Ignacio Salas
email: ignacio.salas@mines-nantes.fr

Gilles Chabert
email: gilles.chabert@mines-nantes.fr

Packing Curved Objects

This paper deals with the problem of packing twodimensional objects of quite arbitrary shapes including in particular curved shapes (like ellipses) and assemblies of them. This problem arises in industry for the packaging and transport of bulky objects which are not individually packed into boxes, like car spare parts. There has been considerable work on packing curved objects but, most of the time, with specific shapes; one famous example being the circle packing problem. There is much less algorithm for the general case where different shapes can be mixed together.

 and the algorithm we propose here is an extension of their work. Martinez et al. use a stochastic optimization algorithm with a fitness function that gives a violation cost and equals zero when objects are all packed. Their main idea is to define this function as a sum of n 2 elementary functions that measure the overlapping between each pair of different objects. However, these functions are ad-hoc formulas. Designing adhoc formulas for every possible combination of object shapes can be a very tedious task, which dramatically limits the applicability of their approach. The aim of this paper is to generalize the approach by replacing the ad-hoc formulas with a numerical algorithm that automatically measures the overlapping between two objects. Then, we come up with a fully black-box packing algorithm that accept any kind of objects.

Introduction

The packing problem consists in placing n items inside a given space, such that no two items overlap. A large number of real-world applications like warehousing, logistics or parallel computing gives a great relevance to this problem. Packing has been well studied in either academic or industrial contexts, but each time considering specific shapes like bins [Lodi et al., 2002], circles [Stephenson, 2005] or polytopes [START_REF] Stoyan | [END_REF]Egeblad et al., 2009]. A lot of problems are actually studied for polytopes, some being closely related to packing, like the collision detection [START_REF] Moore | [END_REF]Brochu et al., 2012;Mainzer and Zachmann, 2015] or the calculation of the penetration depth [Cameron and Culley, 1986;Dobkin et al., 1993;Kim et al., 2002;[START_REF] Kim | [END_REF].

Our goal is to deal with the more general situation where different shapes can be mixed, including non-convex and curved shapes.

In this context, one has to develop a generic approach for solving the packing problem. One such approach has been recently proposed in [START_REF] Martinez | [END_REF]. The approach splits the problem in three parts. First, they build a violation function f ij for each pair (i, j) of objects. This function takes as argument the position o = (x, y) and the orientation α for both objects and gives a measure of "how much" the two objects overlap. We shall call such function an overlapping function.

Second, all these violations are summed up to form a global violation cost f : f ( o 1 , α 1 , . . . , o n , α n ) := i>j f ij ( o i , α i , o j , α j ).

Third, the function f is minimized using a generic blackbox optimization software called CMA-ES [Hansen and Ostermeier, 2001]. The experimental results they obtain are very encouraging. However, there is a serious bottleneck in their approach: the overlapping functions are ad-hoc formulas for every possible combination of object shapes. For instance, in a problem where objects are either circles, squares or triangles, one must build a formula for the 6 combinations: circle-circle, circle-square, circle-triangle, square-square, square-triangle and triangle-triangle. Some formulas are easy to obtain, like for the circle-circle case. Indeed, given two circles of radii r i and r j , the overlapping can be defined as the distance between the two centers and 0 if this distance exceeds the sum of the radii, that is:

f ij ( o i , α i , o j , α j ) := max(0, r i + r j -o i -o j ).
This simple formula is rather an exception. Mixing shapes considerably complicate the task -even the circle-rectangle case is not so simple-. One should be convinced that, in general, these ad-hoc formulas are difficult to write. And their number grows in O(n 2 ).

We show in this paper how to replace these formulas by an algorithm. This algorithm has also been implemented and this paper reports the preliminary experimental results we have obtained on original packing problems.

The paper is organized as follows. We first formalize the problem in §2, that is, the way shapes are represented and which overlapping function we actually consider. Then, we detail in §3 and §4 an algorithm for this overlapping function. Experimental results are reported in §5, followed by concluding remarks.

Formalism

Object definition

We consider objects described by nonlinear inequalities. This representation choice encompasses basic geometric shapes like half-planes or ellipses. We also accept the logical and/or operators in the shape description, which means that these geometric primitives can be combined to define more complex shapes like a polygon, a half-circle or a horseshoe (see Figure 1). Note that non-linear inequalities give also freedom to represent a lot of fancy shapes but real-world objects are rather obtained by combining basic primitives.

With that representation, the shape of Object i can be seen as a regular constraint c i ( p). This constraint means that the point p belongs to the object when the latter is placed at the origin of the frame and with a null rotation angle (see Figure 1). We will identify in the sequel a shape to a constraint and talk about "an object of shape c".

The non-overlapping constraint

An object can be translated or rotated. If Object i is placed at a position o i and rotated by an angle α i , we will denote by q i and call the parameters of Object i the vector:

q i := ( o i , α i ) = (x i , y i , α i ).
The size of q is the number of degrees of freedom. If only translation is considered, the angle is dropped and q has only two components. Otherwise, it belongs to R 3 .

Given a vector of parameters q i , the points p that belong to the translated-rotated Object i are the ones satisfying

c i (R -αi ( p -o i )), (1)
where R is the usual rotation matrix. Therefore, Objects i and j overlap iff

overlap ij ( q i , q j ) ⇐⇒ ∃ p ∈ R 2 , c i (R -αi ( p -o i )) ∧ c j (R -αj ( p -o j )). (2)
The non-overlapping constraint between two objects is just the negation of the latter relation.

The packing problem

The packing problem is a set of pairwise non-overlapping constraints between n objects and an additional constraint that all objects must be packed inside some container (like the enclosing bin in the bin packing).

One appealing aspect of our representation is that we don't actually need a specific treatment for this additional constraint. This is based on a very simple property. An object is inside the container iff this object does not overlap the complementary of the container. And the complementary of a shape is simply obtained with the negation of the underlying constraint. So, in our system, the packing problem consisting in placing n objects of shapes c 1 , . . . , c n inside a container c is reformulated as a non-overlapping constraint between n+1 shapes c 1 , . . . , c n , ¬c.

Note that, for obvious symmetry breaking reasons, one has to fix the origin and the orientation of one object. A natural choice is then to fix that of the container.

The overlapping function

The overlapping function f ij takes as input two vectors of parameters q i and q j and gives a measure of "how much" Objects i and j overlap. The output must fulfill two properties. First, it has to be 0 iff the two objects are disjoint. Second, it has to decrease when the two objects gets more distant.

We define the overlapping function as the distance between the current vector of parameters for Object j, and the closest vector of parameters that satisfies the non-overlapping constraint. This way, the function f ij can be seen as a "distance to satisfaction" for the Object j, the parameters of Object i being considered as fixed. In fact, this function exactly matches the concept of penetration depth in the realm of computational geometry [Cameron and Culley, 1986;Dobkin et al., 1993;Kim et al., 2002;[START_REF] Kim | [END_REF]. Definition 1 (Overlapping Function).

f ij ( q i , q j ) = min{ q j -q j , ¬ overlap ij ( q i , q j )}.

Note that the distance . can be either in 2 or 3 dimensions, depending if rotation is considered or not.

As already said, the previous definition introduces a distinction between the two shapes. Object i is called the reference object and Object j the moving object. Without rotation, the roles are symmetric in the sense that:

f ij ( q i , q j ) = f ji ( q j , q i ).
However, the previous equality does not hold with rotation. Henceforth, we will always assume Object i (resp. j) to be the reference (resp. moving) one.

It is important to notice that our definition of the overlapping function does not coincide with the overlapping surface, as Figure 2 shows. Furthermore, taking the overlapping surface as a measure of overlapping would not be appropriate since the surface can reach a local minimum in a situation where the two objects still overlap (see Figure 2). In contrast, our overlapping function only reaches a minimum at 0, that is, when the objects are disjoint. This is an important difference as this function is eventually called by a local search that can be trapped in local minima.

Figure 2: Overlapping surface versus overlapping function. The first object is the rectangle frame in blue, the second the rectangle in green. In this situation, the overlapping surface, in red, reaches a local minimum while the objects still overlap.

The overlapping region

The implementation we will propose for the overlapping function is based on the concept of overlapping region, a concept that was already introduced in [Salas et al., 2014]. We recall this concept and then give the theoretical result that makes the connection with our definition of overlapping function.

The overlapping region simply corresponds to the set of parameters for Object j that make the objects i and j overlap, the parameters of Object i being all set to 0: Definition 2 (Overlapping Region). Given Objects i and j (resp. the reference and moving ones), the overlapping region is

S ij := { q j ∈ R d | overlap ij ( 0 R d , q j )}.
This concept is depicted in Figure 3.

Figure 3: Overlapping region. For visibility, the dimension here is 2 (rotation is not considered). (left) Objects i (in blue) and j (in green). (right) The overlapping region S ij (in red). Represents all the positions where Object j overlaps Object i, placed at the origin. The distance between the point r( q i , q j ) and the boundary of S ij is the value of the overlapping function f ij (Proposition 1).

We show now the relation that connects the overlapping function to the overlapping region. Proposition 1. Given two vectors of parameters q i and q j , let r( q i , q j ) :

= (R -αi ( o j -o i ), α j -α i ). (3)
Then f ( q i , q j ) = min{ q -r( q i , q j ) , q ∈ S ij }.

Proof. By definition, f ( q i , q j ) = min{ q j -q j , ¬ overlap ij ( q i , q j )}.

Let us introduce the variable

q = ( o , α ) := (R -αi ( o j -o i ), α j -α i ).
We have, equivalently,

q j = (R αi o + o i , α + α i ).
In the one hand:

q j -q j 2 = (R αi o + o i -o j , α + α i -α j ) 2 = R αi ( o -R -αi ( o j -o i )), α + (α j -α i )) 2 = R αi ( o -R -αi ( o j -o i ) 2 + |α + (α j -α i )| 2 = o -R -αi ( o j -o i ) 2 + |α + (α j -α i )| 2
(because rotation preserves distances) = q -r( q i , q j ) 2 .

In the other hand it can be proven (see Proposition 1 in [Salas et al., 2014]) that

overlap ij ( q i , q j ) ⇐⇒ (R -αi ( o j -o i ), α j -α i ) ∈ S ij , that is, overlap ij ( q i , q j ) ⇐⇒ q ∈ S ij .
Hence, the overlapping function maps the current position of Objects i and j to the minimal distance between r( q i , q j ) and the boundary of the overlapping region. This is also depicted in Figure 3.

Main Algorithm

The algorithm we propose for the overlapping function is a direct consequence of Proposition 1. We first calculate (offline) the overlapping region and then obtain (on-line) the value of f ( q i , q j ) from it. More precisely, the two steps are:

1. We entirely calculate the overlapping region S ij . Calculating this region means that an explicit representation of S ij is computed, that is, under the form of numerical data. The data structure we use for S ij is detailed below. 2. We calculate the minimal distance between r( q i , q j ) and S ij using the previous data. The point is that this minimization problem is far more easy to solve than the original minimization problem involved in Definition 1.

Paving of the overlapping region

Let us start with the first step. The data structure we use for S ij is called a paving (see Figure 4). To calculate this paving, we apply the strategy proposed in [Salas et al., 2014]. It ensures that the total surface of the boundary (or, in other words, the precision of the paving) is less than a given value ε. Let us briefly outline this algorithm again, since we will describe an adaptation in the next section.

The algorithm is called a paver [START_REF] Chabert | [END_REF]], a recursive algorithm that starts with an arbitrarily large box [q j ] and alternates three steps:

1. (outer rejection test). If unsatisfiability of ( 2) is proven for all q j ∈ [q j ] ( q i being fixed), then

O := O ∪ {[q j ]}.
The rejection test resorts to an embedded solver that we shall not describe further, since this part has not changed from the paper cited above. 2. (inner inflation). Pick randomly a point qj ∈ [q j ] and check if it belongs to S ij . If it does, then "inflate" qj to a subbox [q j ] proven to entirely lie inside S ij . In contrast, this part has been significantly improved and will be the topic of Section 4. Then, I := I ∪ {[q j ]}. Break the remaining part [q j ]\[q j ] into boxes and perform a recursive call with each box. 3. if [q j ] is small enough, B := B∪{[q j ]}. Otherwise bisect [q j ] into two sub-boxes and perform a recursive call with each boxes.

Distance to the boundary set

Once the paving (I, B, O) is calculated, we first set r := r( q i , q j ) using (3) and then find the closest point to r which is not inside the overlapping region. However, because the boundary of the region is uncertain, we can actually only obtain an enclosure of this minimal distance:

min q ∈∪B q -r ≤ f ij ( q i , q j ) ≤ min q ∈∪O q -r .
Clearly, the accuracy of the enclosure is related to the precision ε the paving has been generated with.

In practice, to calculate a bound, say the upper one, we consider each box [q ] of the set O generated by the paver, and minimize the distance over that box. That is, we calculate:

mindist( r, [q ]) := min q ∈[q ]
q -r .

(5)

Then, we have:

min q ∈∪O q -r = min [q ]∈O mindist( r, [q ]). (6)
The same holds for the lower bound.

Formula ( 5) is nothing different than the distance between a box and a point. This distance can be easily obtained in constant time (whatever is the size of the box), either using interval arithmetics or by a direct geometric argument.

Formula (6) can however be very time-consuming if the boxes in O are stored in a flat unsorted list. One has to systematically scan the whole list to get the minimal distance. This linear complexity can be easily transformed to a logarithmic complexity using a tree-structure representation, as depicted in Figure 4. The structure is then explored using standard global optimization techniques. In particular, pending nodes are stored in a heap, the criterion associated to a node being precisely its distance to r. Nodes with a distance exceeding the current upper bound for the minimum can then be discarded (just like the bounding step in a branch & bound).

A new inflator for the overlapping constraint

The inflation plays a key role in the paving algorithm and thus has a strong impact on the efficiency of our packing approach.We describe in this section an original inflation technique that substantially improves packing performances.

Recall that the inflator builds, from a single vector of parameters qj , a box [q j ] that is inside the overlapping region S ij .

In [Salas et al., 2014], the inflator proposed inflates in all the dimensions of q simultaneously and turns out to be not efficient when the dimension is 3 (i.e., with rotation handled). In fact, this inflator is very sensitive to the number of times a variable appears in Equation ( 1), and the angle α has precisely many occurrences yield by the rotation matrix R -α . It was reported that the time required to calculate 3D pavings with this inflator was prohibitive.

Our new inflator does not suffer from the multiple symbol occurrences introduced by rotation matrices. It splits the inflation in two steps: it first inflates with respect to o and then with respect to α. We describe both steps and then show how the two inflations can be combined.

Translation

The inflation w.r.t. o is based on the following principle (see Figure 5).1 Given initial parameters ( õj , αj ) we first look for a vector p that belongs to both objects, the parameters of Objects i and j being respectively 0 and qj , that is, p satisfies

c i (p) ∧ c j (R -αj (p -õj )).
Then, we inflate p inside Object i using generic inflation technique for inequalities [START_REF] Chabert | [END_REF]Araya et al., 2014]

Rotation

The inflation for the angle works as follows (see Figure 6). Given initial parameters ( õj , αj ) we look for the two angles α and α that force vector p (the same as before) to meet the boundary of Object j (this step is detailed in the next subsection). Then, it is clear that for every angle α j ∈ [α, α] there is an overlapping. However, some caution is required. First, there are generally more than 2 candidate angles (see Figure 6.d), especially if Object j is non-convex. The angle values that form the narrowest interval around αj are kept. Second, there may be no angle at all. This means that Object j can make a complete turn while containing p (or, in other words, p is in the inscribed circle of the object). Third, we have to take into account the 2π-periodicity. Indeed, the search space for angles is a bounded domain like [-π, π] or [0, 2π]. This has the unpleasant consequence to make the inflated domain disconnected, e.g., if αj = π 2 , α = 0 and α = 3π 2 then the inflated interval

[0, 3π 2 ] becomes [-π, -π 2 ] ∪ [0, π].
So, in this case, we only keep the subinterval containing αj , that is,

[-π, -π 2 ].

Shape Boundary

Let us explain now how the angles α and α are calculated. First, we have to generate a constraint c j for the boundary of Object j. In the simplest case of a single inequality,

c j ( p) ⇐⇒ f ( p) ≤ 0,
the boundary of the object is described by the equality f ( p) = 0.

In the case of a conjunction of inequalities,

c j ( p) ⇐⇒ (f 1 ( p) ≤ 0 ∧ . . . ∧ f n ( p) ≤ 0).
The boundary is described by a disjunction of n systems: In the most general case of a disjunction of conjunctions of inequalities,

c j ( p) ⇐⇒ (f 1 ( p) = 0 ∧ f 2 ( p) ≤ 0 . . . ∧ f n ( p) ≤ 0) ∨ . . . ∨ (f 1 ( p) ≤ 0 ∧ . . . ∧ f n-1 ( p) ≤ 0 ∧ f n ( p) = 0). (a) (b) (c) (d)
c j ( p) ⇐⇒ (f 11 ( p) ≤ 0 ∧ . . . ∧ f 1n1 ( p) ≤ 0) ∨ . . . . . . ∨ (f m1 ( p) ≤ 0 ∧ . . . ∧ f mnm ( p) ≤ 0).
The boundary is described by a disjunction of (n 1 +. . .+n m ) conjunctions, each block of conjunctions being obtained with the previous pattern (see Figure 7). The boundary constraint c j can therefore be automatically generated by a simple symbolic processing.

The angle α and α are then two solutions of the system c j (R -α (p -õj )) which is a disjunction of 1-dimensional equations (of variable α), i.e., something very easy to solve.

Note that if the shape is a disjunction with two subparts that plainly intersect (contrary to the figure), the boundary is only a subset of the generated system, in which case the angle inflation may not be optimal.

Global Inflation

We finally show that the two inflations can be combined, that is, the cartesian product of Together, they give

∀o j ∈ [o j ], ∀α j ∈ [α j ], c i (o j +(p-õj ))∧c j (R -αj (p-õj )).
Substituting p to (o j + p -õj ), we obtain:

∀o j ∈ [o j ], ∀α j ∈ [α j ], ∃ p, c i ( p) ∧ c j (R -αj ( p -o j )), that is, ∀o j ∈ [o j ], ∀α j ∈ [α j ], ( o j , α j ) ∈ S ij .

Experimental Results

The goal of these preliminary experiments is to validate the theory and to have a first impression of how the algorithm behaves in practice. The ambition is neither to solve real-world problems nor to make an extensive performance analysis.

Setup

We have created 5 cases of increasing difficulty. In each case, n objects have to be placed with n = 10, 20 and 30. Each time, the size of the container has been adjusted manually so that the density of the resulting packing looks similar.

Case 1. This is a standard circle packing problem. The objectives are, first, to check that our algorithm also succeeds in solving this problem and, second, to measure its overhead as compared to a dedicated approach. In this circle packing instance, the radius of the i th circle is equal to √ i, 1 ≤ i ≤ n. The radius of the enclosing circle is set to 2.1 (n = 10), 2.35 (n = 20) and 2.48 (n = 30).

Case 2. The purpose of this case is to introduce a new shape, that is, for which (to our knowledge) no dedicated solver exists. The problem is otherwise kept as simple as possible: (1) there is only one shape in addition to the enclosing shape, (2) this shape is described by a single inequality and (3) only translation is permitted (no rotation). The shape we have chosen is an ellipse of radii 1 and 0.5. The ellipses have to be packed inside a circle of radius 2.9 (n = 10), 4 (n = 20) and 5 (n = 30).

Case 3. We increase the complexity by allowing rotation in the same ellipse packing instance. With this new degree of freedom, the enclosing circle can be chosen a little bit smaller. The radius of the circle is set to 2.7 (n = 10), 3.9 (n = 20) and 4.9 (n = 30).

Case 4. We increase again the complexity by replacing the ellipse with a non-convex shape which description involves several inequalities. The shape is the horseshoe depicted in Figure 1. The enclosing shape is an ellipse of radii 2/4 (n = 10), 3.9/1.8 (n = 20) and 5/2.9 (n = 30).

Case 5. The last case is a mixed packing problem where n/2 ellipses and n/2 horseshoes have to be packed inside a circle of radius 2.9 (n = 10), 4 (n = 20) and 6 (n = 30). Again, both translation and rotation are considered.

Results

Remind that the packing process is in two steps. The paving step, performed off-line, and the packing step. Since some pavings are used in several cases, paving and packing times are reported separately. We also only give the average time for the 13×12 circle-circle pavings calculated in Case 1 and for the other pavings involving the enclosing shape (since this shape changes with n). Tables 1 and2 summarize the paving and packing times. Figure 8 shows pictures of the results. Every paving has been calculated with a precision ε set to 0.1. First, we can see that the circle packing problem has been solved in a few seconds only. However, the enclosing circle is not optimal. In [START_REF] Martinez | [END_REF], the circles are packed in a circle of radius 1.95 (n = 10), 2.15 (n = 20) and 2.27 (n = 30). This gap is due to the precision ε. Using this precision for paving amounts to consider that objects to pack are "enlarged by" ε. The radius we have set for the enclosing circle is actually the smallest one for which packing is successful, given the precision ε of 0.1.

Packing for the other cases has been done in the order of the minute. Above all, we see in Figure 8 that the holes introduced by non-convex objects like the horseshoe are not lost. The paving time is in the order of the hour.

Conclusion

We have proposed a generic algorithm for packing non convex curved objects. The approach is not limited to polyhedral shapes and does not make convex approximation. Cavities introduced by non-convexity are used to place objects.

The approach is split in two steps. The paving step calculates off-line the set of relative positions and orientations of one object with respect to a second one that make them overlap. The packing step minimizes a fitness function which is a sum of distances between points and boundaries of the previously calculated sets. These distances are fast to calculate thanks to a tree-structured representation of the sets.

The other contribution is a new inflator for the overlapping constraint, i.e., the paving step. This new inflator uses separate techniques for inflating with respect to the position and the angle, which leads to a much larger inflation (and therefore a better packing time) than with previously existing techniques.

Preliminary experimental results are encouraging. One possible future experiment could be to see how both paving and packing time increases as ε decreases and to mix this numerical approach with the available ad-hoc formulas.

Figure 1 :

 1 Figure 1: Half-circle and horseshoe. The constraint for the half-circle of radius 1 is c( p) ⇐⇒ ( p ≤ 1 ∧ y p ≥ 0). The constraint for the horseshoe is c( p) ⇐⇒ ( p ≤ 1 ∧ p ≥ 0.75 ∧ y p ≥ 0).

Definition 3 (Figure 4 :

 34 Figure 4: Paving of an ellipse. Boxes of I (resp. B, O) are painted in red (resp. blue, green).

  . Let u := p -õj . The resulting box [ p] can be translated to õj in the sense that if o j moves inside [ o j ] := [ p ] -u then o j + u is both inside [ p ] and inside the Object j with parameters ( o j , αj ).

Figure 5 :

 5 Figure 5: Inflation w.r.t. translation. (a) The shapes of Objects i (in blue) and j (in green). (b) A point p at the intersection of the two objects when Object j has parameters ( o j , αj ). (c) Inflation of p inside Object i. (d) Inflation of o j .

Figure 6 :

 6 Figure 6: Angle inflation. (a) Point p obtained with initial parameters (translation step). (b) The smallest angle greater than αj that makes p meet the boundary of object j. (c) The greatest angle lower than αj with the same property. (d) Another angle with the same property (there are 4) that is discarded.

Figure 7 :

 7 Figure 7: Boundary of an object.

  [ o j ] and [α j ] = [α, α] is a valid inflation of qj . The inflation w.r.t. translation produces a box [ o j ] that satisfies ∀o j ∈ [o j ], c i (o j + (p -õj )). The inflation w.r.t. rotation produces a box [α j ] that satisfies ∀α j ∈ [α j ], c j (R -αj (p -õj )).

Figure 8 :

 8 Figure 8: Packing results.

Table 1 :

 1 Paving time (in seconds)

	d.o.f.	Object i		Object j	Time
		circle		circle	0.01 (avg)
	translation	ellipse		ellipse	0.16
		enclosing circle	ellipse	0.81 (avg)
		ellipse		ellipse	300
	translation and rotation	enclosing circle ellipse horseshoe enclosing circle	ellipse horseshoe horseshoe horseshoe 8311 (avg) 2245 (avg) 1740 3780
		enclosing ellipse horseshoe 10909 (avg)
	#objects case 1 case 2 case 3 case 4 case 5
	10	<0.1	4.5	66	103	53
	20	54	32	235	372	194
	30	203	53	559	1005	395

Table 2 :

 2 Packing time (in seconds)

This part is similar to the old inflator applied in the

2D case.


Emmanuel Wiernik

Bruno Pannier

Sébastien Czernichow

Hermann Nabi

Olivier Hanon

Tabassome Simon

Jean-Marc Simon

Frédérique Thomas

Kathy Bean

Silla M Consoli

Nicolas Danchin

Dr Cédric Lemogne
email: cedric.lemogne@egp.aphp.fr

OCCUPATIONAL STATUS MODERATES THE ASSOCIATION BETWEEN CURRENT PERCEIVED STRESS AND HIGH BLOOD PRESSURE: EVIDENCE FROM THE IPC COHORT STUDY

Although lay beliefs commonly relate high blood pressure to psychological stress exposure, research findings are conflicting. This study examined the association between current perceived stress and high blood pressure and explored the potential impact of occupational status on this association. Resting blood pressure was measured in 122,816 adults (84,994 men), aged ≥30 years (mean age ± standard deviation: 46.8±9.9 years), without history of cardiovascular and renal disease and not on either psychotropic or antihypertensive drugs.

High blood pressure was defined as systolic blood pressure ≥140 mmHg or diastolic blood pressure ≥90 mmHg. Perceived stress in the past month was measured with the 4-item perceived stress scale. A total of 33,154 participants (27.0%) had high blood pressure (151±14/90±9 mmHg). After adjustment for all variables except occupational status, perceived stress was associated with high blood pressure (odds ratio for a 5-point increase:

1.06, 95% confidence interval: [1.03-1.09]). This association was no longer significant after additional adjustment for occupational status (odds ratio: 1.01 [0.99-1.04]). There was a significant interaction (p<0.001) between perceived stress and occupational status in relation to blood pressure: perceived stress was negatively associated with high blood pressure among individuals of high occupational status (odds ratio: 0.91, [0.87-0.96]) but positively associated among those of low status (odds ratio: 1.10, [1.03-1.17]) or unemployed (odds ratio: 1.13, [1.03-1.24]). Sensitivity analyses yielded similar results. The association between current perceived stress and blood pressure depends upon occupational status. This interaction may account for previous conflicting results and warrants further studies to explore its underlying mechanisms.

epidemiology; hypertension; interaction; moderation; occupational status; socioeconomic position; stress.

Introduction

Hypertension is a leading risk factor for cardiovascular mortality worldwide and has several known risk factors such as obesity, smoking, excessive alcohol or salt intake [START_REF] Chobanian | Seventh report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure[END_REF] .

However, most patients with hypertension lend great importance to psychological stress in the regulation of blood pressure (BP) and in the need for taking antihypertensive drugs [START_REF] Marshall | Lay perspectives on hypertension and drug adherence: systematic review of qualitative research[END_REF] .

Although acute psychological stress is associated with a transient BP elevation [START_REF] Chida | Chronic psychosocial factors and acute physiological responses to laboratory-induced stress in healthy populations: a quantitative review of 30 years of investigations[END_REF] , epidemiological studies do not consistently show chronic psychological stress to affect BP in the long-term [START_REF] Sparrenberger | Does psychosocial stress cause hypertension? A systematic review of observational studies[END_REF] . Some studies found positive associations between psychological stress and hypertension [START_REF] Levenstein | Psychosocial predictors of hypertension in men and women[END_REF][START_REF] Markovitz | Increases in job strain are associated with incident hypertension in the CARDIA Study[END_REF] , while others showed no [START_REF] Sparrenberger | Stressful life events and current psychological distress are associated with self-reported hypertension but not with true hypertension: results from a cross-sectional population-based study[END_REF][START_REF] Fauvel | Neither perceived job stress nor individual cardiovascular reactivity predict high blood pressure[END_REF] or even negative associations [START_REF] Winkleby | Self-reported stressors and hypertension: evidence of an inverse association[END_REF][START_REF] Suter | Relationship between self-perceived stress and blood pressure[END_REF] .

Reasons for these conflicting results can be potentially attributed to differences in the evaluation of psychological stress (e.g. objective measures versus subjective measures) or outcomes (e.g. BP threshold values, use of antihypertensive drugs). As regards psychological stress, objective measures quantify the exposure to several kinds of stressors (e.g. stressful life events, occupational stress) whereas subjective measures, such as perceived stress, quantify the psychological impact of these stressors [START_REF] Cohen | A global measure of perceived stress[END_REF] . Previous studies may also have overlooked the potential confounding role of some psychosocial variables. For instance, few studies have included a measure of depression, which is linked to stress but may be associated with a lower BP [START_REF] Hildrum | Anxiety and depression lowers blood pressure: 22-year follow-up of the population based HUNT study, Norway[END_REF][START_REF] Nabi | Trajectories of depressive episodes and hypertension over 24 years: the Whitehall II prospective cohort study[END_REF] . Another important factor is socioeconomic status (SES), which is negatively associated with hypertension, especially in high-income countries [START_REF] Colhoun | Socio-economic status and blood pressure: an overview analysis[END_REF][START_REF] Dyer | Ten-year incidence of elevated blood pressure and its predictors: the CARDIA study. Coronary Artery Risk Development in (Young) Adults[END_REF] .

However, studies adjusting for SES do not usually test for a moderation hypothesis (i.e. that stress may relate to hypertension to a different extent according to SES). Although there is preliminary evidence that job strain may relate to BP at work site to a different extent across certain occupational categories [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF] , little is known about the role of the SES in moderating the association of hypertension with perceived stress from a broader perspective, as well as among non-working individuals.

The aim of this study was to examine the association between perceived stress and BP, and to explore the potential moderating role of occupational status on this association.

Occupational status is a useful proxy for SES as it integrates the educational achievements, the skills required to obtain a job, the long-term associated rewards (including, but not limited to, income) and several job characteristics, such as working conditions and decision-making latitude.

Methods

Participants

All participants were recruited at the "Investigations Préventives et Cliniques" (IPC) Center (Paris, France). This medical Center, which is subsidized by the French national health care system, offers all working and retired individuals and their families a free medical examination every five years. It carries out approximately 25,000 examinations per year for people living in the Paris area. Our target population was composed of all subjects who had at least one health checkup at the IPC Center in the period from January 1996 to December 2007. All clinical and biological parameters were evaluated on the same day at the examination. In the case of participants who benefited from more than one examination, only data from the first examination were considered. Eligibility criteria were: 30 years of age or more (owing to the low prevalence of hypertension in younger individuals), able to fill out the study questionnaires, and with no missing data for selected variables (see below). To minimize potential biases, individuals with a history of cardiovascular or renal disease and those who reported using antihypertensive or psychotropic drugs were not included in the first set of analyses but were included in sensitivity analyses. The IPC Center received authorization from a local ethics committee and from the "Comité National d'Informatique et des Libertés" to conduct these analyses. All subjects gave their informed consent at the time of the first examination. The procedures followed were in accordance with institutional guidelines. The data were rendered anonymous before analysis.

Blood pressure and outcome

After a 10-minute rest period, supine brachial systolic and diastolic BP were measured 3 times by trained nurses in the right arm using a semi-automated mercury sphygmomanometer. A standard cuff size was used, but a large cuff was utilized if necessary.

The mean of the last 2 measurements was considered as the BP value. The primary outcome of the present study was a high BP, defined as systolic BP ≥ 140 mmHg or diastolic BP ≥ 90 mmHg [START_REF] Chobanian | Seventh report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure[END_REF] .

Psychological variables

Perceived stress was measured with the French version of the 4-item Perceived Stress Scale (PSS-4) [START_REF] Cohen | A global measure of perceived stress[END_REF][START_REF] Lesage | Psychometric properties of the French versions of the Perceived Stress Scale[END_REF] . Each item is rated on a 0 to 4 scale (please see Appendix 1 in the online Data Supplement at http://hyper.ahajournals.org). The PSS-4 total score has a one-factor structure and a satisfactory internal consistency (α=0.73). It measures the degree to which situations in one's life over the past month were appraised as stressful (e.g. "In the past month, how often have you felt it was difficult to control the important things in your life?").

In order to obtain meaningful odds ratios, the variable was rescaled using the difference between the 25 th and the 75 th percentile as the unit.

Depressive mood was measured with a French 13-item questionnaire (QD2A, Questionnaire of Depression 2 nd version, Abridged) [START_REF] Pichot | Un questionnaire d'autoévaluation de la symptomatologie dépressive, le Questionnaire QD2: I. Construction, structure factorielle et propriétés métrologiques[END_REF][START_REF] Pichot | Un questionnaire d'autoévaluation de la symptomatologie dépressive, le questionnaire QD2. II: Forme abrégée QD2A[END_REF] . Building on previous questionnaires, this 13-item questionnaire was specifically designed for depression screening in community studies and has a high internal consistency (α=0.91). Participants had to give a yes/no answer to each item as regards their current emotional state (e.g. "I am disappointed and disgusted with myself," "I'm sad these days," "I feel hopeless about the future"). The number of "yes" answers is summed, a total score ≥ 7 indicating a high probability of major depression. The QD2A has been found to predict suicide in the IPC Cohort Study [START_REF] Lemogne | Heart rate and completed suicide: evidence from the IPC cohort study[END_REF] .

Occupational status

Occupational status was categorized in 6 classes: (1) high (e.g. managers); ( 2 6) others (e.g. artisans). Retired participants were assigned to their last occupational category. The distinction of three categories among working participants is standard among occupational cohorts examining the relationships between psychosocial variables and physical health outcomes [START_REF] Kivimäki | Job strain as a risk factor for coronary heart disease: a collaborative meta-analysis of individual participant data[END_REF] .

Other covariates

Others covariates included age, gender, living status (living alone or not), smoking status (non-smoker, ex-smoker, current smoker of 1-10 cigarettes/day, 11-20 cigarettes/day, >20 cigarettes/day), at-risk alcohol intake (more than 2 glasses/day for women or 3 glasses/day for men), and regular physical activity (i.e. estimated equivalent to at least one hour/day of walking). Personal history of cardiovascular or renal disease, and family history of hypertension were self-reported (yes, no), as well as current medications including diuretics, antihypertensive drugs (other than diuretics), medications "to sleep" or "for anxiety or depression." Among participants reporting taking diuretics, only those that reported doing this "to lower BP" were considered as taking an antihypertensive drug. Perceived health status was collected with a 10-point scale (with 10 considered to be "excellent health").

Body mass index (BMI) was calculated and categorized in 4 classes (<18.5; 18.5-24.99; 25-29.99; ≥30 kg/m 2 ). Resting heart rate (HR) was measured in beats per minute with a 10-cycle electrocardiogram (HR = 60 / RR interval in seconds) and fasting glycemia in mmol/L.

Statistical analysis

Statistical analysis was carried out with the PASW Statistics software (version 18.0).

Except for BP, all variables were analyzed as continuous when available as such. Participants who had high BP and those who did not were first compared with respect to each variable with binary logistic regressions. Odds ratio (OR) and 95% confidence interval (CI) were then computed in three multivariate models. Model 1 included all variables except the occupational status which was added in model 2. Model 3 further included the interaction between stress and occupational status. A stratified analysis was conducted whenever the interaction was significant. Sensitivity analyses were also carried out, including participants who reported using antihypertensive drugs (considered as having high BP) or those with a history of cardiovascular or renal disease.

Results

Study population selection is described in Figure S1 (available in the online Data Supplement at http://hyper.ahajournals.org). The final study population consisted of 122,816 participants (84,994 men and 37,822 women) with a mean age of 46.8±9.9 years. The mean perceived stress score was 3.7±2.9 with a 5-point difference between the 25 th and the 75 th percentile. Mean perceived stress scores among participants with high, medium and low occupational status were 2.9±3.9, 3.9±2.9 and 4.4±3.0, respectively (p for linear trend <0.001). Mean perceived stress scores among other categories were 4.9±3.2 in unemployed participants, 4.2±3.0 in participants with unpaid occupation and 3.7±2.8 in other participants.

A total of 33,154 participants (27.0%) had high BP (mean systolic / diastolic BP: 151±14 / 90±9 mmHg). Table S1 (available in the online Data Supplement at http://hyper.ahajournals.org) shows the characteristics of study participants. Multivariate models are displayed in Table 1. The association between perceived stress and high BP was significant after adjustment for all variables except occupational status (i.e. model 1), but this association was no longer significant after further adjustment for occupational status (i.e. model 2).

The interaction between perceived stress and occupational status was significant in model 3 (Table 1), suggesting that the association between perceived stress and high BP should be examined in each occupational category, separately. Adjusting for all other variables (i.e. model 1), perceived stress was negatively associated with high BP among participants with high occupational status (OR: 0.91; 95% CI: 0.87-0.96) whereas this association was positive for those with low occupational status (OR: 1.10; 95% CI: 1.03-1.17) and those who were unemployed (OR: 1.13; 95% CI: 1.03-1.24) (Figures 1 and2). The association between perceived stress and high BP was not significant among other occupational categories. In a post hoc analysis, we examined whether the OR significantly increased from high to low occupational status. Including only participants with high, medium or low occupational status, there was a significant interaction between perceived stress and occupational status taken as a linear variable (p<0.001).

In sensitivity analyses, similar patterns of results were obtained when participants who reported using antihypertensive drugs (considered as having high BP) or those with a history of cardiovascular or renal disease were included. Finally, we computed general linear models to examine the association of perceived stress with systolic and diastolic BP as continuous variables in each occupational category, separately. After adjustment for all variables, perceived stress was negatively correlated with systolic and diastolic BP among high status participants (regression coefficients : -0.531, p<0.001 and -0.226, p=0.019, respectively), positively with systolic and diastolic BP among low status participants (regression coefficients: 1.014, p=0.001 and 0.383, p=0.008, respectively), and positively with systolic BP among unemployed participants (regression coefficient: 0.750; p=0.007).

Discussion

Summary of findings

This cross-sectional study aimed to examine the association between perceived stress and high BP and to explore a potential moderating effect of occupational status. To our knowledge, this is the largest study to address this question and the first one to show that occupational status moderates the association between perceived stress and high BP outside the context of work site. Perceived stress was associated with high BP after adjustment for all variables except occupational status. Although this association disappeared after additional adjustment for occupational status, the interaction between occupational status and perceived stress was significant. In analyses stratified by occupational categories, perceived stress was negatively associated with high BP among participants of high occupational status, but positively associated among those of low occupational status and among the unemployed. Not surprisingly, perceived stress was higher among these two categories, thus explaining the lack of association between perceived stress and high BP when adjusting for occupational status. However, these differences may not account for the significant interaction between occupational status and perceived stress, as main effects were included in the model.

Explanatory hypotheses

Some neurobiological pathways may partially account for the links between current perceived stress, occupational status and high BP [START_REF] Pickering | Cardiovascular pathways: socioeconomic status and stress effects on hypertension and cardiovascular function[END_REF] . The neural underpinnings of BP regulation are strikingly similar to those of emotion regulation [START_REF] Kober | Functional grouping and cortical-subcortical interactions in emotion: a meta-analysis of neuroimaging studies[END_REF][START_REF] Lemogne | Negative affectivity, self-referential processing and the cortical midline structures[END_REF] , including (but not limited to) the serotonin system [START_REF] Kaumann | 5-hydroxytryptamine receptors in the human cardiovascular system[END_REF] and the interplay of the medial prefrontal cortex with the insula and the amygdala [START_REF] Gianaros | A review of neuroimaging studies of stressor-evoked blood pressure reactivity: emerging evidence for a brain-body pathway to coronary heart disease risk[END_REF] . This overlap may account for the well-known association between acute mental stress and a transient BP elevation [START_REF] Chida | Chronic psychosocial factors and acute physiological responses to laboratory-induced stress in healthy populations: a quantitative review of 30 years of investigations[END_REF] . Interestingly, perceived social rank is associated with both anatomical and functional changes within this brain network [START_REF] Mcewen | Central role of the brain in stress and adaptation: links to socioeconomic status, health, and disease[END_REF] . For instance, low perceived social rank is associated with increased amygdala reactivity to social stressors [START_REF] Mcewen | Central role of the brain in stress and adaptation: links to socioeconomic status, health, and disease[END_REF] .

The amygdala has a critical role in gating the sensitivity of the baroreflex through projections that inhibit the nucleus tractus solitarius and that activate the rostroventrolateral medulla [START_REF] Gianaros | A review of neuroimaging studies of stressor-evoked blood pressure reactivity: emerging evidence for a brain-body pathway to coronary heart disease risk[END_REF] .

This provides a neurobiological model of how occupational status may moderate the association between perceived stress and high BP. This neurobiological model may underlie or complete other higher-level hypotheses.

First, perceived stress may partially result from work-related stress, which obviously relates to occupational categories. Occupational categories may differ in terms of exposure to job strain, which combines high job demands with low control at work [START_REF] Kivimäki | Job strain as a risk factor for coronary heart disease: a collaborative meta-analysis of individual participant data[END_REF] , tends to be associated with both high BP and lower occupational category [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF][START_REF] Rosenthal | Occupational stress and hypertension[END_REF] . Thus, among individuals of high occupational status, increased perceived stress may indeed relate to a more favorable ratio between job demands and job decision latitude, the contrary being true among their lower status counterparts. Higher decision latitude may thus overcome the impact of increasing stress on BP in the former, while lower decision latitude may even worsen it in the later. Although the detrimental effects of low decision latitude on health are well established [START_REF] Kivimäki | Job strain as a risk factor for coronary heart disease: a collaborative meta-analysis of individual participant data[END_REF] , recent evidence suggests that high decision latitude may have positive effects on the biological underpinnings of stress such as cortisol level [START_REF] Sherman | Leadership is associated with lower levels of stress[END_REF] . Beyond job strain, perceived stress may also relate to exposure to occupational stressors that are specific to certain occupational categories (e.g. noise, cardiotoxic chemicals) and differently associated with the risk of hypertension [START_REF] Tomei | Occupational exposure to noise and the cardiovascular system: a meta-analysis[END_REF][START_REF] Poreba | Ambulatory blood pressure monitoring and structural changes in carotid arteries in normotensive workers occupationally exposed to lead[END_REF] .

Second, participants with lower occupational status might have been less likely to deal with stress with adaptive health behaviors (e.g. physical activity) and more with detrimental ones (e.g. alcohol consumption) [START_REF] Siegrist | Work stress and health risk behavior[END_REF] . However, it is noteworthy that multivariate models were adjusted for health behaviors. Nevertheless, we cannot rule out this hypothesis regarding other unmeasured health behaviors such as dietary patterns or salt consumption.

However, these hypotheses do not account for the apparently "protective effect" of perceived stress among persons of high occupational status. A third hypothesis may build on the construct of emotional awareness, which is one's ability to represent, discriminate, and elaborate one's own emotional state [START_REF] Lane | Levels of emotional awareness: a cognitive-developmental theory and its application to psychopathology[END_REF] . From a theoretical point of view, the perceived stress score results from two components: stress per se and the ability to be aware of it. A large body of evidence suggests that the former may have detrimental effects on health, whereas the latter may have beneficial effects [START_REF] Lane | Neural substrates of implicit and explicit emotional processes: a unifying framework for psychosomatic medicine[END_REF] . Higher occupational status is associated with greater emotional awareness [START_REF] Lane | Sociodemographic correlates of alexithymia[END_REF] . Among individuals of high occupational status, a high perceived stress score may thus partially result from a better emotional awareness, whereas among low status individuals, it could mostly result from higher levels of stress per se. This hypothesis is consistent with evidence for an association between essential hypertension and lower emotional awareness [START_REF] Consoli | Differences in emotion processing in patients with essential and secondary hypertension[END_REF] .

Strengths and limitations

Strengths of the study are the large sample size allowing subsamples analyses and the wide set of covariates considered, including a measure of depressive mood. Some limitations should also be acknowledged. First, owing to the cross-sectional design of this study, no conclusion about causality can be drawn. For instance, hypertension may lead to higher stress due to medical complications. However, results were similar when including or excluding participants with cardiovascular or renal diseases. Furthermore, the first set of analyses excluded participants with treated, allegedly known hypertension. Finally, analyses were adjusted for perceived health status. Second, BP was not measured at several successive consultations. However, high BP has been associated with cardiovascular mortality in the IPC cohort study [START_REF] Thomas | Cardiovascular risk as defined in the 2003 European blood pressure classification: the assessment of an additional predictive value of pulse pressure on mortality[END_REF] as well as in other cohorts. Third, some potential confounders were not examined, such as salt consumption, ethnicity, social support and personality variables.

Fourth, a large sample size ensures statistical power but not clinical significance because even small differences may reach statistical significance. Finally, the IPC cohort may not be representative of the general French population. Study recruitment was limited to the Paris area and two thirds of the participants were men, which potentially limits the generalizability of our results. Compared with Paris area inhabitants, individuals in the IPC cohort were less likely to live alone, and more likely to have a professional activity and a higher occupational status (http://www.recensement.insee.fr/home.action). In addition, they were seeking a preventive medical examination and thus may presumably display increased interest in their own health.

Perspectives

Our results extend to non-working individuals and broader aspects of stress the preliminary evidence that job strain may relate to BP at work to a different extent according to occupational status [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF] . Additionally, they suggest that higher perceived stress among individuals of high occupational status may relate to lower BP. Should these results be replicated with a prospective design, further studies would be warranted to elucidate the mechanisms of this interaction as such knowledge may eventually inform prevention strategies. Finally, our results urge researchers to systematically look for possible interactions with SES when examining the relationships between stress and hypertension. These interactions may account for some negative or equivocal results regarding the links between job strain, perceived stress and BP in previous studies [START_REF] Sparrenberger | Stressful life events and current psychological distress are associated with self-reported hypertension but not with true hypertension: results from a cross-sectional population-based study[END_REF][START_REF] Fauvel | Neither perceived job stress nor individual cardiovascular reactivity predict high blood pressure[END_REF][START_REF] Ducher | Is high job strain associated with hypertension genesis?[END_REF] . Our results may thus constitute an impetus for re-analyzing old datasets.

Novelty and Significance

1) What Is New?

The association of stress with high blood pressure depends on occupational status.

2) What Is Relevant?

Previous conflicting results regarding the association between stress and hypertension may result from overlooking the moderating effect of socio-economic status indicators.

Hypertension might partially explain the association between stress and cardiovascular diseases such as coronary heart disease and stroke among individuals of low occupational status or unemployed.

3) Summary

These results urge to systematically look for possible interactions with socio-economic status indicators when examining the relationships between stress and hypertension and may constitute an impetus for re-analyzing old datasets.

  ) medium (e.g. clerks or first line supervisors); (3) low (e.g. blue collar workers); (4) unemployed participants (i.e. seeking employment); (5) participants without a paid occupation (e.g. housewives); (

Figure 1 .

 1 Figure 1. Association between high BP (i.e. systolic BP ≥ 140 or diastolic BP ≥ 90 mmHg)

Figure 2 .

 2 Figure 2. Prevalence of high blood pressure according to perceived stress and

Figure 1 Figure 2

 12 Figure 1

Table 1 . Associations between high BP (systolic BP ≥140 or diastolic BP ≥90 mmHg) and each variable in multivariate models.

 1 Body Mass Index; BP: Blood Pressure; CI: Confidence Interval; OR: Odds Ratio; PSS-4: 4-item Perceived Stress Scale; QD2A: Questionnaire of Depression 2 nd version Abridged; SD: Standard Deviation. * OR is given per 5-point increment for the PSS-4 and per unit for the other continuous variables. † P<0.001; ‡ P<0.01.

	INDEPENDENT VARIABLES	Model 1	Model 2	Model 3
	CONTINUOUS VARIABLES *	OR [95%CI]	OR [95%CI]	OR [95%CI]
	Age (years)	1.06 † [1.06-1.06] 1.06 † [1.06-1.06] 1.06 † [1.06-1.06]
	Perceived Stress (PSS-4)	1.06 † [1.03-1.09] 1.01 [0.99-1.04] 0.93 ‡ [0.89-0.98]
	Depressive mood (QD2A)	0.97 † [0.96-0.98] 0.98 † [0.97-0.98] 0.98 † [0.97-0.98]
	Perceived health status (10-point scale) 0.98 † [0.97-0.99] 0.99 [0.98-1.00] 0.99 [0.98-1.00]
	Heart rate (beats per minute)	1.05 † [1.05-1.05] 1.05 † [1.05-1.05] 1.05 † [1.05-1.05]
	Fasting glycemia (mmol/L)	1.01 † [1.01-1.01] 1.01 † [1.01-1.01] 1.01 † [1.01-1.01]
	DISCRETE VARIABLES	OR [95%CI]	OR [95%CI]	OR [95%CI]
	Male (vs. female) gender	1.93 † [1.87-2.00] 1.99 † [1.91-2.06] 1.98 † [1.91-2.06]
	Living alone (vs. not living alone)	1.07 † [1.03-1.11] 1.06 † [1.03-1.10] 1.06 † [1.03-1.1]
	BMI
	<18.5	0.77 † [0.67-0.87] 0.76 † [0.66-0.86] 0.76 † [0.66-0.86]
	18.5-24.99	Reference	Reference	Reference
	25-29.99	1.73 † [1.68-1.79] 1.71 † [1.66-1.77] 1.71 † [1.66-1.76]
	≥30	3.38 † [3.23-3.54] 3.27 † [3.13-3.42] 3.26 † [3.12-3.41]
	Smoking status
	Non-smokers	Reference	Reference	Reference
	Ex-smokers	1.00 [0.97-1.04] 1.02 [0.98-1.06] 1.02 [0.99-1.06]
	1-10 cigarettes/day	0.85 † [0.81-0.89] 0.86 † [0.82-0.90] 0.86 † [0.82-0.90]
	11-20 cigarettes/day	0.89 † [0.84-0.93] 0.87 † [0.82-0.91] 0.87 † [0.83-0.91]
	>20 cigarettes/day	1.00 [0.93-1.08] 0.99 [0.92-1.07] 0.99 [0.93-1.07]
	At-risk alcohol intake (see text)	1.50 † [1.43-1.57] 1.51 † [1.44-1.58] 1.51 † [1.44-1.58]
	≥1 hour of walking/day (vs. <1 hour)	0.88 † [0.86-0.91] 0.92 † [0.89-0.94] 0.92 † [0.89-0.94]
	Familial history of hypertension	1.46 † [1.41-1.50] 1.51 † [1.46-1.55] 1.51 † [1.46-1.56]
	Occupational status
	High		Reference	Reference
	Medium		1.22 † [1.17-1.26] 1.15 † [1.09-1.22]
	Low		1.58 † [1.52-1.65] 1.44 † [1.34-1.54]
	Unemployed		1.08 ‡ [1.02-1.14] 0.97 [0.88-1.06]

Acknowledgments

We thank the Caisse Nationale d'Assurance Maladie des Travailleurs Salariés (CNAM-TS, France) and the Caisse Primaire d'Assurance Maladie de Paris (CPAM-P, France) for helping make this study possible.

Sources of funding

None.

Conflict(s) of Interest/Disclosure(s) Statement

Sébastien Czernichow has received consultancy fees from Novo and Sanofi and lectures fees from Servier. Tabassome Simon has received unrestricted research grants from Astra-Zeneca, Daiichi-Sankyo, Eli-Lilly, Glaxo-Smith-Kline, MSD, Novartis, Pfizer, Sanofi-aventis and Servier; speaker and consulting fees from AstraZeneca, Bayer-Schering, Eli-Lilly and Sanofi-

ONLINE SUPPLEMENT OCCUPATIONAL STATUS MODERATES THE ASSOCIATION BETWEEN CURRENT PERCEIVED STRESS AND HIGH BLOOD PRESSURE: EVIDENCE FROM THE IPC COHORT STUDY

Emmanuel Wiernik a,b , Bruno Pannier c , Sébastien Czernichow a,b,d , Hermann Nabi b,e , Olivier Hanon f,g , Tabassome Simon h,i , Jean-Marc Simon j , Frédérique Thomas c , Kathy Bean c , Silla M. Consoli f,k , Nicolas Danchin f,l , Cédric Lemogne a,f,k,m,* ,

Appendix 1. English version of the 4-item Perceived Stress Scale (PSS-4) Instructions

The questions in this scale ask you about your feelings and thoughts during the last month. In each case, please indicate with a check how often you felt or thought a certain way.

1. In the last month, how often have you felt that you were unable to control the important things in your life? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 2. In the last month, how often have you felt confident about your ability to handle your personal problems? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 3. In the last month, how often have you felt that things were going your way? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 4. In the last month, how often have you felt difficulties were piling up so high that you could not overcome them? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often Scoring PSS-4 scores are obtained by reverse coding the positive items, e.g., 0=4, 1=3, 2=2, etc. and then summing across all 4 items. Items 2 and 3 are the positively stated items.