Stoyan Boev
email: stoyan@nbu.bg

Thomas Honold
email: honold@zju.edu.cn

Ivan Landjev
email: i.landjev@nbu.bg

Optimal Arcs in Hjelmslev Spaces of Large Dimension

Keywords: projective Hjelmslev geometry, projective Hjelmslev plane, arcs, optimal arcs, finite chain ring Subject Classifications:51E26, 51E21, 51E22, 94B05

In this paper, we present various results on arcs in projective threedimensional Hjelmslev spaces over finite chain rings of nilpotency index 2. A table is given containing exact values and bounds for projective arcs in the geometries over the two chain rings with four elements.

Introduction

From the point of view of coding theory, arcs in higher dimensional Hjelmslev spaces are of particular interest. The research in the past ten years is focused mainly on plane projective arcs. In this paper, we present some constructions and upper bounds on arcs in the three-dimensional Hjelmslev spaces over the finite chain rings of nilpotency index 2. As a by-product, we obtain new results on non-projective plane arcs. In order to save space, we do not introduce the basic facts on projective Hjelmslev geometries over finite chain rings. A self-contained introduction to Hjelmslev geometries over finite chain rings is given e.g. in [START_REF] Th | Projective Hjelmslev geometries[END_REF].

This note is organized as follows. In section 2, we present several upper bounds on the size of an arc with given parameters in PHG(R t R ). Section 3 contains results on arcs with multiple points in projective Hjelmslev planes. In particular, we give a complete solution for the optimal sizes in the two planes over chan rings with four elements. In section 4, building on the results in the previous two sections, we compile a table with exact values and bounds for the largest arcs in the three-dimensional projective Hjelmslev geometries over the chain rings with four elements.

Upper bounds on the arc size

Denote by m n (R t R ) the largest size of a (k, n)-arc in PHG(R t R ). Theorem 1. Let K be a (k, n)-(multi)arc in PHG(R t R )
, where R is a chain ring with |R| = q 2 , R/ Rad R ∼ = F q , and let x be a point with

K(x) = a. Then k ≤ a + m n-a (R t-1 R ).
Proof. Fix a hyperplane H with x ⌢ ⌣H. Define a projection ϕ from x onto H by ϕ :

P \ {x} → H, y → ∪ L:L∈L,x∈L L ∩ H,
where P is the set of points of PHG(R t R ). If y ∈ P is not a neighbour to x then its image is a point; if y⌢ ⌣x then the image of y is a neighbour class of points in

H ∼ = PHG(R t-1 R ).
The image of an s-dimensional Hjelmslev subspace through x contains an (s -1)-dimensional subspace in H. Conversly, every (s -1)-dimensional subspace in H is contained in the projection of some s-dimensional subspace of PHG(R t R ) through x.

Define a new arc

K ϕ : H → Q via K ϕ (z) := ∑ y:ϕ(y)=z,y ⌢ ⌣ x K(y) + 1 q t-3 ∑ y:ϕ(y)=z,y⌢ ⌣x K(y).
Let H ′ be a hyperplane in PHG(R t R ) containig x, and let F ′ be the unique hyperplane in ϕ(H ′ ). We have

K ϕ (F ′ ) = K(H ′ ) -K(x). Set K([x]) = b. Then K ϕ is a (k -a + q(b -a)
, na)-arc with rational multiplicities of the points. Now define ϕ 0 :

P \ {x} → H, y → L ∩ H,
where L is some arbitrarily chosen line in PHG(R t R ) that contains x and y. Set

K ϕ 0 (z) := ∑ y:ϕ 0 (y)=z K(y).
It is easily verified that

K ϕ 0 (F ′ ) ≤ K ϕ (F ′ ) ≤ n -a. The arc K ϕ 0 is integer-valued (k - a, n -a)-arc in H hence k -a ≤ m n-a (R t-1 R
), which was to be proved.

Corollary 1. Let K be a projective (k, n)-arc in PHG(R 4 R ) where R is a chain ring with |R| = q 2 , R/ Rad R ∼ = F q . Then k ≤ 1 + m n-1 (R 3 R ). Corollary 2. Let K be a (k, n)-(multi)arc in PHG(R t R )
, where R is a chain ring with |R| = q 2 , R/ Rad R ∼ = F q , and let x be a point with K(x) = a, K([x]) = b. Then there exists an arc in PHG(R t-1 R ) with parameters (qk + q 2 bq(q + 1)a, q(na)).

Proof. The arc qK ϕ from the proof of Theorem 1 has the desired parameters.

The next theorem gives better upper bounds in three-dimensional spaces when the values of n are large, i.e. close to the size of a plane.

Theorem 2. Let K be a (k, n)-(multi)arc in PHG(R 4 R ), where R is a chain ring with |R| = q 2 , R/ Rad R ∼ = F q . If there exists a neighbor class of lines [L] with K([L]) = c then k ≤ q(q + 1)(n -⌈ c q ⌉) + c. Proof. Let Π = PHG(R 4 R )
and let [H] be a class of neighbour planes containing [L]. Consider the incidence structure Π ′ having as points the plane segments

H ′ ∩ [x], where H ′ ∈ [H],
x ∈ H, and as planes -the elements of [H]. It is known (cf. [START_REF] Th | Projective Hjelmslev geometries[END_REF][START_REF] Th | On arcs in projective Hjelmslev planes[END_REF]) that Π ′ is isomorphic to the dual affine space AG(3, q) and can be imbedded in PG (3, q). The class [L] in Π ′ is a dual affine plane. The number of lines in a dual affine plane AG(2, q) is q 2 and every point is on exactly q lines. Therefore there exists a line L ′ in Π ′ containing at least ⌈qc/q 2 ⌉ = ⌈c/q⌉ points. This line L ′ consists of the points in a class of neighbor lines in some plane

H ′ ∈ [H], i.e. L ′ = H ′ ∩ [L].
Denote by T 1 , . . . , T q the planes in Π containing

H ′ ∩ [L]. Clearly, K(T i \ (H ′ ∩ [L])) ≤ n -⌈c/q⌉, whence K([H]) ≤ q(n -⌈c/q⌉) + c. Now consider the factor geometry Π/⌢ ⌣ ∼ = PG(3, q). Denote by [H i ], i = 0, . . . , q, the planes through [L] in Π/⌢ ⌣. We have k = q ∑ i=0 K([H i ]) -q • K([L])
≤ (q + 1)(q(n -⌈c/q⌉) + c)qc = q(q + 1)(n -⌈c/q⌉) + c, which is the desired inequality.

It is well-known that there exists a spread of q 2 + 1 lines in the factor geometry Π/⌢ ⌣. Hence we have c ≥ ⌈k/(q 2 + 1)⌉. Applying Theorem 2, we obtain the following corollary.

Corollary 3. Let K be a projective (k, n)-arc in PHG(R 4 R ). Then k ≤ max 1≤c≤q 4 +q 3 min{c(q 2 + 1), q(q + 1)(n -⌈c/q⌉) + c}.
The following theorem is also useful.

Theorem 3. Let F be an ( f , m)-blocking (multi)set in PHG(R 4
R ) with m ≤ q, where R is a chain ring with |R| = q 2 , R/ Rad R ∼ = F q . If there exists a neighbor class with F([x]) = 0 then f ≥ mq(q + 1).

Proof. Consider the projection ϕ from any point in the empty class onto some plane disjoint from [x]. Obviously F ϕ is an ( f , m)-blocking set in an Hjelmslev plane isomorphic to PHG(R 3 R ). The result follows from the trivial upper bound on the size of a plane blocking set.

3 Arcs with multiple points in PHG(R 3 R )

Extensive tables for the optimal sizes of projective arcs in Hjelmslev planes over the small chain rings are given in [START_REF] Th | New results on arcs in projective Hjelmslev planes over small chain rings[END_REF][START_REF] Th | On arcs in projective Hjelmslev planes[END_REF][START_REF] Landjev | Arcs in projective Hjelmslev planes[END_REF]. However, the arcs needed in Corollary 1 are not projective. In this section, we collect results on multiarcs of maximal size in projective Hjelmslev geometries of dimension 3. Let us note that the general bound from [START_REF] Landjev | Arcs in projective Hjelmslev planes[END_REF] applies also for arcs with multiple points. Since this bound is our main tool, we state it explicitly.

Theorem 4. Let K be a (k, n)-arc in PHG(R 3 R ) where |R| = q 2 , R/N ∼ = F q . Suppose there exist a point x with K(x) = a and a neighbour class of points

[x] with K([x]) = b. Then k ≤ (n -a)q 2 + (n -b)q + b.
For some special values of n, the exact value of m n (R 3 R ) is easily found.

Theorem 5. Let R be a chain ring with

|R| = q 2 , R/ Rad R ∼ = F q . Then (a) m sq(q+1) (R 3 R ) = sq 2 (q 2 + q + 1); (b) m sq(q+1)+1 (R 3 R ) = sq 2 (q 2 + q + 1) + 1.
Proof. Part (a) follows easily from (b), so we provide a proof for part (b) only.

Arcs with parameters (sq 2 (q 2 + q + 1) + 1, sq(q + 1) + 1) are easily obtained as the sum of s copies of the complete plane plus an arbitrary point. Now we are goimg to prove that we cannot have an arc with n = sq(q + 1) + 1 and size larger than sq 2 (q 2 + q + 1).

Let K be an (k, n)-arc with n = sq(q + 1) + 1. Assume [x] is a neighbour class of points with K([x]) = sq 2 + α, α > 0. Such a class does exist; otherwise k ≤ sq 2 (q 2 + q + 1) and we are done. Now in each parallel class of [x] (which has the structure of AG(2, q)) we have a line segment of multiplicity at least sq + ⌈α/q⌉. For each line L containing this line segment, we have

K(L \ [x]) ≤ sq(q + 1) + 1 -sq -⌈α/q⌉ = sq 2 + 1 -⌈α/q⌉.

This implies

k ≤ sq 2 + α + q(q + 1)(sq 2 + 1 -⌈α/q⌉) = sq 2 (q 2 + q + 1) + q 2 (1 -⌈α/q⌉) + q(1 -⌈α/q⌉) + α.

If α > q, then q 2 (1 -⌈α/q⌉) + q(1 -⌈α/q⌉) + α < 0. So let 1 ≤ α ≤ q. Clearly, there exists a point in [x] of multiplicity s + β, β ≥ 1. At least one of the line segments through this point has multiplicity more than sq + 2 = sq + 1 + ⌈α/q⌉. Otherwise, we would have

K([x]) ≤ s + β + (q + 1)(sq + 1 -s -β) = sq 2 + (q + 1)(1 -β) ≤ sq 2 ,
a contradiction. Now k ≤ sq 2 + α + q 2 (sq 2 + 1 -⌈α/q⌉) + q(sq 2 -⌈α/q⌉ = sq 2 (q 2 + q + 1) + q 2 (1 -⌈α/q⌉) -q⌈α/q⌉) + α.

We have that for 1 ≤ α ≤ q, q 2 (1 -⌈α/q⌉) -q⌈α/q⌉) + α =) -q⌈α/q⌉) + α ≤ 1, whith equality for α = 1. This proves the theorem.

The next theorem settles the problem of finding the maximal sizes of multiarcs for the two rings with four elements. Theorem 6. Let R be a chain ring with

|R| = 4, R/ Rad R ∼ = F 2 . Then (a) m 6t (R 3 R ) = 28t, (b) m 6t+1 (R 3 R ) = 28t + 1, (c) m 6t+2 (R 3 R ) = 28t + 7 if R = Z 4 ; 28t + 6 if R = F 2 [X]/(X 2 ) , (d) m 6t+3 (R 3 R ) = 28t + 10, (e) m 6t+4 (R 3 R ) = 28t + 16, (f) m 6t+5 (R 3 R ) = 28t + 22,
where t = 0, (c) Let n = 6t + 2. Arcs of cardinality 28t + 7 (resp. 28t + 6) are obtained as a sum of T copies of the complete plane and a (7, 2)-arc (resp. (6, 2)-arc). Now assume there exists a (k, n)-arc with k = 28t + 8. Then there exists a class of points [x] with K([x]) = 4t + 2 and a point in it with K(x) ≥ t + 1. By Theorem 4 we get

k ≤ 4 • (6t + 2 -a) + 2 • (6t + 2 -b) + b = 36t + 12 -4a -b ≤ 28t + 6,
which is a contradiction. Now assume that K is a (k, n)-arc with k = 28t + 7. By the above argument, all classes of points ahve multiplicity 4t + 1 and every point has multiplicity at most t + 1. Moreover, the four points in each neighbour class of points must have multiplicities t + 1,t,t,t since otherwise we get a contradiction by a counting argument. For instance, if the multiplicities of the points in a neighbour class are t + 1,t + 1,t + 1,t -2 then there is a line segment of multiplicity 2t + 2 in each direction and

k ≤ 4t + 1 + 6 • (6t + 2 -2t -2) = 28t + 1.
The other possibility of points of multiplicities t + 1,t + 1,t,t -1 is ruled out by the same argument.

Proof.

  Clearly, (a) and (b) are settled by Theorem 5.

n = 16 :

 16 The upper bound follows by Corollary 1; AHG(R 4 R ) is a (64, 16)-arc. n = 17: The upper bound follows by Corollary 3; for the constructions, add one point to a (64, 16)-arc. n = 18: The upper bound follows by Corollary 3; for the constructions, add two points to a (64, 16)-arc. n = 19: The upper bound follows by Corollary 3; a (76, 19)-arc is obtained by removing an affine (16, 3)-blocking set (four skew lines) from AHG(R 3 R ). n = 20: The upper bound follows by Corollary 3; an (80, 20)-arc is obtained by removing an affine (12, 2)-blocking set (three skew lines) from AHG(R 3 R ). n = 21: The upper bound follows by Corollary 3; an (84, 21)-arc is obtained by removing an affine (8, 1)-blocking set (two skew lines) from AHG(R 3 R ). n = 22: The upper bound follows by Corollary 3; a (92, 22)-arc is obtained by removing an Hjelmslev plane from PHG(R 4 R ). n = 23: The upper bound follows by Corollary 3; a (93, 23)-arc is obtaind by adding a point to the (92, 22)-arc, constructed above. n = 24: The upper bound follows by Corollary 3; in the case R = F[X]/(X 2 ) we get a (99, 24)-arc by removing from PHG(R 4 R ) a copy of PG(3, 2) and a line disjoint from the subgeometry; in the case R = Z 4 we can get a (96, 24)-arc by removing four disjoint lines. n = 25: The upper bound follows from Theorem 3; in the case R = F[X]/(X 2 ) we can get a (105, 25)-arc by removing a copy of PG(3, 2) from PHG(R 4 R ); in the case R = Z 4 we can get only a (102, 25)-arc by removing three disjoint lines. n = 26: The upper bound follows from Theorem 3; a (108, 26)-arc is obtained by removing two disjoint lines from PHG(R 4 R ). n = 27: The upper bound follows from Theorem 3; a (114, 27)-arc is obtained by removing a line from PHG(R 4 R ). n = 28: A (120, 28)-arc is the Hjelmslev geometry PHG(R 4 R ).

Table 1 .

 1 Values of m n (R 4 R ) for Hjelmslev spaces over rings with |R| = 4n/R Z 4 F 2 [X]/(X 2 ) n/R

	Z 4	F 2 [X]/(X 2 )

Acknowledgements. The research of the first and the third author is supported by the Strategic Development Fund of the New Bulgarian University. The research of the first author was supported by the Open Project of Zhejiang Provincial Key Laboratory of Information Network Technology and by the National Natural Science Foundation of China under Grant No. 60872063.

Now we have seven points of multplicity t + 1 and obviously no three of them are collinear. Hence they form a (7, 2)-arc which is known to exist when R = Z 4 and not to exist in case of R = F 2 [X]/(X 2 ).

(d) Obviously, we can construct (28t + 10, 6t + 3)-arcs as a sum of t copies of the plane and a (10, 3)-arc. Assume there is a (k, n)-arc K with k = 28t + 11 and n = 6t + 3. Then there exists a class [x] with K([x]) ≥ 4t + 2 and a point in it, x say, with K(x) ≥ 4t + 1. By Theorem 4, k ≤ 28t + 12. If k = 28t + 11 or 28t + 12 then K([x]) ≤ 4t + 3 for every class [x] and K(y) ≤ t + 1 for every point y in such a class.

Classes of multiplicity 4t + 3 are easily ruled out since they must consist of three (t + 1)-points and one t-point, and must have segments of multiplicity 2t + 2 in every direction. By a similar argument, a class of multiplicity 4t + 2 consists of two (t + 1)points and two t-points. Now the (t + 1)-points form a (κ, 3)-arc with κ = 11 or 12, which is impossible.

The proofs of (e) and ( f ) use similar arguments.

Arcs in three-dimensional Hjelmslev spaces

In this section, we present a table with exact values and bounds on m n (R 4 R ), n ≤ 28, where R is any of the two chain rings with four elements. Let us note that for n = 3, 4, 5 the exact values are computed in [START_REF] Kiermaier | Arcs und Codes über endlichen Ringen[END_REF]. In the remaining cases the bounds are obtained in the way described below. n = 6: The upper bound is obtained by Corollary 2; an (18, 6)-arc is obtained as the union of three (6, 2)-arcs in three planes (non-neighbors) with a common empty line. n = 7: The upper bound follows by Corollary 1; if R = Z 4 a (21, 7) is obtained as the sum of three plane (7, 2)-arcs; if R = F 2 [X]/(X 2 ) a (19, 7) is obtained as the sum of two plane (6, 2)-arcs and a plane (7, 3)-arc.

n = 8: The upper bound is obtained by Corollary 1; for R = Z 4 a (23, 8)-arc is obtained as the union of two (8, 3)-arcs and a (7, 2)-arc in three planes with a common line; for R = F 2 [X]/(X 2 ) replace the (7, 2)-arc by a (6, 2)-arc. n = 9: The upper bound follows by Corollary 1; a (27, 9)-arc is obtained as the union of three (9, 3)-arcs in three planes (non-neighbors) with a common empty line. n = 10: The upper bound follows by Corollary 1; for R = Z 4 a (30, 10)-arc is obtained as the union of three (10, 3)-arcs; for R = F 2 [X]/(X 2 ) a (38, 10)-arc is obtained as the union of one (10, 4)-arc and two (9, 3)-arcs. n = 11: The upper bound follows by Corollary 1; for R = Z 4 a (32, 11)-arc is obtained as the union of two (11, 4)-arcs and a (10, 3)-arc in three planes with a common line; for R = F 2 [X]/(X 2 ) replace the (10, 3)-arc by a (9, 3)-arc. n = 12: The upper bound follows by Corollary 1; a (36, 12)-arc is obtained as the union of three (12, 4)-arcs in three planes (non-neighbors) with a common empty line. n = 13: The upper bound follows by Corollary 1; for the construction delete an affine (16, 3)-blocking set from AHG(R 4 R ). n = 14: The upper bound follows by Corollary 1; for the construction delete an affine (12, 2)-blocking set from AHG(R 4 R ). n = 15: The upper bound follows by Corollary 1; for the construction delete an affine (8, 1)-blocking set from AHG(R 4 R ).


Bertrand Gilles

Laurent Nana

Valérie-Anne Nicolas

Implementing an Automatic Functional Test Pattern Generation for Mixed-Signal Boards in a Maintenance Context

Keywords: Maintenance testing, modeling, mixedsignal boards, automatic test pattern generation

In this paper, we present in the context of mixedsignal board maintenance testing, language and implementation aspects of an automatic functional test pattern generation approach. The goal is to help maintenance test engineers. Our modeling method for mixed-signal boards and their components is presented and the languages proposed for modeling are described. The implementation of our test data generation process based on constraint logic programming is discussed. The application to a simple board is shown and the tool Copernicia developed in the context of our work is presented.

Introduction

Various test methods and techniques have been developed for circuit test [START_REF] Michael | Essentials of Electronic Testing for Digital, Memory and Mixed-signal VLSI[END_REF] at the different stages of product life-cycle, mainly at design and production levels. Nevertheless, few interest has been thrown into maintenance testing which has its own specificity. In particular, functional testing is needed in order to check behaviors. In the case of mixed-signal boards, maintenance testing currently relies mainly on the expertise of test engineers. Methods and tools for automating the full or part of the testing process are needed to help them in their task.

Our work aims at providing such solutions. In this paper, we focus on language and implementation aspects in the approach we proposed in [START_REF] Gilles | Towards a New Modelling of Mixed Signal Boards For Maintenance Testing[END_REF] for the modeling of mixed-signal boards and the automatic generation of test data for such boards.

We first present our modeling approach and focus on the modeling languages and test data generation. The application to a simple board is then shown and the tool we are currently developing is described. The paper ends by conclusions and future works.

Modeling and testing approach

The automation of the testing data generation process of mixed-signal boards requires a structured modeling approach. We consider two hierarchical levels of modeling: Board level and block level [START_REF] Gilles | Towards a New Modelling of Mixed Signal Boards For Maintenance Testing[END_REF]. Both are used for our automatic test pattern generation (ATPG). The board is broken down into a set of interconnected blocks. Each block has an associated functional model which describes its behavior and a test model which specifies how the block can be efficiently tested.

The test of the board is achieved by testing each block individually using its associated test model. Test patterns for a block (BTP) are then generated by carrying out the transition coverage of its test model. Since the block under test is often embedded within the board, without any test access mechanism, the functional models of adjacent blocks are used for forward propagation to primary outputs (PO) and backward propagation to primary inputs (PI). During these propagations, BTP have to satisfy the constraints associated to the adjacent functional models in order to compute the final test patterns. Finally, a board test data set is the union of test patterns for all the blocks of the board.

Modeling languages

We propose three modeling languages: a board description language associated to the board level, a block description language and a transition language associated to the block level. We describe these languages in the next subsections.

Board description language

As mentioned in section 2, the board description language describes the board as a set of interconnected blocks, as depicted in Figure 1. In addition to the building blocks of the board, some external blocks are needed to model connections between the PI/PO of the board and an automatic test equipment (ATE): external sources which supply input signals (blocks S1 and S2) and output measurement points (blocks m1 and m2).

Blocks are analog, digital or mixed-signal, and may have several inputs and outputs. Oriented links denote data exchanges (signals) between components.

Figure 1. A board description

Block description language

The block description language is based on communicating finite state machines (CFSM). CFSM are often used for modeling systems which imply communicating processes and is well suited for our needs. Another advantage of using it is that it is well known by test engineers. Two CFSM interact when one CFSM produces an output that is placed in the input queue of the other.

Each block is described by one or more CFSM, depending on the complexity of the functionality of the As a consequence, the set of blocks of a board is represented by a set of CFSM.

Since communications are involved, CFSM transitions are decorated with labels. A label expresses synchronization conditions with blocking receptions of signals and/or non blocking sending of signals between two or more CFSM. Constraints on signals involved into communications are also expressed. Each label is written with the transition language presented in the next subsection.

Transition language

In this section, we present the grammar of the transition language. The main improvement from previous work is that we can express more complex analog signals and constraints. The grammar is described with production rules. A rule has the following form: A ::= w, where A and w represent respectively the left and the right hand sides of the rule. The rules are given below: The possible values of the attribute appearing into the "signalParam" definition depend on the type of signal. For example, the attributes max, freq and phase are defined for sine signals and correspond respectively to the magnitude, the frequency and the phase of such signals.

Implementation of the test data generation

We have chosen constraint logic programming (CLP) for the test data generation process. CLP is very relevant for test data generation since it makes it possible to represent test data in a symbolic way, using ranges of values. These ranges of values deal efficiently with analog and digital data representations in a uniform way. Ranges of test data are computed for reaching the test requirements. Final test data are instantiated by the user.

The set of CFSM describing the board behavior is translated into Prolog predicates. The labels of transitions expressed in the transition language identify a set of constraints which are associated to the behavior of a block. These constraints are translated and solved using CLP, leading to the test patterns.

On the implementation point of view, communication between CFSM Prolog models is achieved using dynamic predicates. Their use makes it possible to implement the synchronization conditions mentioned in section 3.2 (blocking receptions).

In the next section, we present an application of our modeling and testing approach to a simple case study.

Application to an example of board test

For the sake of simplicity, we consider a board which is a simple first order high-pass analog filter. Figure 2 shows this board modeled by the board description language where block S, MP and F represent respectively an analog source, an analog measurement point and the considered filter.

S F MP

Figure 2. The board level description

Input/output modeling

Figure 3 shows the functional models with internal label representation of the source and the measurement point respectively expressed with the block description language. For each CFSM, the transitions are numbered from 1 to n, n being the total number of transitions. For a given automaton, each transition is internally represented as follows: CFsmName::T#k {L} where k is the k th transition of CFSM CFsmName and L is the label expressed with the transition language. By convention, CFSM name is the same as block name.

Filter modeling

As we explained in section 2, the functional model of the filter describes its behavior. It is depicted in Figure 4. Thus, F::T#1{L1} where L1 is S ? x -> x.type == sine expresses that when a sine signal is received from CFSM S, the corresponding transition is crossed. Then the output signal is sent to CFSM MP thanks to F::T#2{L2} where L2 is

[V == (x.max / sqrt(1 + square (Fc)/square(F0))) && F0 == x.freq && Phi0 == (x.phase + arctan(Fc/F0)) : MP ! sig(sine,V,F0,Phi0)]
Fc is the filter cutoff frequency. The magnitude, the frequency and the phase of the output sine signal are represented respectively by V, F0, and Phi0. The first part of this label (delimited by the ':' symbol) expresses the constraints applied to the output sine signal according to the input signal characteristics. In the second part, the functor sig represents the output signal of the filter sent to CFSM MP. As we can see, this functional model describes the transfer function of the filter (attenuation and phase shift).

Input

Output

F::T#1 {L1} F::T#2 {L2}

Figure 4. The filter functional model

In the other hand, the test model of the filter describes the way it can be efficiently tested (Figure 5). The model integrates test engineers skills. Thus, F::T#1 {L1} and F::T#2 {L2} where L1 is S ? x -> x.type == sine && x.max > 2.0 && x.max < 10.0 && x.freq == (10.0*Fc) && x.phase == 0.0 and L2 is

[F0 == x.freq && V <= (x.max + d1) && V >= (x.max -d1) && Phi0 <= d2 && Phi0 >= -d2 : MP ! sig(sine,V,F0,Phi0)]
describe the filter response in the bandwidth.

Board test data generation

The following code shows the translation of the transition F::T#1{L1} of the filter test model into Prolog predicates, according to the approach described in section 4:

The predicate reception_constraints implements the constraints on the received signal described in section 5.2. The blocking reception of this signal is implemented by the dynamic predicate channel as mentioned in section 4.

Covering the transitions of the test model of the filter as mentioned in section 2, we obtain two test data TD 1 for the bandwidth test and TD 2 for the cutoff frequency test: where In represent the primary input test signal applied and Out the primary output signal when the filter cutoff frequency is equal to 1000 Hz and tolerances d1, d2 in F::T#2{L2} are equal to 0.1.

The tool Copernicia

We are currently developing a tool named Copernicia that implements our modeling and testing approach. The graphical user interface (GUI) is written in C++ with the ILOG Views graphic library. We use CLP and the ECL i PS e solver [START_REF] Cheadle | Eclipse: An introduction[END_REF] for the generation of the test data. Figure 6 shows the modeling of the board considered in section 5. The tool provides two user modes: board modeling and board test data generation. In the main panel, with the modeling mode, the upper left white background window represents the board level modeling and grey background windows represent CFSM at the block level modeling. The lower white background window (output window) gives information about the internal modeling of the board. With the board test data generation mode, the user can in a friendly way click a block to generate its test data that are written in the output window. He can also click a state or a transition on block models in order to generate test data for a particular behavior.

Conclusion and future works

In this paper, we have presented our hierarchical modeling and testing approach for automatic generation of test data for mixed-signal boards. This approach is based on functional testing due to the particularity of maintenance testing.

We have proposed a graphical language for the high level description of the board. The graphical nature of the language aims at making it more convivial for users. We have also proposed a modeling approach based on CFSM and a transition language for the description of board behavior as well as the description of test strategy. CFSM are well known by testing engineers, so the use of the proposed solution should be easy for them. An implementation of our approach using CLP has been presented and its application to a simple board made of a single filter has been shown. The tool Copernicia, which implements the solutions proposed, has been presented. Future works include the application of the approach to more complex boards in order to validate or exhibit its limits.

  Transition : := Recv Constraints Send → | Recv → Send | Constraints Send | Recv | Send Recv : := Recv;RecvFromAutomaton | RecvFromAutomaton RecvFromAutomaton : := Identifier ? SignalIds SignalIds : := SignalIds , Identifier | Identifier Constraints : := Constraints && Constraint | Constraint Constraint : := Arithmetic relOp Arithmetic RelOp ::= <=|>=|<|>|==|!= Arithmetic : := Arithmetic + Term | Arithmetic -Term | Term Term ::= Term * Factor | Term / Factor | Factor Factor ::= Real | (Arithmetic) | Identifier(ParamList) | Identifier | SignalParam ParamList ::= ParamList, Arithmetic | Arithmetic SignalParam : := Identifier . Attribute Send : := [ SendToAutomata ] | ε SendToAutomata ::= SendToAutomata ;SendToAutomaton | SendToAutomaton SendToAutomaton : := Constraints : Identifier ! SignalFunctors | Identifier ! SignalFunctors SignalFunctors : := SignalFunctors , SignalFunctor | SignalFunctor SignalFunctor : := sig ( Identifier, Arg, Arg, Arg ) Arg : := Identifier | Real

S

  S::T#1 {F ! x} MP MP::T#1 {F ? x}

Figure 3 .

 3 Figure 3. The source and measurement point functional models S::T#1{ F ! x } stands for a sending of signal x to CFSM F and MP::T#1{F ? x } stands for the reception of signal x from CFSM F. The number of the transition is 1 in both cases since each CFSM only has one transition.

Figure 5 .

 5 Figure 5. The filter test model

TD 1 =

 1 ( In = sig(sine, 2.0 .. 10.0, 10000.0 .. 10000.0, -0.0 .. 0.0), Out = sig(sine, 1.9 .. 10.1, 10000.0 .. 10000.0, -0.1 .. 0.1)) TD 2 = ( In = sig(sine, 2.0 .. 10.0, 1000.0 .. 1000.0, -0.0 .. 0.0) Out = sig(sine, 1.214 .. 7.271, 1000.0 .. 1000.0, 0.585 .. 0.985))

Figure 6 .

 6 Figure 6. The tool Copernicia


attractor to the system for arbitrary initial conditions [START_REF] Fermann | Selfsimilar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Boscolo | Self-similar parabolic optical solitary waves[END_REF][START_REF] Kruglov | Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters[END_REF]. Experimental demonstrations relying on amplification from either rare-earth doping [START_REF] Fermann | Selfsimilar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF] or Raman scattering [START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF] have been achieved. The unique properties of parabolic pulses have stimulated numerous applications. Experimental interest has been strongly driven by the possibility of fully cancelling the linear chirp and thus generating ultrashort high-power pulses with very low substructures [START_REF] Fermann | Selfsimilar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF][START_REF] Parmigiani | Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating[END_REF]. Moreover, recent fiber lasers which use self-similar pulse shaping in the normal dispersion regime have been demonstrated to achieve high-energy pulses [START_REF] Ilday | Self-similar evolution of parabolic pulses in a laser[END_REF][START_REF] Bale | Strong spectral filtering for a similariton mode-locked fiber laser[END_REF][START_REF] Aguergaray | Experimental realizatin of a mode-locked parabolic Raman fiber oscillator[END_REF]. Parabolic pulses have also enabled new techniques or noticeable improvements of existing techniques of ultrafast all-optical signal processing for high-bit rate telecommunications based on the quasi-instantaneous Kerr response of highly nonlinear optical fibers [START_REF] Finot | Regenerative 40-Gb/s wavelength converter based on similariton generation[END_REF][START_REF] Boscolo | All-optical signal regeneration by temporal slicing of nonlinearly flattened optical waveform[END_REF][START_REF] Boscolo | Time domain all-optical signal processing at a RZ optical receiver[END_REF][START_REF] Parmigiani | Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating[END_REF][START_REF] Hirooka | All-optical 40-GHz Time-Domain Fourier Transformation using XPM with a dark parabolic pulse[END_REF].

Various techniques for the precise synthesis, control and manipulation of the temporal shape of optical pulses have become increasingly important for many scientific applications, including amongst others, ultrahigh-speed optical telecommunications and nonlinear optics.

For instance, various transform-limited pulse shapes were generated by temporal coherent synthesis using a multi-arm interferometer [START_REF] Park | Transform-limited picosecond pulse shaping based on temporal coherence synthesization[END_REF]. Further, the generation of pulses with flattop [START_REF] Petropoulos | Rectangular pulse generation based on pulse re-shaping using a superstructured fiber Bragg grating[END_REF], parabolic [START_REF] Parmigiani | Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating[END_REF][START_REF] Parmigiani | Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating[END_REF] and sawtooth (asymmetric triangular) [START_REF] Parmigiani | An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper[END_REF] shapes has been demonstrated using superstructured fiber Bragg grating technology. Another alternative to achieve the desired temporal optical pulse shaping is to use nonlinear effects in optical fibers. In addition to fiber amplifiers and lasers, parabolic pulses can be generated in passive fibers provided a suitable longitudinal variation of the dispersion is introduced [START_REF] Hirooka | Parabolic pulse generation by use of a dispersiondecreasing fiber with normal group-velocity dispersion[END_REF][START_REF] Finot | Parabolic 13 pulse generation with active or passive dispersion decreasing optical fibers[END_REF]. Recently, a simple approach to the generation of parabolic pulses that uses progressive nonlinear pulse re-shaping and subsequent pulse stabilization in two carefully chosen normally dispersive (ND) fiber segments has been demonstrated [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse re-shaping in a normally dispersive two segment fiber device[END_REF]. Further, the combination of pulse prechirping and nonlinear propagation in a section of ND fiber has been introduced [START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF] and demonstrated [START_REF] Wang | Generation of triangular-shaped optical pulses in normally dispersive fibre[END_REF] as a method for passive nonlinear pulse shaping, which provides a simple way of generating various temporal waveforms including triangular profiles. The simple intensity profile of triangular pulses is highly desired for a range of photonic applications.

Sawtooth pulses have been shown to enhance the performance of wavelength converters based on self-phase modulation (SPM) in fiber and offset filtering [START_REF] Parmigiani | An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper[END_REF]. Time-domain adddrop multiplexing based on cross-phase-modulation (XPM)-induced frequency shifting using saw tooth control pulses was examined [START_REF] Li | OTDM add-drop multiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber[END_REF]. Novel techniques using XPM with triangular pump pulses in a nonlinear Kerr medium were introduced to achieve the doubling of optical pulses in both the frequency and time domains [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF] as well as to realize time-to-frequency mapping of multiplexed signals in high-speed fiber communication systems [START_REF] Bhamber | All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses[END_REF].

One of the most well known examples of pulse re-shaping in the spectral domain is in supercontinuum (SC) generation [START_REF] Alfano | The Supercontinuum Laser Source[END_REF][START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. Typically the broadest spectra are generated when pumping occurs in the anomalous dispersion regime of the fiber, where the broadening mechanism is strongly influenced by soliton dynamics [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. However, solitonic effects and modulation instability lead to decreased temporal coherence and spectral flatness of the continuum generated in the anomalous dispersion regime, especially when picosecond pump pulses are used [START_REF] Gu | Experimental studies of the coherence of microstructure-fiber supercontinuum[END_REF][START_REF] Corwin | Fundamental noise limitations to supercontinuum generation in mi-crostructure fiber[END_REF]. Many applications require the conservation of a short pulse in the time domain while generating a flat and coherent SC spectrum. Previous attempts have been made to overcome this drawback by using dispersion-flattened photonic crystal fibers (PCFs) continuously tapered from anomalous to normal dispersion [START_REF] Mori | Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile[END_REF][START_REF] Genty | Fiber supercontinuum sources (Invited)[END_REF]. However, due to limited availability of such fiber tapers, it is interesting to consider fibers with longitudinally constant dispersion profiles, where a sufficiently broad, stable and flat-top spectrum can still be achieved [START_REF] Chow | Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre[END_REF][START_REF] Heidt | Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers[END_REF][START_REF] Hooper | Coherent supercontinuum generation in photonic crystal fiber with all-normal group velocity dispersion[END_REF]. Such an approach combined with PCFs has recently stimulated experimental attention with several results exhibiting the high coherence and flatness (when plotted on a logarithmic scale) of the resulting supercontinuum.

In this work we qualitatively and numerically show that initially parabolic pulses propagating in a ND fiber with fourth-order dispersion (FOD) can result in temporal triangular intensity profiles and broad, compact, flat spectral profiles. The performance allowed by parabolic pulses is shown to be substantially better compared to more conventional input pulse shapes such as Gaussian or hyperbolic secant pulses. The progressive re-shaping of a parabolic pulse driven by the FOD in a ND nonlinear fiber not only provides an alternative passive method for generating triangular-shaped pulses with respect to that introduced in [START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF], but also allows for remarkably flat spectral intensities at a particular point in the evolution when plotted on a linear scale.

Qualitative Analysis

Our efforts are motivated by the progress of all-normal dispersion PCFs with flattened dispersion characteristics. Indeed, recent theoretical and experimental works have reported the design and use of fibers with all-normal convex dispersion profiles [START_REF] Heidt | Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers[END_REF][START_REF] Hooper | Coherent supercontinuum generation in photonic crystal fiber with all-normal group velocity dispersion[END_REF][START_REF] Tse | Supercontinuum generation at 1.06 µm in holey fibers with dispersion flattened profiles[END_REF][START_REF] Hartung | Design of all-normal dispersion microstructured optical fibers for pulse-preserving supercontinuum generation[END_REF]. Specifically, we will base our discussion on the fiber properties demonstrated in [START_REF] Tse | Supercontinuum generation at 1.06 µm in holey fibers with dispersion flattened profiles[END_REF] with a dispersion profile as shown in Fig. 1 and pumping occurring at the extremum of dispersion.

The flattened dispersion characteristics around the pump wavelength are critical in that the effect of third-order dispersion, which can degrade the self-similar parabolic pulse propagation [START_REF] Bale | Impact of third-order fibre dispersion on the evolution of parabolic optical pulses[END_REF], is significantly reduced below that of FOD. Thus we model pulse propagation by the normalized generalized nonlinear Schrödinger equation [START_REF]Nonlinear Fiber Optics[END_REF]

iu ξ - 1 2 u τ τ + N 2 |u| 2 u = - δ 24 u τ τ τ τ , (1)
where the dimensionless quantities u(ξ, τ ) = U/ √ P 0 , ξ = z/L D and τ = t/T are introduced.

Here, U(z, t) is the electric field envelope in the co-moving system of coordinates, T and P 0 are the characteristic temporal pulse width and the peak power of the initial pulse, respectively, and L D = T 2 /β 2 is the dispersion length, with β 2 being the SOD coefficient. The two normalized parameters in Eq. ( 1) are the energy parameter N 2 = L D /L N L , where L N L = 1/(γP 0 ) is the nonlinear length, and the effective FOD strength parameter δ = β 4 /(β 2 T 2 ).

The parameters β 4 and γ are the fiber FOD and Kerr nonlinearity coefficients, respectively.

Although microstructured fibers has now enabled FOD coefficients where β 4 < 0, in this work we will restrict our analysis to positive FOD coefficients (δ > 0). Indeed high-power picosecond pulses propagating in a PCF with negative FOD coefficient (δ < 0) experience severe pulse degradation arising from fourth-order scalar modulation instability [START_REF] Harvey | Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber[END_REF], which may ultimately lead to the splitting of the pulse into ultrashort solitonic substructures [START_REF] Hammani | Soliton generation in a microstructured fiber by fourth-order scalar modulation instability[END_REF].

To get a qualitative understanding of the re-shaping process, we do a standard decomposition by assuming u(ξ, τ ) = P (ξ, τ ) × e iφ(ξ,τ ) in Eq.( 1), giving

φ ξ = - 1 2
√ P τ τ √ P -Ω 2 +N 2 P + δ 24 R 1 (2a)
P ξ = P Ω τ - δ 12 R 2 (2b)
where Ω ≡ φ τ is the chirp of the pulse and the FOD contributions are

R 1 = 1 √ P √ P τ τ τ τ -6 √ P τ Ω 2 τ +Ω 4 -3Ω 2 τ -4ΩΩ τ τ (3a) R 2 =2 √ P 2 √ P τ τ Ω τ + √ P τ Ω τ τ -P (2Ω 3 -Ω τ τ ) τ . (3b)
For initial high intensity (N 2 ≫ 1) linear chirped parabolic pulses where

P (0, τ ) = (1 -τ 2 )θ(1 -|τ |)
and Ω(0, τ ) = 2C 0 τ with θ(x) being the Heaviside function and C 0 the (normalized) chirp coefficient, the magnitude (in some norm) of each term in Eqs. (2-3) can be analyzed. We find numerically that only certain terms contribute to the main pulse shaping process. Note that we consider C 0 < 0, thus neglecting the transient stage of spectral compression if C 0 > 0 [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF]. Taking the leading order gives an approximate coupled system for the dynamics

φ ξ = 1 2 Ω 2 1 + δ 12 Ω 2 + N 2 P ( 4a
)
P ξ = Ω 1 + δ 6 Ω 2 P τ + Ω τ 1 + δ 2 Ω 2 P. (4b)
For small propagation distances from ξ = 0 the chirp profile is approximately given from Eq. ( 4a)

Ω(ξ, τ ) ∼ 2C 0 τ -2N 2 τ 1 - 2C 2 0 N 2 (1+δC 2 0 τ 2 ) ξ. (5)
Since δ/N 2 ≪ 1, the chirp maintains a nearly linear profile in time, and initial wave-breaking is avoided since the chirp is still monotonous. This nearly linear chirp is coupled to the power evolution in Eq. (4b) and leads to a re-shaping process due to the FOD terms. Specifically, Eq. ( 4b) is a partial differential equation where the power P has the characteristic velocity [START_REF] Whitham | Linear and Nonlinear Waves[END_REF] and power evolution given by

dτ dξ = -Ω 1 + δ 6 Ω 2 (6a) dP dξ = Ω τ 1 + δ 2 Ω 2 P (6b)
Neglecting the δ/N 2 term in Eq.( 5) gives Ω ∼ Ω 0 τ , where

Ω 0 = 2C 0 -2N 2 ξ 0 (1 -2C 2 0 /N 2 )
(with ξ 0 ≪ 1). Without FOD (δ = 0), parabolic self-similar solutions to Eqs. ( 4) have been found [START_REF] Anderson | Wavebreaking-free pulses in nonlinear-optical fibers[END_REF]. In this limit Eq. ( 6) has solutions ξ(τ ) =log(τ /τ 0 )/Ω 0 and P (ξ, τ 0 ) = P (0, τ 0 )×exp(Ω 0 ξ). For relevant initial conditions, C 2 0 ≪ N 2 giving Ω 0 < 0 so that the power P (ξ, τ 0 ) exponentially decays along each characteristic curve. Including FOD in Eq. ( 6) we find the solutions

ξ(τ,τ 0 ) = - 1 Ω 0 log kτ 1 + δ 6 Ω 2 0 τ 2 (7a) P (ξ,τ 0 ) = P (0, τ 0 )e Ω 0 ξ × R(ξ, τ 0 ), (7b)
where

R(ξ, τ 0 ) = δ 6 Ω 2 0 -k 2 e 2Ω 0 ξ δ 6 Ω 2 0 -k 2 3/2 e -3Ω 0 ξ (8)
and k 2 = 1/τ 2 0 + δΩ 2 0 /6. Figure 2 shows an example of the characteristic curves with and without FOD. With FOD the curves deflect away (towards τ = ±∞) from those without FOD. Note that that this deflection is more extreme in the wings of the pulse for short propagation distances, leading to a stretching of the pulse (see grey line and dots in Fig. 2).

Further, Eq. (7b) shows that the exponential decay of the power along each characteristic curve is multiplied by a functional factor R(ξ, τ 0 ) (Eq. ( 8)) which is shown in Fig. 3 for various τ 0 values. Since R(ξ) < 1 for |τ 0 | > 0 the pulse wings undergo an enhanced power reduction when compared to the evolution without FOD. Thus the overall temporal effect of FOD on linearly chirped parabolic pulse propagation is to stretch and enhance the power reduction in the pulse wings, leading to a triangular profile.

In the spectral domain, once the pulse has been stretched towards a triangular shape, the spectral power evolves towards an M-shape due to the SPM effect (N 2 ≫ 1). Indeed, for a triangular pulse where

P (ξ 0 , τ ) ∼ (1 -|τ /τ T |)θ(τ T -|τ |), the nonlinear phase shift acquired upon propagation φ(τ ) = N 2 P (ξ 0 , τ )(ξ -ξ 0 ) leads to a constant, distinct (opposite sign)
chirp induced onto the leading (down-shfited) and trailing (up-shifted) edges of the pulse.

As a result of this, the outer parts of the pulse spectrum are enhanced, whereas the central parts are suppressed and, thus, the spectrum develops an M-shaped structure. An important point is that in the transition from the initially rounded-top (parabolic) spectral power to an M-shaped spectral power, the spectral density becomes flat. This flattened spectrum could be useful in SC generation applications. In the following section, we emphasis the qualitative findings through numerical simulations.

Numerical Simulations

In this section Eq. (1) will be numerically integrated using the standard split-step Fourier method. Following Ref. [START_REF] Tse | Supercontinuum generation at 1.06 µm in holey fibers with dispersion flattened profiles[END_REF] (see also Fig. 1), the fiber parameters are given by β 2 = 2.5 × 10 -3 ps 2 m -1 , β 4 = 2.0 × 10 -7 ps 4 m -1 and γ = 40 × 10 -3 (W m) -1 . To avoid numerical artifacts due to the unphysical compact nature of the parabolic pulse, we use here slight temporal apodization of the intitial condition based on a fifth-order Gaussian shape with a full-width at half-maximum (FWHM) of 1.6T [START_REF] Parmigiani | Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating[END_REF]. The input pulse parameters we use correspond to typical parabolic pulses output from a ytterbium-doped fiber amplifier [START_REF] Fermann | Selfsimilar propagation and amplification of parabolic pulses in optical fibers[END_REF].

They have energy 12 nJ, FWHM pulse duration 2.6 ps and chirp coefficient -3.35π THz 2 .

Both the fiber and initial pulse parameters gives the dispersion length L D = 1.3520 × 10 3 m, energy parameter N 2 = 2.6474 × 10 5 , and FOD parameter δ = 2.3669 × 10 -5 . Propagation in the fiber is considered up to a maximum length of 4L D ×10 -3 which is on the order of meters in physical units (∼ 1000 nonlinear lengths). Note that in this parameter regime N 2 ≫ 1, so that Ω 0 ∼ -2N 2 ξ. Thus we would expect that increasing or decreasing the initial chirp values will not have a significant effect in the re-shaping process.

During propagation in the fiber, the initial pulse will experience significant re-shaping, evolving towards a triangular profile in the time domain. Different approaches are possible to characterize the pulse shape [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse re-shaping in a normally dispersive two segment fiber device[END_REF][START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF][START_REF] Ruehl | Dynamics of parabolic pulses in an ultrafast fiber laser[END_REF]. Here we choose the parameter of misfit M S between the pulse intensity profile and a parabolic or triangular fit of the same energy and FWHM duration:

M 2 S = dt (P -P S ) 2 dt P 2 , S = p, T (9)
where P p (τ ) and P T (τ ) correspond to the respective intensity profiles P p (τ ) = p p (1 -

τ 2 /τ 2 p )θ(τ p -|τ |) and P T (τ ) = p T (1 -|τ /τ T |)θ(τ T -|τ |)
with p S being the pulse peak power.

To examine the impact of FOD on the evolution of the initial parabolic pulse, we compare the evolution obtained with and without FOD. An example of the longitudinal temporal and spectral pulse evolutions is shown in In the absence of FOD, the pulse evolves self-similarly maintaining its parabolic shape both in the time and frequency domains, while undergoing temporal and spectral broadening. Due to the high N 2 value, these results are consistent with previously found results [START_REF] Anderson | Wavebreaking-free pulses in nonlinear-optical fibers[END_REF]. In the presence of FOD, significant temporal and spectral re-shaping of the parabolic pulse occurs over short propagation distances. In the temporal domain the wings of the pulse are stretched, generating a triangular shaped pulse as predicted in the previous section. This shape is maintained over a non-negligible propagation distance. In the spectral domain, the power goes through a re-shaping leading to a spectrum with rapidly falling edges and a ripple-free top over a ∆ω ≃ 578 span (corresponding to approximately 50 THz) at propagation distance ξ = 0.0011. The flat-top spectrum is a transient state and for further propagation distance the spectral power evolves into an M-shape due to the triangular nature of the temporal intensity profile along with SPM. As to be expected, the spectral bandwidth approaches a limit as the pulse undergoes enhanced temporal broadening and faster peak power reduction [START_REF] Finot | Beneficial impact of wave-breaking or coherent continuum formation in normally dispersive nonlinear fibers[END_REF].

The trends shown in Figs. 4-5 are confirmed in Fig. 6, where the misfit parameter (9) has been calculated over the propagation distance. In the presence of FOD, the misfit parameter to a triangular temporal shape reaches a minimum of M T,opt ≃ 0.057 at the distance ξ opt = 0.0028, while being below 0.07 over the range ξ = 0.0021 to the maximum propagation distance considered. Stabilization of the root-mean-square (RMS) pulse spectral bandwidth after the M-shaping of the spectrum has occurred is also evident from Fig. 6. The RMS spectral bandwidth is calculated as ω RM S = dω ω 2 |ũ| 2 / dω |ũ| 2 1/2 , where ũ is the Fourier transform of the field. The two red dots in Fig. 6 symbolize the points in propagation where (Fig. 6a) the temporal profile is most triangular (shown in inset of Fig. 2(b)) and (Fig. 6b) the spectral profile is most flat with a ripple-free top (shown as the green curve in Fig. 3(b)). Note that typically the best triangular shapes are obtained at larger propagation distances than those where the spectrum is maximally flat. Further, the spectrum generated in presence of FOD saturates earlier and exhibits a lower spectral broadening than the spectrum recorded without FOD, which is a direct consequence of the increased broadening experienced when FOD is taken into account.

A. Impact of FOD and energy parameter

As described in Section 2, the overall effect of FOD depends on a term whose coefficient is δΩ 2 0 /6, where Ω 2 0 ∼ 4(N 2 ) 2 ξ 2 0 (see Eq. ( 7)). Thus we expect that increasing either the FOD strength parameter δ (by increasing the physical FOD β 4 ) or the energy parameter N 2

(by increasing the input pulse peak power) will generate a stronger stretching and enhanced power reduction in the wings of the pulse. Indeed, this will effectively lead to pulse re-shaping for shorter propagation distances as well as being able to obtain an improved triangular shape due to a sharpening of the pulse top. This qualitative description is confirmed by numerical simulations. The influence of the FOD (energy) parameter on the temporal evolution can be seen in Fig. 7a(b) where the longitudinal evolutions of the misfit parameter to a triangular temporal shape are shown. For increasing values of δ or N 2 , ξ opt and M T (ξ opt ) decrease

showing that the re-shaping process happens at shorter propagation distances and that more ideal triangular pulse shapes are obtained. In the spectral domain, larger δ and N 2 leads to flattening and subsequent M-shaping of the spectral power earlier in propagation distance, as to be expected. The spectrum settles to a smaller bandwidth at large distances for increased δ where larger N 2 leads to more spectral broadening and larger bandwidths.

Figure 8 illustrates the temporal and spectral power profiles at two significant propagation distances where there is lowest misfit parameter M T (dots in Fig. 7(b)) and maximally flat spectrum.

B. Impact of initial pulse shape

We have investigated the effect of FOD on the evolution of an initially linearly chirped parabolic pulse in a ND fiber. Since in our parameter regime N 2 ≫ 1, we would expect from Eq. (4a) the the initial pulse shape will have a significant effect on propagation. We consider the evolution of both transform-limited Gaussian pulses u(0, τ ) ∼ exp(-τ 2 /(2τ 2 G ))

and a hyperbolic-secant pulses u(0, τ ) ∼ sech (τ /τ S ) and compare to the evolution of a chirpfree parabolic pulse. For all initial conditions, the same pulse energy and FWHM temporal duration are used. The spectro-temporal presentation of the pulses obtained at ξ = 0.0013 in Fig. 9 highlights the distinctly different pulse evolutions. For the initial parabolic pulse, the initial chirp generated is linear (see Eq. ( 5)). Upon propagation, this chirp deviates from a strictly linear behavior because the pulse wings evolve at higher velocity than the central part of the pulse as a result of the FOD. However, the chirp remains monotonic and the pulse does not undergo wave breaking. For the initial Gaussian and hyperbolic secant pulse shapes the last term on the right hand side in Eq. (4a) induces a non-monotonic chirp which leads to wave-breaking. In this case, triangular pulse formation is not observed for such initial pulses. Figure 10 shows the temporal and spectral profiles at ξ = 0.0013 for the three different initial conditions. The high oscillations in the spectrum for the initial Gaussian and hyperbolic secant are induced by wave-breaking. The specific initial parabolic pulse profile avoids wave-breaking and generates a flattening of the spectral power and triangular pulse formation in the temporal domain.

Conclusion

In conclusion, we have investigated the propagation of initially parabolic pulses in a ND fiber in the presence of FOD. For the physical parameter regime explored a qualitative understanding of the re-shaping was obtained and characterized in terms of the parameter

δΩ 2 0 ∼ δN 4 ξ 2 0 .
The overall effect of the FOD is to stretch and enhance the power reduction in the wings of the initial parabolic pulse, leading to a temporal triangular profile. Further, the spectral power is re-shaped from a rounded top profile into a M-shape. At a certain point in this re-shaping process, the spectral power has rapidly falling edges and a ripple-free top over a broad bandwidth. Numerical simulations confirm our qualitative understanding and show that increasing the strength of the FOD or the energy parameter leads to pulse re-shaping for shorter propagation distances as well as being able to obtain an improved temporal triangular profile due to a sharpening of the pulse top. Finally, an initial parabolic pulse profile was shown to be necessary to avoid wave-breaking for such re-shaping dynamics to occur. This work could potentially have impact on applications that rely on temporal triangular pulses as well as those requiring the conservation of a short pulse in the time domain while generating a flat and coherent SC spectrum.

List of Figure Captions

Fig. 1 Convex dispersion profile for PCF under consideration. Pumping occurs at the extremum of dispersion. This profile is similar to that experimentally realized in Ref. [START_REF] Tse | Supercontinuum generation at 1.06 µm in holey fibers with dispersion flattened profiles[END_REF]. The parameters are the same as in Fig. 4 and ω is normalized frequency.

Fig. 2 5 and δ ∼ 2 × 10 - 5 .

 252105 Fig. 2 Characteristic curves (7a) for |Ω 0 | ∼ 2N 2 ξ 0 = 400 and δΩ 2 0 /6 = 0.5 (solid) and δ = 0 (dashed). These values correspond to reasonable values used in simulations of N 2 ∼ 2 × 10 5 and δ ∼ 2 × 10 -5 . The characteristic curves plotted are for |τ 0 | = 0 (black) , 1 (blue), 2 (red) and 3 (green).

Fig. 3

 3 Fig.3Power reduction factor Eq. (8) for Ω 0 ∼ -2N 2 ξ 0 = -400, δΩ 2 0 /6 = 0.5, and |τ 0 | = 0 (black) , 0.5 (cyan) , 1 (blue), 2 (red) and 3 (green).

Fig. 4

 4 Fig. 4 Temporal power profiles (a) without and (b) with FOD at propagation distances ξ = 0 (red), ξ = 0.0011 (green), ξ = 0.0022 (blue), ξ = 0.0033 (cyan), and ξ = 0.0041 (magenta). The parameters are N 2 = 2.6474 × 10 5 , (a) δ = 0 and (b) δ = 2.3669 × 10 -5 . The inset shown in (b) is the temporal profile (black) at ξ = 0.0028, and its fit to a triangular function (red circles).

Fig. 5

 5 Fig. 5 Spectral power profiles (a) without and (b) with FOD at propagation distances ξ = 0 (red), ξ = 0.0011 (green), ξ = 0.0022 (blue), ξ = 0.0033 (cyan), and ξ = 0.0041 (magenta).

Fig. 6 Fig. 7 5 .Fig. 9 Fig. 1 .Fig. 2 .

 675912 Fig. 6 Longitudinal evolutions of (a) The misfit parameter to a parabolic temporal shape with no FOD (dashed) and a triangular temporal shape with FOD (solid) and (b) the RMS spectral bandwidth without (dashed) and with (solid) FOD. The parameters are δ = 0 or δ = 2.3669 × 10 -5 and N 2 = 2.6474 × 10 5 . Fig. 7 Longitudinal evolutions of the misfit parameter to a triangular temporal shape for varying values of (a) δ and (b) N 2 . S = S * /2 (blue), 2S * /3 (cyan), S * (dashed), 3S * /2 (red), 2S * (black), with (a) S * = δ * = 2.3669 × 10 -5 and N 2 = 2.6474 × 10 5 and (b) S * = N * 2 = 2.6474 × 10 5 and δ = 2.3669 × 10 -5 . Fig. 8 (a) Temporal and (b) spectral intensity profiles for N 2 = N * 2 /2, N * 2 , 2N * 2 , with N * 2 = 2.6474 × 10 5 . For N 2 = 2N * 2 a best fit triangular temporal shape is also shown

Fig. 3 .

 3 Fig. 3. Power reduction factor Eq. (8) for Ω 0 ∼ -2N 2 ξ 0 = -400, δΩ 2 0 /6 = 0.5, and |τ 0 | = 0 (black) , 0.5 (cyan) , 1 (blue), 2 (red) and 3 (green).

Fig. 4 .

 4 Fig. 4. Temporal power profiles (a) without and (b) with FOD at propagation distances ξ = 0 (red), ξ = 0.0011 (green), ξ = 0.0022 (blue), ξ = 0.0033 (cyan), and ξ = 0.0041 (magenta). The parameters are N 2 = 2.6474 × 10 5 , (a) δ = 0 and (b) δ = 2.3669 × 10 -5 . The inset shown in (b) is the temporal profile (black) at ξ = 0.0028, and its fit to a triangular function (red circles).

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Spectral power profiles (a) without and (b) with FOD at propagation distances ξ = 0 (red), ξ = 0.0011 (green), ξ = 0.0022 (blue), ξ = 0.0033 (cyan), and ξ = 0.0041 (magenta).The parameters are the same as in Fig.4and ω is normalized frequency.

Fig. 8 .Fig. 9 .

 89 Fig. 8. (a) Temporal and (b) spectral intensity profiles for N 2 = N * 2 /2, N * 2 , 2N * 2 , with N * 2 = 2.6474 × 10 5 . For N 2 = 2N * 2 a best fit triangular temporal shape is also shown (dashed). The temporal profiles shown are taken at the circled points of Fig. 7(b), and the spectral profiles are shown at points of maximum flatness. The parameter δ = 2.3669 × 10 -5 .

Acknowledgements

The authors would like to acknowledge support from the Engineering and Physical Sciences Research Council (Grant No. EP/FO2956X/1), the British Council (Alliance Grant No. 10.002) and the Region Bourgogne (PARI PHOTCOM), the Agence Nationale de la Recherche (ANR SOFICARS, ANR-07-RIB-013-03) and the Minist ére des Affaires étrangeres et européenne (Alliance grant 22836ZJ).


Chiara Taurino

William H Miller

Martin W Mcbride

John D Mcclure

Raya Khanin

María U Moreno

Jane A Dymott

Dr Christian Delles
email: c.delles@clinmed.gla.ac.uk

Anna F Dominiczak

Gene expression profiling in whole blood of patients with coronary artery disease

Keywords: Coronary artery disease, gene expression, microRNA

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

Introduction

Coronary artery disease (CAD) is caused by multiple genetic and environmental factors and the interaction between them. Microarray analysis is a powerful technique for high-throughput, global transcriptomic profiling of gene expression. In addition to identifying new target genes, this method also allows the clustering of genes according to similar patterns of expression and/or function from which gene expression fingerprints can be created. Gene expression fingerprints have become a useful tool in research and diagnosis, with multiple applications, for example in the classification of different tumour types [START_REF] Covell | Molecular classification of cancer: unsupervised self-organizing map analysis of gene expression microarray data[END_REF], in defining the molecular pathogenesis of segmental glomerulosclerosis [START_REF] Schwab | Microarray analysis of focal segmental glomerulosclerosis[END_REF] and detecting exposure to toxic substances [START_REF] Marchant | Toxicogenomics and toxic torts[END_REF].

Microarray studies of human disease are often limited by challenges in obtaining human tissues and by the lack of models that effectively capture clinically relevant disease features. Peripheral blood has recently become an attractive prime tissue for biomarker detection because of its critical role in immune response, metabolism, communication with cells, and the extracellular matrix in almost all tissues and organs in the human body, as well as for the simplicity of sample collection [START_REF] Borovecki | Genome-wide expression profiling of human blood reveals biomarkers for Huntington's disease[END_REF][START_REF] Osman | Novel blood biomarkers of human urinary bladder cancer[END_REF][START_REF] Martin | High-sensitivity array analysis of gene expression for the early detection of disseminated breast tumor cells in peripheral blood[END_REF]. The dynamic and interactive properties of blood give rise to the possibility that subtle changes occurring within the body, such as changes in association with a disease process or in response to an injury, may leave "footprints" in blood. Therefore, transcriptional profiling from whole blood cells might provide an alternative to tissue biopsy in the search for biomarkers genes of cardiovascular disease [START_REF] Liew | The peripheral blood transcriptome dynamically reflects system wide biology: a potential diagnostic tool[END_REF].

MicroRNAs (miRNAs) are small, non-coding RNAs that bind mRNAs at their 3' untranslated regions, stimulating mRNA degradation or inhibiting protein translation [START_REF] Bartel | MicroRNAs: genomics, biogenesis, mechanism, and function[END_REF]. Many miRNAs are up regulated in response to cellular stress [START_REF] Marsit | MicroRNA responses to cellular stress[END_REF] and can modify essential cellular functions of proliferation, differentiation, and programme death [START_REF] Brennecke | bantam encodes a developmentally regulated microRNA that controls cell proliferation and regulates the proapoptotic gene hid in Drosophila[END_REF][START_REF] Chen | MicroRNAs modulate hematopoietic lineage differentiation[END_REF][START_REF] He | A microRNA polycistron as a potential human oncogene[END_REF]. Recent evidence indicates that miRNAs are also implicated in cardiovascular disease [START_REF] Van Rooij | A signature pattern of stressresponsive microRNAs that can evoke cardiac hypertrophy and heart failure[END_REF][START_REF] Sayed | MicroRNAs play an essential role in the development of cardiac hypertrophy[END_REF][START_REF] Thum | MicroRNAs in the human heart: a clue to fetal gene reprogramming in heart failure[END_REF] and have the capacity to create cardiac pathology [START_REF] Zhao | Dysregulation of cardiogenesis, cardiac conduction, and cell cycle in mice lacking miRNA-1-2[END_REF][START_REF] Van Rooij | Control of stress-dependent cardiac growth and gene expression by a microRNA[END_REF][START_REF] Carè | MicroRNA-133 controls cardiac hypertrophy[END_REF].

Given that miRNAs respond to acute changes in cell stress, we hypothesized that combining information from whole blood miRNAs profiles and mRNA signatures could improve the knowledge about pathogenesis of cardiovascular disease. We examined this notion through comprehensive analyses of miRNA and mRNA expression levels in whole blood from patients affected by severe coronary artery disease, also attending a cardiac rehabilitation programme, in comparison with normal subjects.

Methods

Patients

Patients undergoing elective coronary artery bypass graft surgery (N=163) at the Western Infirmary, Glasgow, United Kingdom, were enrolled between October 2006 and August 2008. These patients attended our clinical investigation unit one day before surgery. Control subjects (N=63) were recruited by placing an advertisement in the University of Glasgow Newsletter and in local sport centres. While ethical and funding limitations prevented angiographic investigation of these individuals to absolutely discount the presence of coronary artery disease, these subjects presented as healthy at recruitment and were free of any evidence of coronary artery disease according to their patient history. Related studies [START_REF] Zimmerli | Urinary proteomic biomarkers in coronary artery disease[END_REF][START_REF] Delles | Vascular stiffness is related to superoxide generation in the vessel wall[END_REF][START_REF] Al-Benna | Low-density lipoprotein cholesterol determines oxidative stress and endothelial dysfunction in saphenous veins from patients with coronary artery disease[END_REF] from our group have shown that our healthy control subjects present with normal ECG and no history of arterial hypertension, angina, CAD, or peripheral artery disease. Furthermore we have shown that control subjects have better aortic compliance and lower levels of vascular reactive oxygen species.

Four to six weeks after the CABG procedure 37 of 163 patients were also included in a cardiac rehabilitation programme, consisting of 60 minutes exercise (15 minutes warming up and down and 30 minutes cardiovascular training), twice a week for ten weeks. The initial 4-6 week period following surgery was to allow patients to recover from the procedure to a degree where they were well enough to commence exercise, and also would allow some of the more acute responses to surgery per se (e.g. rises in inflammatory markers) to subside. Although some markers remained elevated (e.g. CRP) this difference was not found to be significant with regard to control subjects. The exercise regimen was adapted to each individual's exercise capacity, assessed by validated step and walking tests [START_REF] Zimmerli | Urinary proteomic biomarkers in coronary artery disease[END_REF][START_REF] Anastasiadis | Diabetes mellitus and coronary revascularization procedures[END_REF][START_REF] Demers | RESOLVD Pilot Study Investigators. Reliability, validity, and responsiveness of the six-minute walk test in patients with heart failure[END_REF] at inclusion and after completion of the rehabilitation programme. For microarray analysis, we selected 12 subjects with severe CAD and 12 control subjects; and 10 patients attending the exercise programme, examined before and after its completion. Details of these patients are given in supplementary tables 1 and 2 in the online supplement.

The study complies with the Declaration of Helsinki and was approved by the local ethics committee. All participants gave written informed consent and used a standardised questionnaire to self-report the health status and medication taken.

Blood collection and RNA isolation

Venous blood samples were drawn via antecubital venipuncture from each subject in a sitting position after three hours fasting and routine biochemistry was assessed in plasma. One aliquot of whole blood (2.5 mL) for each subject was collected directly into PAXgene blood RNA tubes (PreAnalytiX, Hilden, Germany) and stored at -20°C within 24 hours, before RNA extraction. After collection of all samples, total RNA was isolated using the PAXgene blood RNA kit (PreAnalytiX) according to the manufacturer's protocol, and subjected to on-column DNase I treatment with (RNase-free DNase; Qiagen). RNA quantity and quality were determined by a NanoDrop ND-1000 Spectrophotometer (NanoDrop Technologies, Wilmington, DE) and by Agilent 2100 Bioanalyzer (Agilent Technologies, Palo Alto, CA). A total of 400 ng RNA (with RNA Integrity Number >7.5) was used to generate cDNA (Illumina TotalPrep RNA Amplification Kit; Ambion, Austin, TX). Reverse transcription with T7 oligo (dT) primers was used to produce first strand cDNA. The cDNA then underwent second strand synthesis and RNA degradation by DNA Polymerase and RNase H, followed by clean up. In vitro transcription technology, along with biotin UTP, was employed to generate multiple copies of biotinylated cRNA. The labelled cRNA was purified using a filter cartridge and quantified by NanoDrop. The integrity of cRNA was evaluated by Agilent 2100 Bioanalyzer.

Small RNA species (including miRNA) were also extracted by modifying the PAXgene procedure by harvesting the flow-thorough of the PAXgene RNA column, and isolation using the miRNeasy Mini Kit according to the manufacturers' protocols (QIAGEN, Düsseldorf, Germany). Successful isolation of small RNAs was confirmed using Agilent 2100 Bioanalyzer analysis. All mRNA and miRNA expression studies were performed on samples from individual patients (not pooled).

Microarray analysis

Microarray analysis of gene expression was performed on the Illumina Beadstation platform, using Illumina Sentrix humanref-6 beadchips. Labelled cRNA (1.5 ug) was used for hybridization to the array according to the manufacturer's protocol. A maximum of 10 ul cRNA was mixed with 20 uL GEX-HYB hybridization solution. The preheated 30 uL assay sample was dispensed onto each array and incubated for 18 hours at 58°C. Following hybridization, samples were washed and scanned with a BeadArray Reader (Illumina, San Diego, CA). All data passed quality control analysis as assessed by the Illumina on-board software (BeadStudio 3.4; Illumina, San Diego, CA) and Principle Component Analysis (Partek Genomics Suite; Partek, St Louis, MO). Quantile normalisation of gene expression was performed (BeadStudio 3.4), and differential expression determined by Rank Products analysis [START_REF] Breitling | Rank products: a simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments[END_REF], where a 5% false discovery rate (FDR) cut-off value was used. Rank Products is a simulation method based on the average rank each gene has for a pairwise comparison differences compared to what would be expected under the null hypothesis of no differential expression. It has been specifically developed for the analysis of microarray experiments and was shown to be more powerful than Student's ttests [START_REF] Breitling | Rank-based methods as a non-parametric alternative of the T-statistic for the analysis of biological microarray data[END_REF]. Also, in a comparison 11 different methods for microarray analysis [START_REF] Jeffery | Comparison and evaluation of methods for generating differentially expressed gene lists from microarray data[END_REF], Rank Products was shown to have performed well, in particular with small sample sizes. Significance was assessed using the False Discovery Rates (FDR) multiple testing correction method [START_REF] Benjamini | Controlling the False Discovery Rate -a practical and powerful approach to multiple testing[END_REF] with a FDR cut-off of 10% (gene expression) or 5% (miRNA expression). For probe sets with a false discovery rate > 10%, a further 1000 permutations were carried out to obtain more precise P values and FDR values. Molecular interactions between genes were mapped to common cannonical pathways using the Pathway Explorer function within Ingenuity Pathway Analysis software (Ingenuity Systems, www.ingenuity.com).

MicroRNA microarray analysis was performed to determine differential expression in blood-borne miRNAs between (i) healthy individuals and those with CAD (n=7), and (ii) CAD patients pre-commencement of a cardiac rehabilitation exercise programme and 6 weeks after exercise (n = 8). All miRNA expression profiling was carried out by LC Sciences, LLC (Houston, TX) using a proprietary microfluidic array (Chip ID miRHuman_11.0_080411) based on the Sanger miRBase v11.0 database (http://www.sanger.ac.uk/Software/Rfam/mirna/), designed to detect 887 human miRNAs. Proprietary analysis employed a Student's t-test to compare data. An alpha level of 10% was accepted for Student's t-test analysis in order to identify sufficient targets for further evalulation. MicroRNA data were subsequently reanalysed in-house using the Ranked Products method with an FDR of 5% [START_REF] Breitling | Rank products: a simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments[END_REF]. miRNAs achieving significance under both analyses, and which had a signal intensity greater than 500 fluorescence units were considered for further analysis.

Bioinformatic determination of downstream predicted targets for candidate miRNAs was performed as previously described by Selbach et al. [START_REF] Selbach | Widespread changes in protein synthesis induced by microRNAs[END_REF]. Briefly, an unbiased search for sequence motifs (representing positions 2-7 of the miRNA seed region) in a library of genome-wide 3' untranslated regions in human was undertaken for each miRNA. On this basis genes were assigned to either target gene or non target groups. Gene expression was then compared between CAD and controls and pre and post exercise respectively for gene targets and nontargets, and compared with unpaired (CAD v Controls) or paired (Post v Pre Exercise) Student's t-test (P<0.05).

RT-PCR analysis

Quantitative real-time polymerase chain reaction (qRT-PCR) was performed to validate changes in selected genes by IPA software. Purified RNA was converted to cDNA using TaqMan™ Reverse Transcription Reagents in a total reaction volume of 20 uL (Applied Biosystems), containing: 10x TaqMan™ RT buffer (2.0 uL), 25 nM magnesium chloride (4.4 uL), deoxyNTPs (4.0 uL), random hexamers (1.0 uL), RNase inhibitor (0.4 uL) and MultiScribe reverse transcriptase (0.5 uL), using the following cycling parameters: 25°C for 10 minutes, 48°C for 30 minutes and then 95°C for 5 minutes. Appropriate "no template" RT-PCR controls were similarly processed. cDNA was diluted to a final volume of 80uL by addition of A c c e p t e d M a n u s c r i p t nuclease-free water and stored at -20°C until use. Real-time PCR was performed in 5 uL volumes, in a reaction buffer containing 2.5 uL of 2xTaqMan™ Universal PCR Master Mix, 2 uL of cDNA template and 0.25 uL of each TaqMan™ probe / primer mix (supplementary table 3 in the online supplement). Amplification reactions were performed in triplicate, duplexed with GAPDH as an endogenous control. Assay plates were sealed, vortexed, centrifuged (1 minute at 2000 rpm), and subjected to PCR (ABI Sequence Detection System 7900HT), using ABIsuggested cycling parameters: 50°C for 2 minutes, 95°C for 10 minutes and then 40 cycles of 95°C for 15 seconds followed by 60°C for 1 minute. Data were exported using a 0.2 threshold, with 3 to 15 cycles set as baseline. Cycle threshold (Ct) values were normalized by the geometric means of GAPDH. Triplicates were averaged to calculate an expression value for each sample.

Statistical analysis

Using Pounds and Cheng's [START_REF] Pounds | Sample size determination for the false discovery rate[END_REF] power calculation method, using a desired False Discovery Rate of 0.05, a desired average power of 80%, and assuming that 30 of 833 microRNAs to be investigated have a theta value of 1.8 (with the remainder being assumed to be not differentially expressed), we calculated that a a sample size of 12 per group was needed.

Statistical analyses were performed using GraphPad Prism software (GraphPad Software Inc.). Data are expressed as mean ± standard deviation. Paired or unpaired Student's t-test was used to compare data as appropriate. A P-value of less than 0.05 (two-sided) was considered significant. For analysis of microarray expression data a Ranked Products approach was used, with FDR of either 10% or 5% as appropriate.

Results

Gene expression

Between patients with CAD (n=12) and control subjects (n=12) we identified 365 differentially expressed genes: 175 genes were up and 190 genes were down regulated. In patients with CAD studied across an exercise rehabilitation programme (n=10), we found 645 genes differentially expressed between the beginning and completion of the programme with 196 genes up-regulated and 449 genes down-regulated.

Pathway analysis

Biological pathway analysis identified a number of canonical pathways as being significantly and consistently modulated across patients with CAD vs control subjects (Figure 1a), and across the cardiac rehabilitation exercise regime (Figure 1b). From (i) the overall degree of modulation, (ii) the proportion of differentially expressed genes in each pathway, and (iii) biological relevance, further analyses were focused on mitochondrial dysfunction and oxidative phosphorylation pathways (see also supplementary table 4 in the online supplement). Within these pathways, expression of NADH dehydrogenase (ubiquinone) 1 beta subcomplex, 3 gene (NDUFB3), ubiquinol-cytochrome c reductase, complex III subunit VII gene (UQCRQ), cytochrome c oxidase subunit VIIc gene (COX7C), and ATP synthase, H + transporting, mitochondrial F0 complex, subunit E gene (ATP5I) were greater in patients with CAD relative to control subjects. Completion of the rehabilitation period was associated with down regulation of genes in these pathways: NADH dehydrogenase (ubiquinone) 1 alpha subcomplex, 1 gene (NDUFA1), caspase 3 gene (CASP3), COX7C, ATP5I, and ATP synthase, H + transporting, mitochondrial F0 complex, subunit G gene (ATP5L) (Supplementary Figures 1 &2). RT-PCR confirmed up regulation of NDUFB3 and COX7C in patients with CAD and down regulation of CASP3 and ATP5L following the rehabilitation programme (Figure 2).

miRNA expression profiling

The miRNAs hsa-miR-140-3p (median signal intensity, 738 vs 383, P=0.017) and hsa-miR-182 (median signal intensity, 836 vs 549, P=0.093) were differentially regulated between patients with CAD and control subjects (Figure 3a). Accordingly, global analysis of predicted targets within the gene expression dataset demonstrated significantly reduced expression of genes with target regions of hsa-miR-140-3p and hsa-miR-182 (Figures 3b and3c). Within the post surgery exercise groups hsa-miR-92a (median signal intensity, 4712 vs 6250, P<0.01) and hsa-miR-92b (median signal intensity, 1119 vs 2163, P<0.01) which have identical downstream mature sequences, were differentially regulated between beginning and completion of the rehabilitation programme (Figure 4a). Expression levels of predicted gene targets of hsa-miR-92a and hsa-miR-92b were significantly reduced (Figure 4b). Moreover, predicted targets of hsa-miR-92a and hsa-miR-92b include the genes NDUFA1 and CASP3. The observed reduced expression of these genes following the rehabilitation programme is consistent with the increased expression of hsa-miR-92a and hsa-miR-92b (Figure 2).

Discussion

We have successfully used gene expression profiling in whole blood to establish genes that are differentially regulated between patients with CAD and control subjects and whose expression is modified following a cardiac rehabilitation programme. We have, for the first time, identified differentially regulated miRNAs in whole blood in patients with CAD compared to healthy control subjects. Among the biological pathways containing the differentially expressed genes we identified those involved in mitochondrial dysfunction and oxidative phosphorylation being most consistently affected and modulated.

Studies on gene expression in humans are particularly difficult due to the limited availability of tissue from well phenotyped patients, and even more so from healthy subjects. The use of blood as a "surrogate tissue" that can be obtained with a minimally invasive procedure is therefore an attractive alternative. Whilst the majority of studies have been performed in mononuclear cells isolated from peripheral blood [START_REF] Aziz | Peripheral blood gene expression profiling for cardiovascular disease assessment[END_REF] there have been reports on gene expression profiling in whole blood in a number of human diseases including Huntington's disease [START_REF] Borovecki | Genome-wide expression profiling of human blood reveals biomarkers for Huntington's disease[END_REF] and cancer [START_REF] Osman | Novel blood biomarkers of human urinary bladder cancer[END_REF][START_REF] Martin | High-sensitivity array analysis of gene expression for the early detection of disseminated breast tumor cells in peripheral blood[END_REF]. Many different techniques have been proposed to handle peripheral blood samples prior to RNA isolation based on experimental design [START_REF] Feezor | Inflammation and Host Response to Injury, Large-Scale Collaborative Research Program. (2004) Whole blood and leukocyte RNA isolation for gene expression analyses[END_REF]. We utilised the PAXgene Blood RNA system as it provides a way to stabilise RNA immediately after sample collection and facilitates store the samples for a relatively long period of time without compromising RNA integrity [START_REF] Feezor | Inflammation and Host Response to Injury, Large-Scale Collaborative Research Program. (2004) Whole blood and leukocyte RNA isolation for gene expression analyses[END_REF][START_REF] Rainen | Stabilization of mRNA expression in whole blood samples[END_REF][START_REF] Thach | Assessment of two methods for handling blood in collection tubes with RNA stabilizing agent for surveillance of gene expression profiles with high density microarrays[END_REF][START_REF] Chai | Optimization of the PAXgene blood RNA extraction system for gene expression analysis of clinical samples[END_REF]. The straightforward procedure of sample collection is an important condition for future use of whole blood gene expression profiling in clinical practice. Most recently whole blood gene expression profiling has also been applied to cardiovascular disease. Sinnaeve et al. [START_REF] Sinnaeve | Gene expression patterns in peripheral blood correlate with the extent of coronary artery disease[END_REF] showed a relationship between gene expression profiles in whole blood with CAD index as a marker of the extent of disease. Our study adds important new features to this area of research.

In face of the multiplicity of differentially expressed genes both between patients and controls and between pre and post cardiac rehabilitation, we have used pathway analysis to examine patterns of expression rather than individual genes. Using this approach we were able to interpret data on relatively small fold changes in gene expression, which may on their own be of limited importance but in combination represent multiple hits to specific pathways and may thereby affect, for example, mitochondrial function. The finding that we were unable to confirm all differences in gene expression from the microarray experiment with qRT-PCR may again be in keeping with the existence of multiple changes, but of relatively small magnitude, below the two-fold limitation of qRT-PCR. This pattern of numerous small changes of expression could be reasonably expected in complex cardiovascular traits such as coronary artery disease. Of particular note is that the direction of these and other multiple "hits" to mitochondrial pathways, although small in magnitude, reverse following exercise (Figure 2), indicating that this intervention may offer specific benefits in relation to reducing oxidative stress in individuals recovering from coronary surgery.

We have further corroborated our data by simultaneously examining miRNA expression. Again, interpretation of results, especially with regard to targets of particular miRNAs, is challenging [START_REF] Breitling | Rank-based methods as a non-parametric alternative of the T-statistic for the analysis of biological microarray data[END_REF], but the data support our main findings. The data on integration of miRNAs with expression of their predicted targets does indeed show a significant, generalised down-regulation of these targets. This is in keeping with a functional role of these molecules, although such a role of miRNA in blood remains to be elucidated. We have, however, shown that expression of hsa-miR-92a and hsa-miR-92b are increased following cardiac rehabilitation, which occurs in parallel with a confirmed reduction in expression of two of their predicted target genes. Hsa-miR-92 has been found to be modulated in the process of neo-intima formation in a rat model of vascular injury [START_REF] Ji | MicroRNA expression signature and antisense-mediated depletion reveal an essential role of microRNA in vascular neointimal lesion formation[END_REF], but to the best of our knowledge has not yet been clearly implicated in the pathophysiology of human cardiovascular disease. Whilst acknowledging that hsa-miR-92a and hsa-miR-92b are likely to modify a plethora of potential targets, and that the specific interactions with NDUFA1 and CASP3 require further detailed examination, it is nonetheless intriguing to identify physiological candidates which are down-regulated in concert with increased expression of these miRNA. A further level of complexity of the regulation of miRNA expression and activity can be seen by the generation of the identical mature hsa-miR-92 miRNA from two different genetic loci. In relation to target prediction and validation, this study has focused solely on that common mature sequence. However, differential expression of the two pri-miRs (potentially under the control of different up-stream regulators) may occur in a heterogeneous manner in different tissues, cells, or compartments, offering even more exquisite control of their relative activities. The dynamic change of miRNA and gene expression towards a "healthier" profile following cardiac rehabilitation is a unique feature of our study, further supporting the relevance of the identified genes and miRNAs.

We acknowledge that our data only provide initial evidence for a role of the discovered genes and miRNAs. However, these findings are particularly attractive as targets for intervention. Oxidative stress has been implicated by a plethora of studies as playing a key role in a range of cardiovascular diseases [START_REF] Guzik | Coronary artery superoxide production and nox isoform expression in human coronary artery disease[END_REF][START_REF] Paravicini | NADPH oxidases, reactive oxygen species, and hypertension: clinical implications and therapeutic possibilities[END_REF][START_REF] Zalba | Oxidative stress, endothelial dysfunction and cerebrovascular disease[END_REF][START_REF] Delles | Targeting reactive oxygen species in hypertension[END_REF]. Mitochondrial dysfunction and particularly generation of reactive oxygen species by mitochondria is an important yet relatively novel mechanism contributing to the development of oxidative stress and cardiovascular disease [START_REF] Colombo | AMPKalpha1 regulates the antioxidant status of vascular endothelial cells[END_REF][START_REF] Victor | Oxidative stress and mitochondrial dysfunction in atherosclerosis: mitochondria-targeted antioxidants as potential therapy[END_REF]. We have recently shown that a specific mitochondria targeted antioxidant, MitoQ 10 , reduces the increased vascular superoxide generation in stroke-prone spontaneously hypertensive rats and leads to reduction of blood pressure [START_REF] Graham | Mitochondriatargeted antioxidant MitoQ10 improves endothelial function and attenuates cardiac hypertrophy[END_REF]. Our present data indicate that there may be further options for mitochondria targeted therapies in human cardiovascular diseases including CAD.

In regard to limitations of the study, we cannot exclude the possibility that the observations obtained in the cardiac rehabilitation group are not, at least to some extent, due to, e.g. healing of the sternotomy. Nonetheless, significant changes were observed in both gene expression and miRNA profiling in the blood of these patients, and in particular specific pathways were identified as being modified which were also shown to be modified in the CAD versus control comparison. Moreover, in this latter comparison these differences were determined in the CAD patients prior to undergoing surgery, illustrating that dynamic change in gene and miRNA expression can occur without the provocation caused by, for example, significant trauma such as surgery.

Our study is also limited by relatively small sample sizes due to the significant costs involved in microarray studies. However, the careful selection of patients and well matched control subjects from a larger cohort of well characterised individuals reduced "noise" in our expression profiling experiments. We have further addressed this issue by studying a second set of patients before and after completing the rehabilitation programme using paired statistical analysis. We are confident that due to integration of these two datasets and further analysis together with miRNA expression data the identified genes and pathways are not only valid but may also be clinically relevant. It should also be noted that the sample size in our study is equal or greater than that in other human global gene expression profiling studies [START_REF] Guo | Differential expression of microRNA species in human gastric cancer versus non-tumorous tissues[END_REF][START_REF] Sui | Microarray analysis of MicroRNA expression in acute rejection after renal transplantation[END_REF][START_REF] Yi | Differential expression of miRNA patterns in renal cell carcinoma and nontumorous tissues[END_REF]. Clearly the aim of this initial study was to identify such targets. Subsequent validation of these targets in larger studies will be a necessary step to confirm their candidacy as biomarkers and / or functionally relevant molecules in blood.

In summary, mRNA and miRNA expression profiling in whole blood is a promising tool to discover genes that determine cardiovascular phenotypes. The identified genes and pathways are likely to play a key role in the pathogenesis and progression of chronic CAD. Moreover we have identified biological pathways linked to vascular function which appear to be modifiable by an exercise intervention programme. Further elucidation of these mechanisms will contribute to our understanding of disease processes and ultimately lead to new therapeutic and preventative strategies.

Figure 2. Mitochondrial dysfunction pathways.

Up regulation of genes in mitochondrial function pathways in patients with coronary artery disease (CAD) vs healthy control subjects (top boxes) and down regulation of genes in mitochondrial function pathways following the cardiac rehabilitation programme (bottom boxes). All genes were differentially expressed in microarray experiments (for details see supplementary table 4 in the online supplement). P-values and fold changes (FC) in the boxes refer to data from RT-PCR experiments (for details see Figures S1 andS2 in online supplement). NDUFA1 and CASP3 are predicted targets of hsa-miR-92a and hsa-miR-92b, whose expression was increased following the cardiac rehabilitation programme (Figure 4).

Clinical

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Clinical

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Clinical

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Clinical

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Clinical

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Clinical

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Figure 1 .

 1 Figure 1. Graphical representation of significantly affected canonical pathways between (a) patients with coronary artery disease (CAD) and healthy controls; and (b) patients with CAD before and after a cardiac rehabilitation exercise programme, following a coronary artery bypass procedure. Bars indicate the degree of significance (y-axis) to which a given pathway is differentially modulated. "Threshold" indicates the minimally accepted significance level. Squares indicate the relative proportion of constituent members of a pathway significantly differentially expressed. Oxidative Phosphorylation and Mitochondrial Dysfunction pathways were found to be significantly and consistently regulated across both experimental groups.

Figure 3 .

 3 Figure 3. (a) Heatmap analysis of miRNA expression in patients with coronary artery disease versus healthy control subjects (P<0.10, signal intensity >500 intensity units). The analysis shows relative fluorescence from green (2-fold reduction) to red (2-fold increase) of each miRNA per individual (with anonymised identifier at the top of each column). (b) Comparison of log ratios of hsa-miR-140-3p and hsa-miR-182 identified a significant reduction in gene expression levels in predicted target genes versus non-targets.

Figure 4 .

 4 Figure 4. (a) Heatmap analysis of miRNA expression in patients with coronary artery disease before and after completion of the cardiac rehabilitation programme (P<0.10, signal intensity >500 intensity units). The analysis shows relative fluorescence from green (2-fold reduction) to red (2-fold increase) of each miRNA per individual (with anonymised identifier at the top of each column). (b) Comparison of log ratios of hsa-miR-92a and b identified a significant reduction in gene expression levels in predicted target genes versus non-targets.

  Figure 1 a

  Figure 3

  Figure 4

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Acknowledgments

There are no conflicts of interest.

Funding

This work was funded by the British Heart Foundation Chair (AFD) and Programme Grant BHF RG/07/005/23633 (AFD and CD), the European Union's Sixth Framework Programme InGenious HyperCare LSHM-CT-2006-037093 (AFD and CD) and the Welcome Trust Cardiovascular Functional Genomics Initiative 066780/2/012 (AFD). Clinical Science Immediate Publication. Published on 07 Jun 2010 as manuscript CS20100043 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100043

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