semi-Supervised Learning, section 3.13, Adaptive computation and machine learning[END_REF].

Our previously proposed regularization is based on the L 2 -norm of the nonlocal gradient and on the discrete p-Laplacian on graphs. It can be viewed as the discrete analogue of the continuous p-TV isotropic regularization. We have shown that our discrete p-TV isotropic regularization framework leads to a family of linear and nonlinear processing on graphs. In particular, this family of methods includes exact expression of several local and nonlocal filters such as the bilateral filter [START_REF] Smith | SUSAN -a new approach to low level image processing[END_REF][START_REF] Tomasi | Bilateral filtering for gray and color images[END_REF], the TV digital filter, or the nonlocal means filter [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF]. Moreover, for image processing, we have also shown that the continuous local and nonlocal regularizations, based on p-TV models, are the continuous analogue of our discrete regularization approach. Indirectly and naturally, our methodology provides a discrete extension of these continuous regularization methods for any discrete data or functions.

In this paper, we propose to extend our isotropic regularization framework by introduce a new family of discrete anisotropic regularization functionals. This new family is based the L p -norm, and generalize significantly our previous studies. Moreover, it can be considered as the nonlocal discrete analogue of the p-TV anisotropic regularization in the continuous case [START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF].

Let G = (V, E, w) be a weighted graph which consists in a set of vertices V , a set of edges E ⊂ V × V , and a weight function w defined on the edges. The discrete p-TV anisotropic regularization of a function f 0 : V → R is performed by the following energy minimization:

min f :V →R E w (f, f 0 , λ, p) = R w (f, p) + λ 2 f -f 0 2 2 , (1)
where p ∈ [0, +∞[ is the smoothness degree, and λ ≥ 0 is the Lagrange multiplier or fidelity parameter. The first term in [START_REF] Alvarez | Axioms and fundamental equations of image processing[END_REF], is defined as

1 2p v∈V |∇ w f (v)| p p
, where |.| p is the L p norm and ∇ w f is the weighted gradient of f over G. The second term is the fidelity term and it estimates the variations between the functions f 0 and f . For the case of p = 2, one can easily show that Eq. ( 1) corresponds to the Tikhonov regularization. For several values of p, this family of functionals is related to numerous processing methods involve in image restoration, graphs cuts or random walks techniques [START_REF] Grady | A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm[END_REF]. This paper is organized as follows. Firstly we introduce the basic operators needed for the formulation, and for the resolution, of the proposed minimization problem. Then, we propose two different diffusion processes to approximate its solution. Finally, the obtained regularization is applied to filter and to classify images and discrete data.

Discrete operators on weighted graphs

This section recall useful notations on weighted graphs and define the basic discrete operators involved in the proposed regularization framework. Similar definitions and properties have also been used in the context of spectral graph theory [START_REF] Chung | Spectral Graph Theory[END_REF], differential calculus on graphs [START_REF] Bensoussan | Difference equations on weighted graphs[END_REF][START_REF] Mcdonald | Diffusions on graphs, poisson problems and spectral geometry[END_REF], semi-supervised learning [START_REF] Zhou | A regularization framework for learning from graph data[END_REF][START_REF] Zhou | Regularization on discrete spaces[END_REF], or image and mesh processing [START_REF] Bougleux | Image smoothing and segmentation by graph regularization[END_REF][START_REF] Bougleux | Discrete regularization on weighted graphs for image and mesh filtering[END_REF][START_REF] Chan | The digital TV filter and nonlinear denoising[END_REF][START_REF] Osher | Digitized PDE for data restoration[END_REF]. In these works, the discrete operators are usually defined in order to formalize and to solve isotropic regularizations, which leads to diffusion processes based on isotropic graph Laplacians or p-Laplace operators. In the following, we propose to construct analogue operators for the anisotropic framework. In particular, we define the discrete anisotropic p-Laplace operator.

Preliminary definitions

A weighted graph G = (V, E, w) is composed of a finite set V of N vertices, a finite set E ⊂ V × V of edges, and a weight function w : E → R + . An edge of E, which connects two vertices u and v of V , is noted uv. In this paper, graphs are supposed to be connected, with no self-loops or multiple edges, and undirected (for each edge uv ∈ E, vu ∈ E). This implies that the weight function w is symmetric, i.e. w uv = w vu for all edge uv ∈ E. The weight function measures the dissimilarity of two vertices of the graph. When w uv → 0, the two vertices u and v are dissimilar. By convention, we set w uv = 0 if the vertices are not connected by an edge of E.

Let H(V ) be the Hilbert space of real-valued functions on the vertices of a graph G = (V, E, w), with V = {v 1 , . . . , v N }. Each function f ∈ H(V ), f : V → R, assigns a vector f (v i ) to each vertex v i ∈ V . The function f forms a finite N-dimensional space. It can be thought as a column vector f = [f (v 1 ), . . . , f (v N )] T . By analogy with functional analysis in continuous spaces, the integration of f over the graph, is noted V f = V f . Similarly, let H(E) be the Hilbert space of real-valued functions F : E → R, defined on the edges of the graph. These two spaces are endowed with the usual inner products:

f, g H(V ) = u∈V f (u)g(u), F, G H(E) = u∈V v∼u F (uv)G(uv),
where f, g : V → R and F, G : E → R.

Discrete difference, derivative and adjoint

The difference operator of a function f : V → R, noted d : H(V ) → H(E), is defined on an edge uv ∈ E by:

(df )(uv) = f (v) -f (u) µ w (u, v) ,
where µ : V × V → R + is a similarity measure which depends on the weight function w. In the sequel, we restrict ourselves to µ w (u, v) = 1/ √ w uv . This implies the following definition of the difference operator:

(df )(uv) = √ w uv (f (v) -f (u)) (2)
The directional derivative of the function f , at a vertex u, and over an edge uv, is defined to be:

∂ v f (u) = (df )(uv) = √ w uv (f (v) -f (u)). (3)
The difference operator and the directional derivative are antisymmetric, i.e. (df )(uv) = -(df )(vu). They also share the following property with the continuous definition of the derivative of function defined in the Euclidean space:

f (u) = f (v) ⇒ (df )(uv) = 0. (4)
The adjoint operator of the difference operator, noted

d * : H(E) → H(V ), is defined by: df, H H(E) = f, d * H H(V ) , ∀f ∈ H(V ), H ∈ H(E). (5)
Then, from the expressions of the inner products in H(E) and H(V ), one can deduce the expression of the adjoint operator at a vertex u ∈ V :

(d * H)(u) = v∈V √ w uv (H(vu) -H(uv)). (6)
The adjoint is a linear operator which measures the flow of H over the graph. By analogy with continuous differential operators, the divergence operator of the function H is defined as div H = -d * H. Then, one can show that any function H ∈ H(E) has a null flow (divergence theorem):

u∈V (div H)(u) = u∈V v∈V √ w uv (H(uv) -H(vu)) = 0. ( 7
)
Remark. There exists other expressions of the difference operator, depending on the context. For example, the difference operator defined in [START_REF] Zhou | A regularization framework for learning from graph data[END_REF][START_REF] Zhou | Regularization on discrete spaces[END_REF] is formulated as:

(df )(uv) = w uv δ v f (v) - w uv δ u f (u),
where δ u = v∼u w uv is the degree of the vertex u in the graph. One can note that this formulation does not respect the property (4). Moreover, its adjoint operator does not satisfy the divergence theorem (Eq. ( 7)).

Gradient operator and local variations

The gradient operator of a function f : V → R, at a vertex u ∈ V , is the vector operator defined by:

∇ w f (u) = ∂ v f (u) : v ∼ u T = ∂ v 1 f (u), . . . , ∂ v k f (u) T ,
where v 1 , . . . , v k are the neighbors of the vertex u.

The local variation of the function f measures its regularity in the neighborhood of a vertex. Several norms of ∇ w f (u) can be used. In this paper, we propose to use the L 2 -norm:

|∇ w f (u)| 2 = v∼u (∂ v f (u)) 2 1 2 = v∼u w uv (f (v) -f (u)) 2 1 2 , (8)
and more generally the L p -norm:

|∇ w f (u)| p = v∼u |∂ v f (u)| p 1 p = v∼u w p 2 uv |f (v) -f (u)| p 1 p . (9)
The L p -norm of the local variation is a seminorm for p ≥ 1, and it is not a norm for p < 1.

When the set of vertices V represents a set of organized data, such as digital images or meshes, the initial organization, which is a graph by nature, traduces local interactions between the data. On this initial graph, the gradient operator involves these local interactions. A gradient operator, which involves nonlocal interactions, can be obtained by constructing neighborhood graphs from the initial organization. One can remark that in both cases, the expression of the local variation is the same. However, due to the fact that w uv = 0 if uv ∈ E, a totally nonlocal expression of the local variation can be written as :

| N L w f (u)| p = v∈V |∂ v f (u)| p 1 p = v∈V w p 2 uv|f (v) -f (u)| p 1 p . ( 10
)

Anisotropic p-Laplace operator

The anisotropic p-Laplace operator ∆ p w : H(V ) → H(V ), of a function f : V → R, is defined using the difference operator:

∆ p w f = 1 2 d * |df | p-2 df (11)
Its expression, at a vertex u ∈ V , is given by:

∆ p w f (u) = 1 2 v∼u √ w uv |(df )(vu)| p-2 (df )(vu) -|(df )(uv)| p-2 (df )(uv) = v∼u √ w uv |(df )(vu)| p-2 (df )(vu) = v∼u w p 2 uv |f (u) -f (v)| p-2 (f (u) -f (v)). ( 12
)
Two specific cases of the anisotropic p-Laplace operator are given for p = 1 and p = 2. When p = 1, an interesting rewriting of Eq. ( 12) is given by:

∆ 1 w f (u) = v∼u √ w uv sign(f (u) -f (v)). ( 13
) When p = 2, ∆ 2 w f = ∆ w f = 1 2 d * (df )
, which the definition of the combinatorial (isotropic) graph Laplacian. In this case, Eq. ( 12) becomes:

∆ w f (u) = v∼u w uv (f (u) -f (v)). ( 14
)
In order to avoid numerical instabilities when p < 2, the anisotropic p-Laplace operator is regularized as:

∆ p, w f (u) = v∼u w p 2 uv (|f (u) -f (v)| + ) p-2 (f (u) -f (v)), (15)
where > 0 is fixed constant.

Remark. There exists much more discrete expressions of the isotropic p-Laplace operator, defined as

∆ p w f = 1 2 d * | w f | p-2 2 df
. Using the difference operator and its adjoint of Section 2.2, the expression of the isotropic p-Laplace operator is given by:

∆ p w f (u) = v∼u w uv (| w f (u)| p-2 2 + | w f (v)| p-2 2 )(f (u) -f (v)).
See [START_REF] Bougleux | Discrete regularization on weighted graphs for image and mesh filtering[END_REF] for more details.

Discrete anisotropic regularization framework

Let G = (V, E, w) be a weighted graph, and let f 0 : V → R be a function of H(V ). To regularize f 0 , we propose to consider the following variational problem:

min f ∈H(V ) E w (f, f 0 , λ, p) = R w (f, p) + λ 2 f -f 0 2 2 , (16)
where R w (f, p) is the anisotropic p-TV functional defined as:

R w (f, p) = 1 2p u∈V |∇ w f (u)| p p , p ∈]0, +∞[ = 1 2p u∈V v∼u w p 2 uv|f (v) -f (u)| p . (17)
For p ≥ 1, both functionals in the minimizer ( 16) are convex, and the solution of the minimization is unique. This is not the case when p < 1, for which the regularization functional R w (f, p) is not convex, and the uniqueness of the minimization solution is not insured. However, this case is also considered in the following, in order to analyze the behavior of the associated diffusion processes beyond the usual bound p = 1, which have been very few investigated by researchers (see Section 5 and Section 6).

In particular, when p = 1, the regularization functional is the anisotropic total variation of a function f : V → R:

R w (f, 1) = 1 2 u∈V |∇ w f (u)| 1 = 1 2 u∈V v∼u √ w uv |f (v) -f (u)|. ( 18
)
When p = 2, R w (f, 2) is the classical (isotropic) regularization functional:

R w (f, 2) = 1 4 u∈V |∇ w f (u)| 2 2 = 1 4 u∈V v∼u w uv (f (v) -f (u)) 2 , ( 19
)
and the minimization of E w (f, f 0 , λ, 2) is the Tikhonov regularization of the function f 0 on a weighted graph.

Remark. The anisotropic regularization functional R w (f, p) is the discrete weighted transcription of the continuous regularization functional of a realvalued function f : Ω ⊂ R m → R, defined on a bounded domain Ω of the m-dimensional Euclidean space:

J(f, p) = Ω | f | p p dx.
One can note that the resolution of the discrete minimization problem does not require specific boundary conditions. They are naturally encoded by the graph structure.

To establish the solution of the minimization problem [START_REF] Chan | Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[END_REF], on can solve the following system of equations:

∂ ∂f E w (f, f 0 , λ, p) f (u) = ∂ ∂f R w (f, p) f (u) + λ((f (u) -f 0 (u)) = 0, ∀u ∈ V.
Then, if this system has a solution, it is the unique solution of the minimizer ( 16), for p ≥ 1.

Property 1 ∂ ∂f R w (f, p) f (u) = ∆ p w f (u).
PROOF. Let u 1 be a vertex of V . The u 1 th term of the partial derivative of R w (f, p) is given by:

∂ ∂f R w (f, p) f (u 1 ) (17)
= ∂ ∂f

1 2p u∈V v∼u w p 2 uv |f (v) -f (u)| p f (u 1 )
.

The partial derivative depends only on the edges incident to u 1 . Let v 1 , . . . , v k be the vertices of V connected to u 1 by an edge of E. Then, using the chain rule, we have:

2 ∂R w (f, p) ∂f f (u 1 ) = - v∼u 1 w p 2 u 1 v (f (v) -f (u 1 )) |f (v) -f (u 1 )| p-2 + w p 2 u 1 v 1 (f (u 1 ) -f (v 1 ))|f (u 1 ) -f (v 1 )| p-2 + . . . + w p 2 u 1 v k (f (u 1 ) -f (v k ))|f (u 1 ) -f (v k )| p-2 =2 v∼u 1 w p 2 u 1 v (f (u 1 ) -f (v))|f (u 1 ) -f (v)| p-2 (12) = 2∆ p w f (u 1 ).
Based on Property 1, the solution becomes:

∆ p w f (u) + λ(f (u) -f 0 (u)) = 0, ∀u ∈ V. (20)
This latter equation can be considered as the discrete Euler-Lagrange equation associated to the minimization problem [START_REF] Chan | Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[END_REF]. Contrary to the continuous case, it does not involve any PDEs. By replacing the expression of the anisotropic p-Laplace operator ∆ p w in the system [START_REF] Gilboa | Nonlocal linear image regularization and supervised segmentation[END_REF], we obtain the following nonlinear system:

v∼u w p 2 uv|f (u) -f (v)| p-2 (f (u) -f (v)) + λ(f (u) -f 0 (u)) = 0, ∀u ∈ V. (21)
Several methods can be used to solve this system. In particular, for p = 1, commonly used methods are based on graph cut techniques [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF][START_REF] Darbon | Exact optimization of discrete constrained total variation minimization problems[END_REF]. In the sequel, we propose to use two simple methods to approximate the solution of the system [START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF]. These methods leads to semi-discrete and discrete diffusion processes.

Semi-discrete diffusion process. The first method, considered in this paper, is the infinitesimal steepest descent method:

     f (0) = f 0 d dt f (t) (u) = -∆ p w f (u) + λ(f (0) (u) -f (u)), ∀u ∈ V, (22)
where f (t) is the parametrization of the function f by an artificial time. This is a system of ordinary differential equations. Contrary to PDEs methods, no space discretization is necessary. Its solution can be efficiently approximated by local iterative methods, such as the Euler method, given by:

           f (0) = f 0 ∆ = v∼u w p/2 uv (|f (t) (u) -f (t) (v)| + ) p-2 (f (t) (v) -f (t) (u)) f (t+1) (u) = f (t) (u) + τ (∆ + λ(f 0 (u) -f (t) (u))) (23)
In order to satisfy the minimization problem, the stopping criterion, of the above diffusion process, is a convergence condition. Meanwhile a fixed total number of iterations corresponds to an anisotropic semi-discrete diffusion process. When p = 1, an interesting expression of an iteration of the process ( 23) is given by:

f (t+1) (u) = f (t) + τ v∼u √ w uv sign(f (t) (v) -f (t) (u)) + λ(f 0 (u) -f (t) (u)) .
This allows to avoid the use of the regularized anisotropic 1-Laplace operator ∆ p, w . When p = 2, the diffusion process is the classical linear isotropic geometric diffusion based on the combinatorial Laplace operator [START_REF] Chung | Spectral Graph Theory[END_REF]. In this case, an iteration of the process ( 23) is rewritten as:

f (t+1) (u) = f (t) + τ v∼u w uv (f (t) (v) -f (t) (u)) + λ(f 0 (u) -f (t) (u)) .
The behavior of diffusion process ( 23) is illustrated, for several values of p ≤ 2 and graph structures (topologies and weight functions), in the context of image filtering and simplification (see Section 5).

Discrete diffusion process.

To approximate the solution of the minimization problem [START_REF] Chan | Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[END_REF], another method is to linearize the system of equations [START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF], which can be rewritten as:

(21) ⇔ f (u) = λf 0 (u) + v∼u w p 2 uv(|f (u) -f (v)| + ) p-2 f (v) λ + v∼u w p 2 uv(|f (u) -f (v)| + ) p-2
, ∀u ∈ V.

Using the linearized Gauss-Jacobi method, the solution of this system is summarized by the following iterative algorithm:

             f (0) = f 0 γ uv (f (t) ) = w p 2 uv(|f (t) (u) -f (t) (v)| + ) p-2 f (t+1) = λf 0 (u) + v∼u γ uv (f (t) )f (t) (v) λ + v∼u γ uv (f (t) ) , ∀u ∈ V. (24)
Let ϕ be the function defined by:

ϕ uv (f ) = γ uv (f ) λ + v∼u γ uv ) , if u = v, or ϕ uu (f ) = λ λ + v∼u γ uv (f )
.

Then, the diffusion process ( 24) is rewritten as:

     f (0) = f 0 f (t+1) (u) = ϕ uu (f (t) )f 0 (u) + v∼u ϕ uv (f (t) )f (t) (v). ( 25
)
At each iteration of this adaptive process, the new value f (t+1) , at a vertex u, depends on two quantities: the original value f 0 (u), and a weighted average of the existing values in a neighborhood of u. Since ϕ uu (f ) + v∼u ϕ uv (f ) = 1, this diffusion process is a forced low-pass filter. Through the values of the regularization parameter p, it describes a family of iterative filters. In particular, when p = 2, γ uv (f ) reduces to w uv , and the above discrete diffusion process corresponds to the one proposed in [START_REF] Bougleux | Image smoothing and segmentation by graph regularization[END_REF][START_REF] Bougleux | Discrete regularization on weighted graphs for image and mesh filtering[END_REF][START_REF] Osher | Digitized PDE for data restoration[END_REF], in the context of isotropic regularization. The behavior of the diffusion process ( 24) is illustrated in Section 6, for several values of p and graph structures, in the context of semi-supervised image segmentation.

Nonlocal regularization of discrete data and related works

The regularization framework, proposed in the previous section, can be used to process any set of discrete data or function defined on these data. The data can be organized or not. Let V = {v 1 , . . . , v N } be a finite set of data, such that each data v i is a vector in a given metric space. To regularize V , or a function f 0 : V → R, the first step is to construct a graph G = (V, E, w). The edges of E can express different kind of interactions. A particular case consists in generating the fully connected graph, where E = V × V \ {uu, u ∈ V }. In this case, the anisotropic regularization functional R w (f, p) (Eq. ( 17)) takes into account nonlocal interactions:

R w (f, p) = 1 2p v∈V v∈V w p 2 uv |f (v) -f (u)| p . ( 26
)
One can remark that this last expression is always true, whatever the structure of the graph is. This due to the fact that, by definition, w uv = 0 if the vertex u is not connected to the vertex v by an edge of E. This shows that local and nonlocal regularizations have exactly the same expression when they are performed on graphs.

The nonlocal functional [START_REF] Perona | scale-space and edge detection using anisotropic diffusion[END_REF] is the discrete analogue of the continuous nonlocal functional of functions f : Ω ⊂ R m → R, defined in a bounded domain Ω of the Euclidean space:

J w (f, p) = 1 2p Ω×Ω w p 2 xy|f (y) -f (x)| p dydx.
When p = 1, Eq. ( 26) is the discrete analogue of the nonlocal anisotropic functional based on differences proposed by Gilboa and Osher [START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF]:

J w (f, 1) = 1 2 Ω×Ω √ w xy |f (y) -f (x)|dydx.
When p = 2 , Eq. ( 26) is the discrete analogue of the nonlocal isotropic functional proposed in [START_REF] Gilboa | Nonlocal linear image regularization and supervised segmentation[END_REF][START_REF] Kindermann | Deblurring and denoising of images by nonlocal functionals[END_REF]:

J w (f, 2) = 1 4 Ω×Ω w xy (f (y) -f (x)) 2 dydx.
This last functional is the variational interpretation of a family of neighborhood filters, widely used in image processing, such as the nonlocal means filter [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF].

One can remark that one iteration of the discrete diffusion process [START_REF] Mcdonald | Diffusions on graphs, poisson problems and spectral geometry[END_REF], for p = 2, λ = 0 and specific weight functions, corresponds to well-known filters. In particular, we can retrieve the bilateral filter [START_REF] Smith | SUSAN -a new approach to low level image processing[END_REF][START_REF] Tomasi | Bilateral filtering for gray and color images[END_REF] using the following weight function:

w uv = exp - u -v 2 σ 2 V exp   - f 0 (v) -f 0 (u) 2 σ 2 H(V )   , (27)
where σ V and σ H(V ) are the variances respectively in the discrete domain V and in the space of functions H(V ). Similarly, the expression of the nonlocal means filter [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF] is given using the weight function:

w uv = exp - ρ(F f 0 (v), F f 0 (u)) h 2 , ( 28
)
where F f 0 (u) is a feature vector associated to the vertex u ∈ V , which is generally a patch of vertices, centered at u. The function ρ measures the distance between the two feature vectors. With these two weight functions, the anisotropic regularization framework behaves like an iterated bilateral filter or an iterated nonlocal means filter, without updating the weights at each iteration of the diffusion processes.

When λ = 0, and for p = 1 and p = 2, our regularization functional corresponds to the functional used in spectral graph analysis, recently proposed in the context of semi-supervised image segmentation. When p = 1, the minimization problem is solved with graph cut techniques, and when p = 2 with random walks approaches [START_REF] Grady | A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm[END_REF][START_REF] Chung | Spectral Graph Theory[END_REF].

Image filtering and simplification

In this section, the behavior of the proposed anisotropic regularization framework is illustrated in the context of image filtering and simplification, for p ≤ 2 and several graph topologies. An image I : Ω ⊂ Z 2 → R is modelized by a neighborhood graph G = (V, E, w) and a function f 0 : V → R. To each pixel (i, j) of the discrete domain Ω corresponds a vertex u = (i, j) of V . The set E of edges is generated using the nearest neighbors of the vertices, based on the Chebyshev distance. Let u = (i, j) be a vertex of V , its neighborhood is defined by:

N k (u) = {v = (i , j ) ∈ V \ {u} : max{|i -i |, |j -j |} ≤ k, k > 0}.
Then the edge uv is in G k if v ∈ N k (u) (and reciprocally). G 1 is the 8-adjacency graph, and the fully connected graph is noted G ∞ .

To regularize the image I (the function f 0 ), we use the semi-discrete diffusion process. In our experiments, we found that using p ≤ 1, with a local or a nonlocal representation of the image, helps to preserve sharp edges during the regularization. This behavior is illustrated in Fig. 1 on an unweighted graph of 8-adjacency (G 1 ). The results are given for two different values of λ. One can observe the difference between the isotropic regularization with p = 2, and the anisotropic regularization with p ≤ 1. The anisotropic case behaves like a simplification procedure, in which the image discontinuities are more wellpreserved. This is also the case when the graph is weighted (see Fig. 2), or when it is generated from larger neighborhoods (see Fig. 3). In the nonlocal scheme, the main textures are also preserved. The local case, which can be computed efficiently, may be used in simplification and segmentation processes.

Semi-supervised classification

Recent approaches based on diffusion processes show the efficiency of label propagation to achieve manifold semi-supervised classification as in [START_REF] Belkin | Manifold regularization: A geometric framework for learning from labeled and unlabeled examples[END_REF][START_REF] Zhou | Regularization on discrete spaces[END_REF], which use the p-Laplacian diffusion regularization. More recently [START_REF] Grady | A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm[END_REF] uses approaches based on the L p -norm to solve image segmentation problem. In this section, we use our discrete anisotropic regularization framework to address this learning problem for image semi-supervised segmentation and data clus-p = 2, λ = 0.0001 p = 1, λ = 0.0001 p = 0.5, λ = 0.0001 Fig. 3. Regularization of an image, on a the graph (G 4 ), weighted by the function of Eq. ( 28), with a patch of size 5 × 5 as a feature vector.

tering.

Problem formulation

Let V = {v 1 , . . . v N } be a finite set of data, where each data v i is a vector of R m . Let G = (V, E, w) be a weighted graph such that data are connected by an edge of E. The semi-supervised classification of the set V consists in classify the set V into k classes where the number of k classes is given. For this, the set V is composed of labeled and unlabeled data. The objective is to estimate the unlabeled data from labeled ones.

Let c i be the set of vertices which belong to the i th class. The set C = {c i } i=1,...,k is the initial set of labeled data, and the initial unlabeled data belong to the set V \C. This is equivalent to consider k label functions f 0 i : V → R such as

f 0 i (v) =        +1 if v ∈ c i with i ∈ [1, k], ∀c ∈ C -1 otherwise 0 v ∈ V \ C (29)
where each f 0 i , with i = 1, . . . , k, corresponds to a given class. Starting from the labeled data (the f 0 i 's), the classification is accomplished by k regularization processes by estimating the resultant function f i : V → R.

Using our proposed anisotropic regularization framework, this is formalized as follows:

min f i ∈H(V ) R w (f i , p) + λ 2 f i (v) -g i (v) 2 2 , ∀i = 1, . . . , k.
We use the discrete diffusion process [START_REF] Mcdonald | Diffusions on graphs, poisson problems and spectral geometry[END_REF] to compute each minimization. At the end of the label propagation algorithm, one can estimate the class mem-bership probability, for all i ∈ 1, . . . , k, by:

p ω = c i |f (v) = f i (v) i f i (v) ,
and assign to the vertex v the most plausible one by

c(v) = arg max i p ω = c i |f (v) . (30)
The following section shows the application of this classification problem to image segmentation and data clustering.

Semi-supervised image segmentation and data classification

Image segmentation lies in searching relevant image regions or objects. Many successful automatic image segmentation approaches have been proposed in the literature. But, sometimes, their segmentation results are not accurate when image are more complex. Recent interactive image segmentation approaches have became increasingly popular in the image processing community. They reformulate image segmentation tasks into semi-supervised classification approaches. Thus, the segmentation is solved by label propagation strategies. We propose to use our anisotropic regularization framework to address the image semi-supervised task. Recently, similar approaches have also been proposed [START_REF] Grady | A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm[END_REF].

Image semi-supervised segmentation consists in finding objects in images by completing the user initial labels. Commonly, the user marks the desired objects to be segmented, or/and the image background. In the sequel, we directly consider the raw images without any pre-processing.

Grid graph based image semi-supervised segmentation. Fig. 4 shows the behavior of our anisotropic semi-supervised image segmentation based on grid graphs. These experiments show segmentation results for different grid graphs. Figs. 4(c), 4(d), and 4(e) is a local semi-supervised segmentation using a 8-adjacency graph (G 1 ). Figs. 4(f), 4(g), and 4(h) correspond to a nonlocal processing with the graph G 11 , and the weight function [START_REF] Ren | Learning a classification model for segmentation[END_REF], where

F f (v) = f (v).
Figs. 4(i), 4(j), and 4(k) is also a nonlocal processing with the graph G 11 , and the weight function [START_REF] Ren | Learning a classification model for segmentation[END_REF], where F f (v) is defined as a patch of size 5 × 5.

One can note that in the case of 8-adjacency graphs, we obtain a correct segmentation of the object. When we use a larger searching window, the object boundary are less smooth than in the local case, especially for the case where p < 1. This effect is corrected by using patches as observed in the last row of Fig. 4. Fast image semi-supervised segmentation with simplified images. As in Fig. 4, common image semi-supervised segmentation are usually based on a pixel label propagation scheme. Their application becomes difficult when the image is of large size. To avoid this computing restriction, we can consider that image pixels are not the only relevant elements. Then, more abstract structures can be used for the segmentation process, such as image regions or "superpixels" [START_REF] Ren | Learning a classification model for segmentation[END_REF]. We suggest to work, not directly with the image pixel graph representation but, with a reduced version of the image. This simplification can be thought as a graph simplification or a data reduction. To achieve this image pre-processing, any well known image pre-segmentation can be performed such as watershed techniques. In this work, this simplification is performed by an approach based on generalized Voronoï diagram (for more details see [START_REF] Arbeláez | Energy partitions and image segmentation[END_REF][START_REF] Bougleux | Image smoothing and segmentation by graph regularization[END_REF]). The benefits of using a simplified version of the image provides a fast segmentation scheme due to the reduced number of data to be analyze. Hence a minimal number of iterations is needed to obtain a correct segmentation. Nonlocal semi-supervised segmentation with simplified images. In order to illustrate the natural nonlocal expression of our anisotropic regu-lation framework, we perform the semi-supervised image segmentation on a fully connected graph constructed from a simplified images. Fig. 6 shows examples of this processing on color images. The graph is constructed from pre-segmented images (mosaic images of Figs. 6(b), 6(e), and 6(h)). Each vertex v corresponds to an influence zone which is described by its mean RGB color. This semi-supervised segmentation, provides a fast, simple, efficient and uncommon fully nonlocal image scheme as shown in Figs. 6(c), 6(f), and 6(i).

We can quote the following interesting properties:

(1) The fully connected graph contains all the image data information on the weighted edges. (2) Only few labels, as shown in Figs. 6(a), 6(d) and 6(g), are needed to obtain a correct segmentation result. For instance, in Fig. 6(g), the user only marks one nucleus and our fully nonlocal segmentation has found all the others (see Fig. 6(i)). (3) The fully connected graph extends the notion of proximity between two vertices. They can be similar even if they are not spatially close or adjacent. The diffusion process allows to quickly labeled the objects in the same class even if they are not spatially adjacent, as shown in Figs. 6(g) and 6(i). ( 4) The regularization process only needs a minimal number of iterations to compute a correct segmentation result. This approach provides a fast segmentation strategy.

Semi-supervised discrete data classification. To illustrate the flexibility of our anisotropic framework on non-organized data classification. Fig. 7 shows a classical example problem: the two moons. In this example the fully connected graph is computed in order to connect all the data points to each other. The initial data is composed of 99 unlabeled points in each class. Only one label per class is placed as show in Fig. 7(a). Fig. 7(b) shows the obtained result and the correctness of the classification.

Conclusion

In this paper, we have proposed a discrete anisotropic regularization framework on arbitrary graphs, to filter and classify discrete data. The regularization is based on the L p -norm of the graph gradient. It leads to semi-discrete and discrete anisotropic diffusion processes. When the data are naturally organized, such as images, we show that local and nonlocal regularization functionals have the same expression. We apply this framework to perform image and data filtering and clustering. The main ongoing work is to use the proposed anisotropic regularization functional in other discrete variational problems, such as one involves in binary segmentation.

f 0 :Fig. 1 .Fig. 2 .

 012 Fig. 1. Regularization of an image, on a 8-adjacency unweighted graph (G 1 ).

3 Fig. 4 .

 34 Fig. 4. Grid graph based image semi-supervised segmentation. All the images were whitened in order to accentuate the user labels and the segmented boundary. (a) original image (16 960 pixels). (b) original image with user labels.

Fig. 5 (

 5 Fig. 5(b) shows the image simplification of the image of Fig. 4(a). One can note that the approach respect the main image components, notably the edge information (see the energy image Fig. 5(a)). Meanwhile, the data are significantly reduced. Fig. 5 shows the semi-supervised segmentation accomplished with a simplified version of the image Fig. 4(a). Fig. 5(b) is the simplified mosaic image with a mean color value for each zone from the original one. Fig. 5(c) is the mosaic image with the initial labeled zones. Figs. 5(d), 5(e) and 5(f) are the obtained final segmentation with several values of p. In this experiment, the initial conditions are the same as in Fig. 4, for the parameter λ and the weight function. We use the Region Adjacency Graph as the graph representation of the mosaic image, where each vertex corresponds to an image zone.

2 Fig. 5 .

 25 Fig. 5. Semi-supervised image segmentation based on the region adjacency graph of the mosaic image. The images are whitened for accentuate the labels and the segmented boundary. (a) energy image (b) mosaic image (1 927 zones) from Fig. 4(a) (which contains 16 690 pixels) where each zone corresponds to the mean color model of the original image. (c) mosaic image with the user label (d), (e), and (f) segmentation results obtain with the corresponding p parameter.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Semi-supervised image segmentation. First column: original images with user initial labels. (a) and (d) Two classes problem (foreground and background). (g) Three classes problem (nuclei, cytoplasm, and background). Second column: influence zones images. Third column: segmentation results.

(Olivier Lézoray1 This work was partially supported under a research grant of the ANR Foundation (ANR-06-MDCA-008-01/FOGRIMMI) and under a grant of FEDER/Critic3-3273 funds.


Valérie Forest
email: vforest@emse.fr

Lara Leclerc

Jean-François Hochepied

Adeline Trouvé

Gwendoline Sarry

Jérémie Pourchez

Impact of cerium oxide nanoparticles shape on their in vitro cellular toxicity

Keywords: Cerium oxide nanoparticles, nanoceria, morphology, toxicity

Cerium oxides (CeO 2 ) nanoparticles, also referred to as nanoceria, are extensively used with a wide range of applications. However, their impact on human health and on the environment is not fully elucidated. The aim of this study was to investigate the influence of the CeO 2 nanoparticles morphology on their in vitro toxicity. CeO 2 nanoparticles of similar chemical composition and crystallinity were synthesized, only the shape varied (rods or octahedrons/cubes). Macrophages from the RAW264.7 cell line were exposed to these different samples and the toxicity was evaluated in terms of lactate dehydrogenase (LDH) release, Tumor Necrosis Factor alpha (TNF-production and reactive oxygen species (ROS) generation. Results showed no ROS production, whatever the nanoparticle shape. The LDH release and the TNF- production were significantly and dose-dependently enhanced by rodlike nanoparticles, whereas they did not vary with cubic/octahedral nanoparticles. In conclusion, a strong impact of CeO 2 nanoparticle morphology on their in vitro toxicity was clearly demonstrated, underscoring that nanoceria shape should be carefully taken in consideration, especially in a "safer by design" context.

Introduction

Cerium belongs to the lanthanide elements also known as rare-earth metals. Cerium oxides (CeO 2 ) nanoparticles, also referred to as nanoceria, have a wide range of industrial and commercial applications: mainly as fuel additives but also as UV protection (in paints or sunscreens), catalysts, polishing agents, gas sensors, etc. [START_REF] Courbiere | Ultrastructural Interactions and Genotoxicity Assay of Cerium Dioxide Nanoparticles on Mouse Oocytes[END_REF][START_REF] De Marzi | Cytotoxicity and Genotoxicity of Ceria Nanoparticles on Different Cell Lines in Vitro[END_REF][START_REF] Demokritou | An in vivo and in vitro toxicological characterisation of realistic nanoscale CeO₂ inhalation exposures[END_REF][START_REF] Fisichella | Toxicity evaluation of manufactured CeO2 nanoparticles before and after alteration: combined physicochemical and whole-genome expression analysis in Caco-2 cells[END_REF][START_REF] Lin | Toxicity of cerium oxide nanoparticles in human lung cancer cells[END_REF][START_REF] Lord | Cellular uptake and reactive oxygen species modulation of cerium oxide nanoparticles in human monocyte cell line U937[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Peng | Comparative pulmonary toxicity of two ceria nanoparticles with the same primary size[END_REF][START_REF] Pulido-Reyes | Untangling the biological effects of cerium oxide nanoparticles: the role of surface valence states[END_REF][START_REF] Xia | Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties[END_REF].

Recently, they also experienced growing attention for biomedical applications as they were found to exhibit protecting effects against cellular damage induced by toxicants, radiation or in pathological situations such as cardiac or brain ischemia/reperfusion, certain neurological disorders or retinal neurodegeneration [START_REF] Culcasi | EPR spin trapping evaluation of ROS production in human fibroblasts exposed to cerium oxide nanoparticles: evidence for NADPH oxidase and mitochondrial stimulation[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Pulido-Reyes | Untangling the biological effects of cerium oxide nanoparticles: the role of surface valence states[END_REF]. Their potential to behave as anticancer agent has also been explored [START_REF] Gao | Cerium oxide nanoparticles in cancer[END_REF][START_REF] Pulido-Reyes | Untangling the biological effects of cerium oxide nanoparticles: the role of surface valence states[END_REF].

Due to this large use, the risk of CeO 2 nanoparticles release in the environment and exposure to humans (especially through inhalation) are potentially growing while their impact on human health and on the ecosystems is still not fully elucidated. This observation has led the Organization for Economic Co-operation and Development (OECD) to classify since 2010 these nanoparticles among the top priority materials for toxicological evaluations [START_REF] Courbiere | Ultrastructural Interactions and Genotoxicity Assay of Cerium Dioxide Nanoparticles on Mouse Oocytes[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF]OECD Environment, Health and Safety Publications et al., 2010;[START_REF] Peng | Comparative pulmonary toxicity of two ceria nanoparticles with the same primary size[END_REF].

Indeed, the toxicity of nanoceria remains controversial as conflicting results have been reported in the literature. It is generally admitted that cerium oxides have a low toxicity profile [START_REF] Urner | Inflammatory response of lung macrophages and epithelial cells after exposure to redox active nanoparticles: effect of solubility and antioxidant treatment[END_REF] and it has been shown in different models that although internalized by cells CeO 2 nanoparticles do not trigger inflammation or cytotoxicity [START_REF] Fisichella | Toxicity evaluation of manufactured CeO2 nanoparticles before and after alteration: combined physicochemical and whole-genome expression analysis in Caco-2 cells[END_REF][START_REF] Franchi | Cyto-and genotoxic effects of metallic nanoparticles in untransformed human fibroblast[END_REF][START_REF] Xia | Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties[END_REF]. But evidence that cell death can be induced by CeO 2 nanoparticles was also given by Pešić et al. [START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF]. Similarly, regarding oxidative stress, some studies have reported that cerium oxide nanoparticles can be either pro-oxidative [START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF] or on the contrary can exhibit anti-oxidant properties [START_REF] Lord | Cellular uptake and reactive oxygen species modulation of cerium oxide nanoparticles in human monocyte cell line U937[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF][START_REF] Schubert | Cerium and yttrium oxide nanoparticles are neuroprotective[END_REF][START_REF] Xia | Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties[END_REF]. Nanoceria can exert a pro-oxidative effect by producing reactive oxygen species (ROS) responsible for cell damages that can themselves potentially lead to cell death, and also by inducing changes in the intracellular redox status [START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF]. Thus, some studies have shown that nanoceria can induce oxidative stress either in vitro or in vivo [START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF]. On the other hand, cerium oxide nanoparticles were found to behave as direct antioxidants by acting as free radical scavengers (especially by interacting with hydroxyl

radical [OH • ], superoxide radical [O 2 •-
] and hydrogen peroxide [H 2 O 2 ]) and therefore protecting cells from death due to oxidative stress [START_REF] Lord | Cellular uptake and reactive oxygen species modulation of cerium oxide nanoparticles in human monocyte cell line U937[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF][START_REF] Schubert | Cerium and yttrium oxide nanoparticles are neuroprotective[END_REF][START_REF] Xia | Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties[END_REF].

Two main reasons can be evoked to explain these discrepancies. First, no standardized assays exist, making the comparison impossible among the different studies from the literature and therefore making it really difficult to draw firm conclusion on nanoceria toxic potential.

Second and most importantly, it seems that numerous parameters are involved in CeO 2 nanoparticle toxicity: factors from the environment but also the intrinsic physico-chemical features of nanoparticles. Regarding the environmental context, pH seems to play a key role as it can drive either the anti-oxidant or pro-oxidant activity of nanoceria [START_REF] Gao | Cerium oxide nanoparticles in cancer[END_REF][START_REF] Lord | Cellular uptake and reactive oxygen species modulation of cerium oxide nanoparticles in human monocyte cell line U937[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF][START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF]. Indeed, in a neutral pH environment, CeO 2 nanoparticles are cytoprotective and act as antioxidants whereas at acidic pH they behave as oxidases leading to cytotoxic effects. This might be a reason of different influence of nanoceria on normal and cancer cells as these latter are characterized by an acidic pH environment. This leads to another environmental parameter that could greatly influence the response to nanoceria: the cell type in which biological assays are carried out. This was clearly demonstrated by different studies [START_REF] Lanone | Comparative toxicity of 24 manufactured nanoparticles in human alveolar epithelial and macrophage cell lines[END_REF][START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF][START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF] where the biological behavior of cells from various cell lines following incubation with CeO 2 nanoparticles were compared. All these studies concluded to a difference of sensitivity between cell types (both between normal cells and cancer cells but also among the different cell lines).

Nanoceria toxicity can also rely on intrinsic nanoparticle physico-chemical characteristics such as surface chemistry, size, shape, dispersion state [START_REF] Fisichella | Toxicity evaluation of manufactured CeO2 nanoparticles before and after alteration: combined physicochemical and whole-genome expression analysis in Caco-2 cells[END_REF], the synthesis process [START_REF] Lord | Cellular uptake and reactive oxygen species modulation of cerium oxide nanoparticles in human monocyte cell line U937[END_REF], etc, but more particularly the dual oxidation state of CeO 2 nanoparticles. It is well documented that cerium can exist as Ce 3+ or Ce 4+ , this particular configuration allows the unique reduction/oxidation behavior of nanoceria which is responsible for their antioxidant properties [START_REF] De Marzi | Cytotoxicity and Genotoxicity of Ceria Nanoparticles on Different Cell Lines in Vitro[END_REF][START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF][START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF][START_REF] Pulido-Reyes | Untangling the biological effects of cerium oxide nanoparticles: the role of surface valence states[END_REF][START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF].

Among the different nanoparticle physico-chemical features that could be involved in the nanoceria in vitro toxicity, we focused our attention on the influence of the morphology.

Therefore the aim of the present study was to investigate if a relationship existed between the cerium oxide nanoparticles shape and the biological response they induced in a model of macrophages in the perspective of designing safer nanoparticles. Several methods are eligible to produce nanoceria with various and controlled shapes, but the point is to change the shape without risking to change other physico-chemical characteristics that could have an impact on toxicity, for instance by using additives that could be difficult to wash thoroughly and could induce toxicity at trace levels, or by choosing very different methods for each shape. So the ideal case consists in selecting a protocol allowing shape tuning just by changing continuously some physico-chemical conditions as temperature, pH or reaction time, keeping the same chemical reactants. After a literature survey, we considered that protocols published by Florea et al. [START_REF] Florea | Three-Dimensional Tomographic Analyses of CeO2 Nanoparticles[END_REF] were the most relevant. Consequently, the obtained nanoparticles were of identical chemical composition and crystallinity, only the morphology varied (rods or octahedrons/cubes). Macrophages from the RAW264.7 cell line were exposed to these different samples and the in vitro toxicity was evaluated in terms of lactate dehydrogenase (LDH) release, Tumor Necrosis Factor alpha (TNF-production and reactive oxygen species generation.

Materials and methods

Cerium oxide nanoparticle synthesis

The protocols were selected from Florea's article and PhD thesis [START_REF] Florea | Three-Dimensional Tomographic Analyses of CeO2 Nanoparticles[END_REF].

Basically, after mixing at room temperature a solution of Ce(III) salt with the basic solution (soda/ammonia), the precipitate was heated by microwave irradiation at different temperatures and times. Shape tuning (octahedrons, cubes, rods) was expected depending on the conditions. All syntheses were performed with a 10 minutes heating ramp between room temperature and the desired temperature. The experimental conditions are summarized in Table 1. The given time corresponds to the plateau at the desired temperature. Cooling was relatively fast (a few minutes). In one set of conditions, we compared 2 microwave ovens: Monowave (M) and Synthos (S). After cooling, samples were washed 3 times by centrifugations and redispersed in water.

Physico-chemical characterization

Size and shape of particles were examined by transmission electron microscopy (TEM) with a TECNAI 20F microscope operating at 200kV. For cubic or octahedral particles, around 150 particles were measured manually using ImageJ software. Due to their non spherical shape, it was chosen to measure the biggest dimension. For rod-like particles, diameter and length distributions were measured counting around 150 particles. The mode (most represented class) was also determined in each case.

Nitrogen (N 2 ) adsorption/desorption isotherms were measured at 77 K using a Micromeritics ASAP 2010 Analyser. After degassing under vacuum at 80°C, the specific surface area of the powders was determined by applying the Brunauer Emmet Teller (BET) model method on the desorption branch. The equivalent spherical particle diameter was estimated with the following formula D=6000/( x S BET ), where D BET (nm) is the average diameter of the particles, S BET (m².g -1 ) the specific area and  (g.cm -3 ) the theoretical density of the powder.

For rods, neglecting the tip surface as compared to the lateral surface, we used the formula d= 4000/( x S BET ) with d diameter of the rods.  Nanoparticle/cell contacts -Cells were seeded in 96-well-plates (100 000 cells in 50 µL of medium per well) and were allowed to adhere over night. Nanoparticles were diluted in cell culture medium to reach the following final concentrations: 15, 30, 60 and 120 µg/mL.

Nanoparticles were added to cells and further incubated for 24h.

 Lactate dehydrogenase (LDH) release -To evaluate cell membrane integrity, the cellular release in the supernatant of cytoplasmic lactate dehydrogenase was assessed using the CytoTox-96™ Homogeneous Membrane Integrity Assay (Promega, Charbonnières-les-Bains, France) according to the manufacturer's instructions. The optical density of the samples was determined using a microplate reader (Multiskan RC; Thermolabsystems, Helsinki, Finland) set to 450 nm. Three independent experiments were performed, each in quadruplicate and the activity of the released LDH was reported to that of negative control cells (incubated without nanoparticles). A positive control consisted in the maximal cellular LDH released after cells lysis.

 TNF- production -After incubation with nanoparticles, the production of TNF-α was assessed in the supernatant using a commercial ELISA Kit (Quantikine® Mouse TNF-α Immunoassay; R&D Systems, Lille, France) according to the manufacturer's instructions. The optical density of each sample was determined using a microplate reader (Multiskan RC;

Thermolabsystems, Helsinki, Finland) set to 450 nm. A standard curve was established and results were expressed in picograms of TNF- per milliliter of supernatant. Three independent experiments were performed, each in quadruplicate and the production of TNF-α was reported to that of control cells (incubated without nanoparticles).

Results

Physico-chemical characterization

The physico-chemical features of the different nanoceria used in this study are reported in Table 2. TEM. This may be due to size and shape polydispersity, and the fact that rods could be more or less flat, rather than cylindrical. According to synthesis protocols, CeO 2 -R3 was not expected to be rod-like but cubic, in fact the same protocol performed with the Synthos apparatus instead of the monowave produced isotropic particles (CeO 2 -O2), but not purely cubic ones, rather a mixture of cubes, octahedrons and cuboctahedrons. This difference between CeO 2 -R3 and CeO 2 -O2 is probably due to the sensitiveness of the system to temperature and a better temperature probe for the Synthos, but this discussion is beyond the scope of the article. So at the end, we have 2 families, rods and isotropic particles that can be compared. Interestingly, CeO 2 -R2 and CeO 2 -O1/O2 exhibit the same specific surface area, but a very different shape, which is very useful to establish if effects are only due to shape.

We also observed that agglomeration seems rather weak for all samples. As far as zeta potential in water at pH 7 is concerned, values obtained for anisotropic particles are close to -30 mV but with a high standard deviation (especially for CeO 2 -R1 and CeO 2 -R2) whereas the zeta potential of isotropic particles is less negative for CeO 2 -O1 and slightly positive for CeO 2 -O2. As illustrated by Figure 3, the production of TNF- was significantly and dose-dependently enhanced by rod nanoparticles compared to control cells while the other shape of nanoceria did not induce a significant variation in the expression of this pro-inflammatory cytokine.

In vitro toxicity assays

Discussion

Nanoceria toxicity remains controversial and data are still missing to firmly conclude on this issue. Among the numerous parameters that could be involved in the nanoCeO 2 toxicity we have chosen to investigate more particularly the impact of the nanoparticle morphology. For that purpose we used a synthesis mode allowing producing different types of particles of wellcontrolled physicochemical features, varying only in terms of shape (rod versus cube/octahedron). The subsequent study of the biological activity induced in macrophages after incubation with the different nanoceria showed that the extracellular release of LDH (Figure 2) and the production of the pro-inflammatory cytokine TNF- (Figure 3) exhibited similar profiles. Compared to what was observed in control cells, they were significantly and dose-dependently enhanced by rod-like nanoparticles, whereas they did not vary with cubic/octahedral nanoparticles. These results clearly argue for a morphological effect of nanoCeO 2 . It is also noteworthy to mention that no size effect was observed among the rod nanoparticles as the LDH release and TNF- levels were similar between CeO 2 -R1, CeO 2 -R2 and CeO 2 -R3 which greatly vary in size (CeO 2 -R3 rods are almost two times longer than CeO 2 -R1 and CeO 2 -R2, Table 2). Also, we did not observe any correlation between the nanoceria zeta potential and their toxicity. Indeed, the LDH release and TNF- production triggered by CeO 2 -O1 and CeO 2 -O2 were similar whereas they exhibited a very different zeta potential (-17 and +9 mV respectively). This remark also applies to the rods even if the difference between their zeta potential was lower (Table 2). This strengthens our previous conclusion of the impact of the nanoceria shape (and not their charge) on their toxicity.

Finally, we also showed that nanoceria did not induce any ROS production, whatever their morphology (Figure 4).

These results are only partly consistent with the literature. For instance, regarding the production of ROS, as we did, Xia et al. [START_REF] Xia | Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties[END_REF] and Urner et al. [START_REF] Urner | Inflammatory response of lung macrophages and epithelial cells after exposure to redox active nanoparticles: effect of solubility and antioxidant treatment[END_REF] reported that nanoceria did not trigger the production of ROS. On the contrary, Mittal et al. [START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF] and Lin et al. [START_REF] Lin | Toxicity of cerium oxide nanoparticles in human lung cancer cells[END_REF] found that nanoCeO 2 were able to induce a significant oxidative stress. Another discrepancy with these same studies regards the cytotoxicty induced. Indeed, Mittal et al. and Lin et al. (Lin et al., 2006;[START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF], observed that nanoceria reduced cell viability, whereas Xia et al. considered it as non toxic. We were in a more nuanced position as we reported that toxicity depends on the morphology as it was significantly enhanced by rod-like nanoparticles but not by cube or octahedron-like nanoparticles. But even more interesting is the comparison with results from Wang et al. paper (Wang et al., 2015) which study design was quite similar to ours. Although they used another cell type (human hepatocellular carcinoma cells, HepG2), they compared the toxicity of three kinds of nanoCeO 2 with different morphologies (cube-, octahedron-, and rod-like), using similar ranges of nanoceria concentrations as we did. However, unlike us, they showed that all three types of nanoparticles caused cytotoxicity. They further reported that the toxicity of cube-like nanoCeO 2 was the highest, that of the rod-like was the lowest and that of the octahedron-like was intermediate. They linked this toxicity level to the specific surface area as they observed that a smaller surface area was correlated with a higher cytotoxicity.

The comparison of these different studies illustrates how difficult it is to reach firm conclusions on the nanoceria toxicity. This could be partly explained by the fact that the biological response often depends on the cellular context, especially the pH [START_REF] Franchi | Cyto-and genotoxic effects of metallic nanoparticles in untransformed human fibroblast[END_REF] and the cell type. The cell type-specific response to particle exposure has already been well established in different models. For instance, Lanone et al. [START_REF] Lanone | Comparative toxicity of 24 manufactured nanoparticles in human alveolar epithelial and macrophage cell lines[END_REF] evaluated the toxic effect of 24 nanoparticles, including nanoceria, of similar equivalent spherical diameter and various elemental compositions on two human pulmonary cell lines:

A549 and THP-1. Results clearly highlighted the difference of sensitivity between cell types.

Similarly, Pešić et al. [START_REF] Pešić | Anti-cancer effects of cerium oxide nanoparticles and its intracellular redox activity[END_REF] compared the ROS production induced by nanoCeO 2 in human normal and cancer cell lines. Different responses were observed underlining the selectivity towards human cancer cells. In another model, it was also demonstrated that remarkably more ROS were formed in alveolar macrophages compared to alveolar epithelial cells [START_REF] Urner | Inflammatory response of lung macrophages and epithelial cells after exposure to redox active nanoparticles: effect of solubility and antioxidant treatment[END_REF]. Regarding the protective effects of nanoCeO 2 , it was demonstrated that they could protect cells from harmful effects of radiation and oxidative stress, although this protection was cell type specific [START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF]. Also, nanoCeO 2 showed no toxic effect on normal breast epithelial cells and only a slight effect on breast cancer cells, but were able to protect the normal epithelial cells from radiation while in contrast no such protection was observed for the breast cancer cells [START_REF] Rosenkranz | Effects of cerium oxide nanoparticles to fish and mammalian cell lines: An assessment of cytotoxicity and methodology[END_REF].

Another reason explaining why it is so challenging to conclude on nanoceria toxicity is that the underlying mechanisms are particularly complex to investigate. This is especially true for ROS that are unstable and highly reactive species. In this case, the kinetics aspect is of paramount importance and has to be taken into account to determine the optimal detection window. Thus, it has been reported that nanoCeO 2 were able to produce ROS in a concentration and time dependent manner up to 6 hours [START_REF] Mittal | Cerium oxide nanoparticles induced toxicity in human lung cells: role of ROS mediated DNA damage and apoptosis[END_REF]. But level of ROS was found to be decreased at 24 hours. This may be either due to the increase in amount of cell death or generation of ROS stabilized after a certain time period. Furthermore, a production of ROS can be compensated or masked either by the scavenger activity of nanoCeO 2 or by the cell antioxidant defense system. This is why a complete and thorough study should be undertaken to better understand the oxidative stress mechanism induced by nanoCeO 2 . Especially the variation of the antioxidant enzymes level such as the superoxide dismutase (SOD), glutathione (GSH) and catalase should be studied.

It should also be mentioned that care must be taken when concluding on the nanoceria toxicity only on the basis of in vitro experiments. Indeed, although many in vitro studies have concluded that nanoCeO 2 can be regarded as relatively safe nanoparticles, in vivo models allowed underscoring cytotoxicity and inflammation, as well as potential fibrogenicity [START_REF] Demokritou | An in vivo and in vitro toxicological characterisation of realistic nanoscale CeO₂ inhalation exposures[END_REF]. For example, it was reported that after oral or intraperitoneal administration of nanoCeO 2 to CD-1 mouse, even if no lethal effect occurred, nanoCeO 2 could induce inflammation status, revealed by cellular, toxicological, and ultrastructural observations [START_REF] Poma | In vivo inflammatory effects of ceria nanoparticles on CD-1 mouse: evaluation by hematological, histological, and TEM analysis[END_REF]. Similarly, intratracheal instillation of CeO 2 nanoparticles in Sprague-Dawley rats was found to result in liver damage [START_REF] Nalabotu | Intratracheal instillation of cerium oxide nanoparticles induces hepatic toxicity in male Sprague-Dawley rats[END_REF]. Finally, Pulido-Reyes et al. [START_REF] Pulido-Reyes | Untangling the biological effects of cerium oxide nanoparticles: the role of surface valence states[END_REF] studied the effect of five different nanoCeO2 on an aquatic microorganism. They concluded that neither shape, concentration, synthesis method, surface charge, nor nominal size had any influence in the observed biological activity. Discrepancies between in vitro and in vivo toxicity evaluations may be attributed to particle transformations in the microenvironments of cell culture media enriched with serum proteins versus the alveolar lining fluid of the animal lung [START_REF] Demokritou | An in vivo and in vitro toxicological characterisation of realistic nanoscale CeO₂ inhalation exposures[END_REF].

Nanoparticles suspended in physiological media may experience agglomeration and the formation of a protein corona on the particle surface, and these transformations may have a strong impact on subsequent cell/nanoparticle interactions [START_REF] Forest | Electrostatic interactions favor the binding of positive nanoparticles on cells: A reductive theory[END_REF]. Therefore widely used in vitro toxicity assays, such as LDH and MTT assays, may not be predictive of toxicological outcomes observed in vivo. Consequently, further investigations are required.

Conclusion

In the present paper, we clearly demonstrated an impact of cerium oxide nanoparticle morphology on their in vitro toxicity. Indeed, unlike cubic/octahedral nanoparticles, rod-like nanoparticles significantly and dose-dependently enhanced pro-inflammatory and cytotoxicity responses. Although other complementary studies are needed, this strongly argues for the fact that the shape of the nanoceria is an aspect that should not be neglected, especially in a "safer by design" context.



  Reactive oxygen species (ROS) production -A large array of ROS activity can be assessed with the OxiSelect™ ROS Assay Kit (Euromedex, Mundolsheim, France). The assay uses the conversion of a non-fluorescent substrate, 2.7′-dichlorodihydrofluorescein diacetate that can easily diffuse through cell membranes and be converted into a fluorogenic molecule 2′.7′dichlorodihydrofluorescein (DCF) in presence of ROS: fluorescence amount is directly related to ROS level. Fluorescence was detected using a Fluoroskan Ascent fluorometer (Ex: 480 nm, Em: 530 nm, Thermolabsystems) after a 90 min incubation of cells with the nanoparticles. A positive control was included incubating cells with H 2 O 2 (1 mM). Three independent experiments were performed, each in quadruplicate and the generation of ROS was reported to that of the negative control (cells incubated without nanoparticles).  Statistical analysis -Analyses and graphics were performed on Prism 6.0c software (GraphPad, San Diego, CA). Significance was established with two-ways ANOVA tests (p < 0.05) and Tukey post-tests (***: P<0.001 and **: P<0.01). Each data point represents the mean of at least three independent experiments and is presented with the arithmetic standard error of the means (± SEM).

Figure 1 -

 1 Figure 1 -Transmission electron microscope images of the different nanoceria particles (R =

Figure 2

 2 Figure 2 reports the loss of cell membrane integrity as evaluated by the LDH released from

Figure 2 -

 2 Figure 2 -Loss of cell membrane integrity assessed by LDH released in the cell culture

Figure 3 -

 3 Figure 3 -TNF- production measured after a 24h incubation with different nanoceria types.

Figure 4 -

 4 Figure 4 -ROS production assessed 90 min after incubation with the different nanoceria.

Table 1 -

 1 Experimental conditions for the cerium oxide nanoparticle synthesis.C: concentration, V: volume, T: temperature, t: time, M: Monowave, S: Synthos.

	Sample	C(Ce)	V(Ce)	C(NaOH)	V(NaOH)	V(NH3	T(°C) t(min) Microwave
		mol/L	mL	mol/L	mL	28%)
						mL
	CeO 2 -R1 3.8x10 -3	50	12	20	5.4	120	25	M
	CeO 2 -R2 3.8x10 -3	50	12	20	5.4	180	45	M
	CeO 2 -R3	0.1	19	12	20	5.4	180	45	M
	CeO 2 -O1	0.1	24	/	0	1.4	180	60	M
	CeO 2 -O2	0.1	19	12	20	5.4	180	45	S



  Cell culture -The RAW 264.7 cell line derived from mice peritoneal macrophages transformed by the Abelson murine leukemia virus and was provided by ATCC Cell Biology

	Collection (Promochem, LGC, Molsheim, France). Cells were cultured in Dulbecco Modified
	Eagle Medium (DMEM) complemented with 10% of fetal calf serum and 1% of penicillin-
	streptomycin (called DMEMc) at 37°C under a 5% carbon dioxide humidified atmosphere.

Table 2 -

 2 Physico-chemical features of the different nanoceria.

	Nanoparticle Composition	Particle size (nm) TEM	Shape	Mode (nm) TEM BET	Specific surface area (m²/g)	Agglomeration/ aggregation state	Zeta potential (mV) pH7
	CeO 2 -R1	CeO 2	5.5(±1.3) x 44(±19)	Rod	4.5-5.5	3.5	151	Low	-26 (± 11)
	CeO 2 -R2	CeO 2	6.2(±1.1) x 45(±14)	Rod	6.2-7.3	8.5	61	Low	-35 (± 14)
	CeO 2 -R3	CeO 2	8.1(±1.8) x 70(±34)	Rod	6.8-8.3	14	37	Low	-31 (± 7)
	CeO 2 -O1	CeO 2	11.3 (±2.8)	Octahedron	10.0-12.0	13	60	Low	-17 (± 5)
	CeO 2 -O2	CeO 2	16.2 (±3.6)	Cube/octahedron 12.6-15.0	13	61	Low	+9 (± 6)
		As illustrated by Figure 1, the different morphologies of the nanoceria could be clearly
		distinguished using transmission electron microscopy. Samples CeO 2 -R1, CeO 2 -R2 and
		CeO 2 -R3 are nanorods with increasing dimensions. CeO 2 -R1 are thin and short rods, with
		diameters close to 5 nm and lengths close to 40 nm. CeO 2 -R2 samples have more or less the
		same length as CeO 2 -R1, but are slightly thicker rods with diameters around 6 nm, and finally
		CeO 2 -R3 are the biggest rods with diameters close to 8 nm and much longer and polydisperse,
		around 70 nm. BET equivalent diameters exhibit a much sharper variation than expected by

Acknowledgements

The authors would like to acknowledge the funding from Institut Mines Telecom, ROQSTAR grant.


Valérie Forest
email: vforest@emse.fr

Jérémie Pourchez

The nanoparticle protein corona: the myth of average

Keywords: Protein corona, nanoparticles, average analysis

It makes sense that if something is designed based on average measures, it would be appropriate for all, or most of all. But reality is quite the opposite. For example, a seat designed based on the average size of a population will not be suitable for the tallest and the shortest persons. Indeed, by definition an average does not represent extreme values that can be significantly different. The same applies for life sciences as living organisms are made of many cell populations and biological processes are usually investigated globally. Based on a very simplified example, this paper discusses this concept for the particular case of the characterization of the nanoparticle protein corona. Firm conclusions cannot be drawn from the global analysis of samples containing a large number of nanoparticles. It results that in future, different approaches will be required to identify corona subclasses rather than averaging over them. A way to face this challenging task may be through the combined use of different analytical techniques, each shedding light on a different aspect of this complex layer to gain true insight about its nature and dynamics.

Introduction:

The Myth of Average is a paradigm which is prominent in most sciences. It is the belief that we can use statistical averages to understand particular cases. One of the best historical example was perfectly described by Todd Rose concerning the US Air Force [START_REF]The Myth of Average: Todd Rose at TEDxSonomaCounty[END_REF]. In 1952, the Air Force was perplexed because they had good pilots flying better planes that benefitted from the most advanced technologies but their performances were worsening. And they didn't know why. It turned out that the problem was actually with the cockpit ergonomic.

Indeed, these latter were designed based on 10 average dimensions of the pilots (height, shoulders, chest, waist, hips, legs, reach, torso, neck, and thigh). For a long time, it was assumed that if you design something for the average size person, it would fit for most people. But it was wrong and it was demonstrated 60 years ago by Gilbert Daniels, an Air Force researcher who studied over 4000 pilots. He measured them on the 10 abovementioned dimensions of size and he asked how many of these pilots were average on these 10 dimensions. The answer was zero, meaning that the "average pilot" does not exist and that the "average cockpit" that should theoretically be suitable for most people actually fitted nobody.

Biological examples:

The same argument can be extrapolated to many fields, especially biology where organisms are made of many different cell populations. Usually the analysis of a biological process is global and conclusions are drawn from average results. As an example [START_REF] Levsky | [END_REF], gene expression is analyzed in terms of a cell population as information comes from samples containing millions of cells. In other words, the probable state of an average cell in the population is estimated from a complex mixture of cells. But in reality, this supposed "average cell" is a lure and the analysis does not take into account the variation that can be highly significant, among the members of the population. However, recent technological advances have allowed the precise measurement of single-cell transcriptional states to study this variability more rigorously. And it appeared that the way genes are expressed in a population is strikingly different from what was assumed from extrapolating from an average cell.

Another example of increasing importance can be found in the nanotechnology field. It refers to the nanoparticle corona. This dynamic layer of proteins adsorbed at the nanoparticle surface upon contact with biological media alters the initial physico-chemical features and defines the "biological identity" of the nanoparticle. This new interface even impacts the nanoparticle/cell interactions and consequently the response of the biological systems [3,4].

It is therefore of paramount importance to consider the role of this layer in biological assays.

But the protein corona constantly evolves, over time (following a Vroman's effect where more abundant proteins in a medium first adsorb but are progressively replaced by proteins of higher affinity) and also evolves as the nanoparticle travels through different compartments of different composition [5,6]. Consequently, due to this complex and dynamic nature, it is quite challenging to reliably characterize a protein corona. However, many attempts from very simple approaches to much more sophisticated methods have been undertaken, each providing different types of information. For example, nanoparticle size and/or charge changes consecutive to the formation of a protein corona can be detected using dynamic light scattering or transmission electron microscopy. Nanoparticle-protein interactions can be characterized through a wide range of analytical techniques such as isothermal titration calorimetry, fluorescence and ultraviolet-visible spectroscopies, quartz crystal microbalance, surface plasmon resonance, atomic force microscopy, fluorescence correlation spectroscopy, size-exclusion chromatography, circular dichroism spectroscopy, infrared and Raman spectroscopies… [3][4][5][6].

So far, the most commonly used strategy for the nanoparticle-protein corona characterization is a multi-step process consisting in: 1) the incubation of nanoparticles with a biological fluid (often serum, i.e. a mixture of proteins), 2) a centrifugation to remove unbound proteins, 3) the elution of the proteins adsorbed at the nanoparticle surface using denaturing agents, 4) the separation of the isolated proteins by a one or two dimensional gel electrophoresis and finally 5) the protein identification using mass spectrometry [4,7].

Using this proteomic approach, interesting observations were made. For instance, it was demonstrated that out of the 3700 proteins present in human blood plasma, only a few tens could be found in the protein corona. Most interestingly, the most abundant proteins in plasma were not necessarily the most present in the protein corona clearly indicating that the composition of this latter was not the reflect of the surrounding medium [5,[7][8][9]. It was also observed that the nanoparticle physico-chemical features played a key role in the selective binding of proteins as well as the type of serum [6]. The nature of the proteins bound to nanoparticles is crucial as it directly impacts the nanoparticle fate. Indeed, adsorption of opsonins (IgG and complement factors) promotes phagocytosis with removal of nanoparticles from the bloodstream, whereas adsorption of other kinds of molecules can on the contrary favor cellular uptake. For example, adsorbed apolipoproteins might foster transport across the blood-brain-barrier, likely through the interaction with low-density lipoprotein receptors [10]. As the protein corona is the interface interacting with cellular systems rather than the nanoparticle bulk material some authors have proposed a new classification of the nanoparticles based on their protein corona nature [3,5,11]. Others have suggested that as this corona formation is not avoidable, it should be found a way to take advantage of it, especially by a rational design of nanomaterials to optimize biotechnological applications and reduce potential nanohazards. In this regard, quantitative proteomic data are needed as a resource for model building, to predict and simulate the kinetics of proteinbinding to nanosized objects in physiological systems [11].

Huge progress has been made toward the characterization of the nanoparticle protein corona and protein identification has benefited from proteomic approaches and increased sensitivity of apparatus. However, if some trends can be trustworthy it remains difficult to draw absolute conclusions. One of the main reasons why we should be careful in the interpretation of results from such analytical processes is that analyses are global, therefore they are based on average measures. Let's take a (simplified) example to illustrate our point. As shown in Figure 1, let's imagine that the qualitative and quantitative analysis of a sample of 10 nanoparticles results in a protein corona composed of 30% protein A, 30% protein B, 30% protein C and 10% protein D. From this "average point of view", one could imagine different scenarios. It clearly appears that the biological consequences of these different protein corona configurations would not be the same. And it becomes obvious that firm conclusions cannot be drawn from average analyses, pointing out that in future, different approaches will be required to identify such corona subclasses, rather than averaging over them [5].

In addition, because the corona is generally not at thermodynamic equilibrium, there could be statistical fluctuations in its composition and organization from particle to particle within the same sample [5]. Therefore, protein corona is currently studied through "snapshots", its composition being defined at a specific time point. Instead, analyses should tend to real-time observations to follow the protein corona evolution and to be more accurate in the interpretation of results.

Conclusion and perspectives/future directions:

As protein corona directly interacts with biological systems, the isolation and identification of proteins bound to nanoparticle represents a fundamental prerequisite for nanobiology, nanomedicine and nanotoxicology. However it is a quite challenging process. Furthermore, the qualitative and quantitative characterization of the protein corona should be integrated in a broader context as the more abundantly associated proteins will not necessarily trigger the most profound biological response. As a corollary, a less abundant protein with high affinity may instead be a key player [8].

In recent years, many analytical developments have allowed to gain sensitivity and improve the protein corona characterization in vitro. However these techniques still have limitations or pitfalls. Experimental conditions are crucial and their lack of homogeneity makes difficult the comparison between studies from the literature. Furthermore, as the composition of the protein corona on a given nanoparticle, at a given time, depends on the concentrations and kinetic properties of the proteins found in plasma, it is important to not only determine which proteins are adsorbed onto the surface of the nanoparticle, but also understand the binding affinities and stoichiometries [7]. Despite these limitations, interesting results on the protein corona have been obtained in vitro. A new challenge is now to follow it in vivo [5,9] and possibly by performing real-time measurements to monitor the evolution in protein binding to nanoparticles.

Finally, to address the problem of average, nanoparticles should ideally be characterized individually. A parallel can be made with flow cytometry for cell analysis; indeed, this technique allows the individual multi-parametric analysis of cells. In this context the use of nanoparticle tracking analysis (NTA) could represent a useful technique [9]. However, the individual analysis of nanoparticle protein corona may be a utopia due to the complexity of the task. An alternative solution may lie in the development of new methods or in the combined use of different available analytical techniques to analyze simultaneously several aspects of nanoparticle-protein interactions [4,5,12].

Figure legends:

Figure 1 -

 1 Figure 1 -Possible scenarios after the proteomic analysis of the protein corona of a sample of 10 nanoparticles.

Figure 1

 1 Figure 1

Acknowledgements:

The authors would like to acknowledge the French Ministry of the Economy, Finance and Industry, the Région Rhône-Alpes and the Conseil Général de la Loire for the financial support.


From local to global, from past to future, from nature to culture: ▪ For a better consideration and nesting of the different forms of ecological knowledge : take into account

Eco-anthropology, Ethnobiology, Interdisciplinarity, Fisheries, Coastal minorities, Coastal communities, LTK, MTK, western science, Nature/Culture, Finno-Ugric, Baltic Sea.

A multi-sites fieldwork 7 sites in 6 countries of Eastern Baltic Sea ▪ Seven sites qualified as « peripheral maritime regions », with important cross-border issues and the presence of strong fishing and sea hunting communities, all affected by the end of Socialism and decollectivization of fishing kolkhozes. ▪ The existence of five indigenous minorities, Slavic and Finno-Ugric, distributed among the six countries, coupled with the presence of island and coastal communities. ▪ Particular landscape dictated by islands, isthmus and peninsula, estuaries and deltas, but the same Baltic lake croup ecosystem on coastal area (northern forest and dune system). ▪ Different salinity and temperature conditions ; variable sea ice formations during winter time. ▪ Sea hunting and fishing: marine mammals (seals), migratory birds (ducks, geese), fish (herring, sprats, flets, salmon, eels, lamprey,...) . Small-scale coastal and estuarine fishing.

First results

Rich field materials:

▪ The eco-anthropological approach allows for the acquisition of data that is usually not very well collected, but nevertheless very rich. Their analysis requires the implementation of original collection and treatment protocols adapted to coastal and community contexts. ▪ Gigantic local environmental knowledge on fauna, flora, habitats, weather and climate, almost not mobilized in decisions concerning communities.

Changes not about to stop:

▪ Major emerging projects: new pipeline (NordStream 2), new large ports, creation of new MPA without concertation, building of new RME off-shore parks, new rules for fishing and hunting, fleet exit plans, …, leading to a "sea enclosure" and reduced access to common environmental goods. ▪ Continued climatic disturbances, in particular the reduction, or even absence, of winter pack ice (specifically in semi-enclosed marine areas like lagoons and bays).

Strong community willingness to participate:

▪ The communities surveyed do not understand the top-down decisions, which penalize their traditional uses of the environment (like seal hunting ban), still for many food producers. ▪ However, they are open to the explanation and transmission of their knowledge, especially within the decision-making bodies.

Questions & hypothesis

What kind of approach is among the most appropriate to understand changes?

▪ Is an interdisciplinary approach efficient ? ▪ How to take into account traditional local knowledge in local conception (LTK: Ingold, 2012) and western science? ▪ What association is possible for indigenous communities with this research project ? ▪ What kind of ontological benchmarks structures the Baltic coastal communities cosmogonies ?

Work hypothesis:

An eco-anthropological approach based on an immersive field survey with coastal communities of the Eastern Baltic allows the study of their LTK and their cosmology. The study is coupled with an analysis of the data concerning the evolution of the Baltic social ecological system (SES). This interdisciplinary approach leads to a better understanding and apprehension of the changes affecting these coastal indigenous communities, in a perspective of sustainability.

For an eco-anthropological approach to changes affecting fishing communities in the eastern Baltic An ethnographic survey in the communities ▪ Data acquisition through questionnaires, semi-directive interviews and life stories on hunters and fishermen's environmental practices and empirical knowledge, work on archives and museums collections, with ecological scientists.

A social sciences analysis ▪ To apprehend SES and its changes as a whole, work with the conceptual framework of social sciences: socio-anthropology, political science, supplemented by natural sciences analysis.

Eco-anthropology

  ., 2007, Prospects and perspectives of fisheries in the coastal zone of the southern Baltic. -Descola P., 2005, Par-delà nature et culture, Paris, Gallimard, 640 p. -Geistdoerfer A., 2007, "L'anthropologie maritime : un domaine en évolution, hors cadre de l'anthropologie sociale", in Zainak, n°29, pp. 23-38. -Gunda B., (Ed.), 1984, The fishing culture of the world : studies in ethnology, cultural ecology and folklore, Akademia Klado, 2 vol., Budapest. -Ingold T., 2012, "Culture, nature et environnement", in Tracés, Revue de sciences humaines, vol. 22, pp. 169-187. -Mahieddin, E., 2018, "Penser l'anthropologie de la Baltique", in L'Homme, vol. 2, pp. 67-102. -Ojaveer H., MacKenzie B.-R., 2007, "Historical development of fisheries in northern Europe -reconstructing chronology of interactions between nature and man", in Fisheries research, vol. 87, n°2-3, pp. 102-105.

Fishing

  on ice, January 2016, Nida, Curonian lagoon, Lituania. © Danto A. End of fishing, May 2018, Kihnu island, Estonia. © Danto A. Take fish from nets, March 2015, Międzyzdroje, Poland. © Danto A. Map of fieldworks. © Danto A.

  UMR 7372 CEBC : Chizé Centre for Biological Studies (CNRS-ULR) / International research network ApoliMer : political anthropology of the Sea UMR 6051 ARENES : Research center on policy action in Europe (CNRS-UR1-EHESP-IEPR) / Institute of political studies of Rennes

	Anatole DANTO	CNRS PhD candidate	anatole.danto@orange.fr
		An eco -anthropological
		approach of the fieldwork
		Study of the Man/Nature relationship
		▪ Use of human ecology, history, geography and ethnobiology
		tools before, during and after the fieldworks times, with
		participant observation to discover cosmogonies.