Antoine Moinet

Alain Barrat

Romualdo Pastor-Satorras

Generalized voterlike model on

published or not. The documents may come

I. INTRODUCTION

A wide variety of complex physical systems are concerned with the problem of an initially disordered configuration that is able to achieve an ordered state by means of local pairwise dynamical interactions. Examples of such systems range from the formation of an opinion consensus in social systems [1][2][3] to the loss of genetic diversity in evolutionary dynamics [4]. These situations, implying a competition between different alternative states diffusing among the agents, have been modelled with stochastic copying or invasion processes. In these models, each individual is endowed with a state variable and copies or imposes his/her state from or to neighbouring sites until one single state finally dominates the whole system. Among those different frameworks, the voter model [START_REF] Liggett | Interacting Particle Systems[END_REF] was introduced to schematically model the opinion spreading in human populations and has become emblematic for its simplicity and analytical tractability. In this model, the agents possess one of two discrete opinions, and at each time step an individual is chosen and adopts the opinion of a randomly chosen neighbour. On the other hand, in the context of evolutionary dynamics, the Moran process [START_REF] Moran | The Statistical Processes of Evolutionary Theory[END_REF] considers a population of individuals belonging to different species, that reproduce generating an offspring that replaces a randomly chosen nearest neighbor. The voter and Moran models differ thus in the direction in which the state is transferred between pairs of interacting agents.

Recently, it has been acknowledged that the topology in which such ordering dynamics takes place in real systems is often far from homogeneous, and better represented in terms of a complex network [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][START_REF] Newman | Networks: An introduction[END_REF], in which agents are characterized by a number of neighbors (degree) k that is broadly distributed, with a heterogeneous probability distribution P (k) (degree distribution) often schematically described by a power-law-like form P (k) ∼ k -γ [START_REF] Barabási | [END_REF]. This observation has led to an intense research activity in order to unveil the different properties of ordering dynamics in heterogeneous topologies [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][10][11][12][13][14][15], yielding a good understanding of the problem both at the numerical and analytical levels.

These studies have mainly focused on the case of static networks, in which nodes representing agents are connected by a set of edges, standing for pairwise possible interactions, that are fixed in time and never change. However, many networks, and in particular social ones, are dynamic in nature, given by a pattern of connections that evolves in time. Such temporal networks [16,17] have been the subject of an intense research activity, considering in particular their possible impact on the behavior of dynamical processes running on top of them [17][18][19][20][21][22]. Despite their relevance, however, consensus dynamics have seldom been studied in detail in temporal topologies [23][24][25].

Here we contribute to fill this gap by presenting a detailed study of the voter and Moran processes on temporal networks, focusing on a generalization of the recently introduced activity driven temporal network model [26]. In this model [27] agents are assigned an activity parameter a that determines their propensity to establish social interactions with other individuals, and another variable b defining their attractiveness, which in turn determines the probability that they are chosen by an active agent to interact. We provide a full analysis of basic ordering dynamics through a heterogeneous mean-field approach that allows us to describe the dynamics of the process in the limit of a large system size, and in particular to compute the average time to reach consensus starting from a random configuration. Another quantity of interest when studying consensus or invasion processes is the so-called exit probability [28], defined as the probability that a single agent having a discrepant opinion among an unanimous population manages to spread his/her opinion to the whole population. In our work this quantity plays a significant role as it highlights an interesting symmetry between the voter model and the Moran process when comparing the unfolding of these processes either on static heterogeneous networks or on and the activity driven network. In particular, we show how, depending on the type of dynamics, the effect of a node's characteristics on the dynamics can be similar or drastically different when the dynamics runs on a temporal network or on the corresponding static aggregated network.

The paper is organized as follows: In Sec. II we define the variation of the voter model we consider, and the class of activity driven network with attractiveness we use as a substrate for the dynamics. In Sec. III we provide a full description of the dynamics and in particular we exhibit analytical expressions for the exit probability and the average consensus time. In Sec. IV we give more insights for a few particular forms of the joint distribution of the activity and attractiveness η(a, b). Sec. V presents numerical simulations showing agreement with our analytical expressions, which are further studied in some asymptotic limit of the activity distribution in Sec. VI. Finally we conclude our work discussing our results and exploring perspectives in Sec.VII.

II. CONSENSUS DYNAMICS IN GENERALIZED ACTIVITY DRIVEN NETWORKS

A. Activity driven networks with attractiveness

We focus on the class of activity driven temporal networks [26,27,29], which are based in the key ingredient that the formation of social interactions is driven by some innate activity of individuals, which determines their tendency to establish interactions and is empirically observed to be heterogeneously distributed [26]. In this paradigmatic model, a fixed population of N agents is considered, each endowed with an activity a, representing the rate (probability per unit time) at which s/he becomes active and draws edges (interactions) towards other agents. In the original formulation of the model [26], the chosen agents are selected uniformly at random among all peers. In order to make the model more realistic, one can assign to each agent a parameter b, called attractiveness, such that his/her probability of being selected by an active peer is proportional to b [27]. The activity a and attractiveness b are extracted at random for each agent from a given joint distribution η(a, b).

B. Consensus dynamics

Coupling a dynamical process with a temporal network always entails the problem of how to deal with the different time scales inherent in the process and in the evolution of the network. Here we consider the simplest case of a single time scale, imposed by the network evolution. In this way, the state of an agent can only change when s/he interacts with another agent, and is constant in the latency times between interactions. The consensus dynamics are thus defined as follows:

• We start from an initial configuration of states s i ∈ {0, 1}, assigned to each agent.

• In an interval of time δt, an agent i, in state s i , becomes active with probability a i δt, and chooses as peer another agent j (in state s j ) with probability b j b

• The states s i and s j are updated according to the chosen consensus dynamics.

• Time is updated t → t + δt.

We consider three different variations of social dynamics, based on the update dynamics of the state variables. Assuming that, at given time t, agent i becomes active, and chooses agent j to start an interaction, we consider the three different updates:

1. Voter dynamics: s i := s j (i.e., i adopts j's state).

2. Moran dynamics: s j := s i .

3. Mixed dynamics: With probability p, s i := s j ; with the complementary probability 1p, s j := s i .

In what follows, we consider the mixed update rule, as the voter model and the Moran process are particular cases of the latter obtained by setting p = 1 or p = 0 respectively.

III. HETEROGENEOUS MEAN-FIELD ANALYSIS

When agent activation is ruled by a Poisson process, it is possible to tackle the behavior of voter-like dynamics by extending the heterogeneous mean-field [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF]30] approach developed in Refs. [10,31] to study voter dynamics on static networks. This method is based in a coarse-graining of the network, considering that the state of an agent with activity a and attractiveness b depends exclusively on those two quantities. In this way, one considers a fundamental description in terms of the fraction ρ a,b (t) of agents with activity strength a and attractiveness b in the state 1 at time t; in other words, ρ a,b (t) is the probability that a randomly chosen agent with activity a and attractiveness b is in state 1 at time t. The corresponding fraction of agents in state 0 is given by the complementary probability 1ρ a,b (t). The total fraction of agents in state 1, ρ(t), is given by

ρ(t) = a,b η(a, b)ρ a,b (t). (1)
To alleviate notation, we denote the pair (a, b) by the symbol h, writing thus ρ a,b (t) ≡ ρ h (t). The relevant functions defining the dynamics are the transition probabilities R h and L h for respectively increasing and decreasing the number of voters in state 1, among the pool of agents with activity strength a and attractiveness b, in a time interval δt. From these transition probabilities, a differential equation ruling the evolution of ρ h (t) can be derived, as well as information about the exit probability and the average ordering time. In the dynamical rules described in the previous section, agents activate independently so that a priori multiple activations are possible during a single time step. However the use of the transition probabilities R h and L h relies on the implicit hypothesis that only one flip attempt may occur during a single time step, thus in order to ensure the validity of our analysis, we impose that a N δt 1 so that the probability of counting more than one activation during δt is almost zero. This is of course always possible as the time step δt is arbitrary.

Let us now derive the time evolution equation of the fraction ρ h (t) of agents with activity strength a and attractiveness b in state 1 at time t.

A. Evolution equation

In a single time step, the number of agents with activity strength a and attractiveness b in state 1 may either increase by one unit with probability R h , decrease by one unit with probability L h , or stay unchanged with probability 1 -R h -L h . Thus on average the variation δρ h reads

δρ h = (+1) × R h N h + (-1) × L h N h + 0 × 1 -R h -L h N h , (2)
where N h is the number of agents in the state (a, b). In the continuous time limit (for δt 1) we may write

∂ρ h (t) ∂t = R h -L h N h δt . (3)
We consider the mixed process in which every agent, when activated, might either copy the state of his/her peer with probability p, or impose his/her own state to him with probability 1p. The transition probabilities are thus given by

R h = p R V h + (1 -p)R M h (4) L h = p L V h + (1 -p)L V h , (5)
where the rates L X h and R X h refer to the voter (X = V ) and Moran (X = M ) dynamics, respectively.

In the case of the voter dynamics, these transition probabilities take the form

R V h = N h a δt (1 -ρ h ) b ρ h b , (6)
L V h = N h a δt ρ h 1 - b ρ h b . (7)
The origin of these expressions is easy to see. For example, in Eq. ( 6), the probability that the number of agents in state 1, activity a and attractiveness b increases by one unit is proportional to the number of agents in this class in state 0, N h [1ρ h (t)], times the probability that any one of them becomes active in a time interval δt (aδt), times the probability that an active agent generates a link to an agent in state 1, thus copying the state of this last agent. The latter is the sum over all the agents i of the probability that i is chosen and is in state 1, i.e.,

i bi b N s i = b ρ h b
The transition probability L V h can be obtained by an analogous reasoning.

In the case of the Moran process, instead, the probability that in a timestep the state of node i is flipped from 1 to 0 is

P i (1 → 0) = j s i b i (1 -s j ) b (N -1) a j δt = s i b i b δt( a -aρ h ).
(8) Indeed, the probability that the agent i is flipped from 1 to 0 while interacting with j is equal to the probability a j δt(1s j ) that j becomes active and is in state 0, times the probability si bi b (N -1) that i is chosen among all the other agents and is in state 1. We then sum over all the agents j to obtain the total probability. Then, summing over all nodes i with activity a and attractiveness b we get

L M h = N h δt ρ h ( a -a ρ h ) b b . (9)
We obtain in a similar fashion

R M h = N h δt a ρ h (1 -ρ h ) b b . (10)
From these two particular cases we deduce the time evolution equation of the fraction of nodes with activity a and attractiveness b in state 1 in the general mixed case, which is given by

∂ρ h (t) ∂t = pa b ρ h b -ρ h +(1-p) a b b a ρ h a -ρ h . (11)

B. Conservation law

In the case of the voter model on a complete static graph, the total fraction ρ of voters in state 1 is conserved by the dynamics. In our model, it is clear from the previous equation that this in not in general true. Nevertheless, we may look for a conserved quantity of the form

Ω = h λ h ρ h , (12)
where the weights λ h are normalized as h λ h = 1. Using Eq. ( 11), we can check that the condition ∂Ω/∂t = 0 is fulfilled if the functions λ h satisfy the self consistent equation

λ h = η(h) p b [ h a λ h ] + (1 -p) a [ h b λ h ] pa b + (1 -p) a b , (13)
where h aλ h and h bλ h are determined by the normalization of the weights λ h (see details in Appendix):

h aλ h = 1 Q p a 2 ∆ h,p , (14)
h bλ h = a Q p b b 2 ∆ h,p , (15)
where we have defined

∆ h,p = pa b + (1 -p) a b (16)
and

Q p = p a 2 ∆ h,p b ∆ h,p + (1 -p) a b b 2 ∆ h,p a ∆ h,p .
(17) Notice that this last quantity depends only on p.

C. Exit probability

As in the case of the standard voter model [START_REF] Krapivsky | A Kinetic View of Statistical Physics[END_REF], the presence of a conservation law allows us to estimate directly the exit probability E for a single agent with state 1 in a population of agents with state 0, i.e., the probability that all agents finally adopt the state 1. Indeed, the final state with all voters in state 1, corresponding to Ω = 1, takes place with probability E (by definition), while the final state with all voters in state 0, with Ω = 0, happens with probability 1 -E. The conservation of Ω implies that Ω(t = 0) = E × 1 + (1 -E) × 0, from where we immediately obtain

E = h λ h ρ h (0), (18)
which depends exclusively on the initial state in which the system is prepared. For the particular initial conditions consisting of a single voter with variables h = (a, b), i.e., activity a and attractiveness b, in state 1 in a background of voters in state 0, we have that ρ h (0) = δ h ,h N -1 h , which leads to an exit probability

E a,b = λ(a, b) N η(a, b) , (19)
which, using Eq. ( 13) can be more explicitly expressed as

E a,b = 1 N Q p pb a 2 ∆ h,p + (1 -p)a a b b 2 ∆ h,p pa b + (1 -p)b a . (20)
Interestingly, this exit probability is a function of the ratio a b only.

D. Average consensus time

In order to compute the consensus time we can follow [10,31] and apply a one-step calculation to write down the recursion relation for the time T [{ρ h }] to reach consensus starting from a configuration {ρ h }:

T [{ρ h }] = δt + 1 - h (R h + L h ) T [{ρ h }] + h (R h T [{ρ h , ρ h + 1/N h }] + L h T [{ρ h , ρ h -1/N h }]) , (21)
where the notation {ρ h , ρ h ±1/N h } denotes a modification of the configuration {ρ h } by the flip of one agent of variables h (either from state 0 to the state 1, for the + case, or vice-versa for thecase).

This equation essentially amounts to consider that the consensus time for a given configuration is equal to the consensus time at the configuration obtained after a transition taking place in a time δt, weighted by the corresponding transition probabilities, plus δt. Expanding Eq. ( 21) at second order in 1/N h we obtain the backward Kolmogorov equation [START_REF] Gardiner | Stochastic Methods: A Handbook for the Natural and Social Sciences[END_REF]

h v h ∂T ∂ρ h + h D h ∂ 2 T ∂ρ 2 h = -1, (22)
where

v h = pa b ρ h b -ρ h + (1 -p) a b b a ρ h a -ρ h (23)
and

D h = pa 2N h b ρ h b + ρ h -2 b ρ h b ρ h + (1 -p) a b 2 b N h a ρ h a + ρ h -2 a ρ h a ρ h (24)
are the drift and diffusion coefficients, respectively [START_REF] Gardiner | Stochastic Methods: A Handbook for the Natural and Social Sciences[END_REF].

After a transient time depending on the distribution η(a, b), the system reaches a steady state where ρ h = Ω, ∀h. Then we may drop the drift term in Eq. ( 22), and, considering Eq. ( 12), change variable from ρ h to Ω [10,31]

∂T ∂ρ h = λ h ∂T ∂Ω . (25)
Substituting into Eq. ( 22) and simplifying (see details in Appendix), we finally obtain

Ω(1 -Ω) ∂ 2 T ∂Ω 2 = -N b ( h aλ h ) ( h bλ h ) . ( 26
)
This last equation can be directly integrated, yielding the consensus time

T = τ N a (1 -Ω) ln 1 1 -Ω + Ω ln 1 Ω , (27)
where we defined the characteristic adimensional consensus time

τ = a b ( h aλ h ) ( h bλ h ) . ( 28
)
The model is then entirely solved in terms of the previous expressions for the consensus time and the exit probability. These expressions are however quite intricate and it is quite insightful to study particular cases of interest, for given forms of the distribution of the activity and attractiveness η(a, b) and particular values of the mixing probability p. We present this analysis in the following Section.

IV. PARTICULAR CASES

A. p = 1/2

From the definition of ∆ h,p in Eq. ( 16), one obtains, by multiplying this equation respectively by a/∆ h,p and b/∆ h,p and averaging,

b = p ab ∆ h,p b + (1 -p) b 2 ∆ h,p a , (29)
a = p a 2 ∆ h,p b + (1 -p) ab ∆ h,p a . ( 30
)
Thus for p = 1/2, one obtains, eliminating ab/∆ h,p between these two equations,

a b b 2 ∆ h,p = b a a 2 ∆ h,p , (31)
and Eq. ( 17) becomes

Q 1/2 = 1 2 b 2 ∆ h,p a b a ∆ h,p + b ∆ h,p a b = b 2 ∆ h,p a b 2 = 1 a a 2 ∆ h,p . (32)
From here, it follows that h a λ h = a and h b λ h = b , which finally implies that λ h = η h and τ = 1.

In this case, the dynamics becomes identical to the standard link update dynamics of the voter model [START_REF] Suchecki | [END_REF], and it is totally independent on a and b (in terms of the number of flip attempts) because the probability that the total number of voters in state 1 is increased during an update attempt is exactly compensated by the probability that this same number is decreased.

B. Pure voter model

The voter model, corresponding to p = 1, leads in Eq. ( 13) to

λ(a, b) = η(a, b) b a -1 b a -1 . (33)
We also obtain ∆ h,p = a b and Q 1 = a ba -1 / b 2 , leading for the consensus time in Eq. ( 28) to the simple form:

τ = a b a -1 2 b 2 a -1 (34)
The exit probability is also straightforward to derive from Eq. ( 19):

E a,b = b a -1 N b a -1 . (35)

C. Moran process

The Moran process corresponds to p = 0, then Eq. ( 13) reduces to

λ(a, b) = η(a, b) a b -1 a b -1 , (36)
leading, with ∆ h,p = b a and Q 0 = ab -1 / a , from Eq. ( 28) to

τ = b a b -1 2 a 2 b -1 . ( 37
)
The exit probability reads in this case

E a,b = a b -1 N a b -1 . ( 38
)
It is noteworthy that the results for the Moran process are obtained from the ones of the voter model by simply exchanging a and b. In fact, we see from Eq. ( 13) that the dynamics of the mixed process is the same as the dynamics of the symmetrical process (i.e. with p ← 1p) upon exchanging a and b. This is intuitively clear if we examine the process from a stochastic point of view: at each update attempt, the node i is chosen at random with probability ai a and the node j with probability bj b .

Besides, changing p into 1p is equivalent to reversing the roles of i and j, which has no effect if a and b are exchanged. This is however valid only when the time is counted as the number of update attempts, the physical time being multiplied by a b when swapping a and b.

D. Pure Activity Driven Networks

The original activity driven network model [26] does not consider a heterogeneous attractiveness, and this cor-

τ = a 2 [pa + (1 -p) a ] -1 a 2 [pa + (1 -p) a ] -1 , (39)
while the exit probability is given by

E a = 1 N p a τ + (1 -p)a pa + (1 -p) a . ( 40
)
In order to study the behavior of the consensus time, in Fig. 1 we plot the analytical evaluation of τ , Eq. ( 39), for a normalized activity distribution with a power-law form, as empirically observed in Ref. [26],

F (a) = 1 -γ 1 -1-γ a -γ , a ∈ [ , 1]. ( 41
)
where is the minimum activity in the system, imposed in order to avoid divergences in the normalization and moments of F (a). From Fig. 1 we see that the consensus time has a minimum around γ = 2 for the Moran process (p = 0) and a maximum around γ = 1 for the voter model (p = 1). Note that by virtue of the symmetry property discussed above, the dynamics of the pure attractiveness model (setting a i = a 0 , ∀i), taking the same distribution F for b and imposing a 0 = b 0 , is the same upon exchanging p by 1p. In particular, the consensus time is obtained by reversing the p axis in Fig. 1.

E. Independent activity and attractiveness

In the case where a and b are drawn independently from the same distribution F , we have η(a, b) = F (a)F (b). In Fig. 2 we plot the characteristic consensus time τ as a function of p and γ for F given by Eq. (41). For this particular form of the distribution η(a, b) (and in general for any symmetric joint distribution such that η(a, b) = η(b, a)), the dynamics remains the same when changing p into 1p because exchanging a and b has no effect. This is clearly observed in Fig. 2. Additionally, we see that both the voter model and the Moran process have a minimum consensus time for γ 2.25 and a maximum consensus time for γ = 0.75, respectively.

F. Strongly correlated activity and attractiveness

As we previously mentioned, the weight function λ h is the product of η h and a function of the ratio a b . This fact straightforwardly implies that, in the maximally correlated case η(a, b) = F (a)δ a,b , where a = b for every agent, the dynamics is the same as in a fully connected static network, i.e. the average density ρ = ρ h of voters in state 1 is conserved, the reduced consensus time τ is equal to 1, and the exit probability is homogeneous and equals 1/N .

G. Discussion

The results obtained above relate the average consensus time and the exit probability with the moments of the joint distribution η(a, b) and the value of the activity a and the attractiveness b of the initial invading voter. Remarkably, when we compare these results with the ones obtained in the case of static networks with a given degree distribution P (k) [10,31] we observe interesting symmetries between voter and Moran dynamics.

This symmetry is of particular interest when we consider the pure activity driven network (setting b = 1). Let us consider for instance the invasion exit probability, Eqs. ( 35) and (38). In the case of the voter model, this exit probability is inversely proportional to the activity of the node with initial state 1 (E voter a ∝ 1/a), while for the Moran process, it is proportional to the activity (E M oran a ∝ a). This can be understood by the fact that an active node will often change state in the voter model, by contacting other nodes, while in the Moran process it will often spread his/her state towards the other nodes contacted.

These results have to be compared with the result for the voter and Moran processes in static networks, in which the exit probability for a single node of degree k with state 1 is E voter k ∼ k for the voter and E Moran k ∼ k -1 for the Moran process [10,31]. Intuitively indeed, in the case of static networks, in the voter model high degree nodes are chosen to be copied with high probability [35], implying that they are very efficient spreaders of their own state to the rest of the network. Hence, the larger k, the higher the exit probability. In the case of the Moran process, by applying the same argument, high degree nodes are prone to often change state by adopting the state of a neighbor [10,31], and hence the exit probability decreases with the degree.

Let us now recall that, for a pure activity driven network, the aggregated degree of a node with activity a takes the value ka (t) ∼ (a + a )t at time t: nodes with high activity tend to have large integrated degree [26,29]. Putting this in relation with the behavior of the exit probability as a function of activity in temporal networks and of degree in static networks, we thus obtain that the dynamics on the temporal activity driven network yields a completely different and opposite result when compared with the dynamics on the static, integrated network counterpart: High activity nodes are more prone to spread under Moran dynamics, while low activity nodes are more prone under voter dynamics.

This symmetry voter-Moran between pure activity driven networks and their integrated counterpart occurs as well at the level of the average consensus time when we measure it as a function of the update attempts. Considering that a randomly chosen node becomes active with average probability a , we have that, as a function of updated attempts, the convergence time is TN ≡ a T N . We have thus, for homogeneous initial conditions (Ω = 1/2),

T voter N = N a a -1 ln 2, T Moran N = N a 2 a 2 ln 2. ( 42
)
Comparing with the results for static networks [10,31],

T Moran N = N k k -1 ln 2, T voter N = N k 2 k 2 ln 2, ( 43
)
we observe that the formulas for voter and Moran dynamics are indeed mirror images, with the activity distribution a in the temporal representation substituted by the degree distribution in the integrated representation.

Let us now consider instead the pure attractiveness temporal network model (setting a = 1). In that case, the exit probability is proportional to the attractiveness for the voter model, E voter b ∝ b, while for the Moran process, it is inversely proportional to the attractiveness, E M oran b ∝ 1/b. Moreover, the integrated degree of a node with attractiveness b is kb (t) ∼ b b t. Here therefore, we have the same kind of behavior on the temporal and corresponding integrated static network when making an equivalence between attractiveness in the temporal network and degree in the static network. This equivalence between a static network with a degree distribution P (k) and a pure attractiveness temporal network with the same distribution P (b) is also obtained by looking at the consensus time measured as the number of update attempts.

V. NUMERICAL RESULTS

In order to check the analytical predictions made above, we have performed simulations of the mixed process defined earlier on activity driven networks with attractiveness, choosing a marginal activity distribution following a power-law, Eq. ( 41), similar to the distribution observed empirically in some real networks [26,36]. We have performed simulations for network sizes N = 10 2 , 10 3 and 10 4 , averaging over 10 3 realizations.

In Fig. 3 we plot the reduced consensus time τ as a function of γ for three different values of the network size and three different dynamics: voter and Moran processes on a pure activity driven network, and voter model on an activity driven network with attractiveness with a and b independently and equally distributed η(a, b) = F (a)F (b). The curves are compared to the theoretical value given in Eq. (28). We see that for N = 10 4 the dynamics already matches well the expected behaviour in the infinite size limit. We deduce that our heterogeneous mean-field analysis captures efficiently the opinion dynamics on the activity driven network with attractiveness.

VI. ASYMPTOTIC BEHAVIOUR

In Figs. 123, we see that the consensus time presents minima and maxima when γ varies. To investigate this point in more details, we analyse the asymptotic behaviour of the moments of the distribution F (a) when tends to zero. For an activity distributed with Eq. (41), the moments of a take the form In each case, the dashed line corresponds to the analytical expression given by Eq. ( 28).

a n = 1 -γ n + 1 -γ 1 -1+n-γ 1 -1-γ . ( 44
)
The dynamics of the voter and Moran processes on pure activity driven network and on activity driven network with equally and independently distributed a and b depends on the moments a -1 , a and a 2 only. The asymptotic behaviour of these three quantities for → 0 are summarized in Tables I andII, along with the resulting behaviour of the consensus time in the case with a and b independently and equally distributed and p = 1 (or equivalently p = 0), for which

τ = a 3 a -1 a 2 . ( 45
)
For all values of γ, the physical ordering time T ∼ τ / a diverges when tends to zero. This is not surprising if we notice that the consensus of the voter dynamics is strongly limited by the agent with the smallest activity , which activates and copies the opinion of a peer for the first time in a time 1/ on average. However, we observe that for 0 < γ < 1.5 the consensus time τ measured as the number of update attempts, in Eq. ( 45), diverges when goes to zero, and tends instead to zero for 1.5 < γ < 3. The number of update attempts required to reach consensus may either be infinite or null depending on the exponent γ, with a sharp transition between both regimes at γ = 1.5. This non-trivial behaviour reveals the critical importance of the temporal network's topology in this case, and is not easily predictable with qualitative arguments. We also recover the fact that the fastest consensus is reached for γ = 2 and the slowest for γ = 1 and that for 0 < γ < 3 the consensus time exhibits a symmetry with respect to the axis γ = 1.5 : τ (γ) = 1 τ (3-γ) . Finally, for γ 3, the heterogeneity of the distribution of a is no longer significant, so that the dynamics is that of a fully connected static network. In

γ ]0, 1[ ]1, 2[ ]2, 3[ > 3 a -1 O( -γ ) O( -1 ) O( -1 ) O( -1 ) a O(1) O( γ-1 ) O( ) O( ) a 2 O(1) O( γ-1 ) O( γ-1 ) O( 2 ) τ O( -γ ) O( 2γ-3 ) O( 3-γ ) O (1)
TABLE I. Asymptotic behaviour of the moments of the distribution F (a) defined in Eq. ( 41), and resulting asymptotic behaviour of the reduced consensus time τ given by Eq. ( 45) when tends to zero, as a function of the exponent γ. For γ = 0, 1, 2, 3, logarithmic corrections are present, given in Table II.

Fig. 4 we plot the reduced consensus time for the voter dynamics on a network with independent activity and attractiveness as obtained by direct numerical simulations of a voter model on a temporal network, compared with the analytical predictions of Eq. ( 45), for various values of and N = 10 4 . The simulations confirm the predicted asymptotic behaviour of τ when tends to zero. We also observe stronger finite size effects when epsilon tends to zero due to a poorer sampling of the activity distribution given by Eq. (41).

VII. CONCLUSIONS

In this paper, we have studied in detail the properties of consensus processes mixing the voter and Moran models update rules, on temporal network models based on the activity driven paradigm. Through a heterogeneous meanfield approach, we have derived the evolution equation of the average density of voters in state 1 with activity a and attractiveness b. This has allowed us to identify

γ 0 1 2 3 a -1 -ln -( ln ) -1 (2 ) -1 2 3 -1 a 1/2 -(ln ) -1 -ln 2 a 2 1/3 -(2 ln ) -1 -2 2 ln τ -3 8 ln -2 -1 (ln ) -3 -1 2 (ln ) 3 -8 3 (ln ) -1
TABLE II. Asymptotic behaviour of the moments of the distribution F (a) defined in Eq. ( 41), and resulting asymptotic behaviour of the reduced consensus time τ given by Eq. ( 45) when tends to zero, for the specific cases γ = 0, 1, 2, 3.

a conserved quantity of the dynamics, and subsequently to compute the average time to reach consensus and the probability that a single agent with a discrepant opinion among an otherwise unanimous population spreads his/her opinion to the whole network, called the exit probability. Surprising results arise from the study of particular cases of the distribution of the parameters a and b. When the attractiveness is taken to be proportional to the activity a, the dynamics is the same as if the copying process were running on a static complete graph. The average activity a determines the time scale of the dynamics, but otherwise the precise distribution of activity among the agents is no longer relevant. This holds for all values of the probability p determining the state update rule, and in particular for the voter model (p = 1) and the Moran process (p = 0). The same behaviour happens when p = 1/2, regardless of the distribution η(a, b) of activity and attractiveness: surprisingly, the exit probability is equal to 1/N and does not depend of the parameters of the initial invading node.

Interestingly, when the activity and the attractiveness are independent and equally distributed, the dynamics is unchanged when replacing p by 1p. In fact, it appears that by construction, when the time is counted as the number of update attempts, exchanging a and b on all the nodes is equivalent to replace p by 1p in the update rule. One of our main results lies in the observation that the voter model and the Moran process on a pure activity driven network (setting b = 1 for all nodes) are in some sense mirror images of their static network counterparts. Indeed, the dynamics of the voter model on an activity driven network with a distribution F (a) is the same as the Moran dynamics running on top of a static network with a degree distribution P (k) = F (k). The same holds for the Moran process on the activity driven network and the voter model on the static network. This implies that the apparently appealing operation consisting in considering an activity driven network and its integrated counterpart as similar substrates for this kind of opinion dynamics process would be misleading, despite the fact that the degree distribution of the integrated network is practically equal to the activity distribution of the temporal network [26]. On the contrary, a pure attractiveness temporal network (setting a = 1 for all nodes) and and its integrated counterpart are equivalent substrates for the voter and Moran processes. It would be very interesting to check whether similar conclusions hold for other consensus formation processes with more complex update rules like the majority rule process. Our results will hopefully motivate further research in this direction.

  responds to a joint distribution η(a, b) = F (a)δ b,b0 , where F (a) is the activity distribution and b = b 0 , constant. In this case we have Q p = a b 0 [pa + (1p) a ] -1 , and the characteristic consensus time reads

FIG. 1 .FIG. 2 .

 12 FIG. 1. Characteristic consensus time τ for the dynamics on a pure activity driven network, i.e., fixed attractivity b = b0, and a distribution of activities F (a) given by Eq. (41), as a function of p and of the exponent γ of the distribution F . = 10 -3 .

FIG. 3 .

 3 FIG. 3. Voter-like dynamics in temporal activity driven networks with attractiveness. Reduced consensus time as a function of γ for different values of the network size N , for an activity distribution F (a) given by Eq. (41) with = 10 -3 . (a): Voter and Moran processes (equivalent) for η(a, b) = F (a)F (b). (b): Voter model on pure AD network (b = 1). (c): Moran process on pure AD network (b = 1). In each case, the dashed line corresponds to the analytical expression given by Eq. (28).

3 FIG. 4 .

 34 FIG.4. Voter dynamics in temporal networks with independent and equally distributed activity and attractiveness. Consensus time τ as a function of γ for activity and attractiveness distributed according to Eq. (41) and different values of . The analytical expression given by Eq. (45) is shown in dashed lines. The numerical simulations are performed with network size N = 10 4 .

ACKNOWLEDGMENTS

We acknowledge financial support from the Spanish MINECO, under project FIS2016-76830-C2-1-P. R.P.-S. acknowledges additional financial support from ICREA Academia, funded by the Generalitat de Catalunya.

Appendix A: Details of some computations From the expression of the weights λ h in Eq. ( 13) we obtain, by multiplying by a and summing over h (in the next equations we write aλ and bλ as shorthands for h aλ h and h bλ h , respectively):

A1) which gives a relation between aλ and bλ bλ

The normalization of the weights gives

(A3) Combining Eqs. (A2) and (A3) leads to

(A4) Besides, by definition of ∆ h,p we wrote Eq. ( 29) which, after dividing by b gives

Inserting this into Eqs. (A4) and (A2) one recovers the correct expressions given in Eqs. ( 14) and (15).

To derive the expression of the average consensus time, we write, combining Eqs.( 22), ( 24) and ( 25)

We have


Rémy Gauthier

Max Langer

Hélène Follet

Cécile Olivier

Pierre-Jean Gouttenoire

Lukas Helfen

Frédéric Rongieras

David Mitton

Françoise Peyrin
email: peyrin@esrf.fr

Frédéric Rongiéras

3D micro structural analysis of human cortical bone in paired femoral diaphysis, femoral neck and radial diaphysis

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

Introduction

Bone micro-architecture has been largely studied among several years in order to better understand its growth process or its mechanical properties. This hierarchical organization is the locus of a biological activity that will specify the overall quality of the tissue [START_REF] Busse | Decrease in the osteocyte lacunar density accompanied by hypermineralized lacunar occlusion reveals failure and delay of remodeling in aged human bone[END_REF][START_REF] Currey | Cavities in the compact bone in tetrapods and fish and their effect on mechanical properties[END_REF][START_REF] Gourion-Arsiquaud | Spatial variation in osteonal bone properties relative to tissue and animal age[END_REF][START_REF] Hannah | Bimodal distribution of osteocyte lacunar size in the human femoral cortex as revealed by micro-CT[END_REF][START_REF] Nalla | Role of microstructure in the aging-related deterioration of the toughness of human cortical bone[END_REF][START_REF] Pfeiffer | Secondary osteon and Haversian canal dimensions as behavioral indicators[END_REF][START_REF] Seeman | Bone quality--the material and structural basis of bone strength and fragility[END_REF][START_REF] Van Oers | A unified theory for osteonal and hemi-osteonal remodeling[END_REF][START_REF] Varga | Synchrotron X-ray phase nano-tomography-based analysis of the lacunar-canalicular network morphology and its relation to the strains experienced by osteocytes in situ as predicted by case-specific finite element analysis[END_REF].

On the microscopic scale, cortical bone main porosity is related to the osteonal canals forming a network hosting blood vessels and nerves. The osteonal canals in humans are approximately 50 -100 µm in diameter and a few millimeters in length [START_REF] Jowsey | Studies of Haversian systems in man and some animals[END_REF]. Osteonal canals play a major role in the bone remodeling process due to the passage of blood and nutriments. The bone matrix also exhibits a porous network on a finer scale. This system is the lacuno-canalicular network (LCN) that hosts the osteocytes network. The osteocytes, which are the most numerous cells present in bone material, are hypothesized to play a key mechanosensor role in the adaptive capacity of bone in its particular mechanical environment [START_REF] Bonewald | The amazing osteocyte[END_REF][START_REF] Busse | Decrease in the osteocyte lacunar density accompanied by hypermineralized lacunar occlusion reveals failure and delay of remodeling in aged human bone[END_REF]. They are included in osteocyte lacunae that are roughly ellipsoidal [START_REF] Marotti | Osteocyte orientation in human lamellar bone and its relevance to the morphometry of periosteocytic lacunae[END_REF] and are connected by small canals, canaliculi, containing the processes of the osteocytes.

In addition with these porosities , bone also includes defects that are caused by different type of damages occurring throughout the lifetime of the tissue [START_REF] Vashishth | Hierarchy of bone microdamage at multiple length scales[END_REF]. The main type of damage is linear micro-cracks, and the second type is the diffuse micro-damage, which will not be discussed here. The planar micro-cracks, defects which can span hundreds of µm in length and a gap of about 1 µm, are the result of fatigue of the bone material due to external loading. The presence of such damages within cortical bone may be associated to a weakening of the tissue resulting in an increased fracture risk [START_REF] Burr | Microdamage and bone fragility[END_REF][START_REF] Burr | Bone Microdamage and Skeletal Fragility in Osteoporotic and Stress Fractures[END_REF]. This is why the analysis of micro-cracks appears to be relevant. These micro-damages are also assumed to be related to the bone remodeling process. The detection of such defects by the osteocytes may act as a signal for remodeling [START_REF] Burr | Bone remodeling in response to in vivo fatigue microdamage[END_REF][START_REF] Lee | Bone adaptation to load: Microdamage as a stimulus for bone remodelling[END_REF].

The morphology of the osteonal canals has largely been investigated using 2D techniques such as histology or micro-radiography [START_REF] Bell | Structure of the femoral neck in hip fracture: cortical bone loss in the inferoanterior to superoposterior axis[END_REF][START_REF] Jowsey | Microradiographs and autoradiographs of cortical bone from monkeys injected with 90Sr[END_REF]. Both of these techniques facilitate the observation of canals but only on sections of bone. More recently, the three-dimensional vascular network has been largely studied in cortical bone mostly using micro-computed tomography as it allows the investigation on large volumes [START_REF] Bala | Pore network microarchitecture influences human cortical bone elasticity during growth and aging[END_REF][START_REF] Bousson | Cortical bone in the human femoral neck: three-dimensional appearance and porosity using synchrotron radiation[END_REF][START_REF] Chappard | Comparison of synchrotron radiation and conventional x-ray microcomputed tomography for assessing trabecular bone microarchitecture of human femoral heads[END_REF][START_REF] Cooper | Age-dependent change in the 3D structure of cortical porosity at the human femoral midshaft[END_REF]. It seems important to investigate such a network in 3D as it has been shown that the orientation of these canals may be associated to their mechanical environment [START_REF] Hennig | Does 3D orientation account for variation in osteon morphology assessed by 2D histology?[END_REF][START_REF] Heřt | Osteon orientation of the diaphysis of the long bones in man[END_REF]. Studies on this vascular system has shown that its 3D organization may be determinant regarding bone mechanical properties and remodelling process [START_REF] Bala | Pore network microarchitecture influences human cortical bone elasticity during growth and aging[END_REF][START_REF] Perilli | Regional Heterogeneity in the Configuration of the Intracortical Canals of the Femoral Shaft[END_REF][START_REF] Zebaze | Intracortical remodelling and porosity in the distal radius and post-mortem femurs of women: a cross-sectional study[END_REF].

The observation of micro-cracks and osteocytes is usually conducted using imaging methods such as Scanning Electron Microscopy (SEM) [START_REF] Granke | Prevalent role of porosity and osteonal area over mineralization heterogeneity in the fracture toughness of human cortical bone[END_REF][START_REF] Milovanovic | Multi-level characterization of human femoral cortices and their underlying osteocyte network reveal trends in quality of young, aged, osteoporotic and antiresorptive-treated bone[END_REF], visible light microscopy [START_REF] Wang | In situ measurement of solute transport in the bone lacunar-canalicular system[END_REF], and confocal microscopy [START_REF] Lai | The dependences of osteocyte network on bone compartment, age, and disease[END_REF][START_REF] Verbruggen | Strain amplification in bone mechanobiology: a computational investigation of the in vivo mechanics of osteocytes[END_REF] or UV light microscopy [START_REF] Follet | Determinants of Microdamage in Elderly Human Vertebral Trabecular Bone[END_REF]. However, most of these different techniques only provide 2D images of micro-cracks and the LCN. [START_REF] Schneider | Serial FIB/SEM imaging for quantitative 3D assessment of the osteocyte lacuno-canalicular network[END_REF] used a SEM coupled with focused ion beam milling to investigate a three-dimensional (3D) region containing lacunae [START_REF] Schneider | Serial FIB/SEM imaging for quantitative 3D assessment of the osteocyte lacuno-canalicular network[END_REF]. Although this technique can provide high-resolution 3D renderings of osteocyte lacunae, the size of the region of interest (ROI) remains limited to one or two lacunae due to the extensive scanning time. µCT has also been used to investigate the morphometric properties of osteocyte lacunae within a larger region [START_REF] Van Hove | Osteocyte morphology in human tibiae of different bone pathologies with different bone mineral density -Is there a role for mechanosensing[END_REF][START_REF] Vatsa | Osteocyte morphology in fibula and calvaria -Is there a role for mechanosensing?[END_REF]. However, recent studies have shown that the segmentation of osteocyte lacunae from such images might compromise the quantitative analysis because of the need to develop complex algorithms [START_REF] Betts | Osteocyte lacuna segmentation from ultr-high desktop micro-CT images: Low precision reveals limitation of state of the art[END_REF].

Synchrotron radiation (SR) CT was previously used to image and analyze osteocyte lacunae in human femoral cortical bone [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF][START_REF] Hannah | Bimodal distribution of osteocyte lacunar size in the human femoral cortex as revealed by micro-CT[END_REF]. It also facilitates imaging of the LCN [START_REF] Langer | X-Ray Phase Nanotomography Resolves the 3D Human Bone Ultrastructure[END_REF][START_REF] Pacureanu | Nanoscale imaging of the bone cell network with synchrotron X-ray tomography: optimization of acquisition setup[END_REF] and bone micro-cracks [START_REF] Larrue | Synchrotron radiation micro-CT at the Micrometer scale for the analysis of the three-dimensional morphology of microcracks in human trabecular bone[END_REF]. The advantage of this technique compared to standard CT in this context is that it can offer a higher signalto-noise ratio, even at higher spatial resolution [START_REF] Salomé | A synchrotron radiation microtomography system for the analysis of trabecular bone samples[END_REF], thus facilitating the segmentation of lacunae and its quantitative analysis. Synchrotron radiation also provides a high flux, allowing a significant decrease of scanning time.

To the authors' knowledge, no data has been produced describing such properties of the vascular system, micro-cracks and lacunae on other anatomical sites beyond the human femoral diaphysis. As bone architecture is dependent of its mechanical environment during life and because bone fracture behavior is related to its microstructure [START_REF] Zimmermann | Bone as a Structural Material[END_REF], the weight-bearing or nonweight-bearing nature of bone may have an influence on this fracture behavior [START_REF] Maude | One-month spaceflight compromises the bone microstructure, tissue-level mechanical properties, osteocyte survival and lacunae volume in mature mice skeletons[END_REF][START_REF] Mikkola | Genetic and environmental influence on structural strength of weight-bearing and non-weight-bearing bone: A twin study[END_REF][START_REF] Vico | Effects of long-term microgravity exposure on cancellous and cortical weight-bearing bones of cosmonauts[END_REF]. Here, weight-bearing and non-weight-bearing bones are assumed to correspond to the leg's bones and the arm's bones, respectively. This is nevertheless a simplification as even arm's bones are subjected to some load during life.

Thus, the aim of this study is to quantify in 3D the morphology of the multi-scale porosities on paired human femoral diaphysis femoral neck and radial diaphysis based on SR-µCT imaging at the submicrometric scale.

Materials and methods

Sample description

Paired femoral diaphyses, femoral necks, and radial diaphyses were collected from 8 female donors (70 ± 14 y.o., min: 50 y.o., max: 91 y.o.) provided by the Departement Universitaire d'Anatomie Rockefeller (Lyon, France) through the French program on voluntary corpse donation to science. No additional information regarding disease status or medication history was available. Extracted bones were wrapped in gauze soaked with saline to keep them hydrated and then stored at -20°C until sample preparation. Two contiguous rectangular samples were extracted from each bone using a low speed saw with a diamond-coated blade (ISOMET 4000, Buehler, USA). This was done in order to consider the internal variability within the same bone. Samples were extracted from the lateral region of the femoral diaphysis, at nearly 60% of the diaphysis height (0% = distal epiphysis) on each donor. On the femoral neck, the specimens were taken from the inferior region for reasons of cortical availability. On the radius, the samples were extracted at nearly 70 mm from the proximal epiphysis on the anterior region, which corresponds to the region used in a previous study involving ultrasound guided waves [START_REF] Vallet | In Vivo Characterization of Cortical Bone Using Guided Waves Measured by Axial Transmission[END_REF]. The samples were extracted in the middle between the endosteal and the periosteal surface. A delipidation protocol was applied one week before imaging: the samples were immerged in acetone for 30 minutes, then rinsed with water, and finally dehydrated by successive immersion in 70 % and nearly 100 % ethanol baths for a maximum period of 2 days. Samples of 2.09 ± 0.14 mm in width and 0.99 ± 0.07 mm in thickness were extracted. The height was about 12.5 mm in order to facilitate a proper clamping during the acquisition (Figure 1).

Synchrotron radiation micro-computed tomography (SR-µCT)

SR-µCT was performed on beamline ID19 at the European Synchrotron Radiation Facility (ESRF), Grenoble, France. We used a "pink beam" filtered undulator radiation with an effective energy of 31 keV. The detector was composed of a scintillator screen, a visible light microscope, and a CCD camera and was set up to yield an effective pixel size of 0.7 µm. For each sample, 2000 projection images were recorded over a total rotation of 360°, with a counting time of 0.3 s per projection and a total scan time of 14 minutes. Phase contrast was obtained by placing the detector at 40 mm from the sample. The images were reconstructed with and without phase retrieval. Phase retrieval was performed using Paganin's method [START_REF] Paganin | Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object[END_REF], with the δ/β-ratio set to 572. Tomographic reconstructions were performed using a filtered back-projection algorithm yielding 3D images with 2048 3 voxels.

A total of 48 scans were recorded (8 female donors, 3 anatomical locations and 2 volumes of interests (VOIs) for each location). Since the field of view (FOV) on the detector was 1.4x1.4 mm 2 , the samples were not completely included in the FOV for some angles, yielding to local tomography [START_REF] Paleo | Efficient implementation of a local tomography reconstruction algorithm[END_REF]. The size of the reconstructed bone volume of interest (VOI) was approximately 1 x 1.4

x 1.4 mm 3 . Because of some motion problem during the acquisition, 5 scans were not exploitable.

3D image analysis

The analyses were performed on the two contiguous samples. The values retained for the further analysis of the investigated parameters were considered as the average between these two samples (Figure 1). Statistical analyses were performed to check for any difference between these two contiguous samples (cf. 2.4).

Segmentation of canals, lacunae and micro-cracks

The first step consisted of the segmentation of the different structures of interest: osteonal canals, lacunae, and micro-cracks. To this aim, a bone mask was generated by thresholding using Otsu's method after applying a median filter to remove lacunar and other small porosities. The total bone envelope was obtained as the convex hull of the binarized bone image. Subsequently, the canal mask was obtained as the complement of the bone volume in the bone envelope. Lacunae and micro-cracks were then segmented using hysteresis thresholding limited to within the bone mask. Hysteresis thresholding uses a low and a high threshold value as input parameters. The low value facilitates the selection of voxels that are directly labeled as osteocyte lacunae or micro-cracks. The higher threshold permits the refinement of the segmentation by adding voxels with higher intensity values, but only if they are connected to the previously detected voxels. Due to the local CT acquisition in the current study, the reconstructed attenuation coefficient could be shifted from one sample to another.

Therefore, the hysteresis thresholding low and high values were determined manually for each sample.

Finally, a binarized volume containing lacunae and micro-cracks was obtained (Figure 2). Bone volume (BV) was calculated by multiplying the inverse of the final binarized VOI with the canals mask. BV was considered as the sum of the BVs measured on the two contiguous samples.

Osteonal canal analysis

The analysis of the osteonal canals was done using the canal mask image with the automated commercial software (CTAnalyser Software V 1.14.4, Skyscan NV, Kontich, Belgium). On these segmented VOIs (Figure 3), the canal volume fraction Ca.V/TV (%) was evaluated. TV was measured by adding all of the voxels of the analyzed volume. The mean diameter of the canals, Ca.Dm (µm), and the standard deviation Ca.Dm.SD (µm) corresponding to the heterogeneity of canals' size within a same bone sample were also measured. The analysis of the osteonal canal was realized on the complete scanned regions of the two contiguous samples resulting in a total volume of twice (1 x 1.4 x 1.4) mm 3 (Figure 1).

Morphometric parameters on lacunae and micro-cracks

Since the detailed analysis of lacunae and micro-cracks is highly time consuming, it was performed with cropped VOIs of 500 x 500 x 1000 µm 3 . As previously performed, the total volume for the analysis of lacunae and micro-cracks was 2 x (500 x 500 x 1000) µm 3 , the lacunar analysis was thus performed over 10.000 lacunae for each samples (cf. table 3).

From the binary volume of lacunae and micro-cracks, morphometric descriptors of each object were calculated using the method proposed by Dong et al. [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]. The first step was to perform a connected component analysis to label each object (either lacuna or micro-crack). Then, the following parameters were computed on each labeled object (we keep the notations used for lacunae):

-Object Volume (Obj.V): This is the sum of all the voxels within the cell multiplied by the volume of one voxel (0.7 x 0.7 x 0.7 µm 3 ) -Object dimensions (Obj.L1), width (Obj.L2) and depth (Obj.L3) assuming an ellipsoidal shape; L1, L2 and L3 are the three axes of the best fitting ellipsoid, with L1 being the main axis.

-Object thickness (Obj.Th): This measure is assessed using an automated algorithm assigning the value of the diameter of the maximal fitting sphere within the structure [START_REF] Martín-Badosa | A method for the automatic characterization of bone architecture in 3D mice microtomographic images[END_REF].

-Object surface area (Obj.S), object structure model index (Obj.SMI), and object Euler number (Obj.χ): This value is calculated using the intrinsic volumes Obj.V 0 , Obj.V 1 , and Obj.V 2, which are geometrical invariant used in 3D image processing, by applying the following equations [START_REF] Schladitz | Measuring Intrinsic Volumes in Digital 3d Images[END_REF] [START_REF] Di Zenzo | Run-based algorithms for binary image analysis and processing[END_REF]. The SMI's value ranges between 0 and 4 and defines the global shape of the object: SMI = 0 for a pure plate, SMI = 3 for a rod, and SMI = 4 for a sphere [START_REF] Hildebrand | Quantification of Bone Microarchitecture with the Structure Model Index[END_REF].

Separation of osteocytes and micro-cracks

So far, the segmented structures analyzed as described above can be either lacunae or micro-cracks.

Lacunae are denoted by Lc and micro-cracks by µCr. Since these structures are assumed to have quite different shapes, we defined the criteria based on the different descriptors to identify them, similarly to those proposed by Dong et al. [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]. First, objects with a volume smaller than 82 µm 3

were considered as noise incorrectly classified as porosity and were therefore subsequently removed [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]. Objects with a volume larger than 10000 µm 3 were also excluded from the lacuna class, considered as either an incorrectly segmented object or crack. Criteria were also applied to some geometrical ratios; objects with Lc.L1/Lc.Th ratio higher than 15 and Lc.L2/Lc.Th ratio higher than 8

were excluded and considered as micro-cracks. Lc.SMI and Lc.χ were also exploited as exclusion criteria: lacunae were considered as having a SMI higher than 1.6 and an Euler number ranging from 0 to 2. A summary of these criteria is given in Table 1.

To extract micro-cracks (µCr), we selected cells with a volume higher than 500 µm 3 to exclude small objects due to noise. To exclude lacunae, we used the opposite criteria as done previously on Obj.L1/Obj.Th and Obj.L2/Obj.Th ratios, since micro-cracks have relatively small thicknesses compared to other objects. Objects with Obj.L1/Obj.L3 ratios lower than 5 and features with ObjL2/Obj.L3 lower than 3 were considered as ring artefacts, since these incorrectly segmented objects had a relatively short ellipsoid axis L3 compared to cracks. Finally we defined the objects with SMIs lower than 2.5 to exclude other types of artefacts. A summary of these criteria is given in Table 1.

The Euler number (χ) and SMI were only used for the separation step; they were not used for any further analysis. Their values are thus not given.

Lacunae and cracks analysis

After the previously described step, we obtained two distinct binary volumes, including on one hand, the lacunae and on the other hand the micro-cracks. The different geometrical parameters listed above were then recalculated for each of these. Henceforth, as the two different features are investigated separately, lacunae are denoted by Lc and micro-cracks by µCr.

The dimensions of the lacunae were taken as the three axes of the best-fitting ellipsoid, Lc.L1 (µm), Lc.L2 (µm), and Lc.L3 (µm). The mean individual volume and surface, Lc.V (µm 3 ) and Lc.S (µm 2 ), respectively, were evaluated. In order to characterize the curvature at the top of the elliptic lacunae, we introduced Lc.ρ 1 (µm) and Lc.ρ 2 (µm) as follows:

𝐿𝑐. 𝜌 1 = 𝐿𝑐. 𝐿2 2 𝐿𝑐. 𝐿1 𝑎𝑛𝑑 𝐿𝑐. 𝜌 2 = 𝐿𝑐. 𝐿3 2 𝐿𝑐. 𝐿1
A high Lc.ρ i is indicative of a less elliptical lacuna; the lower the radii of curvature are, the more discshaped the ellipsoid is, and the higher they are, the more spherical the ellipsoid is. The lacunar volume fraction, Lc.V/BV (%) was calculated. Finally, the lacunar density, Lc.N/BV (mm -3 ), was measured.

The bone volume (BV) was considered to be the voxels that were not in an osteonal canal. Osteocyte lacunae and microcracks were thus considered to be included in BV. In that way, lacunae and microcracks volume fraction or density were assessed considering a volume (BV) in which we might find one of these features, whereas in the osteonal canal, we know that we would not observe any lacunae or micro-cracks.

Statistical analysis

In order to use an averaged value for the two contiguous samples instead of two distinct values, the two data sets corresponding to the two contiguous samples were arranged randomly, resulting in 8 groups of 2 samples for each anatomical location. Friedman tests were then performed to check for differences; samples with p-value < 0.05 were considered as significantly different. This was done to validate that the results obtained from the two contiguous samples can be averaged to give a unique result considering a larger volume being equal to twice the volume of one sample. This also provides information regarding the variability inside a similar anatomical location.

After averaging the two contiguous samples, Friedman tests were performed to investigate whether there are differences between the locations. In the case of the significant results from the Friedman test (p-value < 0.05), we analyzed the differences between anatomical locations by applying the Wilcoxon tests for paired samples. The mean values and interquartile range (IQR) are given in the tables.

Statistical analysis was performed using StatView (Abaqus, USA).

Finally, relationships between the parameters of the canals and lacunar system between the different anatomical locations were investigated using Spearman's coefficient. Correlations were considered as significant if p-value < 0.05.

Results

Figure 4 presents a 3D rendering of control VOIs for the 3 anatomical locations. No significant correlation with age considering the different investigated structural features was found.

Comparison between contiguous samples.

A significant difference between the contiguous samples was observed for the SD Ca.Dm measured on the femoral neck. No difference was observed for all other parameters, considering both osteonal canals and osteocytes lacunae. From these results, the authors considered the average of the two samples as the result for each location. This implies that the analyses was performed on twice the volume of one sample, meaning 2 x (1.4 x 1.4 x 1) mm 3 for the osteonal canals, and 2 x (500 x 500 x 1000) µm 3 for the lacunae and micro-cracks were analysed. between the canal diameter from the femoral neck and the radius, with canals 27 % smaller in the radius. The standard variation of canal thickness within the tissue was also 47 % higher in the femoral neck and 51 % higher than the femoral diaphysis than in the radius. The femoral neck has a more heterogeneous osteonal canals system.

Osteonal canals

Osteocyte lacunae

Table 3 and Erreur ! Source du renvoi introuvable. present the mean values over the eight subjects of the lacunae morphometric parameter Lc.ρ 1 . Significant differences between paired anatomical locations are also indicated. It can be noticed that individual geometrical parameters of osteocyte lacunae are significantly different between the radius and the femur (diaphysis and neck). The Lc.L1 at the radius is 18 % larger than in the femoral diaphysis and 16 % higher than in the femoral neck, and the Lc.L2 is 8 % smaller than in the femoral diaphysis and 10 % smaller than in the femoral neck.

These differences lead to a sharper shape of lacunae in the radius, with a 27 % and 29 % lower Lc.ρ 1 and a 19 % and 22 % smaller Lc.ρ 2 than in the femoral diaphysis and femoral neck, respectively. The average individual lacuna volume and surface, on the other hand, remain the same in all studied anatomical locations. The Lc.N/BV is 18 % lower in the radial diaphysis than in the femoral neck. No significant difference is measured between the locations for the Lc.V/BV. ) for each sample. The two contiguous samples are distinguished thanks to the colour chart.

Micro-cracks

Table 4 shows the mean values over the 8 subjects for the morphometric parameters of micro-cracks measured in the control VOIs. In contrast to the osteocytes lacunae, micro-cracks appear to have nearly the same morphometric parameters in all three anatomical locations, and no significant difference is measured.

Relationships between the three anatomical locations

Table 5 shows the relationships between the three investigated anatomical locations. Regarding the osteonal canals system, no significant correlation has been found between the three anatomical locations. Significant correlations between all of the three locations are observed for Lc.ρ 1 . Others significant relation regarding lacunar parameters can be observed between two different anatomical locations.

Discussion

It is known that bone architecture depends on its function within the organism. However, little quantitative information is available concerning the influence of the anatomical locations on cortical bone microstructure. The aim of this study was to measure the morphometric parameters of different types of porosity on eight paired human femoral diaphyses, femoral neck, and radial diaphyses.

With regard to the parameters of the osteonal canals system, the volume fraction of vascular porosity measured on the femoral diaphysis in the current study is in the same order of magnitude as values also measured in 3D found in the literature (Ca.V/TV = 8.45 ± 2.80 % here, 12.26 ± 3.15 % in [START_REF] Perilli | Regional Heterogeneity in the Configuration of the Intracortical Canals of the Femoral Shaft[END_REF], 11.48 ± 7.0 % in [START_REF] Wachter | Prediction of cortical bone porosity in vitro by microcomputed tomography[END_REF]7.6 % in (Repp et al., 2017)). Moreover, the canal volume fractions measured at the radius in the current study are in the range of results found in the recent study (4.34 ± 2.46 in the current study, between 2.3 and 13.3 in [START_REF] Minonzio | Bone cortical thickness and porosity assessment using ultrasound guided waves: An ex vivo validation study[END_REF]).

The canal diameter measured in the femoral diaphysis is also in accordance with the data found in the literature (Ca.Dm = 73.01 ± 29.92 µm here, 55.09 ± 14.33 µm in [START_REF] Jowsey | Studies of Haversian systems in man and some animals[END_REF], up to 72 µm in [START_REF] Zimmermann | Age-related changes in the plasticity and toughness of human cortical bone at multiple length scales[END_REF], and 78.24 ± 43.5 in [START_REF] Particelli | A comparison between micro-CT and histology for the evaluation of cortical bone: Effect of polymethylmethacrylate embedding on structural parameters[END_REF]). It is nevertheless known that canals properties depend on the region investigated within cortical thickness and that the porosity significantly increases close to the endosteum [START_REF] Bala | Pore network microarchitecture influences human cortical bone elasticity during growth and aging[END_REF][START_REF] Burghardt | Age-and genderrelated differences in the geometric properties and biomechanical significance of intracortical porosity in the distal radius and tibia[END_REF]. In their study, Bala et al., found a large standard deviation for the averaged canals diameter measured on the whole thickness of human fibula reflecting a high variation from the periosteum to the endosteum (144 ± 69 µm, [START_REF] Bala | Pore network microarchitecture influences human cortical bone elasticity during growth and aging[END_REF]). Regarding these results and the sample used in the current study, the porosity investigated in the current is representative of an intra-cortical area between the endosteum and periosteum.

The differences in Ca.V/TV found between the femur and the radius could be explained by the nonweight-bearing nature of the radius. Structural differences between weight-bearing and non-weightbearing bones have already been stated in the past [START_REF] Vico | Effects of long-term microgravity exposure on cancellous and cortical weight-bearing bones of cosmonauts[END_REF]. The internal variation of canal diameter (SD Ca.Dm) was found to be significantly higher in the femoral neck than in the femoral or radial diaphysis. As femoral neck is subjected to a complex mechanical environment [START_REF] Verhulp | Load distribution in the healthy and osteoporotic human proximal femur during a fall to the side[END_REF], it may be subjected to more remodeling leading to a non-uniform canals network at the time bones were collected. Also, inferior femoral neck cortical thickness is close to 2 mm [START_REF] Treece | High resolution cortical bone thickness measurement from clinical CT data[END_REF]. It is known that the cortical porosity may increase near the transition zone between cortical and trabecular bone [START_REF] Buenzli | Endocortical bone loss in osteoporosis: The role of bone surface availability[END_REF]. It has also been shown in a previous study that femoral neck is a region of complex osteonal remodeling [START_REF] Jordan | Spatial clustering of remodeling osteons in the femoral neck cortex: A cause of weakness in hip fracture?[END_REF]. Different localization of a sample extraction can also occur at the origin of these differences. Samples were extracted from the lateral region of the femoral diaphysis and from the anterior part of the radius. A previous study showed no difference between the osteonal canals diameter measured on the lateral and on the anterior region of the same femoral diaphysis [START_REF] Carter | Normal variation in cortical osteocyte lacunar parameters in healthy young males[END_REF][START_REF] Chan | Variation in cortical bone histology within the human femur and its impact on estimating age at death[END_REF]. This result suggests that the regional localization of the sample might be minor regarding the results reported in the current study on the canals system.

Considering the lacunar morphometric parameters measured on the femoral diaphysis, values of Lc.L1, Lc.L2 and Lc.L3 demonstrate the same order of magnitude as the values found in the literature (Lc.L1 / Lc.L2 / Lc.L3 = 21.69 ± 0.97 / 11.71 ± 0.54 / 5.45 ± 0.59 µm in the current study, 18.9 ± 4.9 / 9.2 ± 2.1 / 4.8 ± 1.1 µm in [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]). The Lc.S and Lc.V values are higher in the current study (Lc.S / Lc.V = 508.71 ± 44.18 µm 2 / 561.48 ± 84.81 µm 3 in the current study, 336.2 ± 94.5 µm 2 / 409.5 ± 149.7 µm 3 in [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]). The differences found in comparison with the literature may come from the different sizes of the VOI or different voxel sizes (the voxel size used by Dong was 1.4µm). Moreover, Lc.V measured in the current study is in the same order of magnitude than that found in [START_REF] Bach-Gansmo | Osteocyte lacunar properties and cortical microstructure in human iliac crest as a function of age and sex[END_REF] in which analysis was performed on the iliac crest (Lc.V = 561 ± 32 µm 3 ) for a population older than 57 y.o. The lacunar density is also comparable ((Lc.N / BV) = 21617 ± 3622 in the current study, 20573 ± 2850 in [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]). We can see that the lacunar volume fraction is a little overestimated in comparison with the values found in the literature (Lc.V / BV = 1.2 ± 0.2 in the current study, 0.9 + 0.1 in [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF]). This slight difference is related to the increased Lc.V in our study. Furthermore, the lacunar density (N.Lc/BV) measured in the current study (Lc.N/BV = 21617.32 ± 3621.85 mm -3 / 26123.55 ± 4990.54 mm -3 / 21191.24 ± 1479.55 mm -3 for femoral diaphysis / femoral neck / radius) is in accordance with the values' range, between 20,000 and 30,000, reported by Buenzli & Sims, who completed a global review of osteocytes lacunae morphometric parameters over the skeleton [START_REF] Buenzli | Quantifying the osteocyte network in the human skeleton[END_REF].

The lacunar geometrical differences between femur and radius might be the consequences of a different nature of the remodeling process related to their different function. Since the femur is a weight-bearing bone and the radius is not, the femur is more often subjected to high levels of loading.

This mechanical loading is assumed to be detected by the osteocytes network, which signals initiation of bone turnover [START_REF] Burr | Bone remodeling in response to in vivo fatigue microdamage[END_REF][START_REF] Lee | Bone adaptation to load: Microdamage as a stimulus for bone remodelling[END_REF]. It has been shown in the past that the geometry of the lacunae cultured in vitro has an influence on their mechanosensitivity: a flatter lacuna appears to be less mechanosensitive than a rounder lacuna [START_REF] Bibliography Bacabac | Round versus flat: Bone cell morphology, elasticity, and mechanosensing[END_REF]. They explained this difference by pointing out that rounder lacunae have been shown to be less stiff than the flatter ones but also because the area of anchorage of these rounder lacunae cultured in vitro was smaller. Reality may be more complex in vivo, a significant difference in the shape of these lacunae has nevertheless been measured, and this could be an hypothesis. As reported by Carter et al. these differences may also be explained by the different localization used in the current study. Carter et al., found that the lacunae were flatter in the lateral part of the diaphysis than in the anterior region [START_REF] Carter | Normal variation in cortical osteocyte lacunar parameters in healthy young males[END_REF]. If these differences in shape might be explained with difference in the loading applied to the bone during life, it is nevertheless currently difficult to investigate the role of external loading on osteocytes' shape [START_REF] Maude | One-month spaceflight compromises the bone microstructure, tissue-level mechanical properties, osteocyte survival and lacunae volume in mature mice skeletons[END_REF][START_REF] Sugawara | The early mouse 3D osteocyte network in the presence and absence of mechanical loading[END_REF].

The geometrical parameters of micro-cracks are difficult to compare with the values found in the literature. Several studies were performed using 2D analysis [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF][START_REF] Follet | Determinants of Microdamage in Elderly Human Vertebral Trabecular Bone[END_REF][START_REF] Vashishth | Hierarchy of bone microdamage at multiple length scales[END_REF][START_REF] Zioupos | Microcracking damage and the fracture process in relation to strain rate in human cortical bone tensile failure[END_REF]. For the baseline case, the existing data on three dimensional morphometric parameters of micro-cracks measured on human cortical bone are very limited. [START_REF] Larrue | Synchrotron radiation micro-CT at the Micrometer scale for the analysis of the three-dimensional morphology of microcracks in human trabecular bone[END_REF] investigated the geometry of micro-cracks observed on native human trabecular bone [START_REF] Larrue | Synchrotron radiation micro-CT at the Micrometer scale for the analysis of the three-dimensional morphology of microcracks in human trabecular bone[END_REF]. Wolfram et al. investigated micro-crack morphology in three dimensions, but only after applying critical loading on human femoral cortical bone samples, which modify the morphology of the micro-cracks [START_REF] Wolfram | Characterizing microcrack orientation distribution functions in osteonal bone samples[END_REF]. We can compare our results with previous 2D studies. Considering that the main axis of the micro-cracks is the same as measured in previous studies, the values presented in the current study are in accordance with previous results (µCr.L1 = 72.94 ± 39.29 µm in the current study, 75 ± 26 µm in [START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF], 72.6 ± 13 µm in [START_REF] Haupert | Non destructive characterization of cortical bone micro-damage by nonlinear resonant ultrasound spectroscopy[END_REF]). We also notice that, in the current study, µCr.L2 ranged between 13 µm and 91 µm over the eight subjects on the femoral diaphysis.

It is interesting to note that there are no differences in the geometrical parameters of micro-cracks between the different anatomical locations. This might indicate that the accumulation of micro-cracks proceeds similarly in all three studied sites. However, it might also be due to the fact that the bone tissue reacts in a similar way between the sites during sample preparation in terms of defect geometry.

Future studies to analyze the effect of the sample preparation protocol on micro-crack morphometric parameters are thus necessary, such as the investigation of different preparation protocol on crack morphology.

The current study presents some limitations. First, the micro-cracks were quantified by considering an ellipsoid fit, which might not accurately relay their actual geometrical parameters. Micro-cracks can undergo some deviation, which may increase their geodesic length but not the length of the best-fitting ellipsoid. However the parameters assessed in the current study are not affected by this method of analysis. Future work could be done in order to improve the investigation of these micro-cracks.

Furthermore, the sample preparation method might have introduced micro-defects post-hoc. The amount of micro-cracks observed in our samples might not be representative of bone tissue in vivo.

The size of the VOIs used in the current study might be a limitation, since it is small compared to other studies found in the literature [START_REF] Dong | 3D osteocyte lacunar morphometric properties and distributions in human femoral cortical bone using synchrotron radiation micro-CT images[END_REF][START_REF] Hannah | Bimodal distribution of osteocyte lacunar size in the human femoral cortex as revealed by micro-CT[END_REF][START_REF] Haupert | Non destructive characterization of cortical bone micro-damage by nonlinear resonant ultrasound spectroscopy[END_REF][START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF]. Due to the large number of volumes (48), we had to reduce the size of the analyzed region to avoid an excessively long processing time. Moreover, to consider the internal variability within the same bone, two samples were investigated on each location, implying a real analyzed volume being multiplied by two. Results showed no significant variation within a same anatomical location with consideration for the volumes used in the current study, except for the heterogeneity in canal diameters as measured in the femoral neck (SD Ca.Dm).

Then, our population involved only eight female subjects (aged 50 to 91 y.o.). It is known that the porosity network is dependent of the age and sex of the subject [START_REF] Bala | Bone micromechanical properties are compromised during long-term alendronate therapy independently of mineralization[END_REF][START_REF] Vashishth | Sexual dimorphism and age dependence of osteocyte lacunar density for human vertebral cancellous bone[END_REF]. In the current study, no correlation was found between age and any measured morphometric parameters, probably due to the limited age range. We note, however, that the samples used are representative of a population of post-menopausal women.

To our knowledge, this is the first study investigating the multi-scale pore network of paired human femoral diaphysis and neck and radial diaphysis using SR-µCT with a voxel size of 0.7 µm. It is known femoral diaphysis, femoral neck, and radial diaphysis demonstrate different fracture behavior under a standard condition of loading [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF]. The microstructural difference observed in the current study may be at the origin of this variation of mechanical behavior from an anatomical location to another, as it is known that porosity can have an influence on cortical bone fracture toughness [START_REF] Ural | Effects of Intracortical Porosity on Fracture Toughness in Aging Human Bone: A microCT Based Cohesive Finite Element Study[END_REF].

Conclusion

In the current study, we assessed the morphometric parameters of human cortical bone on eight paired femoral diaphyses, femoral necks, and radial diaphyses. Results show that the vascular canals are wider in the femoral diaphysis or neck than in the radius. The femoral neck appears to have a more heterogeneous osteonal canals system, and lacunae observed in the radial diaphysis are longer and sharper. The amount of lacunae within the femoral neck is significantly higher than in the femoral or radial diaphysis. These results provide evidence that the internal organization of the tissue depends on the weight-bearing or non-weight-bearing nature of the bone. In future work, analysis of the osteons should be of interest, since they are considered to be the basic bone mechanical units. On a lower scale, understanding the role of the canaliculi may also be of great interest.

Acknowledgments

The authors acknowledge the ESRF ID19 staff for his support for synchrotron data acquisition during the experiment MD923. This work was supported by the LabEx PRIMES (ANR-11-LABX-0063) within the program "Investissements d'Avenir" (ANR-11-IDEX-0007). This study was also partly funded by the Région Auvergne-Rhône-Alpes (14-011125-01) and by the ANR project MUTLIPS (ANR-13-BS09-0006). The authors also want to thank the Fédération INGE'LYSE for its financial support.

Figure 1

 1 Figure 1 Summary of samples preparation from the three paired locations (T: thickness ; W: width ; H: height)

Figure 2

 2 Figure 2 VOI slices from a femoral diaphysis bone sample; a. initial phase contrast image, b. canals mask and c. final binarized image containing osteocytes and micro-cracks. Notice the micro-cracks pointed by the red arrow. Scale bar: 100 µm.

Figure 3 3D

 3 Figure 3 3D Volume rendering of the osteonal canals network segmented on the full VOIs at different anatomical sites on the same subject a. femoral diaphysis, b. femoral neck and c. radial diaphysis. Scale bars: 500 µm.

Figure 4

 4 Figure 4 VOIs of human femoral diaphysis, femoral neck and radial diaphysis in one subject (Woman, 50 y.o.). Top: original phase contrast slice, middle and bottom: 3D volume renderings under two orientations. Lacunae are represented in yellow, and micro-cracks in red.

Figure 5

 5 Figure 5 Values of Ca.Dm (µm) for each sample. The two contiguous samples are distinguished thanks to the colour chart.

  :

	𝑂𝑏𝑗. 𝑆 = 2 𝑂𝑏𝑗. 𝑉 2 ,	𝑂𝑏𝑗. 𝜒 = 𝑂𝑏𝑗. 𝑉 0 ,
	𝑂𝑏𝑗. 𝑆𝑀𝐼 =	12 𝜋 𝑂𝑏𝑗. 𝑉 𝑂𝑏𝑗. 𝑉 1 𝑂𝑏𝑗. 𝑆 2

Where Obj.V represents the object volume. The Euler number is a descriptor of the topological property of the object. It indicates the number of connected components minus the number of tunnels plus the number of cavities (

Table 1 Summaries of exclusion criteria for osteocytes lacunae and microcracks

 1

	Osteocytes lacunae	Microcracks
	Obj.V > 82 µm 3	Obj.V > 500 µm 3
	Obj.V < 10000 µm 3	Obj.L1/Obj.Th > 15 and Obj.L2/Obj.Th> 8
	Obj.L1/Obj.L3< 15 and Obj.L2/Obj.L3 < 8	Obj.L1/Obj.L3 > 5 and Obj.L2/Obj.L3 > 2
	Obj.SMI > 1.6	Obj.SMI < 2.5
	0 < Obj.χ < 2	-

Table 2

 2

and Erreur ! Source du renvoi introuvable. present the osteonal canal morphometric parameters means and values over the 8 subjects in control VOIs. The radius shows a lower canal volume fraction than the femur. Ca.V/TV is 49 % and 56 % lower in the radius than in the femoral diaphysis and femoral neck, respectively. No difference is measured regarding the averaged canals diameter (Ca.Dm) using Wilcoxon test. Friedman statistical test resulted in a significant difference

Table 2 Mean values (IQR) for osteonal canals morphometric parameters measured in the VOIs.

 2

	Morphometric parameters	Femoral diaphysis	Femoral neck	Radial diaphysis
	Ca.V/TV (%)	8.45 (2.80) r*	9.94 (4.40) r*	4.34 (2.46) d* n*
	Ca.Dm (µm)	73.01 (29.92)	86.27 (60.61) r°	62.93 (19.33) n°
	SD Ca.Dm (µm)	28.44 (13.27) n*	57.52 (24.08) d* r*	26.36 (12.39) n*

d different from femoral diaphysis, n different from femoral neck, r different from radial diaphysis

Table 3 Mean (IQR) values for lacunae morphometric parameters

 3

	Morphometric parameters	Femoral diaphysis	Femoral neck	Radial diaphysis
	Lc.L1 (µm)	21.69 (0.97) r*	22.23 (1.76) r*	25.70 (2.19) d* n*
	Lc.L2 (µm)	11.71 (0.54) r*	11.99 (0.53) r*	10.81 (0.65) d* n*
	Lc.L3 (µm)	5.45 (0.59)	5.63 (0.33)	5.35 (0.28)
	Lc.ρ 1 (µm -1 )	6.34 (0.60) r*	6.50 (0.78) r*	4.61 (0.78) d* n*
	Lc.ρ 2 (µm -1 )	1.38 (0.28) r*	1.44 (019) r**	1.13 (0.15) d* n*
	Lc.S (µm 2 )	508.71 (44.18)	537.36 (49.82)	550.50 (46.38)
	Lc.V (µm 3 )	561.48 (84.81)	571.66 (56.20)	589.35 (55.58)
	Lc.N/BV (mm -3 )	21617.32 (3621.85)	26123.55 (4990.54) r*	21191.24 (1479.55) n*
	Lc.V/BV (%)	1.21 (0.21)	1.51 (0.32)	1.26 (0.12)
	Lc.N	10028	11504	9302

d different

from femoral diaphysis, n different from femoral neck, r different from radial diaphysis * p < 0.05 Group 1 Group 2 Figure 6 Values of Lc.N/BV (mm -3

Table 4 Mean (IQR) values for micro-cracks morphometric parameters measured in the VOIs. Morphometric parameters Femoral diaphysis Femoral neck Radial diaphysis

 4

	µCr.L1 (µm)	72.94 (39.29)	67.12 (21.50)	69.69 (50.45)
	µCr.L2 (µm)	28.61 (12.70)	30.04 (10.01)	30.91 (14.50)
	µCr.Th (µm)	1.92 (0.20)	1.99 (0.31)	2.06 (0.21)
	µCr.S (µm 2 )	3181.16 (3144.164) 4445.50 (3626.99)	5183.01 (7347.26)
	µCr.V (µm 3 )	2257.43 (2197.265) 3735.25 (2886.29)	4291.97 (6242.06)
	µCr.V/BV (% x 10 3 )	16.47 (9.49)	23.61 (19.39)	24.32 (17.66)

Table 5 Spearman's coefficients between the parameters assessed on the three anatomical locations.

 5

		Spearman's coefficient
		Femoral diaphysis /	Femoral diaphysis /	Femoral neck /
		Femoral neck	Radial diaphysis	Radial diaphysis
	Ca.V/BV	-0.50	0.36	-0.38
	Ca.Dm	-0.07	0.64	-0.05
	SD Ca.Dm	-0.60	0.60	-0.41
	Lc.L1	0.71	0.38	0.60
	Lc.L2	0.60	0.27	0.05
	Lc.L3	0.31	0.24	0.76 *
	Lc.ρ 1	0.93 *	0.95 *	0.88 *
	Lc.ρ 2	0.64	0.71	0.88 *
	Lc.S	0.76 *	0.00	0.05
	Lc.V	0.83 *	-0.14	0.26
	Lc.N/BV	0.62	0.17	0.50
	Lc.V/BV	0.43	0.69	0.76 *

* p-

value < 0.05