J Morán
email: morancoj@coria.fr

A Fuentes

F Liu

J Yon

FracVAL: An Improved Tunable Algorithm of Cluster-Cluster Aggregation for Generation of Fractal Structures Formed by Polydisperse Primary Particles

Keywords: Tunable Algorithm, Fractal Aggregates, Polydisperse Primary Particles, Cluster-Cluster Aggregation Program Programming language: Fortran 90 Computer: PC Operating system: Windows and Linux RAM: 1.0 Gb Classification: Nature

 for generating fractal aggregates formed by monodisperse spherical primary particles R1.C5 is extended to polydisperse primary particles. This new algorithm, termed FracVAL R1.C5 , is developed by using an innovative aggregation strategy. R1.C0 The algorithm is able to preserve the prescribed fractal dimension (D f ) R1.C4 and prefactor (k f ) R1.C4 for each aggregate, regardless of its size, with negligible error for lognormally distributed primary particles with the geometric standard deviation σ p,geo being as large as 3. In contrast, for polydisperse primary particles the direct use of Filippov et al. 2000 method, as is done by Skorupski et al. 2014, does not ensure the preservation of D f and k f for individual aggregates and it is necessary to generate a large number of aggregates to achieve the prescribed D f and k f on an ensemble basis. The performance of FracVAL is evaluated for aggregates consisting of 500 and 1000 R2.C4 monomers and for fractal dimension variation over the entire range of D f between 1 and 3 and k f between 0.1 and 2.7 R1.C5 . Aggregates consisting of 500 monomers R2.C4 are generated on average in less than 2.4 minutes on a common laptop, illustrating the efficiency of the proposed algorithm.

Introduction

Fractal-like aggregates formed by nearly spherical primary particles are frequently encountered in many applications, such as in colloidal or aerosol systems, combustion systems, and flame synthesis of functional nanoparticles. Such particles have some remarkable and unique characteristics, e.g. large surface area and porous geometry, implying that they have high absorption capacity, good catalyst performance, fast dissolution, light-weight, and require relatively little solid material to occupy large space [START_REF] Babick | Suspensions of colloidal aggregates[END_REF]. These particles are generated by aggregation processes which can be theoretically classified into two categories, particle-cluster [2, 3,4,5,6] (PC) and clustercluster (CC) aggregation. A complete review of different algorithms focused on these aggregation mechanisms is found in [7]. Additionally, optimized R1.C5 version of these algorithms can be found in more recent studies [8,9,10].

Under the idea of fixing the fractal dimension, Thouy and Jullien [11] introduced the first tunable CC aggregation algorithm. Subsequently, several studies are conducted to propose different tunable algorithms for fractal aggregate generation [12,13,14,15,16], though most of these studies preserved only the fractal dimension. One remarkable exception is the algorithm developed by Filippov et al. [13], which is able to preserve both the fractal dimension and the prefactor, paving the way to investigate the individual effect of D f and k f on the morphology and physical properties of fractal aggregates [17,18,19]. Some distinct features of tunable algorithms can be highlighted: [START_REF] Babick | Suspensions of colloidal aggregates[END_REF] they allow systematic studies of the individual effects of either D f or k f on the physical properties of fractal aggregates; (2) a large number of aggregates can be generated numerically with a considerably low computational time; and (3) they allow the generation of fractal aggregates with prescribed D f and k f over a wide range irrespective of the physical aggregation mechanism. Almost all the existing tunable algorithms developed in the literature rely on the assumption of monodisperse primary particles. One exception is the code developed by Skorupski et al. [20], which is based on the method of Filippov et al. R1.C5 [13] and can be used to generate fractal aggregates formed by polydisperse spherical primary particles. However, the direct application of the Filippov et al. R1.C5 [13] algorithm to generate fractal aggregates formed by polydisperse primary particles, as conducted by Skorupski et al. [20], encounters the difficulty that the resultant individual aggrgates do not preserve the prescribed fractal dimension. It is important to overcome this issue and to develop a tunable algorithm to efficiently generate fractal aggregates formed by polydisperse primary particles, which can then be used to fill some existing gaps with regard to the effect of primary particle polydispersity on morphological properties, such as the center of mass, inertia moment, radius of gyration of sub-clusters being aggregated and consequently on the resulting structures [21,22], as well as various physical and optical properties.

Since practical colloid or aerosol aggregates consist of polydisperse primary particles, various studies have been conducted in order to assess its influence on their morphological characterization [23,24,25,21,22], and the physical or chemical properties of such particles, such as kinetics of coagulation and sintering or coalescence, light scattering, mobility and settling [26,27,28,29,30,31,32,33,22,[START_REF] Yon | Proc. Combust. Inst[END_REF][START_REF] Spyrogianni | [END_REF]. In 2012 Eggersdorfer and Pratsinis [21] found found a dependency between aggregates morphology and monomers polydispersity. R1.C0 Based on the above mentioned studies R1.C0 , it is important to develop efficient tunable algorithms to generate fractal aggregates formed by polydisperse PP with known D f and k f . In this study, an improved hierarchical tunable algorithm of CC aggregation is developed based on the algorithm of Filippov et al. [13] to generate numerically fractal aggregates formed by polydisperse PPs.

Theoretical background

Primary particle size distribution

The nearly spherical primary particles constituting fractal aggregates, such as soot, fumed silica and titania, encountered in practice are always polydisperse and the primary particle size distribution (PPSD) can be commonly described by either the normal or lognormal probability density functions [36,37]. Nevertheless, the normal distribution is not practical because for large standard deviations of the distribution it can lead to negative PP radii. Eq. ( 1) presents the lognormal probability density function of PP radii,

f [r p ] = N ln[σ p,geo ] √ 2πr p exp - 1 2 ln[r p ] -ln[r p,geo ] ln[σ p,geo ] 2 , (1)
where f [r p ]dr p is the probability of finding a particle with a radius between r p and r p +dr p , r p,geo and σ p,geo correspond to the geometric mean and geometric standard deviation, respectively. Please note that for a population of N monomers these values are given by the following equations, R2.C1 log(r p,geo ) =

N i=1 log(r i ) N (2a) log(σ p,geo ) = N i=1 (log(r i ) -log(r p,geo )) 2 N (2b)
It can be checked that for monodisperse primary particles the RHS of Eq. (2b) becomes zero and therefore σ p,geo = 1. R2.C1 Although the level of PP polydispersity is often fairly low with σ p,geo ≤ 1.3, highly polydisperse primary particles can be encountered. For example, for flame generated soot aggregates, the geometric standard derivation σ p,geo can be as high as σ p,geo = 2.1 [36,37,38,39], depending on the residence time, fuel type, and flame conditions.

Characterization of fractal aggregates

The mass of a fractal aggregate (m a ) follows the fractal scaling law. For an aggregate consisting of point-touch polydisperse spherical PPs [21],

m a m p = k f R g r p,geo D f , (3)
where m p is the average PP mass, i.e., m a /m p = N , is the number of PP in the aggregate. The exponent D f and the proportionality constant k f in Eq. ( 3) are the fractal dimension and prefactor, respectively. For fractal aggregates consisting of monodisperse monomers (σ p,geo = 1) generated through diffusion-limited cluster aggregation (DLCA), it has been well established that D f ≈ 1.78 and k f ≈ 1.40 [7,40]. However, Eggersdorfer and Pratsinis [21] showed a dependency of both D f and k f on PP polydispersity.

In fact, for aggregates formed in the same molecular regime they found that both D f and k f decrease with increasing PP polydispersity. They obtained D f = 1.68 and k f = 0.98 for σ p,geo = 2.0, and D f = 1.48 and k f = 0.77 for σ p,geo = 3.0. The ranges of these parameters are used for the generation and analysis of fractal aggregates of the present work. Finally, R g is the radius of gyration of the aggregate calculated using the expression proposed by [22],

R 2 g = 1 m a N i=1 m p,i (R i -R c ) 2 + r 2 g,i , (4)
where r p,i and m p,i are respectively the PP radius and mass of the ith spherical primary particle, which is located at a distance R i from the origin of a fixed coordinate system. The term r g,i is the radius of gyration of the ith PP, i.e., r 2 g,i = (3/5)r 2 p,i . R c is the center of mass of the aggregate evaluated as,

R c = 1 m a N i=1 m p,i R i . (5)
It can be easily shown that Eqs. ( 4) and ( 5) degrade to those for monodisperse primary particles given in Filippov et al. [13] when all the primary particles have the same mass.

3. Formulation of the improved tunable algorithm

Main equations

Consider two clusters (aggregates) with mass m 1 and m 2 and radius of gyration R g1 and R g2 , which have the same prescribed D f and k f and are to be aggregated to form a larger aggregate with mass m = m 1 + m 2 and radius of gyration R g and to preserve both D f and k f . To this end, it can be shown that the distance between the mass centers of the two clusters Γ satisfies the following equation,

m 2 R 2 g = m m 1 R 2 g1 + m 2 R 2 g2 + Γ 2 m 1 m 2 , (6)
The consequence and advantage of using Eq. ( 6), as it will be demonstrated later, are that it ensures the preservation of both D f and k f R1.C5 during each step of the aggregation process for each individual aggregate generated. The derivation of this equation is provided in Appendix A. In the special case of two aggregates consisting of N 1 and N 2 monodisperse PPs (N = N 1 + N 2 ), Eq. ( 6) is reduced to,

N 2 R 2 g = N N 1 R 2 g1 + N 2 R 2 g2 + Γ 2 N 1 N 2 , (7)
which is identical to the relationship derived by Filippov et al. [13], i.e., the derived Eq. ( 6) is the generalized form of Eq. ( 7) for aggregates consisting of polydisperse PPs. Eq. ( 6) represents an original contribution of the present study and forms one of the main relations of the present tunable algorithm.

FracVAL: A tunable cluster-cluster aggregation algorithm

The FracVAL (fractal aggregate generation algorithm developed in Valparaíso) algorithm is programmed in a hierarchical manner, i.e., only aggregation between sub-clusters with approximately the same number of primary particles is allowed [11]. The algorithm consists of four main steps corresponding to the flow chart shown in Fig. 6 and it is described as follows.

Step 1: The fractal dimension D f (between 1 and 3) and the fractal prefactor k f (a positive value, typically on the order of unity) of aggregates to be generated are prescribed, which will be preserved during the entire fractal aggregate generation process. In addition, the primary particle radius distribution and the aggregate size N (the number of primary particles contained in the aggregate) are also required. The PPSD is assumed lognormal and characterized by the geometric mean and standard standard deviation, i.e., r p,geo and σ p,geo . To obtain a total of N radii from this lognormal distribution given by Eq. ( 1), a pseudo-random number generator is used [START_REF] Forbes | Statistical distributions[END_REF]. To avoid extremely large and extremely small values of PP radius, which are unrealistic, the selection of PP radii is constrained to the following range [r p,geo /σ 2 p,geo , r p,geo σ 2 p,geo ], which contains 95.5% of the radii.

Step 2: A particle-cluster aggregation algorithm is first used to obtain a total of I t sub-clusters (smaller aggregates) consisting of approximately the same number of primary particles (N sub ), where It N sub = N . This criterion is introduced in order to mimic the hierarchical aggregation of particles that possess the self-similarity of fractal aggregates [42]. In the present work N sub is selected depending on N. For N∈[50,500], each sub-cluster R1.C5 consists of N sub = 0.1N and under the lower limit (N≤ 50 ) a constant N sub = 5 is employed. On the other hand, above the upper limit (N≥ 500) a constant N sub = 50 is R1.C5 used. This type of particle-cluster aggregation algorithms has been found to experience some difficulties for retaining exactly the fractal parameters and therefore small N sub are recommended in the literature in order to avoid effects in the final aggregates [13,20]. In this context, a sensitivity analysis is included in the Appendix C. The main conclusion of this analysis is that for N sub < 0.15N there is not effect on the morphology of final aggregates. Additionally, an variation of ±20% of D f or k f for the initial sub-clusters is expected to have a relatively small effect on the densitydensity correlation function of the final aggregates. R1.C4

Step 3: In this step, the sub-clusters generated in Step 2 are organized into pairs that are able to aggregate to form a larger cluster. This "ability to aggregate" is defined according to the criterion R max,1 + R max,2 ≥ Γ 12 , where R max,i corresponds to the furthest distance of a PP to the center of mass of the ith aggregate (i = 1,2) and Γ 12 is the distance between the centers of mass of the two sub-clusters to be aggregated and calculated as,

Γ 12 = 1 √ m 1 m 2 m 2 R 2 g -m m 1 R 2 g1 + m 2 R 2 g2 1/2 , (8)
where m 1 , m 2 , R g1 and R g2 are the masses and radii of gyration of the two sub-clusters being aggregated. Additionally, m = m 1 + m 2 and R g are the corresponding parameters for the resulting larger aggregate. The radii of gyration R g1 , R g2 and R g are calculated from the scaling law,

R gi = r p,geo n i k f 1/D f (9)
where n i is the number of monomers of the ith aggregate (here for both the sub-clusters and the resultant larger aggregate) and D f and k f are the prescribed fractal parameters.

Step 4: This is the main part of the algorithm. In this step, the two sub-clusters belonging to the same pair determined in Step 3 are aggregated to form a larger aggregate. It is based on the idea that there are two subclusters, from here onwards they will be called sub-cluster A1 (containing N 1 monomers) and sub-cluster A2 (containing N 2 monomers), that can be aggregated in many different ways, each of these ways associates a distance between the center of mass of aggregates A1 and A2, but there is only one distance (represented by Γ 12 in Eq. ( 8)) that ensures the preservation of both D f and k f . Therefore, knowing this distance, our goal only consists in finding one of the possible paths to combine the clusters by keeping the distance between them fixed. R2.C3 To better describe this step, it is divided into 6 sub-steps as shown in Fig. 6.

• Sub-step a: Select a pair of sub-clusters A1 and A2 R2.C3 from the pairs determined in Step 3.

• Sub-step b: Create a binary matrix a ij . Let d i1 and d j2 be respectively the distance from the center of monomer i (in A1, i = 1,2,...,N 1 ) and j (in A2, j = 1, 2,..., N 2 ) to the center of mass of A1 and A2, respectively.

Assign a ij = 1 when: The restriction defined by Eq. ( 10) is fulfilled, this restriction ensures that both monomers are close enough to be in point-touching,

A1 A2 Monomer i Monomer j CM 1 CM 2 r pi r pj
D i1,+ + D j2,+ ≥ Γ 12 , (10)
where

D i1,+ = d i1 + r pi , D i1,-= d i1 -r pi , D j2,+ = d j2 + r pj and D j2,-= d j2 -r pj .
Additionally, there is a lower limit defined by 3 possible cases described as follows,

-Case 1: When the spheres of radius D i1,+ and D j2,+ can be intersected (see Fig. 2), this means that

|D i1,+ -D j2,+ | ≤ Γ 12 .
The two following cases are associated with the intersection of the spheres with radius D i1,-and D j2,-.

-Case 2: When the sphere of radius D i1,+ is too big containing the sphere of radius D j2,+ , i.e. when (D i1,+ -D j2,+ ) > Γ 12 with an R1.C5 upper limit D i1,-≤ Γ 12 + D j2,+ .

-Case 3: Analogously, when the sphere of radius D j2,+ is too big containing the sphere of radius D i1,+ , i.e. when (D j2,+ -D i1,+ ) > Γ 12 with an upper limit D j2,-≤ Γ 12 + D i1,+ .

Please note that Case 2 and Case 3 are theoretically and mathematically possible, and they are considered in FracVAL. Please see the Appendix D for a further explanation of Cases 2 and 3. R1.C1 However, Case 1 is considered the most common one, and therefore it is used to explain the subsequent steps of the algorithm.

These restrictions ensure that ith monomer belonging to A1 is able to aggregate with the jth monomer belonging to A2; otherwise, assign a ij = 0. For the pair A1 and A2, Γ 12 is evaluated using Eq. (8). • Sub-step c: Loop over the elements of the binary matrix a ij . If a ij = 0, continue the loop. If a ij = 1, assign the ith primary particle in A1 as s1 and the the jth primary particle in A2 as s2.

CM 1 PP i D i1,+ CM 2 PP j Γ 12 D i1,- D j2,+ D j2,-
The goal of Step 4 is to aggregate the sub-clusters A1 and A2. Specifically, the algorithm needs to find the locations of the mass centers of A1 and A2 and the proper orientations of A1 and A2 upon aggregation to satisfy the following restrictions:

First, the distance between the centers of mass of A1 and A2 is Γ 12 given by R1.C5 Eq. ( 8). Secondly, the selected primary particle candidates s1

(belonging to A1) and s2 (belonging to A2) are in point-touch. Thirdly, there is no overlapping between any primary particles in A1 and A2.

The above mentioned 3 restriction will be satisfied progressively by 3 stages described as follows, R1.C2

-Stage 1: To satisfy restriction 1 we retain the center of mass of A1 (referenced as CM1 = (X cm,1 , Y cm,1 , Z cm,1 )) fixed while the center of mass of A2 (referenced as CM2 = (X cm,2 , Y cm,2 , Z cm,2 )) is placed at a distance Γ 12 from CM1. This displacement is made along the unitary direction defined by CM1 and the center of s1. At this point we already fulfil restriction 1. This is shown in Fig. 3. -Stage 2: Next, we will rotate A1 and A2 in the following manner to fulfill restrictions 2 and 3. Firstly, A1 is rotated, to this end we calculate the distances d s1 and d s2 from monomers s1 and s2 to CM1 and CM2, respectively, and place the monomer s1 at a tangential point defined by the interception of two of the following spheres as illustrated in Fig. 4.

R1.C2 CM 1 s1 Γ 12 A1 A2 CM 2
R1.C2 D 2 s1,+ = (X -X cm,1 ) 2 + (Y -Y cm,1 ) 2 + (Z -Z cm,1 ) 2 , ( 11a
)
D 2 s2,+ = (X -X cm,2 ) 2 + (Y -Y cm,2 ) 2 + (Z -Z cm,2 ) 2 , (11b)
This random point can be determined analytically by considering the following procedure (Here the reader may go directly to Stage 3 without loss of continuity). As shown in Fig. 5, the sphere-sphere where c and ρ are the coordinates of the geometric center and the radius of this circle, respectively. Additionally, î and ĵ are

CM 1 s1 Γ 12 A1 A2 CM 2 s2 D s1,+ D s2,+
Step b:

Matrix of candidates

Step d: Aggregation process Overlapping?

Step e: Rotation YES NO

Rotations >MAX? YES

Step 1: Parameters

Step 2: Sub-clusters (PC)

Step 3:

Generate pairs

Step 4:

Aggregate pairs

Step c: Select a pair s1-s2

Step a: Select a pair A1-A2

Step 4 details -Stage 3: The location of s2 is determined by finding a random point in the interception of the two spheres, R1.C2

(r s1 + r s2 ) 2 = (X -X s1 ) 2 + (Y -Y s1 ) 2 + (Z -Z s1 ) 2 , ( 13a
)
d 2 s2 = (X -X cm,2 ) 2 + (Y -Y cm,2 ) 2 + (Z -Z cm,2 ) 2 , ( 13b
)
where Eq. (13a) ensures that both candidates s1 and s2 will be in one point touch and Eq. (13b) ensures that candidate s2 will retain their relative distance to the center of mass of aggregate A2 and therefore restrictions 1 and 2 mentioned above will be fulfilled. This is illustrated in Fig. 7. R1.C2 One point in the in- tersection of these two spheres is found using the same procedure described by equation (12). Subsequently the whole aggregate A2 is rotated based in the rotation of s2 by using the Euler-Rodriguez model [43]. At this point, restrictions 1 and 2 are ensured. Restriction 3 is not necessarily ensured, so the algorithms continue with sub-step d to check it. R1.C2

s1 Γ 12 A1 A2 CM 2 s2 d s2 r s1 +r s2
• Sub-step d: Eventually, in the current orientation of A1 and A2 overlapping between PP exists. In this case the orientation of A2 is modified by random rotation by using the Euler-Rodriguez formula [43] and using the solution of the intersection of spheres given by Eq. (13a) and Eq. (13b), i.e. monomer s2 is rotated around monomer s1.

In case that overlapping still exists (following a maximum number of iteration defined by the user), the algorithm return to sub-step c and it picks another pair of monomers s1 and s2. Finally at this point, restrictions 1, 2 and 3 are ensured.

Density-density correlation function

It is expected that the spatial distribution of the mass of an aggregate should follow a specific behavior to be considered as fractal [44]. The density-density correlation function is usually examined to test if the generated aggregates possess the expected fractal behavior in addition to the scaling law [45,13]. For the particular case of aggregates formed by R2.C5 polydisperse primary particles, Bushell and Amal [24] suggested to use the partial distance distribution function, which unfortunately is not practical when dealing with lognormally distributed primary particles, this because there are too many different radii to obtain a partial distribution function as proposed in the mentioned reference. Therefore, we developed a new alternative to evaluate it. Let f (r) be the density-density correlation function evaluated at a distance r from the mass center of the aggregate. This value is calculated in an iterative manner for radius in the range r ∈ [R min , R max ],

where R min = r p,geo /10 and R max = δR g , i.e., δ times the radius of gyration of the aggregate. Normally, a value of δ = 3.5 is considered sufficiently large.

At the kth iteration, radius r is discretized as follows,

r (k) = R min (R max /R min ) k/(n it -1) . ( 14
)
where k ∈ [1, (n it -1)], with n it being a total number of discretized radii considered. For a given k, a "copy" of the aggregate is generated and displaced at a distance r (k) from the mass center of the original aggregate in a random direction (i.e., a random point in a sphere). Then the volume of intersection of both aggregates

V (k)
int is analytically determined based on the concept of spherical cap [46]. Hence, this process is repeated until a total of n or orientations is evaluated. Finally, the density-density correlation function is calculated by averaging the volumes of intersection over all orientations and dividing by the total volume of the aggregate V a , and using the algorithm developed by Skorupski et al. [20] are compared in Fig. 10. It is noticed that these aggregates are generated from monodisperse primary particles.

f (r (k) ) = 1 nor nor V (k) int V a , (15) 4
The density-density correlation function f (r) corresponds to the spatial mass distribution of the aggregate, and it is supposed to exhibit a slope of As can be seen in Fig. 10, the agreement in f (r) for aggregates generated from both algorithms is very good, especially for large N . In addition, the expected slope of the decay of f (r) with r for sufficiently large N , i.e., D f -d, is progressively displayed with increasing N , consistent with the findings of

Aggregates formed by polydisperse primary particles

In Fig. 11 polydispersity [21]. As expected, the agreement is better for N = 1024 rather than N = 100, consistent with the results shown for monodisperse primary particles in Fig. 10.

Preservation of prescribed fractal parameters

An important feature of tunable algorithms is to ensure each individual aggregate generated accurately satisfies the scaling law expressed in Eq. ( 9), regardless of its size. Specifically, each individual aggregate generated by a tunable algorithm possesses the same D f and k f as the prescribed values.

This requirement is naturally built into the algorithm through the application of Eq. ( 9) during the generation.

To probe the preservation of fractal parameters (D f and k f ) on a global sense, i.e., to examine an ensemble of aggregates of different sizes, plots of log(N ) -log(R g /r p,geo ) for different prescribed fractal dimension and pref- As can be seen in Fig. 13(a) for all thee three sets of aggregates formed by monodisperse primary particles, the fractal scaling law is very accurately preserved and the fractal dimension based on the slope of the linear fit deviates from the prescribed value by less than only 1%. The same observations can be made for aggregates formed polydisperse primary particles as shown in Fig. 13(b). It is worth pointing out that these pairs of D f and k f are selected to represent the most extreme values under which FracVAL is able to generate the fractal aggregates for the evaluated PP polydispersity (σ p,geo = 2).

The results shown in Fig. 13 confirm that the ensemble of aggregates gener-ated by FracVAL strictly preserves the prescribed fractal parameters under different conditions in terms of fractal parameters (D f and k f ) and the level of primary particle polydispersity as long as the fractal aggregates can be generated by FracVAL. Once these aggregates are generated, numerical projections, like numerical TEM images [46], could be generated in order to relate the projected density-density correlation function as calculated by Nelson et al. [47] for numerically generated aggregates in comparison with experimentally measured ones. This feature will be available in a future version of FracVAL. R1.C3

Discussion

To demonstrate the advantage of FracVAL algorithm over the classical A total of 200 aggregates is generated using the method of Filippov et al. [13] for each set of parameters (D f , k f , N , and σ p,geo ). As can be seen, when primary particles are polydisperse, the classical Filippov et al. [13] does not preserve exactly the fractal parameters (D f and k f ) for each individual aggregate, though the fractal parameters are preserved on an ensemble basis.

Therefore, a large population of aggregates should be generated if one uses the Filippov et al. [13] R1.C4 method or the FLAGE code to generate aggregates formed by polydisperse primary particles. It is interesting to observe that the scatter of log(R g /r p,geo ) at a given N becomes increasingly smaller and the agreement between Filippov et al. [13] and FracVAL R1.C5 is improved with increasing N . This agreement is employed in previous studies presented in the literature [46,32]. Finally, the computational performance of the FracVAL algorithm is evaluated. The computations are carried out using a laptop equipped with a 2.40

GHz CPU and 8 Gb of memory. The reported CPU times are averaged over a population of 100 aggregates for different combinations of D f and k f within the range of D f = 1.0 -3.0 and k f between 0.1 and 2.4. The computational time is dependent on the number of monomers, the level of primary particle polydispersity, and the combination of D f and k f . Table 1

Conclusions

A novel tunable cluster-cluster aggregation algorithm, FracVAL, is developed in this study to numerically generate fractal aggregates formed by polydisperse primary particles. The algorithm represents an improvement R2.C4 of the tunable cluster-cluster aggregation algorithm of Filippov et al. [13] for generation of fractal aggregates formed by monodisperse primary particles.

The algorithm is validated by comparing the density-density correlation functions of fractal aggregates generated by FracVAL and a literature method, FLAGE [20], and excellent agreement is observed. The prescribed fractal parameters D f and k f are found accurately preserved for each individual aggregate consisting of either monodisperse or polydisperse primary particles.

The fractal properties of an ensemble of aggregates of different sizes are also found to accurately preserve the prescribed D f and k f with less than 1% deviation for different combinations of D f and k f , regardless of the degree of primary particle polydispersity characterized by σ p,geo up to 3.

Since the positions of the two aggregating primary particles and the mass centers of the two aggregating clusters are found using analytical expressions by using the intersection of two spheres and random parameters, FracVAL • Please note that the intersection of spheres 2 and 4 is not possible and therefore it does not lead to any possible solution.

• Scenario 1: When Γ 12 ≥ |D s1,+ -D s2,+ |. In this case (considered

Figure 1 :

 1 Figure 1: Sub-clusters A1 and A2 selected to be aggregated.

Figure 2 :

 2 Figure 2: Example of intersection between the spheres of radius D i1,+ and D j2,+ can be intersected.

Figure 3 :

 3 Figure 3: Displacement of A2 to fulfill restriction (1).

Figure 4 :

 4 Figure 4: Location of monomer s1 given by the intersection of spheres given by Eq. (11a) and Eq. (11b).

Figure 5 :

 5 Figure 5: Sphere-sphere intersection and circle parametrization.

Finish Step 4 NOFigure 6 :

 46 Figure 6: Flow chart of FracVAL cluster cluster aggregation algorithm.

Figure 7 :

 7 Figure 7: Intersection of spheres given by Eq. (13a) and Eq. (13b).

Figure 8 Figure 8 :

 88 Figure 8 shows the histogram of primary particle radii for an aggregate consisting of N = 1024 primary particles sampled from a lognormal distribution with σ p,geo = 2.0 and r p,geo = 15.0 nm. It is evident that the specified PPSD iss well preserved. When the Kolmogorov-Smirnov goodness-of-fit test (D) is applied, a value of D = 0.03 is obtained. A significance of α = 0.01 corresponds to a critical value of C α=0.01 = 0.05. Since D < C α=0.01 , the radius of the primary particles of the aggregate effectively satisfies the specified lognormal size distribution.

.2. 1 .

 1 Aggregates formed by monodisperse PPIn Fig.9samples of the generated aggregates consisting of monodisperse primary particles are shown. These aggregates are generated for three different fractal dimensions of D f = 1.40, 1.79, and 2.40 and four aggregate sizes of N = 20, 50, 500, and 1024. It is evident that a larger fractal dimension leads to a more compact structure. As a way to validate our algorithm, the density-density correlation function f (r) is calculated as described above considering a total of n or = 300 orientations for each case.The calculated density-density correlation functions f (r) for different aggregate sizes from N = 20 up to N = 1024 generated using FracVAL R1.C5

Figure 9 :

 9 Figure 9: Selected aggregates generated by FracVAL with monodisperse PP (σ p,geo = 1.0) for D f = 2.40, k f = 0.80 (top row), D f = 1.79, k f = 1.40 (middle row), and D f = 1.40, k f = 1.80 (bottom row).

FilippovFracVALFigure 10 :

 10 Figure 10: The density-density correlation functions of aggregates formed by monodisperse primary particles with D f = 1.79, k f = 1.40. The solid lines correspond to the aggregates generated by using R1.C5 the algorithm of Skorupski et al. 2014 and symbols are for aggregates generated by R1.C5 using FracVAL. d stands for the Euclidean dimension of space d = 3.

  Fig. 12(a) and Fig. 12(b) show the density-density correlation function of fractal aggregates of N = 100 and 1024, respectively, generated using FracVAL for polydispersity levels ranging from σ p,geo = 1.0 (monodisperse) to σ p,geo = 3.0 (extremely polydisperse) and for different fractal properties as shown in the figure legend. As can be seen, the expected power law behavior becomes more evident, i.e., the linear decay of f (r) with r in the log-log scale plot, especially for N = 1024. It is interesting to notice that the power law behavior seems to extend over a larger range of r for σ p,geo = 2. Moreover, the slope of the calculated density-density correlation function is in good agreement with the theoretical slope of (D f -d) for all the three combinations of D f and σ p,geo , i.e., D f = 1.79 and σ p,geo = 1.0, D f = 1.68 and σ p,geo = 2.0, and D f = 1.48 for σ p,geo = 3.0. These values of D f correspond to those of DLCA aggregates when considering different levels of primary particle

Figure 11 :

 11 Figure 11: Sample aggregates generated by FracVAL from polydisperse PPs R1.C5 with σ p,geo = 2.0 and specified fractal parameters of D f = 2.60, k f = 0.25 (top row), D f = 1.68, k f = 0.98 (middle row), and D f = 1.30, k f = 1.50 (bottom row) R1.C5 .

Figure 12 :

 12 Figure 12: The calculated density-density correlation functions of aggregates consisting of N = 100, N = 1024 primary particles of different levels of polydispersity. The imposed fractal parameters are: D f = 1.79, k f = 1.40 for σ p,geo = 1.0, D f = 1.68, k f = 0.98 for σ p,geo = 2.0, and D f = 1.48, k f = 0.77 for σ p,geo = 3.0. d stands for the Euclidean dimension of space with d = 3.

Figure 13 :

 13 Figure 13: Preservation of fractal parameters for aggregates consisting of (a) monodisperse (σ p,geo = 1) and (b) polydisperse (σ p,geo = 2) primary particles for different combinations of fractal parameters (D f and k f ). Continuous lines represent the log-log fit of the points, corresponding parameters are reported in the legend of the figures.

Filippov

  et al.[13] one, Fig.14compares the plots of log(N ) as a function log(R g /r p,geo ) in log-log scale for aggregates generated using FracVAL and a direct application of the Filippov et al.[13] R1.C4 algorithm for polydisperse primary particles. The prescribed parameters are D f = 1.68, k f = 0.98, and σ p,geo = 2.0 and for aggregate sizes from N = 20 to 1024. It is noticed that in the Filippov et al.[13] method considered Eq. (7) instead of the derived Eq. (6), which is used in FracVAL. Therefore, the effect of primary particle polydispersity on aggregate center of mass and radius of gyration is not taken into account in the formulation of Filippov et al..[13]. R1.C4 It is also worth pointing out that the FLAGE algorithm recently developed by Skorupski et al.[20] also made a direct application of the Filippov et al.[13] R1.C4 method for aggregates formed polydisperse primary particles and the results of Filippov et al.[13] R1.C4 shown in Fig.14R1.C4 should be identical to the results of FLAGE.

Figure 14 :

 14 Figure 14: Comparison of results by using Filippov et al. [13] method and FracVAL. Aggregates consisting of D f = 1.68 and k f = 0.98 and σ p,geo = 2.0.

Figure C. 15 :

 15 Figure C.15: Sensitivity analysis of density-density correlation function for the initial subclusters consisting of monodisperse (a and b) and polydisperse PPs (Figs. c and d). Figs. (a) and (c) illustrate the sensitivity of the initial sub-clusters size (N sub ), and Figures (b) and (d) show the sensitivity to fractal parameters (D f and k f ). All aggregates consist of N = 100 monomers, fractal parameters D f = 1.79 and k f = 1.40 for monodisperse (σ p,geo = 1) and D f = 1.68 and k f = 0.98 for polydisperse PPs (σ p,geo = 2).

Figure D. 16 :FigureFigure D. 18 :

 1618 Figure D.16: Limiting candidate for Case 2 (D i1,+ is large to enclose the sphere of radius D j2,+ ). This is analogous to Case 3.

Table 1 :

 1 presents examples of average time required to generate an aggregate consisting of N = 500 for monomers polydispersity from σ p,geo = 1.0 to 3.0. All examples of aggregates consisting of 500 PPs R2.C4 are generated in less than 2.4 minutes.Meanwhile aggregates consisting of 1000 PPs are generated in less than 14 minutes. Generation of aggregates formed by polydisperse PPs is generally more time-consuming than those formed by monodisperse PPs. R2.C4 It is important to remark that the majority of times is in the order of the minimum values reported in Table1, while the maximum values are achieved only for combinations of extremely different D f and k f pairs and also these maximum times exhibit a much larger standard deviation than the minimum values. R2.C4 Assessment of the average computational time (in seconds) required by FracVAL.

	σ p,geo N = 500 R2.C4 N = 1000 R2.C4
	1.00	1.1-70.1	5.3-84.6 R2.C4
	1.45	1.9-73.6	8.9-408 R2.C4
	2.00	3.9-145.8	28.2-791.1 R2.C4
	2.50	5.6-15.1	37.6-113.7-R2.C4
	3.00	7.0-9.4	38-74.6 R2.C4

  is considerably computationally efficient for generating fractal aggregates for different combinations of D f , k f , and σ p,geo for σ p,geo up to 3, taking in average less than 2.4 minutes for N = 500 and 14 minutes for N = 1000 PPs R2.C4 on a regular laptop, as long as the pair of D f and k f falls in the valid range where it is possible to generate such fractal aggregates.

Finally, it is demonstrated that the direct use of the tunable clustercluster aggregation algorithm Filippov et al.

[13]

to generate aggregates formed by polydisperse primary particles, such as implemented in the FLAGE code, does not preserve the prescribed fractal properties (D f and k f ) for individual aggregates, though it does so globally, i.e., the fractal properties of an ensemble of aggregates recover the prescribed values. This observation suggests that caution should be taken when a certain property of individual aggregates formed by polydisperse primary particles generated by FLAGE is investigated, since the fractal dimension or prefactor may be different from the expected value. On the other hand, the algorithm developed in this study, FracVAL, overcomes the drawback of the Filippov et al.

[13]

R1.C4 method and the FLAGE code.

Acknowledgments

This work is supported by the Chilean CONICYT Research programs under Grant FONDECYT project 1161453 and partially by PIA-Anillo CyT project ACT172095.

Appendix A. Derivation of Eq. (6) The resulting center of mass (R c ) of an aggregate formed by the aggregation of two clusters of N 1 and N 2 PPs with mass m 1 and m 2 , respectively is,

where N 1 + N 2 = N and m 1 + m 2 = m. Therefore,

Let's Γ be the vector that stands for the distance between the center of mass of the two sub-clusters,

getting m 1 and m 2 from the relation m = m 1 + m 2 and replacing their values in Eq. (A.2) we can obtain,

On the other hand, from the radius of gyration of the original aggregate calculated from Eq. ( 4) we obtain,

where the terms:

combining this result with Eq. (A.4a) and Eq. (A.4b) and introducing R g1

and R g2 as the radius of gyration of both sub-clusters we get,

if we consider the particular case of aggregates consisting of monodisperse PPs with mass m p , consequently m = N m p , m 1 = N 1 m p and m 2 = N 2 m p , then Eq. ( 6) turns to,

which is equivalent to the relationship derived by Filippov et al. [13].

Appendix B. Sphere-sphere intersection

Lets consider two spheres in 3 with equations,

Lets d be the distance between the center of the two spheres. Based in the law of cosines,

The intersection of these two spheres is a circle contained in the plane of intersection. Lets ρ be the radius of this circle, then ρ = r 1 sin[α], considering Eq. (B.2) and the identity sin

On the other hand, the plane of intersection is obtained by subtracting Eq. (B.1a) and Eq. (B.1b) by considering A = 2(x 2 -x 1 ), B = 2(y 2 -y 1 ),

The equation of the line that connect the centers of the two spheres is,

The intersection of this line and the plane given by Eq. (B.4) bring us the value of the constant t as following,

replacing in Eq. (B.5) we obtain the coordinates c of the center of the circle of intersection between the two spheres. Finally, with the coordinates of the center c and the radius ρ, the equation of the circle of intersection of the two spheres can be parametrized as follows,

where î and ĵ are two perpendicular unit vectors belonging to the plane given by Eq. (B.4).

Appendix C. Sensitivity analysis to initial sub-clusters morphology and size

We intend to answer the following question: how important are the ini- Appendix D. Further explanation of cases 2 and 3 of section 3.2

In the context of Section 3.2 in sub-step 4b we identified 3 possible cases of aggregation of two sub-clusters A1 and A2. The main part of the manuscript is devoted to explain Case 1 for being considered as the most common case.

This appendix intends to provide a detailed explanation of Cases 2 and 3.

• Case 2: When the sphere of radius D i1,+ is large to enclose the sphere of radius D j2,+ , i.e., when (D i1,+ -D j2,+ ) > Γ 12 with an upper limit • Case 3: Analogously, when the sphere of radius D j2,+ is large to enclose the sphere of radius D i1,+ , i.e., when (D j2,+ -D i1,+ ) > Γ 12 with an upper limit D j2,-≤ Γ 12 + D i1,+ .

In the context of sub-step 4c, there are three different scenarios for joining the sub-cluster A1 and A2 by selecting the candidate primary particles s1 (belonging to A1) and s2 (belonging to A2). Therefore, we will be concerned about the following spheres, The reason that many spheres appear is that the possible solutions, i.e., the intersections between s1 and s2, correspond to the intersection of two


Solenne Louison

Julie Blanc

Charlotte Pallot

Seydou Alassane

Andréa Praudel

Alain-Marie Bron

Catherine Creuzot-Garcher

Prof Alain Bron

Visual outcomes and complications of congenital cataract surgery 1 Résultats fonctionnels et complications dans une série de cataractes congénitales

Keywords: Congenital cataract, Visual acuity, Postoperative complications, Intraocular lens Résumé Cataracte congénitale, acuité visuelle, complication post-opératoire, implant intra-oculaire

Purpose. -To describe the visual results and postoperative complications of congenital cataract surgeries performed between 2005 and 2016.

Methods. -A retrospective consecutive case series of congenital cataract surgeries was carried out at the Dijon University Hospital in France. Intraocular lens implantation was primary or secondary according to age. Pre-and post-operative assessment was performed through ophthalmologic consultations with orthoptic measurements. Visual function and adverse events were recorded.

Results. -Fifty-six consecutive procedures were evaluated in 37 infants. Overall, 26.8% of patients had unilateral cataracts, and 73.2% had bilateral cataracts. Median age at surgery was 1.0 years [IQR (interquartile range): 0.3-5.2] and 2.7 years [IQR: 0.4-9.5] for unilateral and bilateral cataracts, respectively. Median best-corrected visual acuity (BCVA) at the last follow-up was 0.5 logMAR [IQR: 0.2-0.8] and 0.1 logMAR [IQR: 0.0-0.8] for the unilateral and bilateral group, respectively. Posterior capsule opacification (PCO) was the primary postoperative complication: 60.0% in unilateral cataract and 46.3% in bilateral cataracts (P = 0.019). Median follow-up was 2.0 years [IQR: 1.0-5.0].

Conclusion.

-Congenital cataract surgery appears to be safe and effective when combined with early, conscientious amblyopia treatment. Bilateral congenital cataracts had better visual outcome than unilateral cataract.

Introduction

Congenital cataract is one of the main causes of preventable childhood blindness. In France, the incidence of congenital cataract is estimated at 1-3/10000 births [START_REF] Daien | Incidence and Characteristics of Congenital Cataract Surgery in France from 2010 to 2012: The EPISAFE Program[END_REF]. In the developed world, the etiological distribution of congenital cataract has changed over time, mainly because of the introduction of a rubella vaccination program. In more recent studies, the percentage of cases of unknown origin is generally reported to be at least 50.0%. The percentage of genetically based cataracts varies from 27.0% to 39.0%, the majority of these being hereditary [START_REF] Wirth | Aetiology of congenital and paediatric cataract in an Australian population[END_REF][START_REF] Rahi | Congenital and infantile cataract in the United Kingdom: underlying or associated factors. British Congenital Cataract Interest Group[END_REF][START_REF] Fakhoury | Epidemiologic and etiological characteristics of congenital cataract: study of 59 cases over 10 years][END_REF].

Great advances have been made in the management of paediatric cataracts over the last several decades. These surgical improvements have contributed to a decrease in the incidence of postoperative complications such as posterior capsule opacification (PCO), postoperative inflammation and glaucoma [START_REF] Hutcheson | Application of new ophthalmic technology in the pediatric patient[END_REF][START_REF] Nihalani | Technological advances in pediatric cataract surgery[END_REF]. In the most recent studies, visual improvements were better in bilateral cataracts with best-corrected visual acuity (BCVA) between 0.40 to 0.50 logMAR than unilateral cataracts with BCVA between 0.91 to 1.00 logMAR [START_REF] Borisovsky | Results of congenital cataract surgery with and without intraocular lens implantation in infants and children[END_REF][START_REF] Lu | Visual results and complications of primary intraocular lens implantation in infants aged 6 to 12 months[END_REF][START_REF] Vera | Visual outcomes and complications of cataract surgery with primary implantation in infants[END_REF]. To date, early cataract removal and replacement with an intraocular lens (IOL) is considered the most appropriate treatment to avoid irreversible amblyopia [START_REF] Wilson | Pediatric cataract surgery and intraocular lens implantation: practice styles and preferences of the 2001 ASCRS and AAPOS memberships[END_REF].

Amblyopia treatment is of paramount importance. Indeed, patient compliance for patching therapy is a key factor for good visual outcome. The indication and selection of IOL continues to be a controversial subject. Recent reports by the Infant Aphakia Treatment Study (IATS) group showed no significant difference between the median BCVA of operated eyes in children who underwent primary IOL implantation and those left aphakic [START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF]. However, there was a lower rate of adverse events and additional intraoperative procedures in the contact lens group (contact lens, 56.0% vs IOL, 81.0%; P = 0.020) [START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF].

In the present study, we report on the visual results and postoperative complications of congenital cataract surgeries performed between 2005 and 2016 in Dijon University Hospital.

Materials and methods

The medical records of children who underwent congenital cataract surgery operated between July 2005 and December 2016 at the Dijon University Hospital were retrospectively reviewed. Demographic parameters and clinical characteristics were as follow: diagnosis, type of cataract, pre-and postoperative BCVA, intraocular pressure (IOP), type of surgery, type and power of the implanted IOL, pre-and postoperative complications. Traumatic cataracts were excluded.

Surgical procedure

Visual acuity measurements, fundus examination, retinoscopy, keratometry and axial length measurement by immersion A-scan were performed pre-operatively, and, when the retina was not visible, B-scan ultrasound was used. IOL power was calculated with the Sanders-Retzlaff-Kraff II formula. The target postoperative refraction was based on the age and the biometry of patient. We planned initial undercorrection as recommended by Sachdeva et al [START_REF] Sachdeva | Validation of Guidelines for Undercorrection of Intraocular Lens Power in Children[END_REF]. They suggested a postoperative refractive goal from +6 diopters (D) for a 1 year-old to -1 to -2 D for patients >8 years of age (rule of seven).

The surgeries were all performed under general anaesthesia and by the same surgeon (AMB). First, infants under anaesthesia were examined for assessment of corneal diameter, axial length, biomicroscopy, aplanation tonometry, keratometry and cyclopegic refraction. Surgery consisted of 2 millimeters corneal micro-incisions, anterior capsulorhexis and phacoaspiration. Implantation was primary or secondary according to age. A foldable hydrophobic acrylic IOL was implanted. Posterior capsulotomy and anterior vitrectomy were performed at the discretion of the surgeon. Patients with bilateral cataracts generally had bilateral surgery during the same procedure. Postoperatively, all patients received topical corticosteroids and antibiotics.

During follow-up, if children presented PCO, they underwent either surgical membranectomy or YAG laser depending on age and patient cooperation.

Amblyopia Treatment

After surgery, amblyopia treatment was started as soon as possible in order to achieve a satisfactory visual outcome. Amblyopia treatment began immediately after surgery. It consisted of refractive correction under cycloplegia with timely updates and patching duration. The occlusion began with 1 hour per month of age per day for up to 100% of waking hours. Amblyopia treatment was continued for several years according visual recovery and, when appropriate, patching duration was reduced slowly with regular monitoring.

Functional Outcomes

BCVA was tested on a logarithm progression chart adapted to age (for children of at least 2 years old) and was converted to the logarithm of the minimum angle resolution (logMAR) for statistical purposes. BCVA was evaluated both at 4 weeks and at the end of the follow-up.

Final BCVA of ≤ 0.1 logMAR was considered a very good visual outcome, logMAR between 0.1 and 0.5 was considered good visual acuity and poor visual acuity was defined as BCVA > 0.5 logMAR. Adverse events such as uveitis, glaucoma or PCO were recorded during follow-up visits.

Refractive Error

The value of refraction at 3 months postoperatively were considered for analysis. The spherical equivalent (SE) of the residual refractive error was calculated in the following manner: spherical equivalent (D) = sphere power (D) -cylinder power (D).

Statistical analysis

All analysis were performed using SAS software (version, SAS Institute, Inc., NC, USA).

Descriptive statistics are given as medians (with interquartile range) because the distribution of continuous variables was not normal according to the Shapiro-Wilk test. Numbers (with percentages) are given for categorical variables. The Wilcoxon test was used to compare the longitudinal change of visual acuity. The Mann-Whitney test was used to compare continuous variables between groups. The chi-square test or the Fisher exact test were used to compare categorial variables between groups. The tests were two-tailed and P-values of less than 0.05 were considered statistically significant.

Results

Baseline Characteristics

Fifty-six congenital cataracts were operated in 37 children between 2005 and 2016. Patient characteristics are summarized in Table 1. Median follow-up was 2.0 years [IQR: 1.0-5.0]. There were no differences in the age at diagnosis or the gender between the unilateral and bilateral groups. The main symptom which led to diagnosis in the two groups was leukocoria (39.9%). Unilateral cataracts presented significantly more ocular abnormalities coexisting than bilateral cataracts (P = 0.046). The most commons were microphthalmos (n = 6) and persistent foetal vasculature (PFV) (n = 3).

Surgical Procedure

Details of surgery are displayed in Table 2. Median period between diagnosis and surgery in the bilateral group was 0. Germany)). The surgical procedure was statistically different in the two groups with more posterior capsulorhexis and anterior vitrectomy in the unilateral group (P < 0.001 and P < 0.001, respectively). We did not have peroperative posterior capsule rupture in this series

Refractive Error

The median SE at 3 months postoperatively were +0.

Visual Outcomes

The results are displayed in Table 3. In the bilateral cataract group, 53.3% reached a final BCVA ≤ 0.1 logMAR versus 20.0% of those in the unilateral cataract group. There was not any statistical difference between the unilateral and bilateral groups for BCVA > 0.5 which was defined as poor visual acuity (Figure ).

Pseudophakia or aphakia were corrected with different types of optical correction including unifocal, bifocal or progressive spectacles and contact lenses. The optical correction was statistically different for the two groups (P = 0.013). There was statistically more postoperative amblyopia treatment in the unilateral group than in the bilateral group (73.3% versus 29.2%, P = 0.014).

Adverse Events

Adverse events are summarized in Table 4. During follow-up, PCO was the most common complication in both groups (50.0% of all eyes). PCO was statistically more frequent in unilateral than bilateral cataracts (60.0% versus 46.3%, P = 0.019).

Discussion

We report on a retrospective case series of congenital cataracts managed in an University Center where paediatrics is not the first subspecialty. In our analysis, median postoperative BCVA was higher in the bilateral cataract group than in the unilateral cataract group, 0.1 logMAR [IQR: 0.0-0.8] vs 0.5 logMAR [IQR: 0.2-0.8], respectively. This is in agreement with other studies [START_REF] Borisovsky | Results of congenital cataract surgery with and without intraocular lens implantation in infants and children[END_REF][START_REF] Lu | Visual results and complications of primary intraocular lens implantation in infants aged 6 to 12 months[END_REF][START_REF] Vera | Visual outcomes and complications of cataract surgery with primary implantation in infants[END_REF]. In a previous French study, Vera et al. reported a median BCVA of 1.00 logMAR in the case of unilateral surgery and 0.40 logMAR in the case of bilateral surgery at the last follow-up examination [START_REF] Vera | Visual outcomes and complications of cataract surgery with primary implantation in infants[END_REF]. Lu et al. obtained a mean BCVA of 0.98 ± 0.18, 0.50 ± 0.14 and 0.61 ± 0.25 logMAR for unilateral, bilateral and all eyes, respectively, at last follow-up [START_REF] Lu | Visual results and complications of primary intraocular lens implantation in infants aged 6 to 12 months[END_REF].

There are three possible reasons for better outcomes in bilateral than unilateral cataracts. First, amblyopia develops more quickly and with more intensity in unilateral cataract because of uneven visual deprivation. After surgery, there are more anisometropia, strabismus and nystagmus in unilateral cataract. Therefore patching therapy is longer and more challenging in unilateral cataract. Secondly, in our study, we found a difference of age at diagnosis and at surgery between the two groups. Children tended to be older in the bilateral group, and they had better visual results despite a relatively late extraction. Their cataracts allowed for a visual experience which diminished the cortical effect of visual deprivation. Thirdly, our unilateral cataract group had more eye disorders such as microphthalmia or PFV than the bilateral cataract group. Previous studies have showed that unilateral cataracts occurred more frequently in cases of PFV or anterior ocular disorders, while bilateral cataracts were usually associated with syndromes or systemic disorders [START_REF] Traboulsi | Associated systemic and ocular disorders in patients with congenital unilateral cataracts: the Infant Aphakia Treatment Study experience[END_REF][START_REF] Mullner-Eidenbock | Persistent fetal vasculature and minimal fetal vascular remnants: a frequent cause of unilateral congenital cataracts[END_REF][START_REF] Cassidy | Congenital cataract and multisystem disorders[END_REF]. Associated ocular abnormalities could thus be one of the reasons for the poorer outcomes in the unilateral cataract group.

The management of amblyopia has not been specifically studied in this work.

However, all children benefited from full refractive correction and occlusion with patching therapy after surgery. Occlusion is the oldest, simplest, and one of the most effective method to treat amblyopia [START_REF] Repka | A randomized trial of patching regimens for treatment of moderate amblyopia in children[END_REF]. The most recommendable occlusion consists to cover the fixing eye. Patching on lenses has the disadvantage of allowing vision from the side of the eyeglasses and the possibility for the child to take it off. Pharmacological [START_REF] Repka | Treatment of severe amblyopia with atropine: Results from 2 randomized clinical trials[END_REF] or optical penalization is another therapeutic possibility for amblyopia. More recent techniques such as perceptual learning or drug therapy (levodopa, citicoline, phenylethylamine) have been proposed.

Regarding IOL implantation, 73.2% of patients benefited from primary implantation. This is higher than the study of Repka et al. using the Pediatric Eye Disease Investigator Group (PEDIG) registry [START_REF] Repka | Cataract Surgery in Children from Birth to Less than 13 Years of Age: Baseline Characteristics of the Cohort[END_REF]. Among the 1266 eyes included in this study, 59.0% were implanted. There was a significant effect of age on the decision to implant an IOL, with 8.0% of eyes implanted before 1 year of age and 90.0% of eyes implanted after 2 years of age. At 3 months postoperatively, our refractive error was +0.4 D [IQR: -0.5; +1.5]. This target for IOL calculation was probably not enough hyperopic. Indeed, myopic shift that would be anticipated from earlier studies after cataract surgery would require higher hyperopic SE in the early postoperative follow-up [START_REF] Plager | Refractive change in pediatric pseudophakia: 6-year follow-up[END_REF].

PCO was the major adverse event in our study. It affected 50.0% of all eyes in our study which is higher than existing reports of children undergoing cataract surgery with implantation during infancy [START_REF] Borisovsky | Results of congenital cataract surgery with and without intraocular lens implantation in infants and children[END_REF][START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF][START_REF] Cassidy | Congenital cataract and multisystem disorders[END_REF][START_REF] Repka | Cataract Surgery in Children from Birth to Less than 13 Years of Age: Baseline Characteristics of the Cohort[END_REF]. Borisovski et al. reported 16.0% occurrence of PCO [START_REF] Borisovsky | Results of congenital cataract surgery with and without intraocular lens implantation in infants and children[END_REF]. In the IATS series presented by Plager et al., PCO occurred in 4.0% of aphakic cases and 40.0% of pseudophakic cases [START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF]. Firstly, among our patients, some children did not receive posterior capsulotomy and anterior vitrectomy during the first surgery whereas all children in the two studies mentioned above had posterior capsulotomy and anterior vitrectomy [START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF][START_REF] Plager | Refractive change in pediatric pseudophakia: 6-year follow-up[END_REF]. This surgical procedure may prevent PCO [START_REF] Trivedi | Opacification of the visual axis after cataract surgery and single acrylic intraocular lens implantation in the first year of life[END_REF]. Vasavada et al. suggest posterior capsulorhexis with anterior vitrectomy as a strategy for posterior capsule management in children younger than 3 years. Posterior capsulorhexis without anterior vitrectomy may be considered in children between 3 years and 6 to 7 years. In older children, it is recommended to maintain an intact posterior capsule [START_REF] Vasavada | Posterior capsule management in congenital cataract surgery[END_REF]. In our study, there were more surgeries for PCO in the unilateral group and more laser procedures for PCO in the bilateral group. The infants in the unilateral group were younger making it more challenging to perform a laser procedure. Moreover, according to previous studies, patients 1 year-old or less had a more significant spread of lens epithelial cells across the visual axis after surgery [START_REF] Plager | Complications in the first 5 years following cataract surgery in infants with and without intraocular lens implantation in the Infant Aphakia Treatment Study[END_REF][START_REF] Tadros | Primary versus secondary IOL implantation following removal of infantile unilateral congenital cataract: outcomes after at least 5 years[END_REF].

In our series, two glaucoma post cataract surgeries occurred in the two eyes of a single patient whose bilateral cataracts were diagnosed at 1 month. This patient had surgery 7 weeks after birth without implantation of IOL and, to our knowledge, he did not have glaucoma before surgery. A number of reports have been published on the risk factors of glaucoma after congenital cataract surgery which include age at detection of cataract [START_REF] Chak | Incidence of and factors associated with glaucoma after surgery for congenital cataract: findings from the British Congenital Cataract Study[END_REF] or comorbidities such as microcornea or nanophthalmos [START_REF] Borisovsky | Results of congenital cataract surgery with and without intraocular lens implantation in infants and children[END_REF][START_REF] Vasavada | Posterior capsule management in congenital cataract surgery[END_REF]. On the other hand, this patient did not have primary IOL implantation. This has been shown to decrease the risk of glaucoma development postoperatively [START_REF] Chak | Incidence of and factors associated with glaucoma after surgery for congenital cataract: findings from the British Congenital Cataract Study[END_REF][START_REF] Freedman | Infant Aphakia Treatment Study G (2015) Glaucoma-Related Adverse Events in the First 5 Years After Unilateral Cataract Removal in the Infant Aphakia Treatment Study[END_REF][START_REF] Asrani | Does primary intraocular lens implantation prevent "aphakic" glaucoma in children?[END_REF][START_REF] Astle | Surgical outcomes of primary foldable intraocular lens implantation in children: understanding posterior opacification and the absence of glaucoma[END_REF]. Surgical treatment of glaucoma was not required in our case.

Optical correction was statistically different in the two groups (P = 0.013). In This study has several limitations. Firstly, the average follow-up period was 2 years, and many young infants were not eligible for visual acuity testing. Secondly, the characteristics of patients in each group varied including: age at baseline, age at surgery and BCVA before treatment. Third, our case series was small.

To conclude, patients with bilateral cataracts had better postoperative BCVA than those with unilateral cataracts. The age at diagnosis and surgical treatment appear to be a major criterion for good BCVA. Amblyopia treatment is the hallmark for a successful outcome.

  4 years [IQR: 0.1-1.4]. In the unilateral group, children benefited from surgery more quickly than the bilateral group with a median period between diagnosis and surgery of 0.1 year [IQR: 0.1-0.4]. When IOL implantation was performed, the median IOL power was 25.0 D [IQR: 20.5-28.0] in the bilateral group and 28.0 D [IQR: 24.0-30.0] in the unilateral group. There were 35 unifocal IOL (AcrySof MA50BM (Alcon Inc., TX, USA) or CT LUCIA (Carl Zeiss Meditec AG, Jena, Germany)), 6 toric IOL (AT TORBI (Carl Zeiss Meditec AG, Jena, Germany)) and 7 multifocal IOL (AT LISA (Carl Zeiss Meditec AG, Jena,

  4 D [IQR: -0.1; +2.0] and +0.3 D [IQR: -2.8; +1.1] in unilateral and bilateral cataracts, respectively.

  unilateral cataract, bifocal then unifocal lenses were most common. Progressive lenses were more frequent in patients with bilateral cataracts because of the older age in this group at follow-up. Péchereau et al. suggest unifocal lenses before 12 months, bifocal lenses after 12 months and progressive lenses after 4-6 years [27].

Table 1

 1 Baseline characteristics of the study population. Categorical variables are expressed as number (%) Continuous variables are expressed as median [IQR (interquartile range)] P value reaching statistical significance are in bold

	Main characteristics	Unilateral cataract Bilateral cataract P value
	Number of eyes (n = 56)	15 (26.8)	41 (73.2)
	Gender (female)	10 (66.7)	24 (58.5)	0.178
	Age at diagnosis (years)	0.7 [0.1-2.4]	0.5 [0.3-5.8]	0.421
	During of follow-up (years)	2.0 [1.0-5.0]	1.0 [1.0-3.0]	0.053
	Ocular disorders	6 (40.0)	8 (19.5)	0.042
	Microphthalmia	6 (40.0)	8 (19.5)
	Persistent foetal vasculature	3 (20.0)	1 (2.4)
	Strabismus
	Esotropia	4 (26.6)	10 (24.4)	0,354
	Exotropia	3 (20.0)	11 (26.8)	0,276
	Nystagmus	2 (13.3)	20 (48.8)	0,167

Table 4

 4 Postoperative complications after surgeries for congenital cataract (n = 56). Categorical variables are expressed as number (%) Continuous variables are expressed as median [IQR (interquartile range)] P value reaching statistical significance are in bold

		Unilateral cataract Bilateral cataract P value
	Posterior capsule opacification	9 (60.0)	19 (46.3)	0.019
	Laser	1 (6.7)	3 (7.3)	< 0.001
	Surgical treatment	5 (33.3)	11 (26.8)	< 0.001
	Uveitis	0	3 (7.3)
	Glaucoma	0	2 (4.9)

Disclosure of interest

The authors declare that they have no competing interest related to this work.