S Rivollier
email: rivollier@emse.fr

J Debayle
email: debayle@emse.fr

J.-C Pinoli
email: pinoli@emse.fr

SHAPE DIAGRAMS FOR 2D COMPACT SETS -PART II: ANALYTIC SIMPLY CONNECTED SETS

Keywords: Analytic simply connected compact sets, Geometrical and morphometrical functionals, Shape diagrams, Shape discrimination

. It is focused on convexity discrimination for analytic and discretized simply connected compact sets.

Introduction

The Santalo's shape diagrams [START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF] allow to represent a 2D compact convex set by a point in the Euclidean 2D plane from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF]. The axes of each shape diagram are defined from a pair of geometric inequalities relating these functionals. Sometimes, the two geometric inequalities provide a complete system: for any range of numerical values satisfying them, there exists a compact convex set with these values for the geometrical functionals (in other words, a point within the 2D Santalo shape diagram describes a 2D compact convex set). This is not valid for all the Santalo shape diagrams.

This paper deals with the study of shape diagrams for a wide range of 2D nonempty analytic simply connected compact sets. The first part [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] of this study focused on the compact convex sets. This second part generalizes to the simply connected compact sets. The considered simply connected compact sets are mapped onto points in these shape diagrams, and through dispersion and overlapping quantifications, the shape diagrams are classified according to their ability to discriminate the simply connected compact sets.

Shape functionals

In this paper, the non-empty analytic simply connected compact sets in the Euclidean 2-space E 2 are considered. A set will be called analytic if its boundary is piecewise defined by explicit functions in such a way that the geometrical functionals enumerated below can be calculated. These geometrical functionals are determinated in order to characterize the sets. They are related by the so-called geometric inequalities, which allow to define morphometrical functionals.

Geometrical functionals

For a simply connected compact set in E 2 , let A, P, r, R, ω, d, denote its area, its perimeter, the radii of its inscribed and circumscribed circles, its minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF], respectively. Figure 2.1 illustrates some of these geometrical functionals. For a simply connected compact set, these six geometrical functionals are greater than zero. The line curves provide null values for A, r and ω, and the points for P, R and d. The sets with an infinite perimeter (the fractal sets) are not considered. P must be computed by a line integral.

Geometric inequalities

For a simply connected compact set in E 2 , the relationships between these geometrical functionals are constrained by the geometric inequalities [START_REF] Bonnesen | Theorie der konvexen körper[END_REF][START_REF] Hillock | Inequalities for lattice constrained planar convex sets[END_REF][START_REF] Osserman | The isoperimetric inequality[END_REF][START_REF] Osserman | Bonnesen style isoperimetric inequalities[END_REF][START_REF] Scott | A family of inequalities for convex sets[END_REF][START_REF] Scott | Inequalities for convex sets[END_REF][START_REF] Siegel | An isoperimetric theorem in plane geometry[END_REF][START_REF] Yaglom | Convex figures[END_REF] referenced in the second column of Table 2.1. Some geometric inequalities are restricted to convex sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], that are not considered in this paper. These inequalities link geometrical functionals by pairs and determine the so-called extremal simply connected compact sets that satisfy the corresponding equalities (Table 2.1, fourth column). Futhermore, they allow to determinate morphometrical functionals.

Morphometrical functionals

The morphometrical functionals are invariant under similitude transformations (consequently, they do not depend on the global size of the simply connected compact set) and are defined as ratios between geometrical functionals. In these ratios, the units of the numerator and the denominator are dimensionally homogeneous and the result has therefore no unit. Moreover, a normalization by a constant value (scalar multiplication) allows to have a ratio that ranges in [0, 1]. For each morphometrical functional, the scalar value depends directly on the associated geometric inequality. These morphometrical functionals are referenced in the third column of Table 2.1. These morphometrical functionals are classified according to their concrete meanings: roundness, circularity, diameter constancy and thinness [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

Table 2.1 synthesizes the geometrical and morphometrical functionals, the geometric inequalities and the extremal 2D analytic simply connected compact sets.

Geometrical Geometric Morphometrical Extremal functionals inequalities functionals sets r, R r ≤ R r / R C ω, R ω ≤ 2 R ω /2 R C A, R A ≤ π R 2 A /π R 2 C d, R d ≤ 2 R d /2 R Y r, d 2 r ≤ d 2 r / d C ω, d ω ≤ d ω / d W A, d 4 A ≤ π d 2 4 A /π d 2 C R, d √ 3 R ≤ d √ 3 R / d Z

Shape diagrams

From these morphometrical functionals, 2D shape diagrams can be defined. They enable to represent the morphology of any analytic simply connected compact sets in the Euclidean 2D plane from two morphometrical functionals (that is to say from three geometrical functionals because the two denominators use the same geometrical functionals).

Definition

Let be any triplet of the considered six geometrical functionals (A, P, r, R, ω, d) and (M 1 , M 2 ) be some particular morphometrical functionals valued in [0, 1] 2 (Table 3.1). A shape diagram D is represented in the plane domain [0, 1] 2 (whose axis coordinates are the morphometrical functionals M 1 and M 2 ) where any 2D compact set S is mapped onto a point (x, y). Note that if M 1 or M 2 is in {π r 2 / A, 2 r / ω}, the line curves can not be mapped onto a point because they provide null values for A and ω. In other terms, a shape diagram D is obtained from the following mapping:

D : K(E 2 ) → [0, 1] 2 S → (x, y)
where K(E 2 ) denotes the compact sets of the Euclidean 2D plane. Using the morphometrical functionals listed in Table 2.1, twenty-two shape diagrams are defined, and by preserving the notation and indexing used in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], they are denoted (D k ) k∈ 1,30 \ ( 7,10 ∪ 17,20 ) , respectively. Some geometric inequalities, and consequently some shape diagrams, are restricted to convex shapes [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], that are not considered in this paper.

Shape diagrams

Axes coordinates The property stated in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] proving the non-injectivity and non-surjectivity of the mapping that associates a point in a shape diagram (D k ) k∈ 1,30 \ ( 7,10 ∪ 17,20 ) to a compact convex set in E 2 , is necessarily truth for the 2D simply connected compact sets. The proof uses the same examples as in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

D 1 : (ω, r, R) x = ω /2 R y = r / R D 2 : (ω, A, R) x = ω /2 R y = A /π R 2 D 3 : (r, A, R) x = r / R y = A /π R 2 D 4 : (A, d, R) x = A /π R 2 y = d /2 R D 5 : (ω, d, R) x = ω /2 R y = d /2 R D 6 : (r, d, R) x = r / R y = d /2 R D 11 : (ω, r, d) x = ω / d y = 2 r / d D 12 : (ω, A, d) x = ω / d y = 4 A /π d 2 D 13 : (r, A, d) x = 2 r / d y = 4 A /π d 2 D 14 : (A, R, d) x = 4 A /π d 2 y = √ 3 R / d D 15 : (ω, R, d) x = ω / d y = √ 3 R / d D 16 : (r, R, d) x = 2 r / d y = √ 3 R / d D 21 : (ω,

Complete systems of inequalities

In this subsection, only 2D compact convex sets are considered. A system of (two) geometric inequalities associated to a shape diagram is complete if and only if for any range of geometrical functionals values satisfying those conditions, a 2D compact convex set with these geometrical functionals values exists [START_REF] Hernández Cifre | The missing boundaries of the Santaló diagrams for the cases (d,ω,R) and (ω,R,r)[END_REF][START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF]. In other words, such a system is complete if and only if the mapping which associates a 2D analytic compact convex set in E 2 to a point in a shape diagram (D k ) k∈ 1,30 \( 7,10 ∪ 17,20 ) can be surjective by restricting the arrival set. For a shape diagram, each of the two associated inequalities determines a part of the convex domain boundary (the domain in which all compact convex sets are mapped). These two inequalities determine the whole boundary of the convex domain if and only if they form a complete system. The compact convex sets mapped onto the boundary points are the extremal compact convex sets of each considered inequality.

For sixteen among the twenty-two shape diagrams (D 3 illustrates several of these twenty-two shape diagrams, chosen according to the results synthetized in section 6. Remember that the shape diagrams are included in [0, 1] 2 . For a better visualization of the shapes drawn on a point of abscissa or ordinate equal to 0 or 1, the shape diagrams are illustrated in [-0.06, 1.04] 2 . Whatever the morphometrical functional, the extremal value 1 is reached for extremal simply connected compact sets. Thus, in each shape diagram, there is at least one simply connected compact set mapped to a point of abscissa or ordinate equal to 1. Moreover, the extremal value 0 is not always reached.

The dispersion of simply connected compact set locations within each shape diagram will be studied after the analysis of similarities between shape diagrams.

Similarity

The fact that ω = 2 r for some compact convex sets implies that shape diagrams (ω, x 1 , x 2 ) are similar to shape diagrams (r, x 1 , x 2 ) where x 1 ∈ {d, R, P, A} and x 2 ∈ {d, R, P}, that is to say

D 2 ∼ D 3 , D 5 ∼ D 6 , D 12 ∼ D 13 , D 15 ∼ D 16 , D 22 ∼ D 23 , D 25 ∼ D 26 , D 28 ∼ D 29
where ∼ denotes a strong similarity between shape diagrams.

In the same way, the fact that d = 2 R for some simply connected compact sets implies that:

• shape diagrams (x 1 , x 2 , R) are similar to shape diagrams (x 1 , x 2 , d) where x 1 ∈ {ω, r, A} and x 2 ∈ {r, A, P} (D 1 ∼ D 11 , D 2 ∼ D 12 , D 3 ∼ D 13 ); • shape diagrams (x 1 , R, P) are similar to shape diagrams (x 1 , d, P) where x 1 ∈ {ω, r, A} (D 24 ∼ D 27 , D 25 ∼ D 28 , D 26 ∼ D 29 ).
An algorithm of hierarchical classification [START_REF] Diday | Croisements, ordres et ultramétriques[END_REF] based on distances between shape diagrams allows to justify many of these similarities and to find other ones. Let k 1 , k 2 ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ), the distance between shape diagrams D k 1 and D k 2 , based on the Euclidean distance d E , is defined by Equation 4.1.

d E (D k 1 , D k 2 ) = 1 19 i∈ 1,19 d E (P i,k 1 , P i,k 2 ) (4.1) For all k 1 ∈ 1, 29 \ ( 7, 10 ∪ 17, 20 ) and k 2 ∈ k 1 + 1, 30 \ ( 7, 10 ∪ 17, 20 ), the distances d E (D k 1 , D k 2 )
are computed. Among them, the minimum distance value gives the best similarity between two shape diagrams. From these two shape diagrams, a mean shape diagram is built. To each step of the algorithm, two shape diagrams are similar up to the distance computed and they are gathered to build a mean shape diagram. The algorithm can be run until all the shape diagrams are gathered. Figure 4.4 shows the first fifteen steps of the hierarchical tree resulting from this algorithm. The remaining steps are not shown because the distance values are too high and do not present an interest in the study of similarities. For instance, if the algorithm is stopped before the distance value reaches 0.2, the following classification of shape diagrams is obtained:

• D 1 , D 11 , D 2 , D 12 • D 3 , D 13 , D 23 • D 4 , D 6 • D 24 D 27 D 26 D 29 • D 5 D 15 • D 14 D 16 • D 21 D 22 • D 25 D 28 • D 30

Dispersion quantification

For each shape diagram, the dispersion of the locations of 2D analytic simply connected compact sets of the family F sc 1 is studied. The spatial distribution of simply connected compact sets locations in each shape diagram is characterized and quantified from algorithmic geometry using Delaunay's graph (DG) and minimum spanning tree (MST) [START_REF] Boissonnat | Algorithmic geometry[END_REF]. Some useful information about the disorder and the neighborhood relationships between sets can be deduced. From each geometrical model, it is possible to compute two values from the edge lengths, denoted µ (average) and σ (standard deviation) for DG or MST. The simple reading of the coordinates in the (µ, σ)-plane eanables to determine the type of spatial distribution of the simply connected compact set (regular, random, cluster, . . . ) [START_REF] Raby | Quantitative cell dispersion analysis: new test to measure tumor cell aggressiveness[END_REF]. The decrease of µ and the increase of σ characterize the shift from a regular distribution toward random and cluster distributions, respectively. Following the two models, the shape diagrams D 4 , D 5 , D 6 , D 14 , D 15 , D 16 and D 30 have a low average µ. In these shape diagrams, the simply connected compact sets are located within a restricted domain in [0, 1] 2 . Their shape discrimination is not strong, especially for D 5 and D 15 that have a higher standard deviation σ, which is translated visually by several sets at the same locations.

The shape diagrams D 1 , D 2 , D 11 , D 12 , D 21 , D 22 , D 24 , D 26 , D 27 and D 29 have the highest averages. In these shape diagrams, the simply connected compact sets are located within a large domain in [0, 1] 2 and are well spaced from each other.

Finally, these statements are in agreement with those obtained for the similarities between shape diagrams.

A shape diagram with a strong dispersion (both high µ and σ values), for both DG and MST, guarantees a strong discrimination of shapes: the simply connected compact sets are located within a large domain in [0, 1] 2 and are well spaced from each other. However, remember that d = 2 R and ω = 2 r for some simply connected compact sets. This explains that some simply connected compact sets of the family F sc 1 are mapped onto a point on a line for some shape diagrams, namely:

• diagonal line from (0, 0) to (1, 1) on shape diagrams (x 1 , x 2 , x 3 ) where (x 1 , x 2 ) ∈ {(ω, r), (d, R)} and

x 3 ∈ {A, R, d, P} (D 1 , D 11 , D 21 , D 30 ) • horizontal line from (0, 1) to (1, 1) on shape diagrams (x 1 , d, R) where x 1 ∈ {ω, r, P, A} (D 4 , D 5 , D 6 ) • horizontal line from (0, (3)/2) to (1, (3)/2) on shape diagrams (x 1 , R, d)
where x 1 ∈ {ω, r, P, A} (D 14 , D 15 , D 16 ) For a shape diagram, the fact that some simply connected compact sets are mapped onto a point on a line does not yield to allow a strong discrimination of the simply connected compact sets as quantified and illustrated in Figure 4.5.

Following this quantification, the shape diagrams D 24 , D 25 , D 26 , D 27 , D 28 and D 29 , have a strong dispersion. During the study restricted to convex sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], they appear more moderate. The other shape diagrams have dispersions quite similar to those obtained with the convex sets.

5 Shape diagrams overlapping quantification

Shape diagrams for simply connected compact sets with one degree of freedom

Let two 2D analytic simply connected compact sets of the family F sc 1 , and the simply connected compact set class allowing to switch from one to the other using one degree of freedom. For example, the semi-circle goes to the semi-disk through semi-rings whose the cavity radius decreases. Therefore, several classes of "simply connected compact sets with one degree of freedom" could be defined. Thus, a curve denoted C i,k from each simply connected compact set class C i is created in each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). This process is used for some pairs of simply connected compact sets of the family F sc 1 . The tendency of various curves are observed.

These analytic simply connected compact sets verify shape properties which are preserved only under similitude transformations and under the variation of one parameter t ∈ R. For example, the cavity radius of semi-rings varies between 0 and the radius of the semi-disk. In other terms, the degree of freedom is the parameter t ∈ R.

Nineteen analytic simply connected compact set classes, gathered in three families, are considered: There are:

• family F sc 2.1 ⊇ {C i } i∈
• Family F sc 2.1 : C 1 -Semi-rings: They are the complementary sets of a semi-disk in a larger semi-disk. The ratio between the radius of the small and the large semi-disks varies in [0, 1]. When it reaches the lower and upper bounds, the semi-ring becomes a semi-disk and a semi-circle, respectively. C 2 -Semi-disks minus ungula: They are the complementary sets of an ungula [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] in a semi-disk. The ratio between the height of the ungula and the radius of the semi-disk varies in [0, 1]. When it reaches the lower and upper bounds, the semi-disk minus ungula becomes a semi-disk and a semi-circle, respectively. C 3 -Semi-disks minus semi-symmetrical disks with two peaks: They are the complementary sets of a semi-symmetrical disk with two peaks [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] in a semidisk. The ratio between the height of the semi-symmetrical disk with two peaks and the radius of the semi-disk varies in [0, 1]. When it reaches the lower and upper bounds, the semi-disk minus semi-symmetrical disk with two peaks becomes a semi-disk and a semi-circle, respectively. C 4 -Arrows: They are the complementary sets of an isosceles triangle in an equilateral triangle. The ratio between the height of the isosceles and the equilateral triangles varies in [0, 1]. When it reaches the lower and upper bounds, the arrow becomes an equilateral triangle and a shape with two line segments forming an angle whose the value is equal to π/3, respectively.

• Family F sc 2.2 : C 6 -Dual "Yamanouti" squares: They are defined in a similar way as the "Yamanouti" squares [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF]. The extremal simply connected compact sets are the square and the dual "Reuleaux" square. C 8 -Four-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (square). C 10 -Six-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (regular hexagon). C 12 -Eight-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (regular octogon, not illustrated here). C 14 -Even-stars, minimal filiformity: Their area value is null. They are asterisks with a branch even number. C 16 -Regular even polygrams: They are regular polygrams (the filiformity coefficient is "perfect", involving parallelism between the edges) with a positive even edges number. In the shape diagrams, the real positions of these sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve, and this is drawn. C 18 -Even-stars, maximal filiformity: They are regular even polygons [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF]. C 19 -Crosses: They are symmetrical (center) crosses whose the four branch length varies in [0, +∞[. When this length value reaches the lower bound, the resulting simply connected compact set becomes a square. When it tends to infinity, the resulting simply connected compact set tends to the asterisk-4.

• Family F sc 2.3 : C 5 -Dual Yamanouti triangles: They are defined in a similar way as the Yamanouti triangles [START_REF] Yamanouti | Notes on closed convex sets[END_REF]. The extremal simply connected compact sets are the equilateral triangle and the dual Reuleaux triangle. C 7 -Three-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (equilateral triangle). C 9 -Five-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (regular pentagon). C 11 -Seven-stars: the filiformity coefficient varies between 0 (A = 0) and 1 (regular heptagon, not illustrated here).

C 13 -Odd-stars, minimal filiformity: Their area value is null. They are asterisks with a branch odd number. C 15 -Regular odd polygrams: They are regular polygrams (the filiformity coefficient is "perfect", involving parallelism between the edges) with a positive odd edges number. In the shape diagrams, the real positions of these sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve, and this is drawn. C 17 -Odd-stars, maximal filiformity: They are regular odd polygons [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF]. For the simply connected compact sets of each class, the morphometrical functionals are computed. In each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ), a simply connected compact set class i ∈ 1, 19 is represented by a parametric curve C i,k (t) since the class is infinite and bounded by the two extremal sets. For the considered simply connected compact set classes, the extremal simply connected compact sets are sets that would be able to be in the family F sc 1 (Figure 4.1). For example, up to a similitude transformation, an infinite class of semi-rings exists, from the semi-circle to the semi-disk.

Figures 5.4, 5.5 and 5.6 illustrate some of these shape diagrams, chosen according to the results synthetized in section 6.

Overlapping quantification

An overlapping of curves is visible in some shape diagrams. Its quantification is based on a discretization of the spatial domain [0, 1] 2 of shape diagrams, depending on the discretization level n ∈ N * [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

Equation 5.1 quantifies (by a measurement ranging between 0 and 1) the overlapping of all curves for each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). A high (resp. low) value for this ratio means a strong (resp. weak) overlapping. When the computation is done for various discretization parameter n values, not only the curves overlapping is considered but also the curves proximity (small n value).

Overlapping n (D k ) = 1 - (x,y)∈ 0,n 2 D max k (x, y) (x,y)∈ 0,n 2 D sum k (x, y) (5.1)
where D max k : 0, n 2 → {0, 1} and D sum k : 0, n 2 → N are defined ∀(x, y) ∈ 0, n 2 by: [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

D max k (x, y) = max i∈ 1,23 C i,k (x, y) D sum k (x, y) = i∈ 1,23 C i,k (x, y) C i,k denoted the discretization of the curve C i,k
Figure 5.7 illustrates the discretized shape diagrams (n = 100) with overlapping intensities, and Figure 5.8 shows the quantification of curves overlapping (representing the simply connected compact set classes) for some shape diagrams, according to the n values 100 and 1000.

Several informations can be extracted from these graphs:

• The graphs representing the 2D analytic simply connected compact set classes of the family F sc 3 of 2D analytic simply connected compact sets with one degree of freedom and an odd number (strictly greater to 1) of symmetrical axes.

C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings C 1,k :Semi-rings
C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 2,k :Semi-disks minus ungula C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C 3,k :Semi-disks minus semi-symmetrical disks with two peaks C Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles Semi-circles 2 of analytic simply connected compact sets with one degree of freedom and an odd number of symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6). 3 of analytic simply connected compact sets with one degree of freedom and an even number strictly greater to 1 of symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6). of simply connected compact sets with one degree of freedom and an even number of symmetrical axes. of simply connected compact sets with one degree of freedom and an odd number strictly greater to 1 of symmetrical axes. Generally speaking, the results obtained with this quantification are quite different from those obtained during the study restricted to convex sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

Synthesis

To obtain a strong discrimination of 2D analytic simply connected compact sets, it is necessary to have both a strong dispersion and a weak overlapping. This analysis is summarized in Table 6.1. 6.1: Shape diagrams classification according to their quality of shape discrimination of analytic simply connected compact sets and according to the completeness of associated systems of inequalities.

Complete system

In this paper, only some shape diagrams have been illustrated. The choice was based on the results of shape discrimination (dispersion and overlapping studies) and on the results of similarities between shape diagrams (subsection 4.2). The aim was to illustrate dissimilar shape diagrams with different qualities of shape discrimination, and shape diagrams with different completeness of associated systems of inequalities. The framed shape diagrams of Table 6.1 are those illustrated throughout this paper.

Conclusion

This paper has dealed with shape diagrams of 2D non-empty analytic simply connected compact sets built from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. Each such a set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. From existing morphometrical functionals for these sets, twenty-two shape diagrams can be built. A detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these shape diagrams. It is based on the dispersion and overlapping quantifications from simply connected compact set locations in diagrams. Among all the shape diagrams, nine present a strong shape discrimination of sets, five are based on complete system of inequalities. Among these five diagrams, the shape diagram D 24 : (A, R, P) is retained for its representation relevance and discrimination power.

This paper reports the second part of a general comparative study of shape diagrams. The focus was placed on analytic simply connected compact sets. The first part [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] was restricted to the analytic compact convex sets. The main difference between the study of shape diagrams for the simply connected compact sets and those for the compact convex sets lies in the fact that some are not defined in the simply connected case. Moreover in this case, if the shape diagram is defined, the zone gathering the sets locations is necessarily larger in the simply connected case. The third part [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part III: convexity discrimination for analytic and discretized simply connected sets[END_REF] of the comparative study is published in a following paper. It focuses on convexity discrimination for analytic and discretized simply connected compact sets.

Figure 2 . 1 :

 21 Figure 2.1: Geometrical functionals of a simply connected compact set: radii of inscribed (r) and circumscribed (R) circles, minimum (ω) and maximum (d) Feret diameters.

  r, P) x = π ω / P y = 2π r / P D 22 : (ω, A, P) x = π ω / P y = 4π A / P 2 D 23 : (r, A, P) x = 2π r / P y = 4π A / P 2 D 24 : (A, R, P) x = 4π A / P 2 y = 4 R / P D 25 : (ω, R, P) x = π ω / P y = 4 R / P D 26 : (r, R, P) x = 2π r / P y = 4 R / P D 27 : (A, d, P) x = 4π A / P 2 y = 2 d / P D 28 : (ω, d, P) x = π ω / P y = 2 d / P D 29 : (r, d, P) x = 2π r / P y = 2 d / P D 30 : (d, R, P) x = 2 d / P y = 4 R / P

Figure 4 . 1 :

 41 Figure 4.1: Family F sc 1 of 2D analytic simply connected compact sets.

Figure 4 .

 4 Figure 4.3 illustrates several of these twenty-two shape diagrams, chosen according to the results synthetized in section 6. Remember that the shape diagrams are included in [0, 1] 2 . For a better visualization of the shapes drawn on a point of abscissa or ordinate equal to 0 or 1, the shape diagrams are illustrated in [-0.06, 1.04] 2 . Whatever the morphometrical functional, the extremal value 1 is reached for extremal simply connected compact sets. Thus, in each shape diagram, there is at least one simply connected compact set mapped to a point of abscissa or ordinate equal to 1. Moreover, the extremal value 0 is not always reached.The dispersion of simply connected compact set locations within each shape diagram will be studied after the analysis of similarities between shape diagrams.

3 Asterisks- 4 4 Asterisks- 7 7 Asterisks- 8 1 D 1 D 1 DFigure 4 . 3 :

 34477811143 Figure 4.3: Family F sc1 of analytic simply connected compact sets mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

1 11 2 Figure 4 . 4 :

 244 Figure 4.4: The first fifteen steps of an algorithm of hierarchical classification based on distances between the shape diagrams. To each step, two shape diagrams are similar up to the distance value, whose the scale is indicated on the right.

Figure 4 .Figure 4 . 5 :

 445 Figure 4.5: Two dispersion quantifications for all shape diagrams applied on the family F sc 1 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ) of the shape diagrams D k are located according to their µ and σ values.

1 , 4 :

 14 four classes of simply connected compact sets with one symmetrical axis (Figure 5.1), • family F sc 2.2 ⊇ {C i } i∈{6,8,10,12,14,16,18,19} : eight classes of simply connected compact sets with an even number of symmetrical axis (Figure 5.2), • family F sc 2.3 ⊇ {C i } i∈{5,7,9,11,13,15,17} : seven classes of simply connected compact sets with an odd number strictly greater to 1 of symmetrical axis (Figure 5.3).

2 . 1

 21 show a strong overlap (from n = 1000) for the shape diagrams D 1 , D 2 , D 4 , D 5 , D 6 , D 11 , D 12 , D 14 , D 15 , D 16 , D 25 , D 28 and D 30 .

Figure 5 . 1 :

 51 Figure 5.1: Family F sc 2.1 of 2D analytic simply connected compact sets with one degree of freedom and one symmetrical axis.

Figure 5 . 2 :

 52 Figure 5.2: Family F sc 2.2 of 2D analytic simply connected compact sets with one degree of freedom and an even number of symmetrical axes.

Figure 5 . 3 :

 53 Figure 5.3: Family F sc 2.3 of 2D analytic simply connected compact sets with one degree of freedom and an odd number (strictly greater to 1) of symmetrical axes.

1 D 1 D 1 DFigure 5 . 5 :

 11155 Figure 5.5: Family F sc 2.2 of analytic simply connected compact sets with one degree of freedom and an odd number of symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

C 5 ,

 5 k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles C 5,k :Dual Yamanouti triangles :Stars three parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 9,k :Stars five parts C 11,k :Stars seven parts

Figure 5 . 7 :

 57 Figure 5.7: "Intensity" discretized shape diagram D sum k (n = 100) with the nineteen curves representing the simply connected compact set classes with one degree of freedom.

  the nineteen classes of simply connected compact sets with one degree of freedom.

Figure 5 . 8 :

 58 Figure 5.8: Overlapping quantification for the twenty-two shape diagrams D k , k ∈ 1, 30 \( 7, 10 ∪ 17, 20 ), with n = 100 in red, and n = 1000 in blue.

•

  The shape diagrams D 5 and D 15 are excluded due to their weak dispersion and overlapping results, whatever the considered simply connected compact sets. • In the shape diagrams D 4 , D 6 , D 14 , D 16 and D 30 , only the family F sc 2.3 shows a weak overlapping. Futhermore, their dispersion quantification is weak. • In the shape diagrams D 25 and D 28 , the family F sc 2.1 shows a strong overlapping. Futhermore, their dispersion quantification is moderate. • In the shape diagrams D 1 , D 2 , D 11 and D 12 , the family F sc 2.1 shows a strong overlapping. Futhermore, their dispersion quantification is strong. • In addition to their strong dispersion, the shape diagrams D 3 , D 13 , D 21 , D 22 , D 23 , D 24 , D 26 , D 27 and D 29 provide a weak overlapping of the simply connected compact set classes considered in this paper. Futhermore, among the shape diagrams D 3 , D 13 , D 21 , D 22 , D 23 , D 24 , D 26 , D 27 and D 29 that obtain the best results for dispersion and overlapping quantifications, only D 3 , D 22 , D 23 , D 24 and D 26 are based on known complete systems of inequalities. Observing in details the representation of quantifications for these five shape diagrams, D 24 is retained for shape discrimination of analytic simply connected compact sets.

D 3 ,

 3 D 22 , D 23 , D 24 , D 26 D 13 , D 21 , D 27 , D 29 discrimination Moderate D 1 , D 11 , D 12 , D 28 D 2 , D 25 discrimination Weak D 4 , D 5 , D 6 , D 14 , D 15 , D 16 , D 30 discrimination Table

Table 2 .

 2

1: Shape functionals for simply connected compact sets. A, P, r, R, ω, d, denote the area, perimeter, radii of the inscribed and circumscribed circles, minimum and maximum Feret diameters

[START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF]

, respectively.

Table 3 .

 3 1: Axes coordinates of the twenty-two shape diagrams for 2D simply connected compact sets.

  1 , D 3 , D 4 , D 5 , D 6 , D 11 , D 12 , D 14 , D 15 , D 16 , D 22 , D 23 , D 24 , D 26 , D 28 , D 30 ), the completeness of systems of inequalities has been proved [3, 7, 8, 9, 10, 18].Figure 4.2 illustrates the convex domain boundary for seven of them.

	4 Shape diagrams dispersion quantification
	4.1 Shape diagrams for simply connected compact sets	For the family
	1 of nineteen sets (Figure 4.1), the morphometrical functionals are straightfor-F sc wardly computed. Each simply connected compact set i ∈ 1, 19 is represented by one point denoted P i,k , in each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). These 2D analytic simply connected compact sets are (Figure 4.1):
	-line segments;	-"1/2" semi-rings;	-equilateral triangles;
	-asterisks-3;	-semi-disks;	-squares;
	-asterisks-4;	-disks;	-regular pentagons;
	-asterisks-7;	-Reuleaux triangles [5, 15];	-regular hexagons;
	-asterisks-8;	-dual Reuleaux triangles;	-regular pentagrams;
	-semi-circles;	-dual "Reuleaux" squares;	-regular hexagrams;
	-regular crosses: they are symmetrical (center) crosses whose the four branch
	length and width are equals.


S Rivollier

J Debayle

J.-C Pinoli

SHAPE DIAGRAMS FOR 2D COMPACT SETS -PART III: CONVEXITY DISCRIMINATION FOR ANALYTIC AND DISCRETIZED SIMPLY CONNECTED SETS

Keywords: Analytic and discretized simply connected compact sets, Convexity discrimination, Geometrical and morphometrical functionals, Shape diagrams, Shape discrimination

. They focus on analytic compact convex sets and analytic simply connected compact sets, respectively. The purpose of this paper is to present the third part, by focusing on the convexity discrimination for analytic and discretized simply connected compact sets.

Introduction

The Santalo's shape diagrams [START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF] allow to represent a 2D compact convex set by a point in the Euclidean 2D plane from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF]. The axes of each shape diagram are defined from a pair of geometric inequalities relating these functionals. Sometimes, the two geometric inequalities provide a complete system: for any range of numerical values satisfying them, there exists a compact convex set with these values for the geometrical functionals (in other words, a point within the 2D Santalo shape diagram describes a 2D compact convex set). This is not valid for all the Santalo shape diagrams.

This paper deals with the study of the convexity discrimination for shape diagrams of 2D non-empty analytic and discretized simply connected compact sets. The two first parts [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF] of this study focus on the analytic compact convex sets and analytic simply connected compact sets, respectively. This third part presents an analysis of the convexity discrimination and extends the previous work to the discretized simply connected compact sets. The considered discretized simply connected sets are mapped onto points in these shape diagrams, and through dispersion quantification and convexity discrimination, the shape diagrams are classified according to their ability to discriminate the simply connected compact sets.

Shape convexity

The following study on the convexity discrimination first requires the definition of the shape convexity. A set is convex when the line segment which joins any two points in it lies totally within the set. In other terms, the shape convexity could be quantified with the probability that two points in the set lies totally within it.

Convexity parameters are commonly used in the analysis of shapes. The measurement value of the shape convexity of any set ranges between 0 and 1 (it is a probability). A convex set gives the value 1. Futhermore, the less the parameter value is high, the less the shape is convex. The convexity measurement can be computed, for instance, by the ratio A / A C where A C is the area of the convex hull of the set, but this is not sensitive to boundary small variations (Figure 2.1). A second convexity parameter is the ratio P C / P where P C is the Euclidean perimeter of the convex hull of the set. Futhermore, a suitable convexity parameter is particularly required for discretized sets. For that purpose, the convexity parameter of Zunic and Rosin [START_REF] Zunic | A new convexity measure for polygons[END_REF] will be used for the following study on the convexity discrimination. For all polygon S of the Euclidean 2D plane E 2 , it is defined by:

c(S) = min α∈[0,2π] P 2 (R(S, α)) P 1 (S, α) (2.1)
P 1 (S, α) denotes the perimeter of the set S, rotated by the angle α with the origin as the center of rotation, in the sense of the l 1 metric (l 1 (e) equals the sum of the projections of the straight line segment e onto the coordinate axes). P 2 (R(S, α)) denotes the Euclidean perimeter of the minimal bounding rectangle R with edges parallel to the coordinate axes which includes the rotated set of S by the angle α.

This convexity parameter c has the following desirable properties:

• its value is always a number within (0, 1]

• its value is 1 if and only if the measured set is convex • there are sets with its value arbitrarily close to 0

• it is invariant under similitude transformations • there is a simple and fast procedure for computing it.

Convexity discrimination for analytic simply connected sets

Observing the 2D compact set locations in the shape diagrams D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ) for the families F c 1 and F sc 1 [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF] (Figures 3.1 and 3.2), some shape diagrams seem to stronger discriminate the convex shapes from the non-convex shapes than others. Equation 3.1 quantifies (by values ranging between 0 and 1) this convexity discrimination for each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). A high (respectively low) resulting value means a weak (respectively strong) convexity discrimination.

Convexity_discrimination(D

k ) = 1 # {F c 1 ∪ F sc 1 }   #F c 1 i=1 |c(i) (3.1) -c (argmin {d E (i, j)|j ∈ F c 1 })| + #F sc 1 i=1 |c(i) -c (argmin {d E (i, j)|j ∈ F sc 1 })|  
where c(i) denotes the convexity value of the shape indexed by i in Figure 5.1, and d E is the Euclidean distance. Figure 3.3 shows the results of this quantification for every shape diagrams D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). The stronger convexity discrimination appears in the shape diagrams D 24 , D 26 , D 27 , and D 29 , that is in agreement with the visual interpretation [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], and the weaker discrimination appears for the shape diagrams D 4 and D 14 . These results have to be confirmed with more general sets. This is not easy to conclude with the restriction to analytic sets. For this reason, this study is extended to the discrete case.

Shape functionals for a discretized simply connected set

In this paper, the non-empty discretized simply connected compact sets in the discrete Euclidean 2-space E 2 d are considered. They are represented by points. The inter-point distance is the step discretization. The discretization is fine enough to preserve the simple connectivity [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF]. For the characterization of these sets, six geometrical functionals are used allowing to define morphometrical functionals from geometric inequalities.

Geometrical functionals

For a discretized simply connected compact set in E 2 d , let A, P, r, R, ω, d, denote the estimations of its area, its perimeter, the radii of its inscribed and circumscribed circles, its minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF], respectively. Figure 4.1 illustrates these geometrical functionals for a discretized simply connected compact set represented in a point lattice [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF]. For all non-empty discretized simply connected compact sets, these geometrical functionals are greater or equal than zero.

Geometric inequalities and morphometrical functionals

Geometric inequalities and morphometrical functionals for analytic simply connected compact sets are referenced in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]. There exists special cases where a discretized set has geometrical functional values that do not verify a geometric inequality, due to the estimation error (because of the discretization). This means that this set is a discretization of an extremal set for the associated geometric inequality. Consequently, for practical reasons, the morphometrical functional value greater than one is reduced to one. Thus, the twenty-two shape diagrams referenced in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF] can be used for discretized simply connected compact sets.

5 Shape diagrams for a basis of various discretized simply connected compact sets

Shape diagrams

Figure 5.1 illustrates seventy-eight discretized patterns constituting the family F dsc 1 . It is assumed that the discretization process is fine enough such that each discretized pattern is a discretized simply connected compact set on the points lattice [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF]. Thus, the morphometrical functionals are computed and located in each shape diagram.

The discretized patterns are numerated from one to seventy-eight. The pattern number are mapped to its proper point in each shape diagram. Figure 5.2 illustrates several of these twenty-two shape diagrams, chosen according to the synthesized results of [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]. The color of the number is related to the convexity parameter value c of the associated set (dark red for a high c value, dark blue for a low c value). Moreover, the black curves indicate the convex domain boundary, if it is known [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF].

Sometimes, a dark red number appears slightly outside this domain. This is mainly due to the estimation error (discretization) of the geometrical functionals. This convexity range will enable to analyze the convexity discrimination within shape diagrams. of discretized simply connected compact sets mapped into eleven shape diagrams (chosen according to the results synthetized in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]). of discretized simply connected compact sets mapped into eleven shape diagrams (chosen according to the results synthetized in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]).

Dispersion quantification and convexity discrimination

A wider family F dsc 2 of 1370 discretized patterns is considered (Kimia database [START_REF] Sharvit | Symmetry-based indexing of image databases[END_REF], not illustrated here). These are twenty deformations of each of the sixty-eight first sets of the family F dsc 1 , plus the ten last sets of F dsc 1 ; section 6 shows two examples of these deformations). For each discretized pattern representing a discretized simply connected compact set, the morphometrical functionals are computed and located by a point in each shape diagram (Figure 5.3). The point color is associated to the convexity parameter value c, and the convex domain boundary, if it is known, is illustrated with black lines.

Dispersion quantification

For each shape diagram, the dispersion of the locations of the 2D discretized simply connected compact sets of the family F dsc 2 is studied.

The spatial distribution of discretized simply connected compact sets locations in each shape diagram is characterized and quantified from algorithmic geometry using Delaunay's graph (DG) and minimum spanning tree (MST) [START_REF] Boissonnat | Algorithmic geometry[END_REF]. Some useful information about the disorder and the neighborhood relationships between sets can be deduced. From each geometrical model, it is possible to compute two values from the edge lengths, denoted µ (average) and σ (standard deviation) for DG or MST. The simple reading of the coordinates in the (µ, σ)-plane eanables to determine the type of spatial distribution of the discretized simply connected compact set (regular, random, cluster, . . . ) [START_REF] Raby | Quantitative cell dispersion analysis: new test to measure tumor cell aggressiveness[END_REF]. The decrease of µ and the increase of σ characterize the shift from a regular distribution toward random and cluster distributions. As in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], the DG model allows to divide the shape diagrams into two groups, mainly according to the average µ value: the shape diagrams D 4 , D 5 , D 6 , D 14 , D 15 , D 16 and D 30 have a weak average µ, thus the discretized simply connected compact sets are located within a small domain in [0, 1] 2 . In [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], the MST model confirmed this tendency for the average. Figure 5.4 shows also a distribution of these two groups according to the standard deviation σ. Note that the constitution of each of the two groups is the same as in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF].

Convexity discrimination

Observing the colors dispersion in the shape diagrams of Figure 5.3, some shape diagrams seem to stronger discriminate the convex shapes from the non-convex shapes than others. This stronger discrimination is visually highlighted by a color gradient, from dark red to dark blue. Within a shape diagram, if the colors are irregularly distributed according to the color range, this means that convex sets and non-convex sets can not be discriminated. Equation 5.1 quantifies (by values ranging between 0 and 1) this discrimination for each shape diagram D k , k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ). A high (resp. low) resulting value means a weak (resp. strong) convexity discrimination.

Convexity_discrimination(D k ) = (5.1) 1 #F dsc 2 #F dsc 2 i=1 c(i) -c argmin d E (i, j)|j ∈ F dsc 2
where c(i) denotes the convexity value of the shape indexed by i in Figure 5.1, and d E is the Euclidean distance.

Figure 5.5 shows the results of this quantification for every shape diagrams. The stronger convexity discrimination appears in the shape diagrams D 24 , D 25 , D 26 , D 27 , D 28 and D 29 , that is in agreement with the visual interpretation. The weaker discrimination appears for the shape diagrams D 4 , D 6 , D 14 and D 16 . Even if the shape diagram D 30 is not strong for the general shape discrimination [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], it is not weak for the convexity discrimination. In fact, these two discriminations are independant. 6 Shape diagrams for similar discretized simply connected compact sets

This section focuses on the discrimination of shapes that are visually similar enough, so that they can be considered as the same global shape. The shape diagrams D 4 , D 5 , D 6 , D 14 , D 15 , D 16 and D 30 provide a weak shape discrimination, whatever the visual similarity of the sets. Thus, they are not considered from this section and until the end of this paper. It remains the fifteen shape diagrams D k , k ∈ 1, 29 \ ( 4, 10 ∪ 14, 20 ).

Let be three families F dsc 3 , F dsc 4 and F dsc 5 of discretized patterns representing triangles (Figure 6.1), disks (Figure 6.2) and crosses (Figure 6.3), respectively, that have undergone minor transformations, modifications, deformations. The morphometrical functionals are computed, and the patterns are located by a point in each shape diagram (Figures of twenty 2D discretized simply connected compact sets with 'triangle' shape, mapped into nine shape diagrams (chosen according to the results synthetized in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]). The color of the number is related to the convexity parameter value. of twenty 2D discretized simply connected compact sets with 'disk' shape, mapped into nine shape diagrams (chosen according to the results synthetized in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]). The color of the number is related to the convexity parameter value. of twenty 2D discretized simply connected compact sets with 'cross' shape, mapped into nine shape diagrams (chosen according to the results synthetized in [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF]). The color of the number is related to the convexity parameter value. The results obtained allows to reveal that the shape diagrams D 2 and D 12 do not discriminate the sets in each family F dsc 3 , F dsc 4 and F dsc 5 . This discrimination appears little stronger for the shape diagrams D 1 and D 11 . From a global vision, the shape diagrams D 21 and D 22 strongest discriminate these sets.

Synthesis

To obtain a strong discrimination of 2D discretized simply connected compact sets, it is necessary to have both a strong dispersion and a strong convexity discrimination. Futhermore, among the shape diagrams D 24 , D 25 , D 26 , D 27 , D 28 and D 29 that obtain the best results for dispersion quantification and convexity discrimination, only D 24 , D 26 and D 28 are based on known complete systems of inequalities. Observing in details the representation of quantifications for these three shape diagrams, D 24 retained for shape discrimination of analytic simply connected compact sets, is also retained for shape discrimination of discretized simply connected compact sets.

•

This analysis is summarized in 7.1: Shape diagrams classification according to their quality of shape discrimination of discretized simply connected compact sets and according to the completeness of associated systems of inequalities.

Global synthesis for the three parts of this study

For each part of this study [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], the synthesis gives the shape diagrams that provide the strongest shape discrimination. These are D 12 and D 24 . Table 9.1 gathers the syntheses.

Conclusion

This paper has dealed with shape diagrams of 2D non-empty analytic and discretized simply connected compact sets built from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. Each set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. From existing morphometrical functionals for these sets, twenty-two shape diagrams can be built. A detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these shape diagrams. It is based on the dispersion quantification and convexity discrimination from compact set locations in diagrams. Among all the shape diagrams, six present a strong convexity discrimination of sets, three are based on complete system of inequalities. Among these three diagrams, the shape diagram D 24 : (A, R, P) is retained for its representation relevance and discrimination power.

The purpose of this paper was to present the third part of a general comparative study of shape diagrams. The focus was placed on convexity discrimination of analytic and discretized simply connected compact sets. The two first parts [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF] was restricted to the analytic compact convex and simply connected compact sets, respectively. For an analytic set, the geometrical functionals were accurately calculated. For a discretized set, they are estimated. Thus, in the discrete case, the shape diagrams are based on estimated morphometrical functionals.

Actually, the authors work on the case of hollowed sets (analytic and discretized) which is not specifically considered in this paper.

Complete system of inequalities Non-complete system of inequalities Analytic Analytic Analytic Analytic compact simply connec-Convexity compact simply connec-Convexity convex sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] ted compact sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF] discrimination convex sets [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF] ted compact sets 9.1: Shape diagrams classification according to their quality of shape discrimination and according to the completeness of associated systems of inequalities.

1 Figure 2 . 1 :

 121 Figure 2.1: A / A C gives result values equal to 1 for the set (a) and close to 1 for the sets (b) and (c). Nevertheless, the set (b) is far to check the convexity definition (the probability that two points in the set lies totally within it is low). P C / P gives result values equal to 1 for the set (a) and far from 1 for the sets (b) and (c). Nevertheless, the set (c) is not very far to check the convexity definition.

Figure 3 . 1 :

 31 Figure 3.1: Family F c 1 of 2D analytic compact convex sets [8].

Figure 3 . 2 :

 32 Figure 3.2: Family F sc 1 of 2D analytic simply connected compact sets [9].

Figure 3 . 3 :

 33 Figure 3.3: Convexity discrimination for the families F c 1 and F sc 1 .

Figure 4 . 1 :

 41 Figure 4.1: Geometrical functionals of a discretized simply connected compact set: radii of inscribed (r) and circumscribed (R) circles, minimum (ω) and maximum (d) Feret diameters. The area A is given by the point number, and the perimeter is estimated thanks to the Cauchy-Crofton-Poincaré formula [2, 3, 6].

Figure 5 . 1 :

 51 Figure 5.1: Family F dsc 1 of seventy-eight 2D discretized simply connected compact sets (discretized patterns). The color of each pattern number is related to the convexity parameter value using non-linear color-tones.

D 3 :

 3 (r, A, R)

D 4 :

 4

DDDD

  Figure 5.2: Family F dsc 1

1 D 1 D 1 DFigure 5 . 3 :

 11153 Figure 5.3: Family F dsc 2

Figure 5 .Figure 5 . 4 :

 554 Figure 5.4: Two dispersion quantifications for all shape diagrams applied on the family F dsc 2 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 30 \ ( 7, 10 ∪ 17, 20 ) of the shape diagrams D k is located according to its µ and σ values.

Figure 5 . 5 :

 55 Figure 5.5: Convexity discrimination for the family F dsc 2 .

Figure 6 . 1 :

 61 Figure 6.1: Family F dsc 3

Figure 6 . 2 :

 62 Figure 6.2: Family F dsc 4

Figure 6 . 3 :

 63 Figure 6.3: Family F dsc 5

Figure 6 . 4 :

 64 Figure 6.4: Two dispersion quantifications for all shape diagrams applied on the family F dsc 3 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 29 \ ( 4, 10 ∪ 14, 20 ) of the shape diagrams D k is located according to its µ and σ values.

Figure 6 . 5 :

 65 Figure 6.5: Two dispersion quantifications for all shape diagrams applied on the family F dsc 4 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 29 \ ( 4, 10 ∪ 14, 20 ) of the shape diagrams D k is located according to its µ and σ values.

Figure 6 . 6 :

 66 Figure 6.6: Two dispersion quantifications for all shape diagrams applied on the family F dsc 5 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 29 \ ( 4, 10 ∪ 14, 20 ) of the shape diagrams D k is located according to its µ and σ values.

  The shape diagram D 4 , D 5 , D 6 , D 14 , D 15 and D 16 are excluded due to their weak dispersion and convexity discrimination results. • The shape diagram D 30 presents a strong convexity discrimination although its weak dispersion result. • The dispersion quantification of the shape diagrams D 1 , D 2 , D 3 , D 11 , D 12 , D 13 , D 21 , D 22 , D 23 , D 24 , D 25 , D 26 , D 27 , D 28 and D 29 gives strong values. • The convexity discrimination is stronger for D 24 , D 25 , D 26 , D 27 , D 28 and D 29 . • The similar sets discrimination appears stronger for D 21 and D 22 .

D 3 , 23 D 3 ,D 1 , 30 D 4 ,

 32331304 D 12 , D 22 , D D 22 , D 23 , D 24 , D 26 D 24 , D 26 , D 28 D 2 , D 13 D 13 , D 21 , D 27 , D 29 D 25 , D 27 , D 29 discrimination D 1 , D 7 , D 9 , D 11 , D 18 , D 24 , D 26 , D 28 D 1 , D 11 , D 12 , D 28 D 3 , D 11 , D 12 , D 22 , D 23 D 8 , D 17 , D 19 , D 21 , D 25 , D 27 , D 29 D 2 , D 25 D 2 , D 13 ,D 21 Moderate discrimination Weak D 4 , D 5 , D 6 , D 10 , D 14 , D 15 , D 16 , D 20 , D 30 D 4 , D 5 , D 6 , D 14 , D 15 , D 16 , D D 5 , D 6 , D 14 , D 15 , D 16 , D 30 D 31 discrimination Table

  [START_REF] Poincaré | Calcul des Probabilités[END_REF].1, 6.2 and 6.3). The color of the number is related to the convexity parameter value c, and the convex domain boundary, if it is known, is illustrated with black lines. Figures 6.4, 6.5 and 6.6 give the dispersion quantification representation of the fifteen shape diagrams for each model, DG and MST (method described in subsubsection 5.2.1).

		0
		0.2
		0.28 0.35
		0.4 0.45
		0.49 0.53
		0.57 0.6
		0.63 0.66 0.69		1				2			3	4	5	6	7	8	9	10
		0.72 0.75
		0.77
		0.8
		0.82
		0.85 0.87 0.89 0.92		11				12			13	14	15	16	17	18	19	20
		0.94 0.96
		0.98 1
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

Table 7

 7 D 26 , D 28 D 25 , D 27 , D 29 discrimination Moderate D 1 , D 3 , D 11 , D 12 , D 22 , D 23 D 2 , D 13 , D 21 discrimination Weak D 4 , D 5 , D 6 , D 14 , D 15 , D 16 , D 30 discrimination

	.1.
	Complete system	Non-complete system
	of inequalities	of inequalities
	Strong D 24 , Table


Lara Leclerc
email: leclerc@emse.fr

Delphine Boudard

Jérémie Pourchez

Valérie Forest

Odile Sabido

Valérie Bin

Sabine Palle

Philippe Grosseau

Didier Bernache-Assollant

Michèle Cottier

D Bernache

Quantification of microsized fluorescent particles phagocytosis to a better knowledge of toxicity mechanisms

Keywords: Phagocytosis quantification, macrophages, fluorescent microsized particles, flow cytometry, confocal microscopy, in vitro toxicity evaluations

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

INTRODUCTION

The toxicological assessment of fine and ultrafine particles (and particularly nanoparticles) represent a considerable current issue for environmental science, biosciences or nanomedicine [START_REF] Oberdorster | Nanotoxicology: an emerging discipline evolving from studies of ultrafine particles[END_REF][START_REF] Sayes | Assessing toxicity of fine and nanoparticles: comparing in vitro measurements to in vivo pulmonary toxicity profiles[END_REF][START_REF] Warheit | Comparative pulmonary toxicity assessment of single-wall carbon nanotubes in rats[END_REF]. In order to better understand the toxicity of particles it seems of high interest to determine the number of internalized particles into the cells after exposure [START_REF] Clift | The impact of different nanoparticle surface chemistry and size on uptake and toxicity in a murine macrophage cell line[END_REF][START_REF] Haberzettl | Actin plays a crucial role in the phagocytosis and biological response to respirable quartz particles in macrophages[END_REF].

Different particle entry pathways exist in the human body: skin, gastro-intestinal tract, respiratory tract or by injection. We particularly focus our attention on the respiratory way and its main target cells: the macrophages (MA). These cells are major performers of phagocytosis, a highly conserved complex process, classically defined as the internalization and destruction of particles greater than 0.5 µm in size. It is a receptor-mediated and actindriven process. Phagocytic uptake involves actin dynamics including polymerization, bundling, contraction, severing and depolymerization of actin filaments [START_REF] Aderem | Mechanisms of phagocytosis in macrophages[END_REF][START_REF] Fenteany | Cytoskeletal remodeling in leukocyte function[END_REF]Niedergang andChavrier 2004, 2005;[START_REF] Ravetch | Phagocytic cells[END_REF]. The physicochemical characteristics of particles may also influence the phagocytic process [START_REF] Serda | Quantitative mechanics of endothelial phagocytosis of silicon microparticles[END_REF]) and they have also been shown to be involved in the mechanisms of toxicity [START_REF] Clift | The impact of different nanoparticle surface chemistry and size on uptake and toxicity in a murine macrophage cell line[END_REF][START_REF] Fubini | Surface chemistry and quartz hazard[END_REF][START_REF] Fubini | Relationship between the state of the surface of four commercial quartz flours and their biological activity in vitro and in vivo[END_REF]).

It appears of great interest to quantify the phagocytic process, and compare these data to the toxicity of particles regarding their physicochemical characteristics. Preliminary results have shown that it is impossible to accurately quantify phagocytosis using two dimension (2D) images acquired from cell smears where internalized particles can hardly be distinguished from particles placed at the outer surface of the plasma membrane.

In this context, our work aimed at developing a simple, direct, cost effective, reproducible and easy to perform phagocytosis quantification technique on an in vitro MA cell line (RAW 264.7). The MA were incubated with microsized model particles which are fluorescent beads whose physicochemical characteristics (i.e. size and chemical coating) and observations are more easily controlled [START_REF] Champion | Role of Particle Size in Phagocytosis of Polymeric Microspheres[END_REF][START_REF] Hasegawa | Exact determination of phagocytic activity of alveolar macrophages toward polymer microspheres by elimination of those attached to the macrophage membrane[END_REF][START_REF] Jones | The inhibition of phagocytosis of respirable microspheres by alveolar and peritoneal macrophages[END_REF].

Preliminary confocal observations allowed to distinguish the localization of particles without an accurate quantification of the phagocytic process.

To perform a phagocytosis quantification, we adapted an existing analysis using a trypan blue (TB) quenching in flow cytometry (FCM) [START_REF] Hed | The use of fluorescence quenching in flow cytofluorometry to measure the attachment and ingestion phases in phagocytosis in peripheral blood without prior cell separation[END_REF][START_REF] Nuutila | Flow cytometric quantitative determination of ingestion by phagocytes needs the distinguishing of overlapping populations of binding and ingesting cells[END_REF][START_REF] Thiele | Evaluation of particle uptake in human blood monocyte-derived cells in vitro. Does phagocytosis activity of dendritic cells measure up with macrophages?[END_REF][START_REF] Van Amersfoort | Evaluation of a flow cytometric fluorescence quenching assay of phagocytosis of sensitized sheep erythrocytes by polymorphonuclear leukocytes[END_REF]. In addition to its principal function as an exclusion dye of dead cells, TB is known for its ability to "turn off" the green fluorescence emitted by the beads outside the cells. This process allowed us to distinguish internalized beads from those just adhering to the plasma membrane. This approach has been improved by combining Side Scatter measurement with fluorescent mean intensity of beads without and with TB.

Due to the role played by actin filaments for the initiation phase of the phagocytic process, we used different actin polymerization inhibitors such as Cytochalasin-D (Cyt-D) [START_REF] Haberzettl | Actin plays a crucial role in the phagocytosis and biological response to respirable quartz particles in macrophages[END_REF][START_REF] Kanno | A murine scavenger receptor MARCO recognizes polystyrene nanoparticles[END_REF][START_REF] Moller | Ultrafine particles cause cytoskeletal dysfunctions in macrophages[END_REF], Latrunculine-A (Lat) (de [START_REF] De Oliveira | Latrunculin A is a potent inhibitor of phagocytosis by macrophages[END_REF][START_REF] Defife | Disruption of filamentous actin inhibits human macrophage fusion[END_REF][START_REF] Oliveira | Effects of latrunculin A on immunological phagocytosis and macrophage spreading-associated changes in the Factin/G-actin content of the cells[END_REF][START_REF] Oliveira | Latrunculin A is a potent inducer of aggregation of polymorphonuclear leukocytes[END_REF], Jasplakinolide (Jasp) [START_REF] Odaka | Jasplakinolide induces apoptosis in various transformed cell lines by a caspase-3-like protease-dependent pathway[END_REF] to block cytoskeleton rearrangements and thus phagocytosis. Cells were also fixed with paraformaldehyde (PFA) to prevent phagocytosis.

Moreover, the toxicity of particles was evaluated using different parameters: Tumor Necrosis Factor alpha (TNF-α) production (to assess the inflammatory response), Lactate Dehydrogenase (LDH) release (reflecting integrity of the cell membrane) and the production of total Reactive Oxygen Species (ROS), especially H 2 O 2 [START_REF] Bruch | Variation of biological responses to different respirable quartz flours determined by a vector model[END_REF][START_REF] Catelas | Cytotoxicity and macrophage cytokine release induced by ceramic and polyethylene particles in vitro[END_REF][START_REF] Donaldson | The pulmonary toxicology of ultrafine particles[END_REF][START_REF] Sayes | Assessing toxicity of fine and nanoparticles: comparing in vitro measurements to in vivo pulmonary toxicity profiles[END_REF]. This part aimed at assessing if the two types of beads (1 µm amine or 2 µm carboxylate) led to different levels of toxicity.

5

Following this methodology we looked for a relationship between toxicity response and the level of phagocytosis determined by confocal microscopy and FCM analysis.

MATERIAL AND METHODS

In vitro model

Macrophage cell culture

The RAW 264.7 cell line was provided by ATCC Cell Biology Collection (Promochem LGC) and derives from mice peritoneal macrophages (MA) transformed by the AMLV virus (Abelson Murine Leukemia Virus). Cells were cultured in DMEM medium (Dulbecco's Modified Eagle's Medium, Gibco) complemented with 10% of fetal calf serum (FCS, Gibco), 1% penicillin-streptomycin (penicillin 10 000 units/mL, streptomycin 10 mg/mL, Sigma) and incubated at 37°C under a 5% carbon dioxide humidified atmosphere.

Fluorescent polymer particles (beads)

 P beads: fluorescent polystyrene particles of 2 µm in diameter with carboxylate coating (Fluoresbrite®YGcarboxylate microspheres, Polyscience). These particles have an excitation peak at 441 nm, and an emission peak at 486 nm (green fluorescence) with a large spectrum.

 I beads: fluorescent polystyrene particles of 1 µm in diameter with amine coating (YGFluoSpheres®amine-modified microspheres, Invitrogen). These particles have an excitation peak at 505 nm, and an emission peak at 515 nm (green fluorescence).

Bead exposure conditions

For the two types of beads, an arbitrary ratio of 10 beads for one cell (10/1¢) has been defined on the basis of the MA size comparatively to bead sizes.

Moreover in order to work with similar surface area for the two types of beads, we used the ratio 40 I beads/1¢ (surface of P beads being four times more important than surface of I beads).

Actin inhibitors and blocking of phagocytosis

Three actin inhibitors were used: Cyt-D (Sigma), Lat (VWR), Jasp (VWR). They were directly added to cell cultures and incubated at 37°C for 60 minutes, before addition of beads. These molecules remained in the culture medium throughout all the experiments. The concentrations used (5 µM Cyt-D, 0.1 µM Lat and 1 µM Jasp) were defined depending on literature data or preliminary assays.

To block completely the phagocytic process cells were fixed in a 4% PFA solution (Sigma) for 10 minutes before adding P or I beads [START_REF] Moon | A simple method for combined fluorescence in situ hybridization and immunocytochemistry[END_REF].

Confocal microscopy phagocytosis assay

Exposure conditions

Cells were grown in 6 well plates (1 million cells in 4 mL) containing coverslips and beads were incubated for 16 hours.

Actin cytoskeleton fluorescent labeling

After the particle exposure, the coverslips were fixed in wells with 4% PFA for 10 minutes (except for the cells already fixed). The wells were washed with Phosphate Buffered Saline (PBS, VWR) to remove beads that did not adhere to the cell membrane or were not internalized. Then actin cytoskeleton was labeled with AlexaFluor®594 phalloidin (Invitrogen; diluted 1:40 in PBS) for 20 minutes and rinsed with PBS. Thereafter cell nuclei were stained with 10 µM of Hoechst 33342 solution (Sigma), during 15 minutes and rinsed with PBS.

Analysis

Microscopic analyses were performed on a confocal microscope TCS-SP2 AOBS (Leica®).

Images were obtained from the fluorescence emitted by the beads (ex: 488 nm; em: 499-540 nm), AlexaFluor®594 phalloidin (ex: 543 nm; em: 615-693 nm) and Hoechst (ex: 350 nm; em: 408-443 nm). Confocal microscopy allowed to make stacks of images. Indeed, for a given area of analysis, it was possible to acquire images in different planes of the Z axis (depth of field), to obtain a series of images to achieve XY projections and XZ and YZ section stacks of the selected area of interest.

Quantification of phagocytosis by flow cytometry

FCM exposure conditions and trypan blue quenching

Cells were prepared in 25 cm 2 culture flasks (2 millions cells in 4 mL). A control flask without particles was used to assess the autofluorescence of cells. Some flasks were directly exposed to beads, others were pre-treated with actin inhibitors or PFA before adding P or I beads.

After 90 minutes or 24h of exposure, culture medium was removed, cells were harvested with a scraper and aliquots containing 500 000 cells were centrifuged (10 minutes, 1500 rpm).

Cells were resuspended either in 500 µL of DMEM or in 250 µL of DMEM + 250 µL of TB (0.4%, Sigma).

Cytotoxicity was also assessed by adding 1 mg/mL of propidium iodide (Sigma) in the tubes without TB [START_REF] Jacobs | Use of propidium iodide staining and flow cytometry to measure anti-mediated cytotoxicity: resolution of complement-sensitive and resistant target cells[END_REF][START_REF] Riccardi | Analysis of apoptosis by propidium iodide staining and flow cytometry[END_REF][START_REF] Steinkamp | Flow cytometric, phase-resolved fluorescence measurement of propidium iodide uptake in macrophages containing phagocytized fluorescent microspheres[END_REF] This evaluation allows to quantify the percentage of dead cells. An exclusion window was made on propidium iodide negative cells to analyze only living cells and quantify the dynamic process of phagocytosis.

Cytometric analysis

Each condition was analyzed twice on 10 000 cells. Acquisitions and analyses were performed using a cytometer analyzer-cell-sorter FACS Diva™ (BD Biosciences, CA) equipped with the software Cell Quest™ Pro.

The fluorochromes used were excited through an Enterprise II Argon Laser (wavelength set at 488 nm). The fluorescence emitted by P or I beads after excitation at 488 nm was collected through a band pass filter 530/30 nm (FL1 channel) and the fluorescence emitted by excitation of propidium iodide at 488nm was collected through a band pass filter 695/40 nm (FL3 channel). For each acquisition, the size level of detection corresponding to the Forward Scatter (FSC) was set above the size of cellular debris to detect only well preserved cells.

Three FCM parameters were quantified for each sample:

-the average Side Scatter (SSC), morphological parameter reflecting granularity and intracellular complexity, in relation to the presence of both adherent and phagocytosed beads, and quantified by the geometric mean (Y Geo Mean).

-the total Mean Fluorescence Intensity (MFI) corresponding to the total fluorescence emitted by the P or I beads (internalized and adhered) that we called total MFI. These values were obtained for the condition without TB on FL1 channel.

-the percentage of cells with internalized P or I beads that we called "percentage of phagocytic cells" detected on the FL3 channel after TB quenching.

Toxicity assessment

For each experiment, cells were prepared in 96-well plates (100 000 cells/well for TNF-α and LDH assays, and 300 000 cells/well for ROS and H 2 O 2 parameters) in 25 µL of complete DMEM. Suspensions of beads in a volume of 75 µL of DMEM were added to the culture and incubated for 90 min or 24h at 37°C in a 5% CO 2 atmosphere.

TNF-α production

After incubation with beads, released TNF-α was assessed in the supernatant using a commercial ELISA Kit (Quantikine® Mouse TNF-α Immunoassay, R&D Systems) according to the manufacturer's instructions. The optical density of each well was determined using a microplate reader (Multiskan RC, Thermolabsystems) set to 450 nm. A standard curve was established and results were expressed in pg/mL of TNF-α.

LDH release

The activity of the lactate dehydrogenase (LDH) released from cells with damaged membranes was assessed using the CytoTox-ONE™ Homogeneous Membrane Integrity Assay (Promega) according to the manufacturer's instructions. Detection was performed on a fluorometer (Fluoroskan Ascent, Thermolabsystems), using excitation/emission wavelengths at 530/590 nm. The activity of the released LDH was reported to that of total cellular LDH (measured after control cells lysis) and was expressed as a percent of the control.

Oxydative stress

Total ROS: a large array of ROS activity can be assessed with the OxiSelect™ ROS Assay Kit (Cell Biolabs). The assay employs the conversion of a non fluorescent substrate, 2', 7'-Dichlorodihydrofluorescin diacetate (DCFH-DA), which can easily diffuse through cell membranes, into a fluorogenic molecule DCF (2', 7'-Dichlorodihydrofluorescein) which is highly fluorescent and proportional to total ROS level. DCF production was detected using a Fluoroskan Ascent fluorometer (Thermolabsystems) using excitation and emission wavelengths of 480 nm and 530 nm respectively and expressed as nM per hour. 11 Specific Hydrogen Peroxide (H 2 O 2 ): after incubation, the cells were stimulated to release the hydrogen peroxide produced by addition of 100 ng/ml phorbol-12-myristate-13-acetate (PMA, Sigma). The release of H 2 O 2 was measured as described by De la Harpe (De la Harpe and Nathan 1985). Briefly, KRPG buffer containing a mixture of scopoletin (30 µM), NaN3

(1 mM) and horseradish peroxidase (1 unit pupurogallin/ml HPO) was added to the cells. The horseradish peroxidase catalyzed the oxidation of the fluorescent scopoletin which was measured over a period of 90 min (Fluoroskan Ascent, Thermolabsystems) using excitation/emission wavelengths at 355/460 nm. Results are given as arbitrary units of the fluorescence.

Statistical analysis

Analysis and graphics were performed on Prism 5.0 software (GraphPad, San Diego, CA). Significance was established with ANOVA test (p<0.05).

RESULTS

Confocal microscopy phagocytosis assay

We first verified that the beads were internalized using confocal microscopy imaging (Fig. 1). Figure 1A and 1B confirm that the phalloidin labeling was highly efficient to label actin cytoskeleton (red), beads were easily viewable (green) and Hoechst specifically labeled the nuclei (blue).

Thanks to the XZ and YZ sections of confocal stacks illustrated in Figure 1 (C-a) and (Da), we demonstrated that beads were internalized beyond question. Indeed beads were surrounded by the phalloidin labeling of actin. On the contrary in Figure 1 (C-b) and (D-b), beads observed at the intersection of the white lines were just on the external side of the cell membrane showing the first step of internalization. Therefore, confocal microscopy allowed us to discriminate beads position (inside/outside).

The same types of results were seen with I beads (data not shown).

FCM phagocytosis quantification assay

The first parameter evaluated in FCM was the Side Scatter (SSC) reflecting morphological cellular complexity (Y Geo Mean). Figure 2 illustrates SSC cytograms (representative of three independent experiments) obtained for cells incubated alone or in presence of P or I beads. The SSC parameter dramatically increased for cells in contact with P beads (446±7 vs. 140±18 for cells alone) (Fig. 2A), whereas it didn't vary significantly for cells in contact with I beads (139±12 vs. 140±18). To verify that beads were internalized by phagocytosis, we used different inhibitors of this process (Cyt-D, Lat, Jasp) and studied the impact on the SSC parameter as shown by Fig. 2B. For cells in contact with P beads, Cyt-D and Lat led to a discrete decrease of the SSC parameter (446±7 vs. 245±2 and 243±2). In presence of Jasp or PFA the SSC parameter was significantly reduced (446±7 vs. 64±2 and 40±2). The same types of results were seen for cells in contact with I beads, Jasp and PFA induced a decrease of the SSC parameter (respectively 139±12 vs. 56±2 and 38±3) though with Cyt-D and Lat there was no effect on the SSC parameter (139±12 vs. 143±1 and 127±3).

In an attempt to quantify the phagocytic process we adapted to our model an existing analysis using trypan blue (TB) quenching to differentiate internalized beads from those just adhering to the cell membrane surface. First, cells were incubated without TB and the total Mean Fluorescence Intensity (MFI) reporting the fluorescence emitted by both ingested and membrane-associated (not-yet-ingested) beads was measured. Free beads have been eliminated by washing steps and were not counted as part of the MFI values. MFI measured for I beads was significantly lower than that measured for P beads (respectively 390±44 and 782±63) for the same ratio of beads (10/1¢). MFI decreased significantly for P beads in presence of Cyt-D (782±63 vs. 564±56) but was still equivalent for I beads (352±21). The MFI decreased sharply for both types of beads in presence of PFA (782±63 vs. 75±5 and 390±44 vs. 96±5). For the analysis with TB, the percentage of phagocytic cells was more important after incubation with P beads than with I beads (49±11% and 18±5% of phagocytic cells respectively). Treatment with Cyt-D resulted in a decrease in the percentage of phagocytic cells (16±3% vs. 49±11% for P beads and 10±1% vs. 18±5% for I beads).

According to these data ranges, figure 3 illustrates the results of one among the four independent experiments. Figure 3A shows results without TB (total MFI) and figure 3B presents values after TB quenching.

The effect of these inhibitors and that of Lat and Jasp are summarized in Figure 4. For P beads, Cyt-D and Jasp induced a decrease of the percentage of phagocytic cells close to 50%, whereas Lat seemed to have no impact. For I beads, all inhibitors seemed to have no effects.

Finally, whatever the beads employed, a PFA fixation significantly reduced the percentage of phagocytic cells < 2±0.6% (p<0.0001) (Figure 3B). 13 Moreover, we verified that the inhibitors (Cyt-D, Lat and Jasp) were not toxic at the concentrations used: the amount of dead cells determined by propidium iodide incorporation was found between 8 and 15% (data not shown). Therefore these inhibitors did not appear to be cytotoxic under these experimental conditions.

Toxicity assessment

The four toxicological parameters evaluated in this study are presented on Figure 5.

Fig 5A clearly

shows that the only significant variation in TNF-α concentration was observed when P beads were incubated for 24h with MA. There was no effect when cells were incubated with I beads, neither for 90 min nor for 24h, even for the 40 I beads for one cell ratio. These data could be correlated with the quantification of phagocytosis summarized in Table 1. P beads were more phagocytosed after 24h of incubation (74±5 vs. 49±11) and globally more phagocytosed than I beads (18±5 or 45±6). I beads at a 40/1¢ ratio for 24h were as phagocytosed as P beads at a 10/1¢ ratio after a 24h contact.

No effect was observed concerning the percentage of released LDH showed on Fig 5B .   All values were at the same level than the negative control cells alone compared to the positive control (lysed cells) which was significantly different.

The DCF production (nM/h) reflecting the amount of global ROS generated (Fig 5C )   indicates a significant enhancement after a 24h incubation for the two types of beads.

Correlated with

DISCUSSION

This study aimed at observing and quantifying phagocytosis using confocal microscopy coupled to FCM. Using these approaches we were able to distinguish entirely engulfed fluorescent micro-sized particles from those just adhering to the cell membrane and FCM appeared as a powerful tool to perform a multiparametric analysis of this cellular process.

Confocal microscopic experiments were very useful to visualize bead internalization. This complementary approach confirms the efficiency of the actin labeling with phalloidin which leads to a precise localization of beads throughout cells. A limit to this technique is that we could not quantify beads on images. Image analyzing techniques could be adapted to this type of acquisition but the area analyzed would be limited. Another technique allowing to quantify the physical process of internalization is necessary. This is the reason why we developed FCM in this study.

The multiparametric FCM approach we developed was shown to be a powerful tool for the quantification of phagocytosis. This technique allows to analyze an important number of cells without requiring the physical observation of events. Different steps have been necessary to validate this approach especially the use of different phagocytosis inhibitors. FCM also confirmed the effectiveness of the "TB quenching" for quantifying phagocytosis. The results obtained after TB quenching are in good agreement with the dynamics of the phagocytic process and validated the use of TB quenching for quantification in FCM. We have jointly observed that green fluorescence shut down by TB was detected on the FL3 channel (red fluorescence). Indeed, the TB was excited at 488 nm and emitted in the red spectrum. This red fluorescence quenching could be allowed by a process similar to FRET (fluorescence resonance energy transfer) [START_REF] Haas | The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurements[END_REF][START_REF] He | Monitoring caspase activity in living cells using fluorescent proteins and flow cytometry[END_REF].

SSC parameter reflects cellular complexity inside cells but also at the membrane surface, it appears to be a good indicator of cellular dynamics during the process of bead internalization. SSC values increased with the amount of internalized beads (P beads). Moreover, we observed SSC changes after treating cells with actin inhibitors used as controls.

This FCM technique allowed us to observe significant differences between the two types of beads which size and coating were different (P beads had a diameter of 2 µm and carboxylate surface groups while I beads had a diameter of 1 µm and amine surface groups).

We observed that with P beads the SSC parameter was strongly increased compared to control cells, we quantified 60% of phagocytic cells and the effect of actin inhibitors was important. This suggests that P beads are highly phagocytosed, much more than a same ratio of I beads (10/1¢) for which SSC values were equivalent to that of the control cells, only 20% of phagocytic cells were observed and for which the inhibitors had no effect.

We have shown that Cyt-D, Jasp and Lat did not completely block the internalization of the beads (inhibition of the phagocytosis of P beads up to 50% and no effect on I beads), suggesting that other internalization pathways may be involved and particularly a non actindriven process [START_REF] Aderem | Mechanisms of phagocytosis in macrophages[END_REF][START_REF] Castellano | Actin dynamics during phagocytosis[END_REF][START_REF] Etienne-Manneville | Rho GTPases in cell biology[END_REF][START_REF] Gratton | The effect of particle design on cellular internalization pathways[END_REF][START_REF] Haberzettl | Actin plays a crucial role in the phagocytosis and biological response to respirable quartz particles in macrophages[END_REF][START_REF] Lanzetti | Actin in membrane trafficking[END_REF]Niedergang andChavrier 2004, 2005;[START_REF] Yeung | Lipid signaling and the modulation of surface charge during phagocytosis[END_REF]. It would be interesting to explore other pathways such as energy-dependent phagocytosis, clathrin-mediated endocytosis, caveolaemediated endocytosis or macropinocytosis [START_REF] Gratton | The effect of particle design on cellular internalization pathways[END_REF]. Finally, the study of membrane receptors such as the Scavenger Receptors (SR) MARCO (Macrophage Receptor with collagenous structure) and the complement receptor 3 (CR3) implicated in these pathways could also bring further information [START_REF] Arredouani | The scavenger receptor MARCO is required for lung defense against pneumococcal pneumonia and inhaled particles[END_REF][START_REF] Arredouani | MARCO is the major binding receptor for unopsonized particles and bacteria on human alveolar macrophages[END_REF][START_REF] Groves | Molecular mechanisms of phagocytic uptake in mammalian cells[END_REF][START_REF] Hamilton | MARCO mediates silica uptake and toxicity in alveolar macrophages from C57BL/6 mice[END_REF][START_REF] Kanno | A murine scavenger receptor MARCO recognizes polystyrene nanoparticles[END_REF][START_REF] May | Involvement of the Arp2/3 complex in phagocytosis mediated by FcgammaR or CR3[END_REF][START_REF] Palecanda | Receptors for unopsonized particles: the role of alveolar macrophage scavenger receptors[END_REF][START_REF] Sulahian | Signaling pathways required for macrophage scavenger receptor-mediated phagocytosis: analysis by scanning cytometry[END_REF].

Taken together, these results underline the importance of the physicochemical properties of the particles for the phagocytic process (size and surface groups) [START_REF] Ahsan | Targeting to macrophages: role of physicochemical properties of particulate carriers-liposomes and microspheres--on the phagocytosis by macrophages[END_REF].

Further investigations are needed to determine if these two parameters have the same influence or if one prevails and which one (using beads of different size with the same surface groups or conversely beads of same size but carrying different surface groups like carboxylate, amine and polyethylene glycol).

Similarly, these characteristics could have a major impact on the biological toxicity.

Toxicological evaluations show that in a global manner, ROS and LDH productions do not depend on the size and chemical composition of beads. Generation of global ROS seems more dependant of time incubation and amount of internalized beads. On the contrary, these parameters play a significant role in TNF-α inflammatory response: carboxylate 2 µm beads (P beads) dramatically increase TNF-α production [START_REF] Wilson | Nanoparticle interactions with zinc and iron: implications for toxicology and inflammation[END_REF], whereas amine 1 µm beads (I beads) seem to have no impact (10/1 and 40/1 ratio). These results fit adequately since we demonstrated that P beads were more phagocytosed than I beads (10/1 ratio) but as much phagocytosed as I beads for a 40/1 ratio reflecting equivalence in surface area. Thus relationship between surface chemistry and TNF-α production is clear.

Conclusion

We established a multiparametric method of analysis of phagocytosis coupling flow cytometry to confocal microscopy. This approach allowed to highlight that physicochemical characteristics of beads (size, surface area and surface coating) were key factors of the internalization process and that they also have an impact on toxicity.

DECLARATION OF INTEREST

The authors report no declarations of interest.

FIGURE CAPTIONS

Finally H 2

 2 O 2 production did not show significant differences among the different conditions of exposure (Fig 5D). After a 24h exposure ROS are generated probably due to H 2 O 2 which are likely the source of other organo-peroxydes or lead to formation of hydroxides (Fig 5C).

Figure 1 :

 1 Figure 1: Confocal microscopy images of MA incubated with beads (10/1¢ ratio) for 16 hours. A-B) XY projection stack mean of the interest area with phalloidin and Hoechst labeled MA in contact with P beads. C-D) XZ and YZ section stacks with observed beads at the intersection of the white lines (a: bead inside, b: bead outside).

Figure 2 :

 2 Figure 2: SSC parameter analyzed for MA alone or after a 90 minutes incubation with P or I beads (10/1¢ ratio) (A), and in presence of inhibitors of phagocytosis (Cyt-D, Lat and Jasp) or after PFA fixation (B). The value indicated on each cytogram corresponds to the Y Geo Mean which represents cellular complexity (representative of three independent experiments).

Figure 3 :

 3 Figure 3: FCM analysis of cells without (A) or with trypan blue (B) after a 90 minutes incubation with P and I beads (10/1¢ ratio) in presence or absence of Cyt-D and PFA. A) Total MFI is represented on the FL1 channel (green fluorescence of all beads). B) FL1 channel detects the green fluorescence of beads internalized by MA. After TB quenching non internalized beads emit a red fluorescence that can be detected on the FL3 channel allowing to distinguish phagocytic (FL1+/FL3-) from non phagocytic (FL3+) cells (illustration representative of four independent experiments).

Figure 4 :Figure 5 :

 45 Figure 4: Effects of inhibitors of phagocytosis (Cyt-D, Lat and Jasp) and PFA fixation on the MA in contact with P or I beads (10/1¢ ratio) analyzed in FCM after 90 min of exposure. The percentage of phagocytic cells is normalized with the control (n=4, *p<0.0001).

  le

Figure 1 :

 1 Figure 1: Confocal microscopy images of MA incubated with beads (10/1¢ ratio) for 16 hours. A-B) XY projection stack mean of the interest area with phalloidin and Hoechst labeled MA in contact with P beads. C-D) XZ and YZ section stacks with observed beads at the intersection of the white lines (a: bead inside, b: bead outside).

Figure 2 :

 2 Figure 2: SSC parameter analyzed for MA alone or after a 90 minutes incubation with P or I beads (10/1¢ ratio) (A), and in presence of inhibitors of phagocytosis (Cyt-D, Lat and Jasp) or after PFA fixation (B). The value indicated on each cytogram corresponds to the Y Geo Mean which represents cellular complexity (representative of three independent experiments).

Figure 3 :

 3 Figure 3: FCM analysis of cells without (A) or with trypan blue (B) after a 90 minutes incubation with P and I beads (10/1¢ ratio) in presence or absence of Cyt-D and PFA. A) Total MFI is represented on the FL1 channel (green fluorescence of all beads). B) FL1 channel detects the green fluorescence of beads internalized by MA. After TB quenching non internalized beads emit a red fluorescence that can be detected on the FL3 channel allowing to distinguish phagocytic (FL1+/FL3-) from non phagocytic (FL3+) cells (illustration representative of four independent experiments).

Figure 4 :

 4 Figure 4: Effects of inhibitors of phagocytosis (Cyt-D, Lat and Jasp) and PFA fixation on the MA in contact with P or I beads (10/1¢ ratio) analyzed in FCM after 90 min of exposure. The percentage of phagocytic cells is normalized with the control (n=4, *p<0.0001).

Figure 5 :

 5 Figure 5: Toxicity assessment for the different conditions : 10/1¢ ratio for P and I beads (after 90 min or 24h contact with RAW 264.7 cells) and 40/1¢ ratio for I beads (after 24h contact) in terms of inflammation assessed by TNF-α production (pg/ml) (A), cell damage measured by the amount of LDH released (B), total ROS (C) and H2O2 specific production (D) (n=3, *p<0.0001).

Table 1 :

 1 Summary data of quantification parameters obtained in FCM (SSC, total MFI and percentage of phagocytic cells) for the different exposure conditions : 10/1¢ ratio for P and I beads after a 90 min or 24h contacts with RAW 264.7 cells and 40/1¢ ratio for I beads after a 24h contact.

Table 1 ,

 1 these data indicate that ROS generation is linked to the amount of phagocytosis, and incubation time (24h).