Romain Gérard
email: romain.gerard@hotmail.fr

Souhila Kaci
email: kaci@cril.univ-artois.fr

Henri Prade
email: prade@irit.fr

Ranking Alternatives on the Basis of Generic Constraints and Examples -A Possibilistic Approach

The paper presents and discusses a method for rank-ordering alternatives on the basis of constraints induced by generic principles (expressing for instance the relative importance of criteria), or by examples of orderings between particular alternatives, without resorting to the use of an aggregation operation for evaluating the alternatives. The approach, which remains qualitative, is based on the minimal specificity principle of possibility theory in order to complete the constraints. It is compared on an illustrative example to an aggregationbased approach using Choquet integral. The way constraints expressed in the Choquet integral setting translate into constraints in the proposed approach is discussed.

Introduction

A classical way for comparing alternatives is to use multiple criteria for evaluating them in an absolute manner, using linearly ordered scales. These scales are often numerical, and under the hypothesis that are commensurable, different aggregation procedures that reflect various combination attitudes can be applied in order to build a complete preorder for rankordering the alternatives on the basis of the global evaluations that are obtained. However, in many practical problems (such as multiple criteria analysis, flexible constraints satisfaction problems), a numerical scale, such as [0,1] is too rich for being used, and more qualitative scales having a finite number of levels have to be preferred. But, the internal operations that can be defined on these latter scales (e.g., [START_REF] Mas | [END_REF]Fodor, 2000]) have a limited discriminating power since they take values on a finite range.

The problem thus amounts to compare alternatives represented by vectors of qualitative criteria evaluations without aggregating them. Apart from the Pareto partial preorder that should constrain any complete preorder between alternatives, one may have some generic rules that further constrain these complete preorders. For instance, one may state that some criterion is more important than, or equally important as, other criteria (maybe in a limited context). One may also have at our disposal some examples of preferences between fully specified alternatives. The problem addressed here, is then to complete the Pareto partial preorder in agreement with the constraints in a way that is as little arbitrary as possible.

The paper is organized as follows. The next section introduces the problem formally, while Section 3 provides a short background on a general family of aggregation functions that can be defined under the form of a Choquet integral, which will be used in the paper as a comparison landmark. Section 4 describes the proposed approach that uses a minimal commitment principle for building a complete preorder in agreement with the constraints. This principle expresses that an alternative is good as much as there is no other alternative that are considered to be better. Section 5 applies this approach to constraints that directly mirror the way the comparative importance of criteria is stated when using a Choquet integral aggregation. This enables a comparison between the two approaches. Section 6 reviews related works, and points out the new features of the proposed approach.

Framework

It is assumed that objects to be rank-ordered are vectors of satisfaction levels belonging to a linearly ordered scale S = {s 1 , • • • , s h } with s 1 < • • • < s h , each vector component referring to a particular criterion. Thus, it is supposed that there exists a unique scale S on which all the criteria can be estimated (commensurateness hypothesis).

Preferences are expressed through comparisons of such vectors

u i = {a i 1 , • • • , a i n } (written a i 1 • • • a i n for short), where a i j ∈ S, under the form of constraints a 1 • • • a n a 1 • • • a n expressing that u = a 1 • • • a n is preferred to (or is more sat- isfactory than) u = a 1 • • • a n .
Let U be the set of all possible vectors u = a 1 • • • a n , called also alternatives, such that a j ∈ S for all j = 1, • • • , n. is a pre-order on U if and only if it is a reflexive and transitive relation. u u means that u u holds but not u u. u = u means that both u u and u u hold, i.e. u and u are equally preferred, and u ∼ u means that neither u u nor u u hold, i.e. u and u are incomparable. is said to be complete (or total) if all pairs of alternatives are comparable.

Some components may remain unspecified when comparing alternatives. They are replaced by a variable x j if the jth component is free to take any value in the scale. This allows to express generic preferences as for example Pareto ordering written as

∀x i ∀x i , x 1 • • • x n x 1 • • • x n if ∀i, x i ≥ x i and ∃k, x k > x k .
In any case, Pareto ordering is always assumed to hold. Besides, other generic constraints of particular interest include those pertaining to the expression of the relative importance of criteria. The greater importance of criterion j with respect to criterion k can be expressed under different forms. One way to state it is by exchanging x j and x k and writing

x 1 • • • x j • • • x k • • • x n x 1 • • • x k • • • x j • • • x n when x j > x k .
One may think of other ways of expressing that j is more important than k. For instance, one may restrict the above preferences to extreme values of S for the x i 's such that i = j and j = k, since weights of importance in conjunctive aggregation can be obtained in this way for a large family of operators (e.g., [Dubois et al., 2001]). A more drastic way for expressing relative importance would be to use a lexicographic ordering of the vector evaluations based on a linear order of the levels of importance for the criteria. Then, the problem of ordering the vectors is straightforward. Note that the first above view of relative importance, which is used in the following, is a ceteris paribus preference of subvector (x j , x k ) with respect to (x k , x j ) for x j > x k , where the first (resp. second) component refers to criterion j (resp. k), which expresses preferential independence.

Another way to relate criteria is to express an equal importance between them. It can be expressed by stating that any two vectors where x j and x k are exchanged, and otherwise identical, have the same levels of satisfaction. Formally,

x 1 • • • x j • • • x k • • • x n = x 1 • • • x k • • • x j • • • x n .
It is worth noticing that transitivity is required between alternatives only and not between generic constraints. More precisely if it holds that u u and u u with respect to some generic constraints then we necessarily have u u . ). However acb is not preferred to bca since we cannot reach bca from acb by transitivity using the generic constraints. Indeed generic constraints require to explicitly express each constraint that we want to have, i.e. X Z in the above example.

Definition of the problem

The problem of rank-ordering the alternatives is decribed as follows. Given a set of constraints C of the form {u i u i |i = 1, • • • , m}, where u i and u i are instantiated on S, our aim is to compute a complete pre-order over U that satisfies all constraints of C. Such a pre-order should not add any additional constraint. One may wonder why we compute a complete pre-order from the set C of partially specified preferences. This is a debatable question and we really believe that the answer depends on the application. If C describes preferences over cars then we may permit that two cars are incomparable. However if C describes preferences over students' grades, as it is the case in our running examples, it is natural to have a complete pre-order over students. The set C may contain generic constraints as described previously but also particular examples of preferences. Note that some pre-orders, such as those induced by the minimum aggregation operator, are excluded as soon as Pareto constraints are considered.

Numerical aggregation by Choquet integral

Aggregation of object attribute values in the presence of interaction between criteria is essential in many decision making problems. Choquet integrals are very popular aggregation operators as they allow to model such interactions and thus to represent preferences that cannot be captured by a simple weighted arithmetic mean [Grabisch, 1995;1996]. Using a particular measure, they aggregate valued attributes describing alternatives into a unique value. A Choquet integral is based on a fuzzy measure defined by: Definition 1 Let A be the set of attributes and I(A) be the set of all possible subsets of A. A fuzzy measure is a function

μ from I(A) to [0, 1] such that: • ∀X, Y ∈ I(A) if X ⊆ Y then μ(X) ≤ μ(Y ). • μ(∅) = 0, μ(A) = 1.
A discrete Choquet integral with respect to a fuzzy measure μ is defined as follows:

Definition 2 Let μ be a fuzzy measure on

A = {a 1 , • • • , a n }.
The discrete Choquet integral w.r.t. μ is defined by:

Ch μ (a 1 • • • a n ) = i=1,•••,n (a (i) -a (i-1) ) * μ A (i) ),
where a (i) indicates that the indices have been permuted so that 0

≤ a (1) ≤ • • • ≤ a (n) , and
A (i) = {a (i) , • • • , a (n) } with a (0) = 0.
Example 1 ( [Grabisch, 1995;1996;Marichal, 1998]) Let A, B and C be three students evaluated with respect to three subjects: mathematics (M), physics (P) and literature (L).

Students' grades are summarized in integral with a fuzzy measure μ, the global grade for each student is computed as follows:

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• student A: Ch μ (A) = Ch μ (18, 16, 10) = 10 * μ MP L + (16 -10) * μ MP + (18 -16) * μ M , • student B: Ch μ (B) = Ch μ (10, 12, 18) = 10 * μ MP L + (12 -10) * μ P L + (18 -12) * μ L , • student C: Ch μ (C) = Ch μ (14, 15, 15) = 14 * μ MP L + (15 -14) * μ P L
, where μ X , μ XY and μ XY Z with X, Y, Z ∈ {M, P, L} denote the values of the fuzzy measure μ for the corresponding set of subjects. The school is more scientifically than literary oriented and it gives the same importance to mathematics and physics. Moreover the school wants to favor well equilibrated students without weak grades so we should have: C preferred to A and A preferred to B, i.e. C A B1 . As indicated before, the fuzzy measure μ models interaction between subjects. Since mathematics and physics have the same importance and they are more important than literature we have μ M = μ P , μ M > μ L and μ P > μ L . Moreover both mathematics and physics are scientific subjects, and thus are considered close, while literature is not. So the interaction between mathematics (resp. physics) and literature is higher than the interaction between mathematics and physics. Thus μ ML = μ P L > μ MP . This gives the following set of constraints on μ:

{μ M = μ P , μ M > μ L , μ P > μ L , μ ML = μ P L , μ ML > μ MP , μ P L > μ MP }.
In addition we consider the constraints Ch μ (C) > Ch μ (A) and Ch μ (A) > Ch μ (B) corresponding to the preference order between students A, B and C. Table 2 gives an example of measure μ satisfying all these constraints [Marichal, 1998]. Using the discrete Choquet integral w.r.t. μ given μ M μ P μ L μ MP μ ML μ P L μ MP L 0.45 0.45 0.3 0.5 0.9 0.9 1

Table 2: Fuzzy measure.

in Table 2 we get Ch μ (A) = 13.9, Ch μ (B) = 13.6 and Ch μ (C) = 14.9.

Let us consider another student D having 15 in physics, 15 is mathematics and 12 in literature. Using discrete Choquet integral with respect to μ given in Table 2 we get Ch μ (D) = 13.5. Then we have the following ordering C A B D. Let us now use another fuzzy measure μ which is equal to μ except for μ P L and μ ML . Instead we have μ P L = μ ML = 0.8. We can check that μ satisfies the set of constraints on μ. Using discrete Choquet integral with respect to μ we have C A D B. So we still have C A B but the ordering over B and D is reversed. This shows that Choquet integral is sensitive with respect to the fuzzy measure μ and may give arbitrary order over alternatives that are not explicitly stated in the set of constraints.

A qualitative ranking approach

Since a scale more refined than S is needed to rank-order the alternatives, we use the interval [0, 1] to encode this order-ing. Indeed S would only offer a finite number of levels for discriminating alternatives. For this purpose, we write our constraints in terms of a possibility distribution π, which is a function from the set of alternatives U to [0, 1], and provides a complete pre-order between alternatives on the basis of their possibility degrees.

General principle

In the possibility theory framework, an elementary preference between alternatives u u is encoded by the constraint π(u) > π(u ). Generally those constraints induce partial pre-orders on the set of interpretations. Our aim is to combine all these partial pre-orders and compute a total pre-order consistent with the set of constraints. In possibility theory we distinguish two completion principles called minimal and maximal specificity principles that respectively generate the largest and the smallest possibility distributions which satisfies the set of constraints. The following defines the notion of specificity between possibility distributions: Definition 3 (Minimal/Maximal specificity principle) Let π 1 and π 2 be two possibility distributions over U. π 1 is less specific than π 2 , denoted π 1 ≥ π 2 , iff ∀u ∈ U, π 1 (u) ≥ π 2 (u). π belongs to the set of the least (resp. most) specific possibility distributions among a set of possibility distributions P if there is no π in P such that π > π (resp. π > π ), i.e. π ≥ π holds but π ≥ π does not (resp. π ≥ π holds but π ≥ π does not). Indeed the minimal specificity principle gives the highest possible degree to alternatives, while the maximal specificity principle gives the lowest possible degree to alternatives. The choice of using minimal or maximal specificity principle depends on the application. The minimal specificity principle used in the following amounts to consider that an alternative is as good as permitted by the constraints. Thus, an unconstrained alternative is good by default.

Before we present our approach it is worth noticing that possibility distributions are purely qualitative here, although they are encoded on the interval [0, 1], and can be represented by a well ordered partition

(E 1 , • • • , E k ) on U such that: • E 1 ∪ • • • ∪ E k = U with E i ∩ E j = ∅ for i = j, • ∀u, u ∈ E i , π(u) = π(u ), • ∀u, u ∈ U, if u ∈ E i and u ∈ E j with i < j then π(u) > π(u ).

Minimal specificity principle-based algorithm

An elementary preference has generally the following form:

π(u) > π(u ), u, u ∈ U. (1)
For example given three criteria X, Y and Z, a relative importance constraint of X over Y is written as:

π(xyz 0 ) > π(yxz 0 ) for x > y, ∀z 0 .
A set of constraints of the form (1) can be written in a compact form as a set of the following constraints:

min{π(u)|u ∈ U 1 } > max{π(u )|u ∈ U 2 }, (2)

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where U 1 and U 2 are subsets of U.

We may also have equality constraints i.e.

π(u) = π(u ), u, u ∈ U. (3)
For example given three criteria X, Y , and Z, stating that X and Y have the same importance is written as:

π(xyz 0 ) = π(yxz 0 ), ∀z 0 .
Algorithm 1 computes the unique least specific possibility distribution satisfying a set of constraints C of the form (2) (constraints (1) being a special case of constraints (2)) and

(3) obtained from generic constraints and/or examples.

Let C = {C i : i = 1, • • • , m}. From C we define L C = {(L(C i ), R(C i )) : C i ∈ C}, where L(C i ) = U 1 and R(C i ) = U 2 for a constraint C i : min{π(u)|u ∈ U 1 } > max{π(u )|u ∈ U 2 }. Let EQ = {π(u j ) = π(u l )}.
Algorithm 1 [Benferhat et al., 2001] is a generalization of Algorithm 1: The least specific possibility distribution.

begin k = 0; while U is not empty do -k ← k + 1 ; -E k = {u|∀(L(C i ), R(C i )) ∈ L C , u ∈ R(C i )} ; α = true while α = true do α = false; for π(u j ) = π(u l ) in EQ such that u j ∈ E k or u l ∈ E k do α = true, E k = E k \{u j , u l } if E k = ∅ then Stop (inconsistent statements); -U = U\E k ; -From C remove (L(C i ), R(C i )) such that L(C i )∩ E k = ∅ ; -From EQ remove π(u j ) = π(u l ) s.t. u j ∈ E k . end return π = (E 1 , • • • , E k )
the possibilistic counterpart of Pearl's algorithm for system Z [Pearl, 1990].

Example 2 Let us consider two subjects "mathematics" and "literature" that are evaluated on a scale a > b > c with "a" for good, "b" for medium and "c" for bad. Thus a student having "ac" is good in mathematics and bad in literature. Pareto ordering forces to have π(xy) > π(x y ) as soon as x > x and y ≥ y or x ≥ x and y > y for x, y, x , y ranging in {a, b, c}. Pareto principle generates the following set of constraints: C = {min{π(aa)} > max{π(ab), π(ba), π(ca)}, min{π(aa), π(ab)} > max{π(ac), π(bb), π(bc), π(cb), π(cc)}, min{π(ac), π(ba)} > max{π(bc), π(cc)}, min{π(bb)} > max{π(cb), π(cc), π(bc)}, min{π(ba)} > max{π(ca), π(bb), π(cb)}, min{π(bc), π(cb), π(ca)} > max{π(cc)}, min{π(ca)} > max{π(cb)}}.

The application of the minimal specificity principle leads to π = ({aa}, {ab, ba}, {ac, bb, ca}, {bc, cb}, {cc}).

Note that letting π(ac) = π(ca) > π(bb) or the converse would lead to express more constraints than what is only specified by Pareto constraints. In fact, it may look a little surprising to get π(ac) = π(bb) = π(ca). However this is justified by the fact that the minimal specificity principle gives to each alternative the highest possible rank (i.e., possibility degree). The alternatives ac, bb and ca cannot have the highest possibility degree since following Pareto ordering, they are strictly less preferred than aa, ab and ba respectively. To ensure that we associate the highest possibility degree to these alternatives, the minimal specificity principle keeps the three pairs of evaluations at the same level, and they are ranked immediately below ab and ba.

The following example extends Example 2 with relative importance constraints:

Example 3 (Example 2 continued) Suppose that mathematics is more important than literature. This is translated by the following relative importance constraint: π(xy) > π(yx) for x > y. The instantiation of this constraint provides a new set of constraints:

C = {π(ab) > π(ba), π(ac) > π(ca), π(bc) > π(cb)}.
Let us now apply Algorithm 1 on C ∪ C , we get: π = ({aa}, {ab}, {ba, ac}, {ca, bb}, {bc}, {cb}, {cc}).

Ordering queries

We may need to compare specific alternatives without computing the whole complete pre-order generated by Algo. 1. An ordering query over two distinct alternatives u 1 and u 2 consists in checking whether u 1 u 2 , u 2 u 1 , u 1 = u 2 , or not, given a set of generic constraints and examples. Algo. 2 gives a way to answer such queries. The idea is to compute the set of alternatives that are less preferred to u 1 w.r.t. generic constraints and examples. Then we compute the deductive closure of this set w.r.t. generic constraints and examples. If u 2 belongs to this set or some alternative in this set is preferred to u 2 following Pareto principle then u 1 u 2 . Let u P u stand for u is preferred to u following Pareto principle, and C be the set of generic constraints and examples.

Example 4 (Example 2 continued) Let us compare the alternatives ac and cb. We have U 1 = {ac} and U 2 = {ca}. We have ca P cb then ac is strictly preferred to cb.

Note that Algorithm 2 returns that ac and bb are incomparable while ac bb w.r.t. Algorithm 1. Indeed, ac is put in the same stratum as ba (that dominates bb in the sense of Pareto). This is the effect of minimal specifity principle that puts any alternatives as high as possible. Thus, using Algorithm 2, one can distinguish the alternatives that are ranked only by virtue of the constraints, from those that also require the application of a default principle (here the minimal specificity principle) to be rank-ordered by application of Algorithm 1. Note that if we are only interested in comparing two alternatives, we may stop Algorithm 1 as soon as the two alternatives are ranked (without in general computing the whole pre-order). Note

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In contrast to Choquet integral, which is sensitive to the numerical values of criteria and coefficients of the fuzzy measure whose adjustment is not obvious, our approach relies on qualitative values of criteria. This qualitative aspect makes that the approach is general, i.e. independent of the values of criteria, which provides more robust results compared to Choquet integral. In fact, constraints over coefficients in a Choquet integral, as well as ranking over specific alternatives can be encoded in our framework by means of generic constraints and examples respectively. Then the application of Algorithm 1 gives a complete pre-order on U which satisfies all generic constraints and examples. In order to make the comparison precise, we encode each inequality between fuzzy measure coefficients in terms of relative importance constraints. Recall that Choquet integral expression writes:

C μ (a 1 • • • a n ) = a 1 × μ X1 + • • • + (a n -a n-1 ) × μ Xn with X n ⊂ • • • ⊂ X 1 and a 1 ≤ • • • ≤ a n . Moreover a i is associated to criterion X i \X i+1 , X n+1 = ∅.
Let us consider Example 1, and use a qualitative scale S = {a, b, c, d, e, f } (with a > b > c > d > e > f) to encode students' grades 18, 16, 15, 14, 12 and 10 respectively given in Table 1. Let x, y, z be students' grades in mathematics, physics and literature respectively so x, y, z ∈ {a, b, c, d, e, f }. We encode the constraints on μ namely μ M > μ L , μ P > μ L , μ ML > μ MP , μ P L > μ MP and μ P = μ M by means of generic constraints on π.

i) M is more important than L: At first sight we encode this constraint by: π(xyz) > π(zyx) for x > z, ∀y.

(mpl) > Ch μ (m p l ) reduces into μ M > μ L when Ch μ (mpl) = p + (l -p) * μ ML + (m -l) * μ M > Ch μ (m p l ) = p + (m -p ) * μ ML + (l -m ) * μ L .
This supposes p ≤ l < m and p ≤ m < l . We put p = p = y, l = m = z and m = l = x. Thus μ M > μ L is encoded in our framework by: π(xyz) > π(zyx) for x > z ≥ y.

(5)

ii) P is more important than L:

The same reasoning can be made for μ P > μ L . It is encoded by:

π(xyz) > π(xzy) for y > z ≥ x. (6)
iii) The interaction between M and L is higher than the interaction between P and M :

The inequality μ ML > μ MP is equivalent to the following inequality between the two Choquet integrals

iv) The interaction between P and L is higher than the interaction between P and M : vii) Lastly C A B is encoded by:

Similarily μ P L > μ MP is encoded by: π(xyz) > π(zyx) for y ≥ z > x. (8
π(dcc) > π(abf ) > π(f ea) (11)

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In sum we have the following set of generic constraints and examples:

C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ π(xyz) > π(zyx) for x > z ≥ y π(xyz) > π(xzy) for y > z ≥ x π(xyz) > π(xzy) for x ≥ z > y π(xyz) > π(zyx) for y ≥ z > x π(xyz) = π(yxz)
for all x, y, z π(xyz) > π(x y z ) for x ≥ x , y ≥ y , z ≥ z , x > x or y > y or z > z π(dcc) > π(abf ) > π(f ea)

Applying Algorithm 1 on C gives a possibility distribution with 26 strata π

= (E 1 , • • • , E 26 ) with E 1 = {aaa}, E 2 = {baa, aba}, E 3 = {caa, aca, aab}, • • •, E 26 = {f f f}.
The alternatives dcc, abf and f ea belong to E 12 , E 13 and E 15 respectively. Using Choquet integral we get 77 different levels:

Ch μ (aaa) = 18, Ch μ (aba) = Ch μ (baa) = 17.8, Ch μ (aca) = Ch μ (caa) = Ch μ (f ea) = 13.6, • • •, Ch μ (f f f) = 10
. Note also that in our approach, cce belongs to E 14 , i.e. we have C A D B.

Related works

This work is based on an idea first presented in [Dubois et al., 2005]. This proposal has been extended in two main directions. First, an algorithm is proposed that directly provides the ranking between any two alternatives (without having to compute the whole complete preorder on the set of all the alternatives). Besides, a comparative discussion on an example suggests that the proposed approach may be more robust, more flexible, and is more transparent to the user (who can control precisely what is expressed by means of the constraints) than an aggregation-based method, which moreover requires the use of a numerical scale in order to have a sufficiently discriminative scale.

Our approach focussed on the particular case where examples and generic constraints are consistent together in order to perform the comparison with the Choquet integral example, which is also based on this hypothesis. However we can deal as well with examples that contradict generic constraints. An algorithm has been proposed in [Dubois et al., 2005] where examples are considered as exceptions. This algorithm computes the complete pre-order associated to generic constraints, which is then modified in order to satisfy the examples provided that Pareto principle is not violated.

The proposed approach may look a bit similar to a topological sorting procedure that computes a linear ordering of the nodes of a directed acyclic graph (DAG). An edge from a node A to a node B means that A should be strictly preferred to B. Then several complete strict orders are generated satisfying all constraints given in the graph. Using our approach on such constraints we generate a complete pre-order that is such that if we would change equalities between alternatives into strict preferences then we would obtain one of the strict orders agreeing with the topological sort.

The recent years have seen the development of important works in Artificial Intelligence on the representations of preferences (e.g., [Wilson, 2004]). However, they start with a set of local conditional preferences and apply a ceteris paribus principle for completing the specifications. In our case, constraints are of a different nature. Indeed they refer to complete specifications of alternatives either in a generic way or by referring to particular situations.

Conclusion

Starting from possibility theory as a framework for representing constraints satisfaction, a qualitative method has been proposed for building a complete preorder that agrees with a set of constraints in a qualitative way. The approach is fairly general, and agrees with the way humans state their preferences in a granular manner, either in terms of generic rules or by means of examples. This is an extrapolation task not to be confused with learning. It would amount to e.g. learn importance relations between criteria from a sufficient number of examples of preferences between complete alternatives.

A topic of interest for further research would be to make a general comparison of the approach with multiple criteria aggregation techniques such as Choquet or Sugeno integrals, and to study to what extent it is possible to extract constraints underlying the way these aggregations handle the assessment of the relative importance of criteria.

  Ch μ (mpl) = p + (lp) * μ ML + (ml) * μ M > Ch μ (m p l ) = l + (pl ) * μ MP + (mp ) * μ M . This supposes p < l ≤ m and l < p ≤ m . Letting p = l = y, l = p = z and m = m = x, then μ ML > μ MP is encoded by: π(xyz) > π(xzy) for x ≥ z > y.

  ) v) M and P have the same importance:π(xyz) = π(yxz) for all x, y, z.(9)vi) As previously said, we suppose that Pareto ordering holds. Namelyπ(xyz) > π(x y z )(10)for x ≥ x , y ≥ y , z ≥ z , (x > x or y > y or z > z ).

Table 1 .

 1 Using Choquet

	student M P	L
	A	18 16 10
	B	10 12 18
	C	14 15 15

Table 1 :

 1 Students' grades.

  However this encoding, apparently natural, is not adequate here since it doesn't recover the ranking on U induced by Choquet integral. Let us consider the following alternatives df e, ef d, ead and dae. Following equation (4) we have π(df e) > π(ef d) and π(dae) > π(ead). However following Choquet integral we have Ch μ (df e) = 12.7, Ch μ (ef d) = 12.4, Ch μ (dae) = 14.8 and Ch μ (ead) = 15.6. So we have well df e preferred to ef d but dae is not preferred to ead. This means that constraint (4) is too weak to encode μ M > μ L . The reason is that the constraint μ M > μ L is more requirying than what it appears. Thus y should be constrained rather than to take any value in S. Let mpl and m p l two vectors.Note that Ch μ

Note that there is no weighted arithmetic mean that gives this order over A, B and C[Marichal, 1998].

Algorithm 2: Ordering queries

and u 2 are incomparable. end also that any strict preference or equality returned by Algorithm 2 is consistent with Algorithm 1.


Anne-Sophie Jourdain

Florence Petit

Marie-Françoise Odou

Malika Balduyck

Perrine Brunelle

William Dufour

Simon Boussion

Elise Brischoux-Boucher

Cindy Colson

Anne Dieux

ORCID Jamal Ghoumid

ORCID Thomas Smol

Elise Brischoux- Boucher

Marion Gérard

Fabienne Giuliano

Alice Goldenberg

Philippe Khau

Van Kien

Daphné Lehalle

Gilles Morin

Sébastien Moutton

Clémence Vanlerberghe

Sylvie Manouvrier-Hanu

Dr Fabienne Escande
email: fabienne.escande@chru-lille.fr

Multiplex targeted high-throughput sequencing in

Keywords: limb malformation, genetics, molecular diagnosis, targeted highthroughput sequencing

come

INTRODUCTION

Congenital limb malformations (CLM) are common defects, affecting approximately 1:500 new-born. They can be isolated or syndromic and more than 1400 syndromes comprising a CLM are described in the literature. Some of them have a genetic basis and, currently, more than 150 genes or regulatory regions are known to be implicated in limb malformations [START_REF] Manouvrier-Hanu | Genetics of limb anomalies in humans[END_REF][START_REF] Manouvrier-Hanu | Limb Development Anomalies: Genetics[END_REF][START_REF] Zuniga | The molecular basis of human congenital limb malformations[END_REF]. In most cases, the responsible genetic anomaly is a single nucleotide variation (SNV) [START_REF] Manouvrier-Hanu | Limb Development Anomalies: Genetics[END_REF]. However, copy number variations (CNVs) involving genes or non-coding regulatory elements are also a major cause of CLM [START_REF] Flottmann | Noncoding copy-number variations are associated with congenital limb malformation[END_REF].

CLM encompass a wide range of conditions displaying a very large clinical and genetic heterogeneity. Taken individually, each group of pathologies is rare and, as with most rare syndromes, clinical and molecular diagnosis of CLM is not easily assessed. Overlapping features, intra-and inter-familial phenotypic variability, nonspecific features and presence of uncommon, unusual or less known phenotypes often characterize CLM, making the diagnosis challenging for the clinical geneticists.

Clinical and molecular diagnostic confirmation is nevertheless very important to allow the establishment of a prognosis, to help in the patient support and in genetic counseling. The former gene-by-gene approach carried out by diagnostic laboratories did not help either, since this strategy was very time-consuming and expensive.

Nowadays, high-throughput sequencing enabling the simultaneous screening of multiple genes is therefore a very attractive option for the clinical practice [START_REF] Abou Tayoun | Sequencing-based diagnostics for pediatric genetic diseases: progress and potential[END_REF]. Two approaches are available: whole exome/genome sequencing (WES/WGS) or targeted high-throughput sequencing of gene panels. For diagnosis purposes, WES/WGS have still significant limitations due to the costs and time and specialized computing infrastructures that are needed for analysis and data storage. Targeted sequencing is a time and cost-saving strategy to screen from dozen to hundreds of genes.

To our knowledge, no study using high-throughput sequencing in CLM patients has been published so far. [START_REF] Furniss | Nail-Patella Syndrome: clinical and molecular data in 55 families raising the hypothesis of a genetic heterogeneity[END_REF] reported the genetic screening of 13 limb genes in 202 CLM patients, causative alterations were identified in 11% of cases (Furniss et al.,

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2009). Prior to 2014, our laboratory carried out the analysis by Sanger sequencing of only 20 genes (BHLHA9, BTRC, GDF5,HOXA13,HOXD13,IHH,LMX1B,NOG,PITX1,PTHLH,ROR2,SALL1,SALL4,SF3B4,TBX3,TBX4,TBX5,TP63,WNT5A,WNT10B). An additional regulatory sequence (ZRS) was also analyzed by Sanger sequencing and copy number determination. The diagnosis yield was 27.5% (data not shown). Since 2016, we had performed next generation sequencing (NGS) by target capture, enabling the simultaneous detection of SNPs and CNVs. We routinely analyzed a panel of 50 genes and 2 regulatory sequences, representing a good compromise for cost and ease of data analysis. We included human genes associated with known Mendelian diseases comprising a CLM (Supp. Table S1: list of genes, associated CLM and MIM#). All demands for genetic testing were carefully assessed clinically. For all patients fitting CLM clinical criteria, genetic testing by NGS was performed. However, in the case of isolated Split Hand/Foot Malformation (SHFM) with or without Long bone Deficiency (SHFLD), first tier analysis was the copy number determination of BTRC and BHLHA9 by quantitative PCR (qPCR). Indeed, copy-number gains of either locus are the major causes for SHFM/SHFLD [START_REF] Carter | Copy-number variants and candidate gene mutations in isolated split hand/foot malformation[END_REF]. If the qPCR was negative, NGS-based panel testing was performed as a second tier analysis, except for patients affected with only one split hand. In the latter, the analyses were discontinued since a genetic cause is unlikely.

In this article, we report on the retrospective analysis of a series of 352 patients presenting CLM and referred to our routine laboratory for diagnosis over a period of 3 years between March 2016 and February 2019. A causal variation was detected in 35.2% of these patients, with a great variability of the diagnosis yield among the different types of limb malformations. We show that the implementation of a gene panel has increased our diagnostic efficiency. In some cases, this allowed us to extend the clinical spectrum or redirect the diagnosis.

MATERIALS AND METHODS

Editorial Policies and Ethical Considerations

Analyses were performed on a diagnosis basis following the bioethics rules of French law.

Patients

In this study, we included a total of 352 unrelated probands with genetically undiagnosed CLM that have been analyzed in our laboratory between 03/01/2016 and Accepted Article 02/28/2019. All patients were referred by clinical geneticists. Clinical and family history details were collected as well as informed consent for genetic studies.

All patients have clinical features suggestive of a disorder associated with one of the 52 targets (genes/regulatory sequences) analyzed, although the features may not be completely typical or fulfill the clinical criteria of a specific syndrome at the time of recruitment.

The clinical data for each case was reviewed by a clinical geneticist expert in limb malformations, in order to classify the CLM in 7 groups (Figure 1

DNA extraction

Genomic DNA was extracted from blood samples using the Chemagic Star (Hamilton, Chemagen) with B1k kit (PerkinElmer) according to the manufacturer's instructions. When only fetal tissue was available, DNA was extracted using the Chemagic Prepito-D with Prepito Cyto Pure kit (PerkinElmer).

Quantitative PCR

For patients with isolated SHFM or SHFLD, quantitative real time PCR (qPCR) assays in the BHLHA9 and/or BTRC genes were performed. Likewise, each CNV identified by NGS (BMP2 enhancer, NOG, SF3B4, RBM8A and TP63) was validated by qPCR. The reaction was performed with the PowerSyber Green technology

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according to the manufacturer's protocol (ThermoFischer Scientific) on QantiStudio 7 (Thermofischer Scientific). Quantification of the target sequences was normalized on 2 control genes. The relative copy number was determined on the basis of the comparative ∆∆Ct method using a normal control DNA as the calibrator.

Targeted Sequencing workflow and analysis

Library preparation and sequencing

Genomic DNA have been quantified using the Quant-iT PicoGreen® dsDNA assay kit (Thermofischer scientific) on the Xenius SAFAS Monaco and normalized in nuclease free water at 25ng/µL in a final volume of 5µL. Libraries were prepared as described in the manufacturer's protocol SureSelect QXT (Agilent). Paired-end sequencing (2x150bp) was performed on the MiSeq sequencer (Illumina), on a standard FlowCell.

Targets

Clinical panel was developed through a collaborative process involving both the molecular diagnosis laboratory and the clinical experts in CLM at our institution.

Genes were selected for their implication in limb malformations based upon a review of the medical literature and public databases like OMIM (www.omim.org) and PubMed (www.ncbi.nlm.nih.gov/pubmed). With SureDesign (Agilent TM ) software, we targeted all exons of the reference transcripts from the Human Gene Mutation Database (HGMD) to design the SureSelect QXT probes. The content of the panel was continuously updated and, over time, 3 versions of the panel have been developed (V1, V2, V3), detailed in Supp. Table S1.

Bioinformatics analysis

The FastQ data were aligned on the human genome reference (GRCh37-hg19) using the Burrows-Wheeler Aligner algorithm (BWA, v0.7.15-r1140) and variant calling was performed using Genome Analysis Tool Kit (GATK, v3.7). The variant call format (VCF) was annotated using Varscan (v2.3.6) and integrated in a homemade database called DVD (Bioinformatics, CHU-Lille). The open source softwares CNVkit v0.8.5 [START_REF] Talevich | CNVkit: Genome-Wide Copy Number Detection and Visualization from Targeted DNA Sequencing[END_REF] and CANOES (Backenroth This article is protected by copyright. All rights reserved. et al., 2014) were used to infer and visualize copy number from targeted DNA sequencing data. Bioinformatic filters were applied to prioritize the variants' type, frequency in public and in in-house polymorphic databases. Variants outside exons +/-25 bp intron sequence were excluded. Only variants resulting in non-synonymous amino acid changes (missense, nonsense, exonic insertion/deletion) and intronic splice variants were analyzed. Frequency data was based on 1000 Genomes (http://www.1000genome.org), Exac Browser (http://exac.broadinstitute.org), GnomAD (https://gnomad.broadinstitute.org/) or dbSNP (Build 137, NCBI).

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Pathogenicity predictions for missense variants were made using the prediction programs Polymorphism Phenotyping v2 (Polyphen-2), Sorting Tolerant from Intolerant (SIFT) and Mutation Taster through Alamut Visual v.2.11.0 (Interactive Biosoftware, Rouen, France) (Supp. S2).

Description and interpretation of variants

Variants were described according to the recommendations of the Human Genome Variation Society (www.hgvs.org/) and submitted to the ClinVar database (SUB5823379). Their interpretation followed the classification system recommended by the American College of Medical Genetics and Genomics (ACMG) [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF].

Confirmation using other methods

Sanger sequencing

Sanger sequencing was used to reanalyze targeted exons with low or no coverage, to confirm the nucleotide variations identified by NGS and for segregation analyzes when familial samples were available. Amplicon fragments were sequenced using the ABI Prism 3730XL Genetic Analyzer (Applied Biosystems, Courtaboeuf, France).

Accepted Article Multiplex Ligation Probe Amplification

Multiplex Ligation-dependent Probe Amplification assay (MLPA) was performed to confirm deletion/duplication in SALL4. We used the commercially available MRC Holland Salsa kit P180 according to the manufacturer's protocol. Results were analyzed using the Coffalyser v.0.1407221 software (MRC-Holland)

RESULTS

Limb malformation panel performance

The SureSelect capture of genes involved in limb malformations (Supp. Table S3) yielded a high coverage in the targeted regions with a mean depth of 937.9X. On average, 98.9% of the targeted regions were covered with a read depth of a least 30X.

Only 1.1% of the targeted regions had coverage below 30X or no coverage. Those are exclusively first exons due to high GC content (DOCK6, FGFR3, HOXA13, NOTCH1, PITX1). One hundred percent of the bases in targeted regions showed an excellent quality value (QC>30). Dependent Depletion) score between 20 and 30 [START_REF] Kircher | A general framework for estimating the relative pathogenicity of human genetic variants[END_REF]. Seven variants are located in candidate non-coding regions: 3 of them in the ZRS (regulatory region of the SHH gene) and 4 of them in the non-coding region of RBM8A gene.

Molecular findings in the limb malformation cohort

To have a better view of the diagnosis yield for each type of malformation, patients were classified into 7 groups according to the clinical entryway, after reviewing by a geneticist specialized in limb malformations. The results are summarized in Table 2. Diagnostic yield was found to be extremely variable between the different phenotypic groups. Among this cohort, it was highest in the patella hypoplasia group (95.2%) and even rose to 100% in the subgroup of patients with Nail Patella syndrome. In contrast, the diagnostic yield for reduction anomalies (42 patients screened) was very low (9.5%).

In 228 cases (64.8%), no definite molecular diagnosis could be confirmed, either because no molecular variation was identified, or a VUS was identified, or only one pathogenic variant could be identified in recessive condition. No evidence for oligogenic inheritance has been found in this cohort.

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Radial anomalies

Overall diagnostic yield was 32.7% for the 101 patients of the radial anomalies group.

This group comprises several different syndromic conditions (Holt-Oram Syndrome, Okihiro Syndrome, Townes-Brocks Syndrome, TAR Syndrome, Nager Syndrome and RECQL4 disorders). However, the diagnostic yield is barely similar (30 to 50%), being highest for TAR and Nager Syndromes (Table 2).

The candidate variants identified were usually found in genes suspected prior the analysis. In 3 cases however, the NGS results allowed to reconsider the initial clinical hypothesis.

For two patients with duplicated thumb together with multiple congenital anomalies, we found variants in the PUF60 gene. This gene encodes a poly-U-binding splicing factor involved in pre-RNA splicing and transcription, through the interaction with

other factors such as SAP49 encoded by SF3B4. It was recently described in patients with multiple congenital anomalies, growth retardation and intellectual disability.

Limb malformations were reported in some cases but not precisely described in the literature. In case NGS226, we initially suspected an Okihiro Syndrome because of an associated congenital heart defect, bilateral coloboma and severe scoliosis. A novel de novo frameshift variant was identified in one of the RNA recognition motif domains of the protein, where all reported variants cluster. In case NGS309, a Townes-Brocks Syndrome was hypothesized because the duplicated thumb was associated with a Sprengel anomaly with fusion of cervical vertebrae, anal anteposition and a branchial sinus (Figure 4). A novel de novo splicing variant was identified in intron 5 of the PUF60 gene, responsible for the out-of-frame insertion of 11bp confirmed by the transcript analysis on lymphoblastoid cell line from the patient.

An Okihiro Syndrome was initially suspected in case NGS228, affected with bilateral hypoplastic thumbs, 2-3 syndactyly of toes, unilateral kidney agenesis, and small dysplastic ears (Figure 4). However, a probable pathogenic missense variant was found in FGFR2 gene, involved in the Lacrymo-Auriculo-Dento-Digital Syndrome and already reported in the literature (Table 2). The variant appeared to be inherited from the mother, who had a flexion defect of both thumbs. No lacrymal duct stenosis or deafness was reported in the proband or his mother.

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We studied 11 patients typical for TAR Syndrome, an autosomal recessive condition due to biallelic variants in RBM8A gene. The diagnosis was confirmed molecularly in 4 of them. In 6 patients, a RBM8A deletion was identified, without any second hit in 2 cases, or with a VUS on the other allele in 4 cases. These candidate VUS are located in the non-coding regions of RBM8A and functional studies are on-going. Therefore, our diagnosis yield of 36.4% may increase as far as 73% after functional validation.

Finally, 5 patients with radial anomalies could not be classified in a known syndromic condition. No candidate variant was identified in these unclassified patients.

Ectrodactyly

In the ectrodactyly group, 50 patients were screened and variants were detected in 21 cases (42%). The most frequent anomalies were TP63 variants, BHLHA9 duplications and 10q24 duplications involving BTRC.

In this cohort, 38 patients (76%) with an isolated CLM first underwent screening by qPCR for BHLHA9 and BTRC copy-number gain. BHLHA9 duplication was found in 6 patients (6/38, 15.8%). Phenotype was highly variable, ranging from bilateral split hand to SHFLD. Also, one patient had an isolated monodactyly of one hand. BTRC duplication was found in 4 cases (4/38, 10.5%) with SHFM. Among the 28 negative cases: 16 had an isolated unilateral split hand and the analyses were discontinued; 12 had ectrodactyly involving at least 2 extremities, and the analyses were continued with the NGS panel.

The NGS panel was performed overall in 24 patients affected with ectrodactyly (12 had isolated CLM with a negative screening for BHLHA9 and BTRC copy-number gain, 12 had syndromic ectrodactyly).

Among the 12 syndromic cases, 9 had a phenotype consistent with EEC Syndrome, while 3 patients had SHFM together with various congenital anomalies (diaphragmatic hernia in case NGS168, duodenal atresia and congenital heart defect in case NGS147, mandibulo-facial dysostosis and deafness in case NGS232). TP63 mutations were identified in 8 patients with EEC syndrome (88.9%, 8/9) but also in two patients with isolated SHFM of the 4 extremities. One proband was a 29 year-old female (NGS279). However, the second proband was a termination of pregnancy at This article is protected by copyright. All rights reserved.

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15 WG (NGS274). Fetal autopsy did not show cleft lip or palate, but ectodermal dysplasia could not be studied. In one case affected with bilateral split foot (NGS276), a nonsense variant in WNT10B, inherited from the asymptomatic mother was identified. While clearly deleterious, we chose to consider that the molecular diagnosis is not established in this proband, since the second hit has not been identified. However, we reported recently that heterozygous WNT10B carriers may be symptomatic [START_REF] Brunelle | WNT10B variants in split hand/foot malformation: Report of three novel families and review of the literature[END_REF]. Therefore it is highly likely that this variant plays a role in the limb phenotype in this case.

No variant in FGFR1, DLX5 or DLX6 was found in this series of 24 patients who underwent the NGS screening.

Brachydactyly

Overall diagnostic yield was 39.7% for the 63 patients of the brachydatyly group, but highly variable between the subgroups (Table 2). The yield is particularly low in type E brachydactyly (BDE), with only 2 positive diagnoses in 19 patients (10.5%).

Interestingly, one patient initially referred for non-syndromic familial BDE (NGS079, Figure 4) carried a likely pathogenic variant in TRPS1 segregating in several affected relatives. Fine clinical evaluation of the affected father, grand-mother and aunt, together with progressive dysmorphic features and hair anomalies in proband's follow-up, allowed to review the diagnosis to Tricho-Rhino-phalangeal Syndrome.

TRPS1 appeared to be the most frequently causative gene in the brachydactylies, together with GDF5 gene (responsible for Type-C Brachydactyly, BDC). A duplication of the BMP2 enhancer was found in two probands. One of them was affected with Type A2 Brachydactyly (BDA2) (NGS027), consistent with the associated phenotype reported in the literature, while the other proband was affected with BDC (NGS050). Surprisingly, no variant was found in PTHLH gene.

In 8 patients, the brachydactyly phenotype could not be classified. The molecular analysis identified the causative gene in only one of them (NGS081). This proband carried a de novo nonsense variant in GLI3 and had short middle and distal phalanges associated with bilateral triphalangeal thumbs. The feet had large halluces and clinodactyly of toes with a median plantar crease (Figure 4). There was no macrocephaly and the corpus callosum was present. This phenotype was intriguing,

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since triphalangeal thumbs had never been reported in other GLI3-mutated patients to our knowledge. After the molecular results, additional investigations showed a bifid epiglottis and a hypothalamic hamartoma, compatible with Pallister-Hall syndrome.

Finally, VUS candidates were identified in 6 patients in IHH, GDF5 and HDAC4

genes. Familial segregation studies may help the interpretation.

Polydactyly

Fifty-one patients affected with different types of polydactyly were screened. The major causative gene was GLI3, with variants found in preaxial, postaxial or mixed polydactylies. The diagnostic yield reaches 70% in the latter subgroup. We identified 12 pathogenic or likely pathogenic variants in GLI3 and one candidate VUS. In the preaxial subgroup, we also found 3 variants in the ZRS (limb-specific SHH enhancer).

These were novel variants affecting highly conserved nucleotides during evolution and unreported in the databases. Due to their location in a non-coding region, their functional consequences will be assessed by enhancer assays. So far, they have been classified in the VUS category. Interestingly, a VUS in GLI2 was found in a female patient presenting with postaxial polydactyly affecting all four extremities (NGS055).

Additionally, the proband had an occult cranium bifidum with a head circumference at +2SD. The VUS segregated in the father and another child who also had postaxial polydactyly of the four extremities. Together, these findings are consistent with the deregulation of the SHH signaling pathway as the main molecular cause for polydactyly.

Furthermore, in the female proband NGS249 and her mother, there was a frameshift variant in TBX3. Both patients had a unilateral postaxial polydactyly, associated with clinodactyly and synostosis of the middle and distal phalanges of the 5 th fingers (Figure 4). TBX3 is responsible for Ulnar-Mammary Syndrome, an autosomal dominant disorder characterized by posterior limb deficiencies or duplications, apocrine/mammary gland hypoplasia and/or dysfunction, abnormal dentition, delayed puberty in males, and genital anomalies. No extra-limb feature was found in this family.

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Synpolydactyly is a particular subgroup where polydactyly is associated with extended syndactylies. The only causal variants found in this sub-group affect the HOXD13 gene in 50% of patients.

Reduction anomalies

As mentioned earlier, it is important to note that this group excluded the radial reduction defects and the SHFM/SHFLD. Among the 42 patients included, the diagnosis yield was low at 9.5%.

In particular, we did not find any causative or candidate variant in longitudinal/transversal defects or in phocomelia/amelia patients. Chromosomal microarrays (CMA) have been performed in 96% of these patients, revealing no candidate CNV. Some patients were affected with sporadic conditions of unknown etiology like Gollop-Wolfgang Complex or Femoral-Facial Syndrome, which may be due to unknown genes so far or to different mechanisms.

Conversely, we found the causative molecular anomalies in 44.4% (4/9) of cases from the Adams-Oliver subgroup. An additional patient carried a candidate missense VUS in NOTCH1 for which the familial segregation is in progress. All 9 patients were typical, affected with scalp defects and terminal transverse defects of the hands and/or feet (Figure 4). For one of them, the cutis aplasia had been overlooked at the first examination, and only noticed when reviewing the patient after the molecular diagnosis. Of the 5 genes known so far to cause Adams-Oliver Syndrome, the most frequently involved was the DOCK6 gene, responsible for an autosomal recessive form.

Fusion anomalies

In this group, 24 patients were screened because of syndactyly (15 patients) or multiple synostosis (9 patients). The fusion anomaly was isolated in 9 cases (37.5%).

Variants were identified in 9 cases: 6 pathogenic or probably pathogenic variants and 3 VUS.

NOG was the major causative gene for multiple synostosis (3/5 patients, of which one case harboring a NOG gene deletion). These patients presented with broad thumbs This article is protected by copyright. All rights reserved.

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and hallux, carpal fusion and synostosis of the 4 th and 5 th fingers phalangeal or metacarpo-phalangeal joints (NGS287, NGS291, NGS296), associated with conduction deafness due to stapes ankylosis in one of them (NGS291, NOG deletion).

We also found a de novo likely pathogenic variant in FGF9 in one case affected with bilateral synostosis of the thumb phalanges, absence of the hallux first phalanges and bilateral dislocation of the radial head (NGS283).

In patients with syndactyly, variants were identified in FGFR2, GLI3 and DLL4

genes. Case NGS282, carrying a de novo likely pathogenic variant in FGFR2, had a typical Pfeiffer syndrome with craniosynostosis and complete membranous syndactyly 2-5 of fingers, 2-3 of toes. In case NGS257, we identified the pathogenic nonsense p.(Ser372*) in GLI3. The patient presented with bilateral 2-3-4 syndactyly of fingers, bilateral 1-2-3-4 syndactyly of toes and head circumference was +2SD

(Figure 4). The variant was inherited from the mother with similar clinical features. In case NGS216, we identified a missense VUS affecting the zinc finger domain of the GLI3 protein. The proband had mesoaxial syndactylies of hands and feet, but normal head circumference. This phenotype segregated in the family, following an autosomal dominant inheritance. Segregation studies may help to reclassify the variant. In both families, an unusual phenotype for GLI3 was observed, with mesoaxial syndactylies and absence of polydactyly.

Interestingly, we found a missense VUS altering a highly-conserved residue of DLL4, in a patient affected with bilateral 3-4 syndactyly of hands and terminal transverse defects with anonychia of the 4 th fingers (NGS086). While these distal reduction defects are consistent with the diagnosis of Adams-Oliver Syndrome, the patient had no scalp defect. Familial segregation study may help to interpret these results.

Patella hypoplasia

The patella hypoplasia group, comprising 21 patients, harbored the highest diagnostic yield of the cohort (95.2%). Note that this group only comprised Nail-Patella and Small Patella Syndromes. The RECQL4-disorders, that may also comprise a patella hypoplasia, have been classified in the radial anomalies group, because this is usually the prominent feature.

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All 18 cases affected with Nail-Patella Syndrome have been resolved, with the identification of a pathogenic or likely pathogenic variant in LMX1B.

Among the 3 Small-Patella Syndrome cases, the same nonsense pathogenic variant p.(Arg261*) was identified in TBX4 confirming the diagnosis in two unrelated probands. One of them was a male patient (NGS281) presenting with bilateral hypoplastic patellae responsible for recurrent dislocations and sandal gaps.

Interestingly, pelvis X-rays were normal, showing no defect of the ischiopubic junction ossification at 15 years of age. There was no pulmonary hypertension. In the third proband (NGS100), a candidate missense variant was identified in the T-Box domain of TBX4, affecting a highly-conserved residue and never reported previously in the databases. This proband had bilateral hypoplastic patellae responsible for dislocations in late childhood, and a defect of the ischiopubic junction ossification at 37 years old. There was no pulmonary hypertension. Familial segregation study may help the interpretation.

DISCUSSION

CLM are a diagnostic challenge due to their variable expressivity, clinical overlap between the different syndromes (e.g. radial anomalies) and their genetic heterogeneity. NGS approaches are increasingly being used for genetic diagnosis in routine clinical practice and many publications report successful use in heterogeneous disorders such as inborn errors of metabolism [START_REF] Yubero | Targeted Next Generation Sequencing in Patients with Inborn Errors of Metabolism[END_REF], neuromuscular disorders [START_REF] Stehlikova | Muscular dystrophies and myopathies: the spectrum of mutated genes in the Czech Republic[END_REF], intellectual disability [START_REF] Martinez | High diagnostic yield of syndromic intellectual disability by targeted next-generation sequencing[END_REF]. To our knowledge, no report of the diagnostic impact in patients affected with CLM using high-throughput sequencing has been published so far. In this study, we report on our data obtained on a series of 352 patients with CLM, using a strategy including quantitative assays and multigenic high-throughput sequencing panel.

The strategy adopted by our multidisciplinary team was used to optimize the turnaround time and diagnostic performance and to maximize the potential for a differential diagnosis without the need for multiple testing, together with keeping the cost down. Since no commercial gene panel was available for CLM, we developed a custom panel of 52 gene/regulatory targets, to provide comprehensive coverage of known CLM-associated genomic regions. We chose to target genes in which variants were known to cause CLM, keeping their number low to facilitate the analysis. We

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selected the genes responsible for isolated or syndromic CLM through the Online Mendelian Inheritance of Man database. In the syndromic conditions, only genes for which CLM is a prominent feature in the phenotype have been selected.

This NGS-based panel testing approach has significantly simplified our workflow in the diagnostic laboratory. Before 2016, Sanger sequencing was performed for up to 21 genes regulatory regions. In most patients, multiple genes had to be screened sequentially given the genetic heterogeneity of most CLM.

Multiple targets can be screened at once, including frequent, rare or differential genes.

The choice of the capture-based enrichment enables us to analyze sequence variants as well as CNVs. With bioinformatic algorithms, CNV detection with NGS is easy and can replace traditional methods such as quantitative PCR or MLPA [START_REF] Yao | Evaluation of copy number variant detection from panel-based next-generation sequencing data[END_REF]. The importance of CNV detection in CLM has been confirmed by numerous studies [START_REF] Carter | Copy-number variants and candidate gene mutations in isolated split hand/foot malformation[END_REF][START_REF] Flottmann | Noncoding copy-number variations are associated with congenital limb malformation[END_REF]. Our strategy allows the detection of CNVs in all 52 targeted regions, when formerly CNV determination was possible in only a very limited number of genes. In the present series, CNVs were identified in 7.4% (26/352) of patients. Among them, 14 CNVs out of 26 identified (54%) would not have been identified with our previous workflow, due to the absence of home-made and commercial tests available (RBM8A, TP63, SF3B4, NOG, BMP2 enhancer). However, we have made the choice to keep on performing q-PCR for BHLHA9 and BTRC as a first-line test in ectrodactyly because of its rapidity and low cost. However, given the low diagnostic rate in this phenotype group, this should be reevaluated. A NGS-first approach would certainly be interesting for these patients as well.

Overall, in this series of 352 patients, 138 pathogenic or likely pathogenic variants were identified, among which 26 CNVs, and 22 variants of uncertain significance. Our approach revealed to be a useful strategy, providing a molecular diagnosis in 35.2% of patients, similar to other heterogeneous conditions (50% for inborn errors of metabolism [START_REF] Yubero | Targeted Next Generation Sequencing in Patients with Inborn Errors of Metabolism[END_REF], 47.3% for myopathies and muscular dystrophies [START_REF] Stehlikova | Muscular dystrophies and myopathies: the spectrum of mutated genes in the Czech Republic[END_REF], 39% for intellectual disability [START_REF] Martinez | High diagnostic yield of syndromic intellectual disability by targeted next-generation sequencing[END_REF], 28.5% for epileptic encephalopathy [START_REF] Kothur | Diagnostic yield of targeted massively parallel sequencing in children with epileptic encephalopathy[END_REF], 28.1% for disorders of sexual development [START_REF] Fan | Diagnostic Application of Targeted Next-Generation Sequencing of 80 Genes Associated with Disorders of Sexual Development[END_REF], 24.5% for developmental eye disorders [START_REF] Patel | The Oculome Panel Test: Next-Generation Sequencing to Diagnose a Diverse Range of Genetic Developmental Eye Disorders[END_REF], 20% for primary arrhythmia This article is protected by copyright. All rights reserved.

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syndrome and cardiomyopathy [START_REF] Robyns | Repeat genetic testing with targeted capture sequencing in primary arrhythmia syndrome and cardiomyopathy[END_REF], 18% for inherited polyneuropathy [START_REF] Wang | Target-enrichment sequencing and copy number evaluation in inherited polyneuropathy[END_REF]). A 7.7% increase was observed in our diagnostic performance, compared to our previous workflow (diagnostic yield 27.5% before 2014, data not shown). This increased yield may be explained by the additional targeted regions as well as by the possibility for CNV detection. Also, we can predict an improvement in assessing differential diagnoses thanks to the panel screening, compared to the candidate gene approach required for the Sanger sequencing.

VUS were identified in 6.2% of patients, with the need for additional investigations (familial segregation, functional assays, in vivo model…). Most of them could potentially be reclassified as likely pathogenic variants if segregation analysis could be performed in the family.

The interpretation of variants needs a tight collaboration between clinical geneticists and molecular biologists, especially when managing CLM patients for whom the molecular cause might be heterogeneous. We observed that the diagnostic yield is highly dependent on the clinical selection of patients

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multidisciplinary discussion was held in an attempt to define further testing. In some cases, phenotyping was limited by the lack of clinical information, X-rays and pictures or the poor quality of the latter.

Limb development is a complex process requiring the precise regulation and inter-connection of multiple signaling pathways, making it highly sensitive to environmental factors (parental exposure to teratogenic agents, intrapartum events…). For 7 patients, an environmental or non-genetic cause was suspected (maternal diabetes, contraceptive or valproate exposition during pregnancy, amniotic bands, vascular cause) and the molecular analyses were motivated to rule out a genetic cause, in the context of genetic counseling.

For certain patients, a molecular diagnosis was subsequently made. Two cases were affected with radial anomalies and cardiac septal defects (NGS087 and NGS290). The first suspected diagnosis was Holt-Oram Syndrome but, given the absence of TBX5 anomaly after the NGS screening, a chromosomal breakage analysis was performed, showing the instability typical for Fanconi anemia disease. In both cases, the diagnosis was able to be confirmed by molecular testing through a highthroughput sequencing panel dedicated for genetic anemia. Given the large genetic heterogeneity (more than 20 genes) and the availability of the chromosomal screening test, we chose not to include the causative genes in our CLM panel. For the same reasons and also because the limb features are not prominent in the clinical picture, the causative genes for Diamond-Blackfan anemia were not included.

Such a high negative rate in individuals with significant congenital anomalies suggests that additional genes may be involved. However, it is unlikely that exome sequencing would significantly increase the diagnostic rate in the negative patients, since trio analyses usually have a similar yield in heterogeneous patient populations (literature review: 34.3% [START_REF] Dragojlovic | The cost and diagnostic yield of exome sequencing for children with suspected genetic disorders: a benchmarking study[END_REF]). Also, this approach would generate more unsolicited findings and VUS, therefore increasing the cost and time spent on the analysis. Furthermore, the technical quality of the NGS panel was high, demonstrating excellent coverage and read depth for most genes, compared to the results obtained by exome sequencing for many genes of the panel (e.g. low coverage for BHLHA9, EOGT, HOXA13, TBX5…). To test this approach, we performed whole exome sequencing in 31 negative patients (17 cases with radial anomalies, 5 with This article is protected by copyright. All rights reserved.

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ectrodactyly, 4 with brachydactyly, 3 with reduction anomalies and 2 with patella hypoplasia). In all cases, no definite molecular diagnosis could be confirmed (data not shown), suggesting the implication of variants located in non-coding regions such as UTRs, in deep intronic or in regulatory regions. Indeed, many CLM are due to variants in non-coding regions that would be missed by exome sequencing (e.g. in the preaxial polydactyly sub-group, 15% of patients carry a variant in the ZRS).

Therefore, genome sequencing that offers the opportunity to explore the full contribution of non-coding variants, including SNV, CNV or structural variants, could be a good choice in the near future, like it has been shown in other pediatric diseases [START_REF] Lionel | Improved diagnostic yield compared with targeted gene sequencing panels suggests a role for whole-genome sequencing as a first-tier genetic test[END_REF].

Genomic disorders may have been insufficiently studied. CMA was performed in 89% of the patients before the NGS screening, while balanced chromosomal rearrangements have not been studied in most of them.

The major concern for the targeted panel is the need for frequent updates, which would not be necessary with the exome sequencing approach. During the past 3 years, we set up 3 different versions of the panel.

In conclusion, we report on the diagnostic rate among all samples sent for congenital limb malformations to a clinical reference laboratory over a 3-year period of time. In more than 300 individuals screened for 52 heritable limb malformation genes/regulatory regions, 37.2% had a pathogenic or probably pathogenic variant, of which 52 were novel variants. Variants of uncertain significance in highly candidate genes were identified in 6% of patients. Fifty-seven percent of the patients (200/352) were tested negative (no variant identified), suggesting other genomic regions or physio-pathological mechanisms involved.

Accepted Article

limb malformations and requiring reconstructive surgery. Journal of medical genetics, 46, 730-735. doi:10.1136/jmg.2009.066027 This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved.

Figures

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  ), further subdivided in several categories: radial anomalies (Holt-Oram Syndrome, Okihiro Syndrome, Nager Syndrome, RECQL4-disorders, Thrombocytopenia-Absent Radius Syndrome, Townes-Brocks Syndrome, unclassified), ectrodactyly (Split hand or Split-Foot, SHFM, SHFLD, Ectrodactyly-Ectodermal dysplasia-Clefting (EEC) syndrome), brachydactylies (Type A, Type B, Type C, Type E, Robinow syndrome, Tricho-Rhino-Phalangeal syndrome, unclassified), polydactylies (prexaxial, postaxial, synpolydactyly, unclassified), reduction anomalies (transversal, longitudinal, phocomelia/amelia, Adams-Oliver Syndrome, unclassified), fusion anomalies (syndactylies, multiple synostosis), patella hypoplasia (Nail-Patella Syndrome, Small-Patella Syndrome). Note that the reduction anomalies group does not comprise the radial reduction defects, which are classified in the radial anomalies, nor the reduction of the central rays of the autopod, which are classified in the ectrodactyly group.

Figure 1 :

 1 Figure 1: Description of the patient series by phenotype groups.

Figure 2 :

 2 Figure 2: Summary of molecular analyses performed, diagnostic yield and variants identified in our patient series.

Figure 3 :

 3 Figure 3: Diagnosis yield in our series of patients affected with CLM. Percentage of patients for whom a pathogenic or likely pathogenic variant has been identified, depending on the phenotype (isolated or syndromic CLM) and the pedigree (sporadic or familial case).

Figure 4 :

 4 Figure 4: Pictures and radiographs of selected patients affected with congenital limb malformations.

Table S1

 S1

						). The classification also included
	reports	in	the	Human	Gene	Mutation	Database	(HGMD,

http://www.hgmd.cf.ac.uk/ac/index.php), OMIM (Online Mendelian Inheritance in Man) data, ClinVar (https://www.ncbi.nlm.nih.gov/clinvar/) and the relevant literature. Effects on splicing were predicted using the splicing module of Alamut v.2.11.0, questioning the following programs: SpliceSiteFinder-like, MaxEntScan and NNSPLICE (Supp. Table

Table 1 :

 1 Likely pathogenic or pathogenic variants identified in our patient series. . This article is protected by copyright. All rights reserved.

	p.(Met11
	7Serfs*5
	5)	4 This report

Isol: isolated; Sd: syndromic; Fam: familial; Spor: sporadic; ACMG: American College of Medical Genetics and Genomics; SNV: single nucleotide variant; CNV: copy-number variation; HMZ: homozygous; HTZ: heterozygous; HemiZ: Hemizygous; ND: not determined; SH: split hand; SF: split foot; SHFM: split hand/foot malformation; SHFLD: split hand/foot malformation with long-bone deficiency; TAR: thrombocytopenia-absent radius; TRPS: Tricho-rhino-phalangeal; EEC: ectrodactyly-ectodermal dysplasia-cleft lip/palate; BDA, BDB, BDC, BDE: brachydactyly type A, B, C or E.

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Table 2 :

 2 Diagnostic yield in the patient series depending on the phenotype. Nb: number; SH: split hand; SF: split foot; SHFM: split hand/foot malformation; SHFLD: split hand/foot malformation with long-bone deficiency; TAR: thrombocytopenia-absent radius; EEC: ectrodactyly-ectodermal dysplasia-cleft lip/palate; BDA, BDB, BDC, BDE: brachydactyly type A, B, C or E.

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Acknowledgements

We gratefully acknowledge the following: Cindy Crunelle, Marthe Cochard and the bioinformatics team of Lille University Hospital for technical support; Anaïs Covain and Laurence Bellengier for administrative support; Pr. Nigel Quayle for manuscript editing. Also, we thank all the clinicians, patients and families who contributed to this work.

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The authors have no conflict of interest to declare.