Nicolas Berthier

Hervé Marchand

Éric Rutten

Symbolic Limited Lookahead Control for Best-effort Dynamic Computing Resource Management

Keywords: Discrete Event Systems, Supervisory Control, Infinite-state Systems, Dynamic Resource Management

We put forward a new modeling technique for Dynamic Resource Management (DRM) based on discrete events control for symbolic logico-numerical systems, especially Discrete Controller Synthesis (DCS). The resulting models involve state and input variables defined on an infinite domain (Integers), thereby no exact DCS algorithm exists for safety control. We thus formally define the notion of limited lookahead, and associated best-effort control objectives targeting safety and optimization on a sliding window for a number of steps ahead. We give symbolic algorithms, illustrate our approach on an example model for DRM, and report on performance results based on an implementation in our tool ReaX.

INTRODUCTION & MOTIVATING EXAMPLE

When dealing with automated resource management or scheduling problems, designers of self-adaptive computing systems often face the need to ensure some quality of service while meeting intrinsic or application-dependent constraints. Following our previous works on the subject [Berthier et al., 2015[START_REF] Berthier | Designing Autonomic Management Systems by using Reactive Control Techniques[END_REF][START_REF] An | Modelbased design of correct controllers for dynamically reconfigurable architectures[END_REF], our approach relieves the designers from this error-prone task by delegating some management decisions to a controlled model of the system issuing said decisions at discrete points in time (also called execution steps). The controlled model consists of a given model that keeps track of the necessary knowledge about the resources at hand, and an associated controller, automatically computed by means of a Discrete Controller Synthesis (DCS) algorithm, and whose goal is to enforce given control objectives. The controlled model is then paired with other pieces of software to form a management system. Yet, the amount of resources to manage may vary during the system's lifetime, yielding the need to handle quantitative, potentially infinite, models.

Dynamic Computing Resource Management

Observe that, while computing systems generally comprise a dynamic number of resources, the latter can often be categorized into a finite number of groups that are of the same kind, sharing some regularity in terms of their status and operations. We say that resources of the same kind belong to the same family. Such families may be the threads in a pool, request handlers, computing resources like CPU cores, communication channels, or some type of objects in memory for instance. A family of resources can be described using a single Resource Automaton (RA), similar to a finite-state automaton where states represent potential statuses of a single one of the resources, and transitions model events (i.e., status change). Transitions that are said controllable feature management decisions (e.g., creating or deleting threads in a pool, powering off a CPU core, resuming or suspending a task). The other events are considered noncontrollable (e.g., end of a task, failure of a CPU core or communication channel). We depict in Fig. 1 a simple example of such an RA for request handlers, that we use as a basis for the examples throughout the paper. Once it has been created (c -i.e., triggered by the arrival of a request), a handler must be started (s) to be able to process the request; it does so after first entering a waiting stage (W) and then being resumed (r ); it can then yield its computation (y) and then be resumed (r ) any number of time. A handler disappears from the managed system once it ends its computation (e). Start-up (s) and resume (r ) must be triggered by some decision of the management system. discrete changes in resource status manifest as variations between these quantities at discrete instants, and management decisions can be deduced from the variations for controllable transitions, chosen by a synthesized controller. The RA of Fig. 1 features two controllable transitions: at each one of its execution steps, the management system decides how many handlers should be started (s) or resumed (r ) based on the other variations it receives as input (c, y, and e). Such modeling results in linear systems, for which we give a symbolic representation in Section 2.

Resource Management Goals as Control Objectives Using our modeling approach, interesting quantitative control objectives about one or more family of resources can be specified. Some management concerns relate to the performance of the overall system, e.g., in terms of power or energy consumption. Such goals can be expressed as optimization objectives, where the role of the controlled system is to minimize some quantity exhibited by the model, such as the number of non-idle CPU cores at any time. Useful qualitative properties can also be specified as safety objectives on the model. For instance, limiting the quantity of resources of a family into a given state expresses some guarantee of service constraints; e.g., stating that the number of threads in a pool does not exceed the (dynamic) number of CPU cores available in the system.

Towards Best-effort Control The DCS algorithms currently available [Berthier andMarchand, 2014, 2015] for logico-numerical systems focus on safety control. Due to the necessary over-approximations that these algorithms involve to terminate and give conservative results, the degree of freedom (i.e., the domain of controllable variables) given to the synthesized controller must be finite; in the case of symbolic systems, this freedom usually corresponds to the domain of some special input variables, said controllable. In some contexts though, strictly considering a safety control objective might be considered too constraining in terms of the controlled systems' behaviors, or computational power involved in enforcing the objective. Hence, in Section 3 we first formally develop the notion of symbolic limited lookahead, inspired by the work of [START_REF] Chung | Limited lookahead policies in supervisory control of discrete event systems[END_REF]. We then detail exact algorithms for best-effort bounded safety control (as opposed to the usual strict safety control), accompanied with a transformation of controllers towards recovery, where the controlled system still does its best to avoid operating states violating a desired invariant when it reaches states from which it might fail in doing so. Algorithms for optimization are also developed, based on the same principles as bounded safety. Sections 4, 5 and 6 give some implementation details and performance results, review related works, and conclude.

CONTROL OF SYMBOLIC TRANSITION SYSTEMS

We first fix some notations, and recall in this section the model of Arithmetic Symbolic Transition Systems [START_REF] Berthier | Discrete Controller Synthesis for Infinite State Systems with ReaX[END_REF] and the associated control problems.

Notations

Let V = v 1 , . . . , v n : D V , with D V = i∈{1,.
..,n} D vi , be a vector (or tuple) of variables v 1 , . . . , v n , each defined on (infinite) domains D v1 , . . . , D vn . V ∩ W is the result of removing from a vector V all variables not belonging to a vector W , and ∅ is the empty vector; V W is the concatenation of V and W , defined iff they contain distinct sets of variables (V ∩ W = ∅).

A valuation ν ∈ D V for each variable in V can be seen as the mapping ν : V → D V . We denote valuations using tuple notations as in ν = (ff, . . . , tt), to actually denote ν = {v 1 → ff, . . . , v n → tt}. Further, given an additional vector W of variables distinct from those in V , and corresponding valuations ν ∈ D V and µ ∈ D W , the union of ν and µ

is (ν, µ) ∈ D V W . Considering two sequences of n ∈ N + valuations ν = a 1 •. . .•a n ∈ D n V and µ = b 1 •. . .•b n ∈ D n W , we denote their union by ν ∪µ = (a 1 , b 1 )•. . .•(a n , b n ) ∈ D n V W .
The empty sequence is noted ε.

Given a vector V x of variables defined on numerical domains (e.g., Integers or Reals), we denote L Vx the set of all linear affine arithmetic expressions involving variables of V x only. Given an additional vector V b of variables defined over finite domains (e.g., Booleans), P V b Vx denotes the set of all propositional predicates defined over variables in V b and a finite set of linear (in-)equalities expressed on variables of V x . Further, L P V b Vx denotes the set of all guarded arithmetic expressions, that are total functions mapping pairwise disjoint predicates of P V b Vx into a finite subset of L Vx .

Note that all numerical expressions involved in predicates and arithmetic expressions are linear, hence any P V is closed under quantifier elimination-i.e., any predicate involving existential or universal quantifiers from P V is equivalent to a quantifier-free predicate also in P V . The process of computing such an equivalent quantifier-free predicate is called quantifier elimination. For succinctness, we denote in the sequel the set of quantifier-free expressions over variables V as E V = P V ∪ L P V . Given e ∈ E V and a valuation ν ∈ D V , e ν is the Boolean or numerical value obtained through the usual evaluation of e after substituting every variable in e with its corresponding value in ν. For a predicate e ∈ P V , one writes ν |= e to denote that ν satisfies e, and thus e ν ≡ (ν |= e).

Arithmetic Symbolic Transition Systems

An Arithmetic Symbolic Transition System (ASTS) comprises a finite set of (state and input) variables, whose domain can be infinite, and evolves at discrete points in time. An update function indicates the new values for each state variable according to the current values of the state and input variables. This model allows the representation of infinite systems whenever the variables take their values in an infinite domain, while it has a finite structure and offers a compact way to specify systems handling data. Definition 1. An Arithmetic Symbolic Transition System is a tuple S = X, I, T, A, Θ 0 where:

• X = x 1 , . . . , x n : D X is a vector of state variables encoding the memory necessary for describing the system's behavior; • I = i 1 , . . . , i m : D I is a vector of input variables; • T ∈ E n X I is the transition function of S, and encodes the evolution of all state variables based on n (welltyped) expressions involving variables in X I;

• A ∈ P X I encodes an assertion on the admissible values of the inputs depending on the current state; • Θ 0 ∈ P X is a predicate encoding the initial states.

An ASTS is linear logico-numerical if its state and input variables are Boolean variables (B) or numerical variables (typically, Z or Q), i.e., it is such that D X = B k ×Q k ×Z k with k + k + k = n (and similarly for I).

Consider an ASTS S = X, I, T, A, Θ 0 . The state reached by S starting from any state x ∈ D X with a sequence of m valuations for its input variables ι ∈ D m I is denoted by [S](x, ι) ∈ D X ∪ {⊥}, defined as

[S](x, ε) def = x [S](x, ι • ι) def = [S](T (x, ι), ι) if (x, ι) |= A, ⊥ otherwise.
(1) [S](x, ι) = ⊥ indicates that the valuation of inputs ι does not satisfy the assertion A, and thus S cannot evolve from state x with input ι: we say that ι is not admissible (by S) in x. We also write [S] i (x) ⊆ D X the set of states reachable by S from state x whatever the sequence of i ∈ N inputs; i.e., [S] i (x)

def = ι∈D i I [S](x, ι) \ {⊥} . Further, the orbit O (S) ⊆ D X of S (also known as its reachable state space) is O (S) def = (n,x0)∈N×Θ0 [S] n (x 0
). Note that the orbit might not be computable due to the infiniteness of the state space. Example 1. Applying the quantitative modeling principles mentioned in the Introduction on the RA described in Fig. 1 provides the ASTS S rh = X, I, T, A, Θ 0 where each integer state variable i, w and a represents the number of handlers in states I, W and A:

X = i, w, a : Z 3 , I = c, s, r , y, e : Z 5 , Θ 0 (X) = (i = 0 ∧ w = 0 ∧ a = 0), T (X, I) = i := i -s + c w := w + s + y -r a := a + r -y -e , and A(X, I) = s 0 ∧ r 0 ∧ y 0 ∧ c 0 ∧ e 0 ∧ s i ∧ r w ∧ y + e a.
The assertion A restricts the domain of admissible inputs and quantities associated with states to natural numbers. Further assumptions on the inputs can be represented in a similar way, e.g., with a conjunction with c 10 if no more than 10 new handlers can be created between two successive execution steps. We show in Example 2 below how to tune S rh to take controllable transitions into account. From the (here single) state x 0 = (0, 0, 0) (i.e., such that Θ 0 (x 0 )), one execution step of S rh with input vector ι 1 = (3, 0, 0, 0, 0) leads to

x 1 = [S rh ](x 0 , ι 1 ) = (3, 0, 0). One fur- ther step with ι 2 = (1, 2, 0, 0, 0) reaches x 2 = [S rh ](x 1 , ι 2 ) = (2, 2, 0). With ι 3 = (0, 0, 3, 0, 0), [S rh ](x 2 , ι 3 ) = ⊥ as (x 2 , ι 3 ) |= A does not hold (due to the falsified condition r w ≡ 3 2).
A longer, admissible execution trace of S rh based on randomly generated inputs satisfying A, is shown in Fig. 2.

Control of ASTSs

A controllable ASTS is an ASTS whose vector of input variables I is partitioned into non-controllable U and controllable variables C (i.e., I = U C). Example 2. Reflecting the controllable transitions of the RA in Fig. 1 on S rh of Example 1 brings the controllable Reactive ASTSs Raw ASTSs shall be reactive, meaning that there always exist admissible valuations for their input variables, whatever the current (reachable) state:

Definition 2. An ASTS S is reactive iff ∀x ∈ O (S) , ∃ι ∈ D I , [S](x, ι) = ⊥. ( 2
)
This notion is very similar to the usual deadlock-freeness, except that we choose to reserve the latter term to controlled ASTSs only; we give a definition for such deadlock-free ASTSs below.

Controlled ASTSs To fulfill its objective, a controller restricts the admissible values for the controllable subset of the input variables of the ASTS it controls. Definition 3. Given an ASTS S = X, I, T, A, Θ 0 with inputs

I = U C, a controller K ∈ P X I provides a controlled ASTS S/ K = X, I, T, K, Θ 0 such that ∀(x, u, c) ∈ O (S/ K ) × D U × D C , [S/ K ](x, u, c) = ⊥ ⇒ [S](x, u, c) = ⊥. (3)
Eq. ( 3) above states that K is at least as restrictive as A, hence does not allow evolutions that are forbidden in the original system; i.e., O (S/ K ) ⊆ O (S), and

∀(x, ι) ∈ O (S/ K ) × D I , ((x, ι) |= K ⇒ (x, ι) |= A) .
In the remainder of the paper, we devise algorithms taking as input a controllable ASTS S = X, I, T, A, Θ 0 and some target objective o, and computing a controller K for S so that a resulting controlled ASTS S/ K fulfills o.

Deadlock-free ASTSs As our aim is to eventually obtain controlled systems that are executable, we devise algorithms producing deadlock-free controlled ASTSs, i.e., for which values for controllable variables always exist whatever the non-controllable inputs admissible by S:

Definition 4. A controlled ASTS S/ K is deadlock-free if S is reactive and ∀(x, u, c) ∈ O (S/ K ) × D U × D C , [S](x, u, c) = ⊥ ⇒ ∃c ∈ D C , [S/ K ](x, u, c ) = ⊥. (4) 3. SYMBOLIC κ-LOOKAHEAD CONTROL
Let us consider given in this Section a controllable ASTS S = X, U C, T, A, Θ 0 and a strictly positive Integer κ ∈ N + . We first explain the core computations of our algorithms for κ-lookahead control in an explicit way-i.e., using set-theoretic notations-and turn to the related symbolic representations and algorithms. We then focus on the use of such tooling for bounded-safety, best-effort, recovery, and optimization control.

3.1 κ-Lookahead Control Potential & Admissible i-paths: R i & A i
As ASTSs' transition functions are deterministic by construction, a set of paths starting in a state x ∈ D X and of a finite number i of transitions, can be identified by relating x and a sequence of i valuations for every variable in I: Definition 5. A (potentially infinite) set of i-paths R i ⊆ D X × D i I , for i ∈ N, relates states of S with sequences of i successive valuations for its inputs.

Note that we define i-paths based on the transition functions only, i.e., both admissible and non-admissible transitions are considered. To take transitions admissibility into account, we further define the sets

A i def = (x, ι • ι) ∈ D X × D i I [S](x, ι) = ⊥ ⇒ [S](x, ι • ι) = ⊥ , for i ∈ N +
, that each are the sets of all i-paths whose ith transitions (if any) are admissible by S.

We can now assume given a set of desirable κ-paths R κ for S; we further detail the computation of such sets in Sections 3.4 and 3.5.

Controllable Prefixes of R κ Given a set R i+1 of (i + 1)paths, the set R i of all its direct controllable prefixes consists of the i-paths whose suffix transitions admissible by S cannot uncontrollably form an (i + 1)-path that does not belong to R i+1 ; i.e., after following any i-path in R i , and given an admissible valuation for all non-controllable variables (U ), one can always find a valuation for the controllable variables (C) so that the system remains on an (i + 1)-path belonging to R i+1 . Formally, R κ being given, we define R i based on R i+1 as

R i def = Prefix c (R i+1 ) (i ∈ {0, . . . , κ -1}) (5)
where

Prefix c : D X × D i+1 I → D X × D i I def = R i+1 → (x, ι) ∈ D X × D i I ∀u ∈ D U , ( c ∈ D C , (x, ι • (u, c)) ∈ A i+1 ) ∪ (∃c ∈ D C , (x, ι • (u, c)) ∈ R i+1 ) with R i+1 def = R i+1 ∩ A i+1 .
The left-hand side of the union in the definition of Prefix c consists in prefixes of (i + 1)paths that are not admissible by S whatever the valuation of the controllable variables at step i + 1. We illustrate this principle in Fig. 3 in which, contrary to i-paths denoted by c, those denoted by a are included into R i as there always exists a choice for valuations c to stay on an (i + 1)path belonging to R i+1 whatever u. i-paths b belong to

a b c ∀u∃cR i+1 ∀u cA i+1 ∪ ∃cR i+1 ∀u cR i+1 d e f g h R i D X × D i I R i+1 D X × D i+1 I R i+1 = R i+1 ∩ A i+1 u ∈ D U c ∈ D C Fig. 3. Direct controllable prefixes R i of R i+1 .
Plain circles represent sets of i-paths (resp. (i + 1)-paths) belonging to R i (resp. R i+1 ); dashed circles represent i-paths (resp. (i + 1)-paths) not belonging to R i (resp. R i+1 ); smaller circles denote transient states, after input values u for the i-th non-controllable variables are given and before the subsequent controllable ones c are chosen; plain (resp. dotted ) arrows denote sets of transitions satisfying admissible (resp. non-admissible) by S. R i as from them valuations u always leading to g are not admissible at step i + 1.

Eq. ( 5) gives an inductive definition for controllable prefixes of any non-negative length i < κ of R κ . We now write R i to denote all i-paths that are controllable prefixes of R κ .

Symbolic Computations

In order to symbolically represent valuations of input variables in κ subsequent future steps, we define indexed versions of all input variables of S. From the vector U (resp. C) we define κ -1 additional indexed versions U 2 . . . U κ (resp. C 2 . . . C κ ), and set U 1 = U and C 1 = C. We additionally define I i def = U i C i for i ∈ {1, . . . , κ}, as well as V i;j def = k∈{i,...,j} V k for any vector of indexed variables V ∈ {U, C, I}, with V i;j def = ∅ if i > j. Lastly, given two vectors of variables of equal lengths V and W , and any expression e ∈ E V Z , e[V /W ] ∈ E W Z denotes the substitution in e of all occurrences of any variable belonging to V with its counterpart in W .

Using the above notations, the symbolic representation of a set of i-paths (for i ∈ {0, . . . , κ}) is a predicate in P X I1;i . More generally, the i-lookahead of e ∈ E X is the expression e| i ∈ E X I1;i encoding the value of e after i successive applications of its transition function T , in terms of the current state (variables from X) and any future input sequence of length i (variables from I 1;i ):

given i ∈ {0, . . . , κ}, e| i is such that ∀(x, ι) ∈ D X × D I1;i , ([S](x, ι) = ⊥) ⇒ ( e| i (x, ι) = e [S](x, ι)) . ( 6
)
Let us now turn to the actual construction of such ilookahead expressions. Assume given an expression e ∈ E V with V ∩ I = ∅. T -1 (e) is the (parallel) substitution in e of each state variable in X with its respective expression in T ; i.e., ∀(x, ι, y)

∈ D X × D I × D V \(X I) ,
T -1 (e) (x, ι, y) def = e (T (x, ι), y).

(7) Proposition 6. e| i can be recursively built as

e| i+1 = T -1 (e| i [I 1;i /I 2;i+1 ]), with e| 0 = e.
Proof. Beyond the base case for i = 0 which is obvious, we show that e| i+1 is properly built if e| i is. Given any (x, ι • ι) ∈ D X × D i+1 I , our recursion hypothesis is e| i (x, ι) = e [S](x, ι) (HR). After substituting e| i+1 by T -1 (e| i [I 1;i /I 2;i+1 ]) and ι by ι • ι in Eq. ( 6), we assume [S](x, ι • ι) = ⊥ (H1) and check the equality

T -1 (e| i [I 1;i /I 2;i+1 ]) (x, ι • ι) ( ? = e [S](x, ι • ι)). = T -1 (e| i ) (x, ι, ι) (Def. of T -1 & I ∩ I 2;i+1 = ∅) = e| i (T (x, ι), ι)
(by Eq. ( 7)) = e [S](T (x, ι), ι)

(by HR) = e [S](x, ι • ι) (by H1 & Eq. ( 1
∧ (a = 42)| 2 = (a 42)| 1 ∧ a + r + r 2 -y -y 2 - e -e 2 = 42 represents all potential paths (x, ι 1 • ι 2 • ι) for which a T (x, ι 1 ) 42 ∧ a T (T (x, ι 1 ), ι 2 ) = 42.
Remark 7. The symbolic computation procedure giving the lookahead of expressions defined on state variables only given in Prop. 6, can be adapted to the case of expressions involving both state and input variables by restricting to i ∈ N + and setting e| 1 = e.

As a result of the above remark, the predicate A| i symbolically describes the set A i , for i ∈ {1, . . . , κ}. Example 4. Considering S rh from Example 2, with future variables

I 2 = c 2 , s 2 , r 2 , y 2 , e 2 : Z 5 , A| 2 (X, I 1;2 ) = s 2 0 ∧ r 2 0 ∧ y 2 0 ∧ c 2 0 ∧ e 2 0 ∧ s 2 i -s + c ∧ r 2 w + s + y -r ∧ y 2 + e 2 a + r -y -e.

Symbolic Computation of Controllable Prefixes

Let R i+1 ∈ P X I1;i+1 be such that R i+1 = ∈ D X × D i+1 I |= R i+1 . Mirroring Eq. ( 5), one can symbolically compute all direct controllable prefixes of the (i + 1)-paths represented by R i+1 using prefix i c : P X I1;i+1 → P X I1;i defined as

prefix i c (R i+1 ) def = ∀ Ui ((∃ Ci A| i+1 ) ⇒ ∃ Ci (A| i+1 ∧ R i+1
)) where ∃ V e (resp. ∀ V e) is the existential (resp. universal) elimination of all variables of V from a predicate e. By extension, prefix j c (R i+1 ), for j ∈ {0, . . . , i}, computes the j-paths that are controllable prefixes of R i+1 . For instance, prefix 1 c (R 3 ) = prefix 1 c • prefix 2 c (R 3 ) denotes all 1-paths that are controllable prefixes of the 3-paths in R 3 .

Building Symbolic κ-lookahead Controllers

We now proceed to define κ-lookahead controllers that enforce bounded safety or κ-optimization objectives. We first advance means for building controllers strictly allowing a given set of desirable κ-paths, and develop controller transformation and composition operations leading to deadlock-free systems with best-effort and recovery control. We detail in Sections 3.4 and 3.5 how to derive desirable κ-paths from target objectives.

State Safety w.r.t. Sets of κ-paths

We say that a state x ∈ D X is safe w.r.t. R κ under non-controllable input u ∈ D U , denoted x u ; R κ , whenever there exists a value for controllable variables admissible by S and forming a 1-path that is a controllable prefix of κ-paths belonging to R κ ; i.e.,

x u ; R κ def = ∃c ∈ D C , (x, u, c) ∈ R 1 ∩ A 1 .
Building Strict κ-lookahead Controllers A κ-lookahead controller implicitly encodes all possible relevant outcomes for the next κ steps and achieves its objective by restricting the admissible values of the controllable inputs for the current step. Definition 8. A strict controller K κ for desirable κ-paths

R κ is such that ∀(x, u) ∈ O (S) × D U , ∃c ∈ D C , (x, u, c) |= K κ ⇔ x u ; R κ . ( 8
)
We say that K κ induces a deadlock with non-controllable input u in every reachable state unsafe w.r.t. R κ under u. Theorem 9. Let R κ ∈ P X I1;κ be a predicate such that

R κ = { ∈ D X × D κ I | |= R κ }. Then K κ = A ∧ prefix 1 c (R κ ) .
Proof. First, K κ is a controller for S by construction, (as ∀P ∈ P X I , (A ∧ P ) ⇒ A, leading to Eq. (3) through Eq. ( 1)). Next, noting that

R 1 = prefix 1 c (R κ ) is such that R 1 = { ∈ D X × D I | |= R 1 } and O (S) ⊆ D X , one can rewrite the left-hand side of Eq. (8) as ∀(x, u) ∈ O (S) × D U , ∃c ∈ D C , (x, u, c) ∈ R 1 ∩ A 1 .
Transformation for Best-effort Control As our aim is to eventually obtain controlled systems that are executable, we now devise a controller transformation producing deadlockfree controlled systems. Definition 10. A best-effort controller be (K) for S obtained from another controller K is such that,

∀(x, u, c) ∈ O (S) × D U × D C , [S/ be(K) ](x, u, c) = [S/ K ](x, u, c) if ∃c ∈ D C , (x, u, c) |= K [S](x, u, c) otherwise. ( 9
)
In plain words, S/ be(K) behaves as S/ K as long as it does not deadlock, or as S otherwise. Proposition 11. be

(K) = A ∧ (∃ C K ⇒ K).
Proof. be (K) is a controller for S for the same reason as for Theorem 9. Then, the rewriting of the definition of be (K) against Eq. ( 9) comes directly.

Although inducing a significant relaxation in terms of the enforced objectives, best-effort controllers bring us deadlock-freeness, hence executable controlled systems: Theorem 12. If S is reactive, then any best-effort controller for S produces a deadlock-free ASTS.

Proof. Let us write Eq. ( 4) as df

(S/ K ) ≡ ∀(x, u, c) ∈ O (S/ K ) × D U × D C , [S](x, u, c) = ⊥ ⇒ ∃c ∈ D C , [S/ K ](x, u, c ) = ⊥.
According to Def. 4, we only need to show df S/ be(K) for any controller K for S. By definition of be (K) (Eq. ( 9) notably), one has

∀(x, u, c) ∈ O (S) × D U × D C , [S](x, u, c) = ⊥ ⇒ [S/ be(K) ](x, u, c) = ⊥ (≡ R1). Noting that O S/ be(K) ⊆ O (S) as be (K) is also a controller for S, then R1 ⇔ ∀(x, u, c) ∈ O S/ be(K) × D U × D C , [S](x, u, c) = ⊥ ⇒ [S/ be(K) ](x, u, c) = ⊥ ≡ df S/ be(K) .
Transformation Towards Best-effort Recovery In addition to the above relaxation of control for strict desirable κpaths R κ , one can design controllers that also recover from failures to stay in safe states w.r.t. R κ under some non-controllable input in case values for non-controllable variables permit such a recovery. Notice we expose here a way to implement recovery using a single step lookahead, yet this approach can be extended to further steps ahead. We denote by Safe (K (x,u,c) |= K the set of states from which a controller K is satisfiable whatever the valuation for the noncontrollable variables. Definition 13. A recovering controller re (K|R κ ) for S obtained from a controller K and a strict controller K κ for desirable κ-paths R κ such that K κ ⇒ K, behaves as S/ K in all states Safe (K κ ), or reaches such a state in one step whenever possible. Theorem 14. re

) def = x ∈ D X ∀u ∈ D U , ∃c ∈ D C ,
(K|R κ ) = ( I E ∧ K) ∨ (∃ U E ∧ A) with E = A ∧ ¬prefix 0 c (K κ ) ∧ prefix 0 c (K κ )| 1 . Proof. Note that x ∈ D X x |= prefix 0 c (K κ ) = Safe (K κ ).
Hence, E represents all 1-paths starting in unsafe states w.r.t. K κ and reaching safe ones in one step (¬prefix 0 c (K κ ) ∧ prefix 0 c (K κ )| 1 ) that are admissible by S. Further, ∃ I E denotes all unsafe states that are also direct predecessors of safe ones, so I E exactly denotes all states that behave according to K in S/ re(K|Rκ) . Also, ∃ U E represents all transitions for which some value for non-controllable variables exist to reach a safe state w.r.t. R κ . At last, K is a controller for S, hence re (K|R κ ) ⇒ A by construction, and re (K|R κ ) is a controller for S.

Control for Bounded-safety

Given the tooling listed above, computing a bounded-safety controller enforcing some invariant Φ over a sliding window of κ steps, boils down to build a κ-lookahead predicate encoding all κ-paths traversing states satisfying Φ. Definition 15. Given a predicate Φ ∈ P X , a strict boundedsafety controller K Φ,κ is a strict controller for the desirable

κ-paths R κ = ∈ D X × D κ I |= i∈{1,...,κ} Φ| i .
Example 5. Considering S rh from Example 2, assume that, for availability purposes, or to accommodate a limited number of hardware resources, one wants to bound the number of already started request handlers at any time. The associated invariant is, e.g., Φ (X) = (w + a 42). Enforcing Φ over a two-step window using a strict controller, one obtains

R 2 = i∈{1,2} (w + a 42)| i = (w + a + s - e 42) ∧ (w + a + s + s 2 -e -e 2 42). The controller is then K Φ,2 = A ∧ prefix 1 c (R 2 ) = A ∧ s + w + a -e 42.
Best-effort and recovering versions of such controllers can be built using K Φ,κ as described in Section 3.3.

Control for κ-optimization

We now briefly address the control of S for the minimization of some accumulated valuation of a guarded arithmetic expression on state variables over a finite number of steps.

Enforcing some optimization objectives over κ steps ahead actually boils down to build the appropriate set of κ-paths ). Each step consists in: values for variables in U are randomly picked so that ∃ C A is satisfied; in turn, values for s and r are also randomly picked so that K ω is satisfied.

R κ ensuring the "best" choice of values for controllable variables at the current step w.r.t. the potential values for the current and future non-controllable ones.

A κ-optimization objective min(Σe) @κ for a guarded linear arithmetic expression e ∈ L P X targets the minimization of its sum over κ steps. In terms of κ-lookahead expressions, the value to be minimized is then E κ = i∈{1,...,κ} e| i , and the set Y ⊆ D X × D κ I of κ-paths for which all valuations of controllable variables are guaranteed to minimize E κ can be defined symbolically as

Y def = ∈ D X × D κ I |= C 1;κ (A κ ⇒ (A κ ∧ E κ < E κ ))
with A κ = i∈{1,...,κ} A| i and e = e[C 1;κ /C 1;κ ], where V = {v | v ∈ V } is the set of primed versions of a vector of variables V . In plain words, all paths in Y are the admissible ones for which there does not exist alternative valuations for current and future controllable variables (C 1;κ ) such that E κ evaluates to a strictly greater value (i.e., they give the lowest possible value for E κ ). Example 6. We continue building up on Example 5. In addition to the limit on started requests, we now consider minimizing the number of handlers waiting to be started, and arbitrarily choose a sliding window of one step for this objective. This previous statement translates into the optimization objective ω = min(Σi) @1 on S rh / KΦ,2 (i.e., now, A = K Φ,2 ). The arithmetic 1-lookahead expression to minimize is

E 1 = i∈{1} i| i = i -s + c. From E 1 , one has E 1 < E 1 = s > s, and then Y = {r ,s } (A 1 ⇒ (A 1 ∧ s > s)) . In the end, the obtained controller is K ω = K Φ,2 ∧ ((s = i ∧ i + w + a -e 41) ∨ (s i ∧ s + w + a -e = 42
)), that actually starts as many allocated handlers (in state I) as possible, as long as the resulting number of started handlers does not exceed 42. We show in Fig. 4 an execution trace of the resulting controlled system. Remark the value chosen for s always equals that of i (which is indeed minimized), except during the two last steps where the value of w + a = 42. The set Y above can actually be empty, if no choice of values for controllable variables at any instant is guaranteed to lead to the minimal outcome after κ steps w.r.t. values for non-controllable variables. In such a case, the resulting system induces deadlocks; yet, a transformation for besteffort control still suffices to eventually obtain a deadlockfree controlled system. Other sets of κ-paths can also be considered to extend the set of desirable κ-paths. One can for instance consider building a controller for desirable κ-paths Y ∪ Z, where

Z def = ∈ D X × D κ I |= U 2;κ (A κ ⇒ (A κ ∧ E κ < E κ ))
with e = e[U 2;κ /U 2;κ ]. Z as defined above consists of the κ-paths for which no future non-controllable input (U 2;κ ) exist that lead to a lower value for E κ .

IMPLEMENTATION IN REAX & EVALUATIONS

We have implemented all the algorithms detailed in this paper in ReaX [Berthier andMarchand, 2014, 2015] 1 , that is a framework suitable for the symbolic manipulation of logiconumerical systems like ASTSs. We use a combination of (multi-terminal) binary decision diagrams [START_REF] Billon | Perfect normal forms for discrete programs[END_REF] and disjunctions of convex polyhedra [START_REF] Cousot | Automatic discovery of linear restraints among variables of a program[END_REF] as symbolic representations of sets of linear arithmetic constraints. This representation allows to implement exact universal and existential eliminations of finite and infinite variables on predicates involving such constraints. Note that our tool fully supports linear logico-numerical systems (i.e., linear ASTSs also involving variables defined on finite domains like Booleans and other enumerated types), although as presented above our quantitative modeling technique does not require this feature. Yet, additional finite-state automata can easily be composed with resource families to model a finite number of elements that are deemed relevant w.r.t. control objectives in the computing system to manage.

We now give some performance evaluation results based on example models of RAs to assess the practicality of our modeling approach and algorithms. Exploiting the scalability of our RA of Fig. 1, we first constructed a series of "Parallel" systems built by assembling N controllable ASTSs S rh using parallel composition, for N ∈ {2, 3, 6, 9} 2 . The resulting systems model as many "kinds" of request handlers to be managed. We also built a second series "Alt" obtained by modeling the RA of Fig. 1 with N states A i and associated additional input/output transitions r i , y i and e i , again for N ∈ {2, 3, 6, 9}. For "Parallel", our example objective is to balance the number of active request handlers of each kind. For "Alt" models on the other hand, our objective is to impose an upper-limit on the number of non-started request handlers, similarly to Example 5. We use best-effort control in all cases, and exercise the recovering synthesis for "Alt". We show in Fig. 5 a simulated execution trace for the 1 http://reatk.gforge.inria.fr/ 2 Building the parallel composition of ASTSs essentially boils down to concatenating their vectors of variables and transition functions; in our case, we obtain systems with variables of S rh suffixed with indexes corresponding to their respective family of request handlers.

3 All results are obtained on a desktop computer equipped with a quad-core Intel i5 (3.3GHz), 16GB of RAM, and a standard GNU/Linux operating system. "Parallel" system with N = 2 controlled κ=1-step ahead.

Focusing on the 5th execution step, one observes that 3 handlers of type 1 are active, 2 are ending their executions, and none yields (a1 = 3, e1 = 2, and y1 = 0); meanwhile, none of the 3 active handlers of type 2 terminates nor yields (a2 = 3, e2 = 0, and y2 = 0). Yet with no handler of type 1 in waiting stage (w1 = 0), no suitable values for the controllable variables (s1, r1, s2 and r2) exist to achieve the safety objective at this execution step (i.e., balancing the number of handlers in active stage). Therefore, the besteffort controller admits any values that allow the system to progress, and the safety objective is thus not considered for this particular instant: 8 handlers of type 1 and 3 handlers of 2 are started (s1 = 8 and s2 = 3). The failure to strictly impose the safety objective occurs only once in this trace. We report synthesis times in Table 1 3 . Observe that ReaX's core symbolic manipulation engine may be less suited for performing the computations required by our algorithms than state-of-the-art SMT solvers for instance. Still, these performance results and our simulations show us that ReaX is able to compute controllers imposing useful objectives on system featuring dozens of numerical variables in reasonable time. Therefore, these preliminary experiments convince us that best-effort control and recovery are worth investigating for dynamic computing resource management, in cases where enforcing strict control objectives is not necessary.

RELATED WORKS

The control of infinite systems has been subject to several studies based on various models: Timed Automata [START_REF] Cassez | Efficient on-the-fly algorithms for the analysis of timed games[END_REF], Vector of Discrete Events Systems [START_REF] Li | Control of vector discrete-event systems-part ii : controller synthesis[END_REF], Petri Nets [START_REF] Kumar | On computation of state avoidance control for infinite state systems in assignment program model[END_REF][START_REF] Holloway | A survey of Petri net methods for controlled discrete event systems[END_REF] or symbolic transition systems with variables [START_REF] Kalyon | Symbolic supervisory control of infinite transition systems under partial observation using abstract interpretation[END_REF]. Inspired by the work of [START_REF] Chung | Limited lookahead policies in supervisory control of discrete event systems[END_REF], we provide here some symbolic devices to compute limited lookahead controllers for logiconumerical reactive systems.

Regarding symbolic modeling for resource control, our quantitative modeling method is loosely related to the one used by [START_REF] Fei | A BDD-Based Approach for Designing Maximally Permissive Deadlock Avoidance Policies for Complex Resource Allocation Systems[END_REF] for modeling Resource Allocation Systems: they handle a specific class of RASs where "process types" (that loosely correspond to resource families in our work) are acyclic, and "resource types" required for the execution of processes, are of finite capacity and constant. They seek maximally permissive non-blocking supervisors for deadlock avoidance by modeling the RASs using Extended Finite Automata [START_REF] Skoldstam | Modeling of discrete event systems using finite automata with variables[END_REF].

Although their approach is well suited for controlling plants and discrete systems that can be modeled using finitestate automata and where limited amounts of resources are involved, this modeling method is not fit for systems featuring dynamic amounts of such resources. [START_REF] Lennartson | Unified Model for Synthesis and Optimization of Discrete Event and Hybrid Systems[END_REF] support dynamic aspects through an encoding similar to ours, and address synthesis for performance optimization (in the sense of make span minimization for Petri Nets), though with a finite alphabet of events and state variables defined on finite domains only.

Few approaches involve discrete event control for selfadaptive computing systems. [START_REF] Dumitrescu | Multicriteria optimal discrete controller synthesis for fault-tolerant tasks[END_REF] focus on finite-state systems equipped with cost functions and final states, and advance an algorithm for multi-criteria optimal discrete controller synthesis; they practice their approach towards fault-tolerance. Berthier et al. [2015] explore the capabilities of over-approximating symbolic DCS algorithms to introduce quantitative aspects into models involved in the control of dynamically partially reconfigurable architectures investigated by [START_REF] An | Modelbased design of correct controllers for dynamically reconfigurable architectures[END_REF]. A common issue of these approaches is the need to enumerate the automata of all individual resource to manage: this aspect challenges the computational cost of DCS algorithms involved, and the modeling task w.r.t. the control problems at hand. We overcome these limits by allowing more powerful models (e.g., involving numerical controllable variables) and relaxing the control objectives (e.g., accommodating non-strict safety with best-effort control, recovery, and limited lookahead optimization).

CONCLUSIONS AND PERSPECTIVES

In this paper, we have exploited the expressivity of logiconumerical models to solve resource management problems occurring in computing systems. Our solution alleviates the need to enumerate the automata of every managed resource, and permits the expression of a wide range of control objectives. To enforce the latter on such systems with infinite control means, we focus on the control over a finite sliding window. We provide exact, effective algorithms for bounded safety and optimization, and illustrate them on an example. We also give means to relax such strict objectives for best-effort control and recovery. We exercise our implementation of the algorithms on some benchmarks. We plan to reuse the resource models put forward in this paper to validate the practicality of the technique, and to transfer them in the framework of ongoing research: on the one side, as a follow-up of the work of [START_REF] An | Modelbased design of correct controllers for dynamically reconfigurable architectures[END_REF] in model-based control of reconfigurable FPGA-based architectures; on the other side, in control of middlewarelevel redeployment of computations and reconfigurations of heterogeneous architectures in the context of smart buildings [START_REF] Sylla | Combining transactional and behavioural reliability in adaptive middleware[END_REF]. Further, best-effort and recovery control on finite sliding windows provides a good trade-off in terms of expressiveness of the models and achievable control objectives, these techniques are thus especially interesting in the case of non-critical systems.

Fig. 1 .

 1 Fig. 1. Example resource automaton for request handlers.Dashed arrows denote controllable transitions; (resp. ) denote resource appearance (resp. disposal).

Fig. 2 .

 2 Fig. 2. An execution trace of S rh . Values for variables in I are randomly picked at each step so that A is satisfied. Each column shows the input values at step i and the new values (for step i + 1) for the state variables; the latter are all null initially. ASTS S rh = X, U C, T, A, Θ o where U = c, y, e : Z 3 and C = s, r : Z 2 . Notice D C is infinite.

Fig. 4 .

 4 Fig. 4. Example execution trace of S rh / Kω (built in Example 6). Each step consists in: values for variables in U are randomly picked so that ∃ C A is satisfied; in turn, values for s and r are also randomly picked so that K ω is satisfied.

  ))

	Example 3. Building upon Example 2, (a 42)| 1 = a +
	r -y -e 42 symbolically represents all potential paths
	(x, ι • ι) (i.e., from state x with successive inputs ι • ι)
	for which the value of the state variable a after one
	step is lower or equal than 42: s.t a T (x, ι)	42. Also,
	(a 42)| 1

Table 1 .

 1 Synthesis Times (in Seconds)

					7	13	14	16	13	13	6	2	3
	i2 9 w1 0	4 6	10 2	11 0	18 8	26 11	25 7	20 13	15 23	10 26	9 30	14 30	19 29
	w2 0	6	3	6	9	12	8	15	24	26	26	26	31
	s1 0 r1 0	6 0	1 5	0 2	8 0	3 0	4 8	5 1	8 1	7 5	10 7	4 5	0 3
	s2 0	6	2	4	3	2	4	6	10	6	6	1	5
	r2 0	0	5	2	0	0	8	5	2	5	7	5	0
	Fig. 5. Example execution trace for two distinct families of
	request handlers (Parallel, κ=1, N = 2). Each column
	shows the input values at step i along with the new
	values (for step i + 1) for the state variables a1, a2,
	i1, i2, w1, and w2 (all null initially).
	N :						2		3	6		9
	Parallel, κ=1				0.04	0.04	0.28		1.03
	Alt, κ=1					0.03	0.04	0.06		0.12
	Alt, κ=2					0.10	0.15	0.49		41.28
	Alt, κ=1, 1-step recovery	0.05	0.09	0.31		0.16
	Alt, κ=2, 1-step recovery	0.14	0.22	0.90		41.32

This work was partially supported by the EPSRC through grant EP/M027287/1. Authors' version.


Rémy Gauthier

Hélène Follet

Max Langer

Evelyne Gineyts

Frédéric Rongieras

Françoise Peyrin

David Mitton
email: david.mitton@ifsttar.fr

Frédéric Rongiéras

Relationships between human cortical bone toughness and collagen cross-links on paired anatomical locations

Keywords: Human cortical bone, fracture toughness, strain rate, inter-sites, collagen cross-links

come   L'archive ouverte pluridisciplinaire

Introduction

Bone fragility still remains difficult to understand as it is associated with several architectural, microstructural and even compositional parameters [START_REF] Seeman | Bone quality--the material and structural basis of bone strength and fragility[END_REF]. Among these, collagen fibers are of great importance in human cortical bone mechanical integrity [START_REF] Oxlund | Reduced concentrations of collagen cross-links are associated with reduced strength of bone[END_REF][START_REF] Wang | Crack propagation in bone on the scale of mineralized collagen fibrils: Role of polymers with sacrificial bonds and hidden length[END_REF][START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Gupta | Intrafibrillar plasticity through mineral/collagen sliding is the dominant mechanism for the extreme toughness of antler bone[END_REF][START_REF] Zioupos | The role of collagen in the declining mechanical properties of aging human cortical bone[END_REF][START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Zioupos | The role of collagen in the declining mechanical properties of aging human cortical bone, 44th[END_REF]. Collagen represents more than 80 % of the organic matter within bone tissue [START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF]. This organic material provides some ductility to bone whereas the mineral phase provides rigidity [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF]. Changes in collagen with age are assumed to be involved in the decrease of bone mechanical properties with aging [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF][START_REF] Granke | Identifying novel clinical surrogates to assess human bone fracture toughness[END_REF][START_REF] Vashishth | Influence of nonenzymatic glycation on biomechanical properties of cortical bone[END_REF].

One of the most important mechanisms of collagen evolution is the formation of cross-links between collagen fibrils [START_REF] Bailey | Molecular mechanisms of ageing in connective tissues[END_REF]. Two different cross-linking processes can be outlined. The first process is a stabilization of collagen fibers that is regulated by lysyl oxidase, which converts the amino groups of lysine and hydroxylysine to form allysine and hydroxyallysine. These molecules can then condensate with some residues of lysine or hydroxylysine to form immature derived divalent ketoimine crosslinks. By undergoing another reaction with lysine or hydroxylysine molecules, there is the formation of pyridinoline (PYD) and deoxypyridinoline (DPD) trivalent mature cross-links [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Bailey | Molecular mechanisms of ageing in connective tissues[END_REF][START_REF] Bailey | Mechanisms of maturation and ageing of collagen[END_REF]. Their relative amounts can be expressed as the ratio between immature and mature cross-links, and are called the CX ratio in the current study (CX = [DHLNL + HLNL] / [PYD + DPD] [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF].

A second mechanism involves a non-enzymatic reaction with glucose. This so-called glycation reaction is at the origin of the presence of Advanced Glycation End-products (AGEs) within bone tissue [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Bailey | Molecular mechanisms of ageing in connective tissues[END_REF]. These non-enzymatic cross-links are thought to be involved in the decline of cortical mechanical properties with age [START_REF] Vashishth | Influence of nonenzymatic glycation on biomechanical properties of cortical bone[END_REF][START_REF] Zimmermann | Age-related changes in the plasticity and toughness of human cortical bone at multiple length scales[END_REF]. Pentosidine (PEN), which is the most investigated AGE, can be accurately quantified in bone tissue using HPLC by natural fluorescence [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Saito | Reductions in degree of mineralization and enzymatic collagen cross-links and increases in glycation-induced pentosidine in the femoral neck cortex in cases of femoral neck fracture[END_REF].

While there are some studies investigating the relationships between collagen crosslinking and cortical bone mechanical properties [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF][START_REF] Vashishth | Influence of nonenzymatic glycation on biomechanical properties of cortical bone[END_REF][START_REF] Zimmermann | Age-related changes in the plasticity and toughness of human cortical bone at multiple length scales[END_REF][START_REF] Poundarik | A direct role of collagen glycation in bone fracture[END_REF], there is little information on the relative difference between different anatomical locations [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF]. In a previous study, our group showed that there was significant difference in bone fracture toughness of weight-and non-weight-bearing bones [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF].

However, the compositional difference between a weight-bearing bone such as the femur, or a nonweight bearing bone, such as the radius, is not well known.

The influence of the CX ratio and PEN on cortical bone mechanical properties has been studied [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF][START_REF] Depalle | Influence of cross-link structure, density and mechanical properties in the mesoscale deformation mechanisms of collagen fibrils[END_REF][START_REF] Willett | In vitro non-enzymatic ribation reduces post-yield strain accommodation in cortical bone[END_REF][START_REF] Tang | Changes in non-enzymatic glycation and its association with altered mechanical properties following 1-year treatment with risedronate or alendronate[END_REF]. PEN is associated with increased fragility of bone tissue in the elderly on weight-bearing bones such as the tibial or femoral shaft [START_REF] Tang | Effects of non-enzymatic glycation on cancellous bone fragility[END_REF][START_REF] Tang | The relative contributions of non-enzymatic glycation and cortical porosity on the fracture toughness of aging bone[END_REF]. However, the effect of the CX ratio remains little studied. A recent study found that the CX ratio was associated with a decrease in post yield properties when comparing fibula extracted from children (10.14 ± 4.56 y.o.) and adult donors (79.00 ± 15.39 y.o.) [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF].

To the authors' knowledge, there is no data on the influence of bone anatomical locations on these enzymatic and non-enzymatic cross-links and their effect on bone mechanical properties, specifically when comparing weight-bearing and non-weight-bearing bones. Moreover, the effect of the CX ratio on cortical bone behavior is not well known for a high-risk population with bone diseases and traumatisms.

The reason for this study is the lack of knowledge on bone fracture behaviour when considering paired weight-bearing and non-weight-bearing bones. While the influence of PEN on bone mechanical response is well known for weight-bearing bones, there is no data regarding the effect of the CX ratio for an adult population at risk regarding bone diseases.

Thus, the aim of this study is to quantify both enzymatic and non-enzymatic collagen cross-links on paired weight-and non-weight-bearing bones (femoral diaphysis (medial and lateral regions), femoral neck and radial diaphysis) in relation to toughness parameters for human cortical bone considering an adult population.

Materials and Methods

Bone samples

This study sampled from thirty-two human cadaver donors (18 women, age 81 ± 12 y.o.; 14 men, age 79 ± 8 y.o.). No additional information concerning eventual disease or medication history was available, except for hepatitis and Human Imunodeficiency Virus status. Whole femurs and radii were collected from these fresh cadaveric subjects (French Ministry of Education and Research, authorization n°DC-2015-2357). Extracted bones were kept hydrated in saline soaked gauze and then stored at -20 °C until sample preparation. Because of the non-availability of some of these bones, the final database was comprised of 2 x 29 femoral diaphysis (both from medial and lateral regions to directly evaluate the influence of anatomical region in the same location), 29 femoral inferior neck, which is the region where cortical thickness is suitable for sample preparation; and 31 radial anterior diaphysis specimens from the mid-diaphysis, which corresponds to the investigated region in a previous study aimed at investigating bone properties using guided waves that was performed on the same population [START_REF] Vallet | In Vivo Characterization of Cortical Bone Using Guided Waves Measured by Axial Transmission[END_REF].

Biomechanical measurements

The biomechanical experiments protocol used here are the same as in a previous study [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF]. A summary of the methodology is summarized below.

Sample preparation

From each bone, from 20 to 25 mm long rectangular notched samples, with a width W = 2.08 ± 0.06 mm and a thickness B = 1.01 ± 0.02 mm, were prepared so that their long side is parallel to the long axis of the diaphysis. The notch was machined in the transverse direction of the bone. As such, a crack will initiate and propagate perpendicularly to the osteons. In order to directly investigate the effect of strain rate, two samples were retrieved as closely as possible, so that we can assume that they are composed of the same material. The first was loaded under a quasi-static strain rate and the other at a representative fall rate. Thus, a total of 236 samples were prepared for this study. The samples were stored at -20°C after preparation, then de-frosted the night before the mechanical tests and kept at 21°C in a physiological solution for 6 h before testing for rehydration.

Biomechanical experiment

Toughness experiments were performed using a three-point bending set up. For the quasi-static loading, which is the standard case, a strain rate of 10 -4 s -1 was applied [START_REF] Zimmermann | Fracture resistance of human cortical bone across multiple length-scales at physiological strain rates[END_REF][START_REF] Yeni | Fatigue damage-fracture mechanics interaction in cortical bone[END_REF][START_REF] Kasapi | Strain-rate-dependent mechanical properties of the equine hoof wall[END_REF]. The fall-like loading condition was defined based on values found in the literature. Thus, the samples were loaded at a strain rate of nearly 10 -1 s -1 [START_REF] Foldhazy | Exercise-induced strain and strain rate in the distal radius[END_REF] to emulate fall conditions. A load-displacement curve was recorded for each mechanical test and was used to obtain different parameters using the methods presented below.

Fracture toughness measurement

The non-linear J-integral introduced by Rice was measured to consider both elastic and plastic properties of human cortical bone [START_REF] Rice | A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks[END_REF]. The elastic strain energy release rate (J el in kN.mm -1 ) and the plastic strain energy release rate (J pl in kN.mm -1 ) were calculated from toughness experiments. The formulae to calculate these mechanical parameters can be found in [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF].

As cross-links may influence cortical bone capacity to undergo plastic deformation before breaking, we decided to divide the non-linear part of the load-displacement curve into two different areas associated to two different non-linear deformation mechanisms [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF]. We assume that the non-linear part of the load displacement curve can be divided into two different fracture mechanisms that represent the development of a process zone and the actual propagation of a crack through the microstructure.

Based on the shape of the curves obtained during the toughness experiment on human cortical bone, we choose to associate a strain energy parameter, J process (kN.mm -1 ), to the area between the yield load and the maximum load that would correspond to the development of a process zone due to rearrangement of the internal organization of cortical bone [START_REF] Vashishth | Crack growth resistance in cortical bone: Concept of microcrack toughening[END_REF][START_REF] Yang | Fracture length scales in human cortical bone: The necessity of nonlinear fracture models[END_REF][START_REF] Willett | The micro-damage process zone during transverse cortical bone fracture : No ears at crack growth initiation[END_REF]. The second parameter, J prop (kN.mm -1 ), is associated with the final part of the curve, from the maximum load until the complete failure, and would correspond to the actual formation of a main crack that would propagate through the cortical bone microstructure until failure. These two parameters can be assessed as follows (Figure 1):

J process = ηA process B(W-a) (1)
J prop = ηA prop B(W-a) ( 2
)
where η is a geometrical factor equal to 1.9 for Single Edge Notched Bending (SENB) specimen, B, W and a are the dimensions of the sample [START_REF]Standard Test Method for Measurement of Fracture Toughness[END_REF]. A process and A prop are the areas under the loaddisplacement curve corresponding to b and c in Figure 1, respectively. We thus have the relationship between the different parameters:

J pl = J process + J prop (3)

Biochemical measurements

Sample preparation

The SENB samples used for biomechanical experiments were not used for biochemical measurements in order to keep them intact for further analysis using other modalities. In that way, cortical bone overages from the previously enounced sample preparation were kept frozen at -20 °C for the biochemical measurements. This means that these measurements were done on a tissue closely surrounding the previously used region during the biomechanical experiments. The samples were cut into small pieces, powered in liquid nitrogen-cooled freezer mil (Spex Centriprep, Metuchen, USA), and demineralized with daily changing of 0.5 M EDTA in 0.05 M in Tris buffer, pH 7.4 for 96 h at 4

°C. The demineralized powder was then extensively washed with deionized water. The samples were then suspended in phosphate buffered saline (0.15 M sodium chloride, 0.1 M sodium phosphate buffer, pH 7.4) and reduced in NaBH 4 at room temperature for 2 h using a reagent/sample ratio of 1:30 (w/w) .

The reduced bone residues were washed, freeze-dried and hydrolyzed in 6 M hydrochloric acid at 110 °C for 20 h. These hydrolysates were used for the measurement of cortical bone cross-links and for the assessment of collagen and hydroxyproline. The availability of matter, the final database for biochemical measurements is composed of 2 x 29 femoral diaphysis (medial and lateral), 25 femoral necks and 30 radial diaphysis specimens.

Cross-links measurements

PYD and DPD mature cross-links and DHLNL and HLNL immature cross-links stabilized by NaBH4 reduction were extracted from the hydrolysates using a solid phase extraction column (Bond Elut Cellulose, Agilent Technologies, Santa Clara, CA, USA) to remove interfering molecules. The DHLNL, HLNL, PYD and DPD cross-links were then investigated using an Alliance 2695 separation HPLC system equipped with a Waters Micromass ® ZQ TM Single Quadrupole Mass Spectrometer, a 2647 Multi λ fluorescence detector and Empower2 chromatography data software (Waters Corp.

Milford, MA, USA). The species were separated on a C18 Atlantis ® T3, 3µm, 4.6×100mm reversedphase column protected by an Atlantis® T3, 3µm, 4.6×20mm guard cartridge (Waters Corp. Milford, MA, USA), maintained at 25 °C, and using a gradient between an aqueous phase composed of 0.12 % of heptafluorobutyric acid in 18 ohms pure water and an acetonitrile organic phase at a flow rate of 1ml/min. Detection of the different molecules was performed by electrospray ionization mass spectrometry in a positive ion mode [START_REF] Gineyts | Quantification of immature and mature collagen crosslinks by liquid chromatography-electrospray ionization mass spectrometry in connective tissues[END_REF]. PYD and DPD calibrators were purchased from Quidel Corp. (San Diego, CA, USA). A PEN calibrator was kindly provided by Dr. M. Takahashi (Hamamatsu University School of Medicine, Shizuoka, Japan). DHLNL and HLNL standards were purified from foetal bovine cortical bone by HPLC and calibrated using the ninhydrin reference method and leucine equivalence [START_REF] Gineyts | Urinary levels of pentosidine and the risk of fracture in postmenopausal women: The OFELY study[END_REF].

The ratio CX = [DHLNL + HLNL] / [PYD + DPD] was used to express the state of cross-links maturation, which assessed collagen matrix maturation [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF].

PEN quantification was performed investigating its natural fluorescence at an emission of 385 nm for PEN and at an excitation of 334 nm [START_REF] Viguet-Carrin | Simple and sensitive method for quantification of fluorescent enzymatic mature and senescent crosslinks of collagen in bone hydrolysate using single-column high performance liquid chromatography[END_REF].

The weight proportion of bone collagen in the sample was analyzed by measuring hydroxyproline using a HPLC assay (Bio-Rad, Munich, Germany) [START_REF] Gineyts | Quantification of immature and mature collagen crosslinks by liquid chromatography-electrospray ionization mass spectrometry in connective tissues[END_REF][START_REF] Viguet-Carrin | Simple and sensitive method for quantification of fluorescent enzymatic mature and senescent crosslinks of collagen in bone hydrolysate using single-column high performance liquid chromatography[END_REF][START_REF] Follet | Effects of preexisting microdamage, collagen cross-links, degree of mineralization, age, and architecture on compressive mechanical properties of elderly human vertebral trabecular bone[END_REF][START_REF] Saito | Single-Column High-Performance Liquid Chromatographic-Fluorescence Detection of Immature, Mature, and Senescent Cross-Links of Collagen[END_REF][START_REF] Eyre | Quantitation of hydroxypyridinium crosslinks in collagen by high-performance liquid chromatography[END_REF][START_REF] Pratt | Automated analysis of the pyridinium crosslinks of collagen in tissue and urine using solid-phase extraction and reversed-phase highperformance liquid chromatography[END_REF] (Figure 2).

Results are given in mmol by mol of collagen (mmol / mol collagen). The amount of collagen was measured by accurately weighing and hydrolysed by HCl 6 M [START_REF] Gineyts | Urinary levels of pentosidine and the risk of fracture in postmenopausal women: The OFELY study[END_REF].

Statistical tests

The normality of the distribution was assessed using Shapiro-Wilk's test. As normality was not obtained for all the parameters, results were analyzed using the Mann-Whitney test, and a nonparametric Wilcoxon test was applied for paired samples. Spearman's correlation tests determined the effect of age on biochemical parameters. These statistical tests were done using R © (The R foundation for Statistical Computing, Austria) and Statview ® (SAS Institute Inc., Campus Drive, Cary, NC 27513). As several statistical tests were applied, only results with a p-value < 0.01 were considered as significant to take into account the multiple statistical tests and to strengthen the conclusions.

Results

Biomechanical experiments

Mean values ± Standard deviation (SD) for the four toughness parameters are presented in Figure 3 and in the Table 1 available in the supplementary materials section.

a b c d

Quasi-static Fall

Linear elastic strain energy release rate

Comparing quasi-static and fall conditions, a significant difference was observed only on the femoral lateral diaphysis and femoral neck: the elastic strain energy release rate increases when the rate is higher. The elastic strain energy release rate (Jel) increases about 27 % for the lateral region of the diaphysis, and 43 % for the femoral neck (see the supplementary materials).

Under a quasi-static loading rate, radius and the femoral lateral diaphysis present a significant higher elastic strain energy release rate than the femoral neck, with a maximum difference between the femoral neck and the radius (-45%).

Under a fall loading rate, no difference was observed between anatomical locations (Figure 3).

Non-linear elastic strain energy release rate

Under fall-like conditions, there is a strain energy release rate (J pl ) decrease for all sites. When distinguishing a process (J process ) and a propagation (J prop ) area on the load-displacement curve, we can observe a significant decrease of J prop for the four locations, from -199 % for the femoral neck to -345 % for the radius; whereas there is no significant difference from the loading condition on J process .

Under both under quasi-static and fall-like loading conditions, the overall non-linear deformation, represented by J pl , appears to be higher than the elastic contribution J el (Figure 4). When defining different regimes within the plastic contribution, under a quasi-static condition, the propagation energy release rate, J prop , is higher than the elastic, J el , and the process, J process , contribution. Meanwhile, under fall-like loading conditions, J process and J prop are comparable (Figure 4).

To summarize, the elastic contribution (J el ) is significantly smaller than the plastic contribution (J pl = J process + J prop ) in the global mechanical response of cortical bone (Figure 4). For some cases (femoral lateral diaphyses and femoral neck) there is a significant increase of the elastic strain energy release rate (J el ) with an increase of strain rate, and the main evolution is due to the significant decrease of J pl for all the anatomical locations. Within this plastic contribution (J pl ) the main contributor is the propagation phase (J prop ) (Figure 4).

When comparing anatomical locations, the radius has a better capacity to resist crack propagation in the non-linear regime, considering both J pl and the distinct J process and J prop . These differences are not more significant under a fall-like loading condition.

Weight-bearing

Non-weight-bearing

J el J process J prop

Biochemical measurements

The mean values (± SD) of enzymatic immature cross-links, enzymatic mature cross-links, CX ratio, Pen and collagen are reported in the Table 2 of the supplementary materials section.

Figure 5 shows the mean values obtained for the CX ratio and PEN measurements on the four anatomical locations. Significant differences were observed depending on the anatomical location. In particular, the CX ratio and PEN are significantly higher in the femoral neck than in femoral and radial diaphyses. The maximum relative difference is noted with the femoral medial diaphysis of about 44 % for the CX ratio and 52 % for PEN (p-value < 0.001). Radius also presents a significant higher CX ratio than femoral diaphysis with a higher difference of about 38 % with the medial region (p-value < 0.0001) and a higher PEN content than in the medial region of the femoral diaphysis of 20 %. There is no difference between femoral lateral and medial diaphysis for CX ratio and PEN.

Relationships between age, mechanical and biochemical properties

For the four anatomical locations, PEN measurements were significantly positively correlated to the age of the donor whereas the CX ratio is negatively correlated to age only for the radius. Results from the correlation tests between age and PEN and CX ratio are given in Table 1. No significant correlation was observed between the CX ratio and biomechanical parameters. A weak relation was found between the CX and J process under a quasi-static loading condition on the medial femoral diaphysis and radius. This indicates a significant trend (Table 2).

Fall -0.08

There was no significant correlation between J prop and CX for the quasi-static condition on all the investigated anatomical locations. A marginal weak relation was observed for the lateral region of the femoral diaphysis under a fall-like loading condition (Table 3). No trend was observed here. There was a significant correlation between the CX ratio and J process when we investigated the gathered population of femoral and radial diaphyses. In a similar manner, this correlation was significant for J prop (Figure 6). Regarding the large differences found concerning the biochemical results between the femoral neck and the other locations, this location was not considered in these correlation tests. These correlations were not significant if we included the femoral neck. Figure 7 shows J process and J prop measured under fall-like loading condition in function of the CX ratio for the gathering population of femoral and radial diaphyses. We can observe here that if there is no correlation between J process and CX, then we found a significant relation between J prop and the ratio.

This correlation is nevertheless weaker than under a quasi-static condition. No relation was found when considering PEN even by gathering the anatomical locations.

Discussion

This study investigated the potential relationships between fracture behavior, both under quasi-static and fall-like loading conditions, and collagen maturation of human cortical bone considering four paired anatomical locations extracted from a population of 32 donors (18 women, age 81 ± 12 y.o.; 14 men, age 79 ± 8 y.o.).

 Biomechanical measurement

The results show that under a quasi-static loading condition, non-linear mechanisms are mainly due to a propagation mechanism. Under a fall-like loading rate, J process and J prop have the same order of magnitude, reflecting an equivalent contribution to the non-linear crack propagation mechanisms. The effect of strain rate on the propagation mechanism, J prop , is the same as previously observed for other parameters [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF]: there is a lower value at higher a rate (Figure 3). For the process zone development mechanism, there is no influence from strain rate. The propagation contribution is the most affected by the increase of rate.

Dealing with differences in the biomechanical parameters between the four anatomical locations, the results are the same for J process and J prop as J pl . Under a quasi-static condition, the radius presents a greater capacity to undergo non-linear mechanisms, both for the process zone development and for the propagation mechanisms. Under a fall-like condition, the femoral lateral diaphysis, femoral neck and radius present the same non-linear capacity to resist crack propagation (Figure 3).

These results show that the radius, which is a non-weight bearing bone, is more able to resist crack propagation in a quasi-static loading condition when considering the non-linear mechanisms. In a previous study, the authors compared the toughness of the radius and the tibia [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF]. The results obtained were the same: the radius has a better capacity to resist crack propagation. The results found in the current study may also be due to the different analyzed regions used between the femur (medial and lateral) and the radius (anterior) as we know that these different regions are not subjected to the same mechanical environment [START_REF] Judex | High-impact exercise and growing bone: relation between high strain rates and enhanced bone formation[END_REF]. In a previous study, Li et al. found significant differences between different regions of the same diaphysis [START_REF] Li | Analysis of fracture processes in cortical bone tissue[END_REF]. Their results showed that the toughness measured on the medial part was significantly higher than the toughness measured on the anterior part (24 %). In the current study, the toughness measured on the medial part (femoral diaphysis, weight-bearing bone)

was significantly lower (50 %) than that measured on the anterior part (radius, non-weight-bearing bone). This suggests a real influence from the weight-or non-weight-bearing nature of the investigated bones.

 Biochemical measurement

The measured biochemical parameters in this study are in the range of results found in the literature [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF][START_REF] Saito | Reductions in degree of mineralization and enzymatic collagen cross-links and increases in glycation-induced pentosidine in the femoral neck cortex in cases of femoral neck fracture[END_REF][START_REF] Follet | Effects of preexisting microdamage, collagen cross-links, degree of mineralization, age, and architecture on compressive mechanical properties of elderly human vertebral trabecular bone[END_REF][START_REF] Saito | Single-Column High-Performance Liquid Chromatographic-Fluorescence Detection of Immature, Mature, and Senescent Cross-Links of Collagen[END_REF][START_REF] Karim | Differences in non-enzymatic glycation and collagen cross-links between human cortical and cancellous bone[END_REF]. While a significant correlation was found between CX and age only for the radius, PEN for each anatomical site correlated with age. This result is in accordance with the fact that PEN is a better biomarker for the ageing of the donor [START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Eyre | Maturation of collagen ketoimine cross-links by an alternative mechanism to pyridinoline formation in cartilage[END_REF][START_REF] Sroga | Glycation of human cortical and cancellous bone captures differences in the formation of maillard reaction products between glucose and ribose[END_REF]. The increase in PEN content is associated with the accumulation of glucose and ribose within an organism due to some cell dysfunction due to ageing, whereas immature and mature cross-links concentration is assumed as in the elderly [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Saito | Single-Column High-Performance Liquid Chromatographic-Fluorescence Detection of Immature, Mature, and Senescent Cross-Links of Collagen[END_REF][START_REF] Stout | Glucose tolerance and ageing[END_REF][46][START_REF] Sell | Structure elucidation of a senescence cross-link from human extracellular matrix. Implication of pentoses in the aging process[END_REF]. The values obtained on the femoral neck appear to be slightly different from the literature [START_REF] Saito | Reductions in degree of mineralization and enzymatic collagen cross-links and increases in glycation-induced pentosidine in the femoral neck cortex in cases of femoral neck fracture[END_REF][START_REF] Karim | Differences in non-enzymatic glycation and collagen cross-links between human cortical and cancellous bone[END_REF]. PEN content measured in the current study (19.83 ± 7.5 mmol / mol collagen) is higher than in [START_REF] Saito | Reductions in degree of mineralization and enzymatic collagen cross-links and increases in glycation-induced pentosidine in the femoral neck cortex in cases of femoral neck fracture[END_REF] (7.82 ± 3.45 mmol / mol collagen) or in [START_REF] Karim | Differences in non-enzymatic glycation and collagen cross-links between human cortical and cancellous bone[END_REF] (~ 10 mmol / mol collagen) measured on the cortical bone of the femoral neck. As a major part of cortical bone on the femoral neck was used to prepare samples for biomechanical experiments in the current study, the remaining part of the cortical bone was thin. Femoral neck is mainly composed of trabecular bone, despite all the experimental precautions it might be possible that this remaining part of the bone contained some region of the trabeculae. It has been shown in the past that the PEN content of trabecular bone is higher than in cortical bone [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Karim | Differences in non-enzymatic glycation and collagen cross-links between human cortical and cancellous bone[END_REF]. Our group also found a significant difference in mature parameters (PYD + DPD) in cortical bone versus trabecular bone [START_REF] Gineyts | Contribution of Enzymatic and Non-Enzymatic Collagen Crosslinks to Femoral Neck Fragility. Comparison Between Osteoporosis and Osteoarthritis[END_REF].

In the present study, we showed that enzymatic cross-link composition and PEN accumulation are highly dependent on the considered anatomical locations. Large significant differences were found between femoral diaphysis, femoral neck and radial diaphysis. The difference found between the lateral and medial part of the femur was relatively small, 8 %, suggesting that cross-link content is not dependent on the region of the femoral diaphysis (Figure 5).

The differences seen between the anatomical locations might be explained by different remodeling mechanisms. The amount of immature and mature cross-links are directly linked to the bone tissue turnover rate [START_REF] Viguet-Carrin | The role of collagen in bone strength[END_REF][START_REF] Bailey | Mechanisms of maturation and ageing of collagen[END_REF]. These differences can be due to the specific function of each bone in the skeleton, and in particular, whether it is a weight-bearing or a non-weight-bearing bone. As hypothesized in a previous study, physiological mechanical loading may stimulate collagen stabilization through the differentiation of immature cross-links to mature cross-links [START_REF] Saito | Collagen cross-links as a determinant of bone quality: A possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF]. This is in accordance with results found in the current study: femoral diaphysis and femoral neck present a lower CX ratio, meaning that they are more mature than the radial diaphysis.

 Relationships between biomechanical and biochemical measurement

The relationships between biomechanical and biochemical properties found in the literature show that the maturation of collagen has an influence on cortical bone ability to undergo non-linear deformation coefficient was higher between PEN and the work of fracture, which is obtained with tensile testing, than between PEN and elastic fracture toughness.

This study has some limitations. The limitations of biomechanical experiments, are discussed in [START_REF] Gauthier | Strain rate influence on human cortical bone toughness: A comparative study of four paired anatomical sites[END_REF].

The new parameters quantified in the current study do not modify these limitations.

Regarding biochemical measurements, the main limitation is that samples were taken from donors, and were then stored at -20 °C until process them. Some studies have shown that the action of freezing may not affect the measured concentration with an HPLC system [START_REF] Bergmann | The influence of sample freezing at -80°C for 2-12 weeks on glycated haemoglobin (HbA1c) concentration assayed by HPLC method on Bio-Rad D-10© auto-analyzer[END_REF]. Considering that all the 236 samples tested in this study followed the same storage and preparation protocol, we can assume that a comparative analysis can be performed and that there is no inter-group bias.

Another limitation is that our studied population can be considered old (79 ± 11 y.o.). It is well known that bone chemical composition changes with age. This point is important to consider when interpreting our results; studies on samples with a wider age distribution may lead to different results.

Nonetheless, the samples used in the current study are representative of the patient demographic with the most bone fractures [START_REF] Nanes | Osteoporosis[END_REF], and as such provides insight into factors that contribute to cortical bone toughness in this age group.

Finally, femoral neck has been shown to have slightly different properties than femoral or radial diaphysis. This can also be seen as a limitation in the current study. Samples prepared from the femoral neck may not be truly cortical samples, but more like a mix between cortical and trabecular.

Nonetheless, femoral neck is one of the anatomical locations that is at greater risk for fracture in the elderly [START_REF] Court-Brown | Epidemiology of adult fractures: A review[END_REF]. The region investigated in the current study is the most compact area on the femoral neck [START_REF] Loveridge | Bone mineralization density and femoral neck fragility[END_REF]. It is thus representative of reality. The femoral neck is a complex bone compared to other locations.

This study on human cortical bone found no correlation between collagen cross-link maturation and bone toughness for an adult population. Further work should be done to investigate the influence of other compositional parameters, such as osteopontin, on this particular sliding mechanism. Future investigations may be performed to evaluate the influence of some structural parameters on human cortical bone fracture behavior as it is know that some structural features, such as the osteons, play a major role in crack propagation mechanisms in human bone [START_REF] Burr | Composition of the cement line and its possible mechanical role as a local interface in human compact bone[END_REF].

Conclusion

To the best of our knowledge, this is the first study dealing with the measurement of enzymatic and non-enzymatic cross-link content on almost 30 paired femoral diaphyses, femoral neck and radial diaphyses. It is also the first study comparing human cortical bone toughness and enzymatic cross-link content. In the current study, we showed that collagen maturation was dependent on anatomical

Figure 1

 1 Figure 1 Typical load-displacement curve obtained with human cortical bone toughness experiments. On the left is the ASTM E-1820 suggestion of division [32]; on the right is the current study choice of division (2 column fitting figure).

Figure 2

 2 Figure 2 On the left is a typical chromatogram of DHLNL, HLNL, PYD and DPD of human cortical bone assessed with a mass spectrometry system. On the right is typical chromatogram of PYD, DPD and PEN of human cortical bone assessed with fluorescence system (2-column fitting figure).

Figure 3

 3 Figure 3 Bar charts values of a:J el ; b: J pl ; c: J process ; and d: J prop which are the strain energy release rates for the elastic, plastic, process zone development and propagation contribution, respectively, under quasi-static and fall-like loading condition. (Significant differences are indicated between groups *** p-value < 0.001; **p-value < 0.01) (2-column fitting figure)

Figure 4

 4 Figure 4 Relative contributions of the different mechanical parameters under quasi-static and fall-like loading conditions for weight-bearing and non-weight-bearing (2-column fitting figure).

FallFigure 5

 5 Figure 5 Bar charts mean values for CX ratio and PEN measurements for the four anatomical locations. (Significant differences are indicated *** p-value < 0.001; ** p-value < 0.01) (2-column fitting figure)

Figure 6 .

 6 Figure 6. J process (left) and J prop (right) under quasi-static loading condition in function of CX. Femoral and radial diaphyses are gathered in a same population for the correlation. Spearman coefficient and p-value are given (2-column fitting figure).

Figure 7 .

 7 Figure 7. J process (left) and J prop (right) under fall-like loading condition in function of CX. Femoral and radial diaphyses are gathered in a same population for the correlation. Spearman coefficient and p-value are given when significant (2-column fitting figure).

Table 1 Spearman's coefficient between age and both CX ratio and PEN ( ** p-value ≤0.01, *** p- value < 0.001, ns: non-significant). (1-column fitting table) Anatomical location n CX PEN Femoral lateral diaphysis 29 ns 0.51 ** Femoral medial diaphysis 29 ns 0.55 **

 1

	Femoral neck	25	ns	0.53 **
	Radial diaphysis	30	-0.64 **	0.67 ***

Table 2 Spearman's coefficient between J process and CX ratio for the four anatomical locations and the quasi-static and fall loading conditions (all non-significant: p > 0.01; * Border line: p < 0.05). (1-column fitting table) Anatomical location n J process -CX ratio

 2

	Femoral lateral diaphysis	29	Quasi-static Fall	0.25 -0.13
	Femoral medial diaphysis	29	Quasi-static Fall	0.53 * 0.06
	Femoral neck	25	Quasi-static Fall	0.04 0.15
	Radial diaphysis	30	Quasi-static	0.52 *

Table 3 Spearman's coefficient between Jprop and CX ratio for the four anatomical locations and the quasi-static and fall loading conditions (all non-significant: p > 0.01; * Border line: p < 0.05). (1-column fitting table) Anatomical location n J prop -CX ratio Femoral lateral diaphysis 29

 3

	Quasi-static	-0.12
	Fall	0.49 *

Acknowledgments

The authors wish to thank Leila Ben Boubaker for her technical support. This work was done in the framework of LabEx PRIMES (ANR-11-LABX-0063). This study was partly funded by the Région Rhône-Alpes and by the ANR project MULTIPS (ANR-13-BS09-0006). The authors also want to thank the Fédération INGE'LYSE for its financial support.

 [START_REF] Berteau | Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula[END_REF]. In this previous study, the donors were from 5 to 99 years old and were separated into two groups (7 children of 10 y.o. in average, 3 adults of 79 y.o., in average). In the current study, performed on an adult population (50 -95 y.o.), the decrease of CX ratio is not correlated to a decrease of J pl , J process or J prop under both a quasi-static and fall-like loading condition, when considering individual locations.

By gathering femoral and radial diaphyses, we found that there was a significant correlation between CX ratio and J process and J prop , thus suggesting that by investigating a larger population of donors, we may find significant correlations between collagen maturation and human cortical fracture behavior under a quasi-static loading condition. However, the correlation coefficients remain small (ρ = 0.60 and ρ = 0.52 for J process and J prop respectively). Considering a fall-like loading condition, a significant correlation was only found with J prop with a smaller coefficient than under quasi-static condition (ρ = 0.33).

The results found in the current study suggest that enzymatic cross-links have a minor influence on cortical bone toughness considering a population older than 50 y.o.. A similar result has already been stated using numerical simulation of mineral collagen fibrils [START_REF] Siegmund | Failure of mineralized collagen fibrils: Modeling the role of collagen cross-linking[END_REF]. In that study, they found that in the presence of enzymatic cross-links, deformation can occur through a sliding between collagen molecules, allowing a low load transfer between collagen and mineral platelets. These results also show that if a minor relation can be found under quasi-static loading, then this disappears under a strain rate representative of a fall. In their study, Zimmerman et al. found that at a higher loading rate, the slope of the fibril strain vs. tissue strain was higher [START_REF] Zimmermann | Fracture resistance of human cortical bone across multiple length-scales at physiological strain rates[END_REF]. Under a fall-like condition, the hypothesis of the stretching of the collagen fibrils is favored over interfibrillar sliding mechanisms.

Considering the effect of PEN on cortical bone mechanical behavior, the current study suggests that between 50 and 95 y.o., PEN is not correlated to any bone toughness variation. Findings in the literature state that an increase in PEN content is related to a decrease in bone mechanical properties [START_REF] Vashishth | Influence of nonenzymatic glycation on biomechanical properties of cortical bone[END_REF][START_REF] Poundarik | A direct role of collagen glycation in bone fracture[END_REF][START_REF] Willett | In vitro non-enzymatic ribation reduces post-yield strain accommodation in cortical bone[END_REF][START_REF] Nyman | Age-related factors affecting the postyield energy dissipation of human cortical bone[END_REF]. In most of these studies, the effect of PEN was assessed by comparing a control population to a ribolysated one. A study by Nyman et al. evaluated the influence of PEN on a population from 49 to 90 y.o. and has shown a significant negative influence of PEN on bone fracture toughness (Pearson's coefficient = -0.602) [START_REF] Nyman | Age-related factors affecting the postyield energy dissipation of human cortical bone[END_REF]. However, another recent study performed on a population from 21 to 98 y.o. found results similar to the current ones, stating that PEN is only weakly correlated with both bone initiation fracture toughness and J-integral fracture toughness (Spearman's coefficient = -0.34 and -0.32, respectively) [START_REF] Granke | Identifying novel clinical surrogates to assess human bone fracture toughness[END_REF]. Nyman et al. performed mechanical tensile testing on "dog bone" type specimens, whereas Granke et al. performed toughness testing on notched samples as in the current study. Standard tensile testing may be more sensitive to PEN than standard toughness experiments. A previous study performed on human femoral bone (from 19 to 89 y.o.) obtained a significant correlation between PEN and bone mechanical properties, including toughness measured following a standard method for toughness measurement [START_REF] Wang | Age-related changes in the collagen network and toughness of bone[END_REF]. Results showed that the correlation location, and more specifically for weight-or non-weight-bearing bones. We also found that the relationship between collagen cross-link maturation and cortical bone toughness is not significant. A weak correlation was obtained between the CX ratio and cortical bone capacity to develop a process zone, implying the need of a better understanding of the different mechanisms involved in this particular mechanical regime. This process zone development is assumed to be associated with the emergence of micro-damage within the tissue.

Conflict of interests

The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.